Computer Aided Chemical Engineering, Volume 21, Part B: 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering

16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING

COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: R. Gani Volume Volume Volume Volume

1: 2: 3: 4:

Volume 5:

Volume 6: Volume 7: Volume 8: Volume 9: Volume 10 Volume 11 Volume 12 Volume 13 Volume 14 Volume 15 Volume 16 Volume 17 Volume 18 Volume 19 Volume 20 Volume 21

Distillation Design in Practice (L.M. Rose) The Art of Chemical Process Design (G.L. Wells and L.M. Rose) Computer Programming Examples for Chemical Engineers (G. Ross) Analysis and Synthesis of Chemical Process Systems (K. Hartmann and K. Kaplick) Studies in Computer-Aided Modelling. Design and Operation Part A: Unite Operations (I. Pallai and Z. Fonyo, Editors) Part B: Systems (I. Pallai and G.E. Veress, Editors) Neural Networks for Chemical Engineers (A.B. Bulsari, Editor) Material and Energy Balancing in the Process Industries - From Microscopic Balances to Large Plants (V.V. Veverka and F. Madron) European Symposium on Computer Aided Process Engineering-10 (S. Pierucci, Editor) European Symposium on Computer Aided Process Engineering-11 (R. Gani and S.B. Jorgensen, Editors) European Symposium on Computer Aided Process Engineering-12 (J. Grievink and J. van Schijndel, Editors) Software Architectures and Tools for Computer Aided Process Engineering (B. Braunschweig and R. Gani, Editors) Computer Aided Molecular Design: Theory and Practice (L.E.K. Achenie, R. Gani and V. Venkatasubramanian, Editors) Integrated Design and Simulation of Chemical Processes (A.C. Dimian) European Symposium on Computer Aided Process Engineering-13 (A. Kraslawski and I. Turunen, Editors) Process Systems Engineering 2003 (Bingzhen Chen and A.W. Westerberg, Editors) Dynamic Model Development: Methods, Theory and Applications (S.P. Asprey and S. Macchietto, Editors) The Integration of Process Design and Control (P. Seferlis and M.C. Georgiadis, Editors) European Symposium on Computer-Aided Process Engineering-14 (A. Barbosa-Povoa and H. Mates, Editors) Computer Aided Property Estimation for Process and Product Design (M. Kontogeorgis and R. Gani, Editors) European Symposium on Computer-Aided Process Engineering-15 (L. Puigjaner and A. Espuha, Editors) 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering (W. Marquardt and C. Pantelides)

COMPUTER-AIDED CHEMICAL ENGINEERING, 21B

16TH EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING AND 9TH INTERNATIONAL SYMPOSIUM ON PROCESS SYSTEMS ENGINEERING Edited by

W. Marquardt RWTH Aachen University, Leiirstulil fur Prozesstechnil<, Aactien, Germany

C. Pantelides Process Systems Enterprise Ltd. & Imperial College London, London, UK

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0-444-52257-3 978-0-444-52257-3(Part A) 0-444-52970-5 978-0-444-52970-1 (Part B) 0-444-52969-1 978-0-444-52969-5(Set) 1570-7946

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Contents Part A Keynotes Innovation in the Chemical Industry: A Growth Engine S. Marcinowski

1

Life Cycle Modelling in the Chemical Industries: Is there any Reuse of Models in Automation and Control /. Bausa and G. Dunnebier

3

Hierarchical Multiscale Model-based Design of Experiments, Catalysts, and Reactors for Fuel Processing D.G. Vlachos, A.B. Mhadeshwar andN.S. Kaisare

9

Process Intensification and Process System Engineering: A Friendly Symbiosis J.A. Moulijn, A. Stankiewicz, J. Grievink and A. Gorak

29

Recent Developments in the Risk Management of Offshore Production Systems D. Averbuch

39

Challenges and Opportunities in Process Innovation L.R. Genskow

45

Recent Developments and Industrial Applications of Data-based Process Monitoring and Process Control M. Kano andY. Nakagawa

57

Model-Centric Technologies for Support of Manufacturing Operations J.A. Romagnoli and P.A. Rolandi

63

The Systems Engineering of Cellular Processes V. Hatzimanikatis and L. Wang

71

Systems Biology and the Silicon Cell: Order out of Chaos H.V. Westerhojf

81

Challenges for Process System Engineering in Infrastructure Operation and Control Z Lukszo, M. P. C. Weijnen, R.R. Negenborn, B. De Schutter and M. Ilic

95

Supply Chain Design, Management and Optimization D. Kassmann and R. Allgor

101

Business Decision Making in the Chemical Industry: PSE Opportunities R. Srinivasan, LA. Karimi and A.G. Vania

107

vi

Contents

Contributed Papers Topic 1 Oral Simulation of Mass Transfer in Reactive Absorption A^. Asprion

119

Integration of Generalized Disjunctive Programming with Modular Process Simulators y.A. CaballerOy A. Odjo and I.E. Grossmann

125

Large-scale Optimization Strategies for Zone Configuration of Simulated Moving Beds Y. Kawajiri and L.T. Biegler

131

Comparison of the Startup of Reactive Distillation in Packed and Tray Towers F. Fomer, M. Meyer, M. Doker, J.-U. Repke, J. Gmehling and G. Wozny

137

Parameter Estimation for Stochastic Differential Equations: Algorithm and Application to Polymer Melt Rheology B. Pereira Lo, A.J. Haslam and C.S. Adjiman

143

A "Targeted" QSPR for Prediction of Properties A^. Brauner, R.P. Stateva, G. St. Cholakov and M. Shacham

149

Global Bounds on Optimal Solutions in Chemical Process Design U.-U. HauSy J. Gangadwala, A. Kienle, D. Michaels, A. Seidel-Morgenstern and R. Weismantel

155

Stochastic Grey Box Modeling of the Enzymatic Biochemical Reaction Network of E. Coli Mutants F.P. Davidescu, H. Madsen, M. Schumperli, M. Heinemann, S. Panke andS.B. J0rgensen

161

Validated Solution of ODEs with Parametric Uncertainties Y. Lin andM.A. Stadtherr

167

Optimal Experimental Design for Ill-posed Problems A. Bardow

173

Dynamic Oil and Gas Production Optimization via Explicit Reservoir Simulation D.I. Gerogiorgis, M. Georgiadis, G. Bow en, C.C. Pantelides and E.N. Pistikopoulos

179

Multi-Scale Modelling and Optimization of Hydrogen Storage Systems Using Advanced Solid Materials E. Kikkinides, M.C. Georgiadis, M. Konstantakou and A. Stubos

185

Contents

vii

Risk Analysis and Robust Design Under Technological Uncertainty R.F. Blanco Gutierrez, C.C. Pantelides and C.S. Adjiman

191

Network of Three Catalytic Reactors with Periodical Feed Switching for Methanol Synthesis: Bifurcation Analysis M. Pota, L. Russo, E. Mancusi and S. Crescitelli

197

CFD Model of a Semi-batch Reactor for the Precipitation of Nanoparticles in the Droplets of a Microemulsion A.A. OncUl, B. Niemann, K. Sundmacher and D. Thevenin

203

Solution of the Population Balance Equation Using the Sectional Quadrature Method of Moments (SQMOM) MM. Attarakih, H.-J. Bart and N.M. Faqir

209

A Global Parametric Programming Optimisation Strategy for Multilevel Problems N.P. Faisca, V. Dua, P.M. Saraiva, B. Rustem and E.N. Pistikopoulos

215

Modelling Deammonification in Biofilm Systems: Sensitivity and Identifiability Analysis as a Basis for the Design of Experiments for Parameter Estimation D. Brockmann, K.-H. Rosenwinkel and E. Morgenroth

221

The Combined-Continuum-and-Discrete-Model (CCDM) for Simulation of Liquid-particle Flows K.F. Malone, B.H. Xu and M. Fairweather

227

Implementation of Efficient Logic-based Techniques in the MINLP Process Synthesizer MIPSYN M. Ropotar andZ. Kravanja

233

Calculation of Three-phase Bubble Columns D. Wiemann and D. Mewes

239

A Framework for Model-based Design of Parallel Experiments in Dynamic Systems F. Galvanin, M. Barolo, F. Bezzo and S. Macchietto

249

OPEN CHEMASIM^^: Breaking Paradigms in Process Simulation H. Hasse, B. Bessling and R. Bottcher

255

Simulation of the Population Balance for Droplet Breakage in a Liquid-liquid Stirred Tank Reactor Using H-matrix Methods /. Koch, W. Hackbusch and K. Sundmacher

261

Simultaneous Dynamic Validation/Identification of Mechanistic Process Models and Reconciliation of Industrial Process Data P.A. Rolandi and J.A. Romagnoli

267

A Model Discrimination Based Approach to the Determination of Operating Regimes for Chemical Reactors A. Yang, E. Martin, G. Montague and J. Morris

273

viii

Contents

A Performance Comparison of some High Breakdown Robust Estimators for Nonlinear Parameter Estimation E.L.T. Conceigdo and A.A.T.G. Portugal

279

Equivalent Dynamic Solution of an Industrial HDPE Slurry Reactor S. Nigam, KM. Moudgalya and A.K. Pani

285

Dynamical and Stationary Analysis of an Electrolyte Diode and Comparison with Experiments Z. Slouka, M. Pfibyl, J. Lindner, D. Snita and M. Marek

291

Stability Analysis of Differential-Algebraic Equations in AUTO_DAE B.C. von Clausbruchy E.C. Biscaia, Jr., and P.A. Melo

297

Application of Particulate Models for Industrial Processes G. Skillas, C. Becker, M. Verduyn and J. Vorholz

303

Optimization of Operating Conditions for Ferrichrome Production in a Membrane Bioreactor Using Ustilago maydis A. Drews, H. Arellano-Garcia, M. Wendt, M. Kraume and G. Wozny

309

Modelling and Simulation of MSP Desalination Process Using gPROMS and Neural Network Based Physical Property Correlation M.S. Tanvir and I.M. Mujtaba

315

A New Operation Mode for Reactive Batch Distillation in Middle Vessel Columns: Start-up and Operation /. Carmona, H. Arellano-Garcia and G. Wozny

321

Towards a Novel Optimisation Algorithm with Simultaneous Knowledge Acquisition for Distributed Computing Environments S. Yang, A. Kokossis and P. Linke

327

Floating Index of Inequality Constrained DAE Systems D.F. de S. Souza, R.C. Vieira and E.C. Biscaia Jr.

333

Predictive Modeling of Ionic Permselectivity of Porous Media L. SedaandJ. Kosek

339

Development of a Multi-Compartment Dynamic Model for the Prediction of Particle Size Distribution and Particle Segregation in a Catalytic Olefin Polymerization FBR G. Dompazis, V. Kanellopoulos and C. Kiparissides

345

Mixing in a T-shaped Microreactor: Scales and Quality of Mixing D. Bothe, C. Stemich and H.-J. Wamecke

351

Direct Modelling of Unit Operations on Molecular Level D. Babic and A. Pfennig

359

Modelling and Simulation of Fe203/Aluminum Thermite Combustion: Experimental Validation L. Durdes, P. Brito, J. Campos and A. Portugal

365

Contents

ix

Topic 1 Poster Modelling of Self-ignition and Process Upsets in Industrial Gaseous Hydrocarbon Oxidation Processes H.J. Pasman and M. Fairweather

371

A Simplex Search Method for Experimental Optimization with Multiple Objectives E. Martinez

377

Automatic Generation of Reduced Reaction Mechanisms for Hydrocarbon Oxidation with Application to Autoignition Boundary Prediction for Explosion Hazards Mitigation R. Porter, M. Fairweather, J.F. Griffiths, K.J. Hughes andA.S. Tomlin

383

Combining HAZOP with Dynamic Process Model Development for Safety Analysis S. Eizenberg, M. Shacham and N. Brauner

389

Validation of a Digital Packing Algorithm for the Packing and Subsequent Fluid Flow Through Packed Columns R. Caulkin, M. Fairweather, X. Jia and R.A. Williams 395 A Hybrid Global Optimization Scheme for Process Design and Dynamic Optimization C.-T. Chen, S.-T. Peng, Y.-J Ciou and C.-L. Chen

401

Parameter Identifiability Analysis and Model Fitting of a Biological Wastewater Model Q. Chai, S.H. Amrani and B. Lie

409

Methodology for Decision Support Among Conflicting Objectives Using Process Simulators A^. Ramzan and W. Witt Grey-box Stochastic Modelling of Industrial Fed-Batch Cultivation J.K. Rasmussen, H. Mads en and S.B. J0rgensen Monitoring and Improving LP Optimization with Uncertain Parameters D. Zyngier and T.E. Marlin

415 421 427

Assessing the Performance of Batch Reactive Distillations Through Conceptual Models J. Espinosa

433

An Integrated Stochastic Method for Global Optimization of Continuous Functions M. Srinivas and G.P. Rangaiah

439

The ProMoT/Diana Simulation Environment M. Krasnyk, K. Bondareva, O. Milokhov, K. Teplinskiy, M. Ginkel and A. Kienle

445

X

Contents

Strategy and Framework for Solving Signal-based MIDO Problems R.H. Nystrom, I. Harjunkoski and R. Franke "Smart Models" - A Framework for Adaptive Multi-Scale Modelling E.S. Fraga, G. Wills, M. Fairweather and T. Ferris

451 457

Process Design Using Ionic Liquids: Physical Property Modeling A. E. Ayala, L D. Simoni, Y. Lin, J. F. Brennecke and MA, Stadtherr

463

Study of Non-linear Dynamics in Reactive Distillation for TAME Synthesis Using Equilibrium and Non-equilibrium Models A.M. Katariya, R.S. Kamath, S.M. Mahajani and K.M. Moudgalya

469

An Agent-Oriented Architecture for Modeling and Optimization of Naphtha Pyrolysis Process X. Gao, B. Chen andX. He

475

On Model Portability H.A. Preisig, T. Haug-Warberg and B.T. L0vfall

483

Utility Systems Operational Planning Optimization Based on Pipeline Network Simulation XL. Luo, B. Hua, B.J. Zhang and M.L. Lu

489

Particle Swarm for the Dynamic Optimization of Biochemical Processes J. Zhang, L. Xie and S. Wang

497

A-priori Identification of Critical Points for the Design and Synthesis of Flexible Process Schemes Z.N. Pintaric and Z. Kravanja

503

Using Water Cascade Analysis to Synthesize Water use Network in Batch Process S. Wang, S. Zheng, X. Yang and Y. Li

509

Multiobjective Optimization of Multipurpose Batch Plants Using Superequipment Class Concept A. Mosat, L. Cavin, U. Fischer and K. Hungerbilhler

515

Integrated Design of Energy-Saving Chemical Process Systems: Strategy, Methods and Implementation G. Ostrovsky, Y. Volin, D. Dvoretsky and S. Dvoretsky

521

Generic Hybrid Models of Solvent-based Reactive Systems Combined with Membrane Separation System P.T. Mitkowski, G. Jonsson andR. Gani

527

On the Numerical Calibration of Discrete Element Models for the Simulation of Bulk Solids T. Groger and A. Katterfeld

533

A Heat Transfer Model of a Scraped Surface Heat Exchanger for Ice Cream P.M.M. Bongers

539

Contents

xi

Computer-Aided Forecast of Catalytic Activity in an Hydrotreating Industrial Process Using Artificial Neural Network, Fuzzy Logic and Statistics Tools F. Jimenez, V. Kafarov and M. Nunez

545

A Framework for Modeling Particle Size Effects in Emulsion Polymerization Systems Using Computational Fluid Dynamics Linked to a Detailed Population Balance Model R.C. Elgebrandt, D.F. Fletcher, V.G. Gomes and J.A. Romagnoli

551

Pricing Utilities for Large-Scale Chemical Production Site K. Hirata, P. Chan, H. Sakamoto and C.-W. Hui

557

Optimal Experimental Design for the Precision of a Subset of Model Parameters in Process Development A. Yang, E. Martin, G. Montague and J. Morris

563

The Complex Distillation Column Network Systematic Optimization by Mathematical Programming S. Choi, H. Kim, C. Han and E.S. Yoon

569

Modelling and Simulation of Coal and Petcoke Gasification in a Co-current Flow Reactor E.M. Lopez, v. Garza and J, Acevedo

577

Simulation of (Electro)Chromatography by means of CFD D.-U. Astrath, T. Schneider and W. Arlt

583

Modeling of Heat Transfer Processes in Particulate Systems Z sale, C Mihdlyko and B.G. Lakatos

589

A Comprehensive Investigation on High-Pressure LDPE Manufacturing: Dynamic Modelling of Compressor, Reactor and Separation Units P. Pladis, A. Baltsas and C. Kiparissides

595

Sensitivity Analysis in the Simulation of Complex Solids Processes D. Schwier, A. PUttmann, E.-U. Hartge, G. Gruhn and J. Werther

601

Identification of Parametric and Structural Models Based on RTD Theory via GAMS Package S. Hocine, L. Pibouleau, C. Azzaro-Pantel and S. Domenech

607

Hybrid Modeling for Continuous Production of Bioethanol E. Ccopa Rivera, I. Mantovaneli, A.C da Costa andR. Maciel Filho

613

Prediction and Estimation Techniques for Modeling Pervaporation Process M.E.T. Alvarez, E.B.Moraes andM.R.W. Maciel

619

Model Discrimination and Parameter Estimation Through Sensitivity Analysis M. Sales-Cruz and R. Gani

625

xii

Contents

Solving MINLP Containing Noisy Variables and Black-Box Functions Using Branch-and-Bound E. Davis andM. lerapetritou

633

Modelling and Simulation of High Pressure Industrial Autoclave Polyethylene Reactor E. Caliani, M. Cavalcanti, FA.N. Femandes and L.M.F. Lona

639

Energy Saving in Distillation Columns: The Linde Column Revisited G. Soave, L. Pellegrini, D. Barbatti, N. Susani and S. Bonomi

645

Computer-Aided Modeling for Hydrodesulfurization, Hydrodenitrogenation and Hydrodearomatization Simultaneous Reactions in a Hydrotreating Industrial Process F. Jimenez, V. Kafarov and M. Nunez

651

Modelling and Dynamic Simulation of Thermal Stresses in Brazed Plate-Fin Heat Exchanger F. Picard, D. Averous, X. Joulia and D. Barreteau

659

ReDrop - An Efficient Simulation Tool for Describing Solvent and Reactive Extraction Columns M. Altunok, T. Gromping and A. Pfennig

665

Numerical Simulation of Micro Roughness Effects on Convective Heat Transfer S. Scholl and W. Augustin

671

Classical Models of Secondary Settlers Revisited R. David, A. Vande Wouwer, P. Saucez and J.-L. Vasel

677

An Approach to Implicit Modelling for Complex Process Optimization XG. Yuan, W.Z, An, YJ. Liu, YQ. Luo and CJ. Liu

683

Topic 2 Oral Structural Design of Polymers for Membrane Based Separation Processes Using Reverse Simulation Approach V. Soni, J. Abildskov, G. Jonsson, R. Gani, N. Karayiannis and V. Mavrantzas Innovative Flowschemes Using Dividing Wall Columns MA. Schultz, D.E. O'Brien, R.K. Hoehn, C.P. Luebke andD.G. Stewart

689 695

On the Rapid Development of New Products Through Empirical Modeling with Diverse Data-bases J.F. MacGregor, K. Muteki and T. Ueda

701

Conceptual Design of Reactive Distillation Flowsheets G. Daniel, P. Patil and M. Jobson

707

Contents

xiii

LCA of a Spent Lube Oil Re-refining Process T.N. Kalnes, D.R. Shonnard and A. Schuppel

713

Effective Process Design Instruction: From Simulation to Plant Design D.R. Lewin, E. Dassau and A. Goldis

719

Developments in the Sequential Framework for Heat Exchanger Network Synthesis of Industrial Size Problems R. Anantharaman and T. Gundersen

725

Linking Experiments to Modeling in Biodiesel Production A.A. Kiss, A.C. Dimian and G. Rothenberg

731

Optimization Studies in Sulfuric Acid Production A.A. Kiss, C.S. Bildea andP.J.T. Verheijen

737

Integrating Advanced Thermodynamics and Process and Solvent Design for Gas Separation E. Keskes, C.S. Adjiman, A. Galindo and G. Jackson

743

Integrated Approach to Crystallization Process Design for Fine Chemicals and Pharmaceuticals C. Wibowo, K.D. Samant andL. O'Young

749

Improved Solutions for Zebra Mussel (Dreissena polymorpha) Control - A Chemical Product Engineering Approach R. Costa, P.M. Saraiva, P. Elliott, D.C. Aldridge and G.D. Moggridge

755

Success Factors for CAPE in the Engineering Practice of a Process Plant Contractor G. Engl and A. Kroner

763

Polyurethane Design Using Stochastic Optimization /. Eslick and K Camarda

769

Real-Time Imaging and Product Quality Characterization for Control of Particulate Processes Y. Zhou, X.-T. Doan andR. Srinivasan

775

An Engineering Company's Approach to Filling "CAPE Gaps" in Process Simulation A. Kroner

781

A Computer-Aided Methodology with Robust Design Criteria for Selection of Solvents for Reactions M. Folic, C.S. Adjiman and E.N. Pistikopoulos

787

Separation of Azeotropes in Batch Extractive Stripper with Intermediate Entrainer V. Varga, E.R. Frits, V. Gerbaud, Z. Fonyo, X. Joulia, Z. Lelkes and E. Rev

793

Conceptual Steady State Process Design in Times of Value Based Management A. Wiesel and A. Polt

799

xiv

Contents

Computer Aided Methods & Tools for Separation & Purification of Fine Chemical & Pharmaceutical Products M.B.C. Afonso, V. Soni, P.T. Mitkowski, L. d'Anterroches, R. Gani and H. Matos

805

Integration Along the Lifecycle of Calcium Fluoride in the Fluorine Industry A. Garea, R. Aldaco, I. Fernandez and A. Irabien

811

Design of Sustainable Processes: Systematic Generation & Evaluation of Alternatives A. Carvalho, R. Gani and H. Matos

817

Systematic Procedure for Designing a Microreactor with Slit-Type Mixing Structure O. Tonomura, T. Takase, M. Kano and S. Hasebe

823

Model-based Optimal Design of Pharmaceutical Formulations F.P. Bernardo, P.M. Saraiva and S. Simoes

829

Scope for Process Systems Engineering Studies in Proton Exchange Membrane Fuel Cells (PEMFC): A Review of Opportunities R. Madhusudana Rao, T. Oh and R. Rengaswamy

835

A Framework for Innovation in Process Development for Heterogeneously Catalysed Gas-phase Reaction Systems D. Montolio-Rodriguez, D. Linke and P. Linke

841

Multi-Objective Optimization of Fixed-Bed Ion Exchange Processes for Phytopharmaceutical Production CM. Silva, A.G. Barreto Jr. and B.C. Biscaia Jr.

847

Computer Aided Methodology for Simultaneous Synthesis, Design & Analysis of Chemical Products-Processes L. d'Anterroches and R. Gani

853

Correlation and Prediction of Drug Molecule Solubihty with the NRTL-S AC Model C.-C. Chen and P.A. Crafts

859

Dynamic Modelling of Complex Batch Distillation Starting from Ambient Conditions S. Gruetzmann, T. Kapala and G. Fieg

865

Topic 2 Poster Genetic Algorithms Approach for Retrofitting Heat Exchanger Network with Standard Heat Exchangers R. Bochenek and J.M. Jezowski

871

A Discrete Interactive Graphical Method for Heat Exchanger Network Synthesis E.S. Fraga and G.W.A. Rowe

877

Contents

xv

Optimization of Nanosized Silver Particles Synthesis via Sequential Pseudo-Uniform Design Method J.-S. Chang and Y.-P. Lee

883

Multiclass Molecular Knowledge Framework for Product and Process Design M. Korichi, V. Gerbaud, P. Floquet, A.-H. MeniaU S. Nacef and X. Joulia

889

Quantitative Structure - Odor Relationship: Using of Multidimensional Data Analysis and Neural Network Approaches M. Korichi, V. Gerbaud^ P. Floquet, A.-H. Menial, S. Nacef and X. Joulia

895

Simulation and Optimization in 1,3-butadiene Process from C4-Cut Using Genetic Algorithm F. Jalali and R. Sajfarl

901

Property Clustering and Group Contribution for Process and Molecular Design F. Eljack, M. Eden, V. Kazantzl and M. El-Halwagi

907

A Combinatorial Formulation for Optimal Sizing, Scheduling and Shift Policy in Designing the Milling Section of a Ceramic Tile Industrial Plant B.P.M. Duarte, L.O. Santos and J.S. Mariano

913

An Automated Method for Synthesizing a Multi-Stream Heat Exchanger Network Based on Stream Pseudo-Temperature D. Yuan, Y. Wang, W. Xiao, P. Yao, X. Luo and W. Roetzel

919

Flexible Heat Exchanger Network Design for Chemical Processes with Operation Mode Changes M. Noda and H. Nlshltanl

925

Molecular Design Based on Enhanced Topological Descriptors A.A. Kiss andM.V. Diudea

931

Effects of Catalyst Activity Profiles on the Scale-up of Polymerization Reactors S. Nemeth, J. Abonyl, B. Fell, P. Arva, J. Tolveth, A. Janecska and G. Nagy

937

Optimization-based Root Cause Analysis E. Dassau and D. Lewin

943

A Synthesis Procedure for the Design of Semicontinuous Reactive Distillation for Specialty Chemicals T.A. Adams II and W.D.Selder

949

Multi-Objective Optimisation of Batch Distillation Processes T.M. Barakat, E.S. Fraga and E, Sorensen

955

Sustainable Production of Industrial Chemical Products from Bioresources J. Seay, M. Eden, R. D'Alessandro and C Weckbecker

961

Cooling Crystallization: A Process-Product Perspective C.B.B. Costa andR. Maclel Filho

967

xvi A Systematic Approach for Automated Reaction Network Generation S.-H. Hsu, B. Krishnamurthy, P. Rao, C. Zhao, S. Jagannathan, J. Caruthers and V. Venkatasubramanian

Contents

973

A Hybrid Methodology for Detailed Heat Exchanger Design in the Optimal Synthesis of Heat Exchanger Networks JM. Garcia, J.M. Ponce and M. Serna

979

Optimal Design of Shell-and-Tube Heat Exchangers Using Genetic Algorithms J.M. Ponce, M. Serna, V. Rico and A. Jimenez

985

Integration of Process Design and Operation for Chemical Product Development with Implementation of a Kilo-Plant Y. Qian,Z. Wu and Y.Jiang

991

Importance of the Selection of Feed Tray Location on the Optimum Design of a Heterogeneous Azeotropic Distillation Column with p-xylene Feed Impurity I-L. Chien, H.-Y. Lee, T.-K. GauandH.-P. Huang

997

Supporting Waste Minimization Studies by Integrating Expert System with Process Simulators /. Halimand R. Srinivasan

1003

Process Intensification for Systematic Synthesis of New Distillation Systems with Less than N-1 Columns B.-G. Rong and I. Turunen

1009

Mixed-Integer Optimization of Distillation Column Tray Positions in Industrial Practice /. Thomas and A. Kroner

1015

A Chemical Process Design Framework Including Different Stages of Environmental, Health and Safety (EHS) Assessment H Sugiyama, U. Fischer, M. Hirao and K. HungerbUhler

1021

Multi-Objective Reactor Network Synthesis for Industrial Mass Transfer Limited Processes F. Neves, D. Silva, N. Oliveira and F. Mendes

1027

Synthesis of Separation Systems for Azeotropic Mixtures: Preferred Distillation Region S.K. Wasylkiewicz

1033

Modeling and Designing Powder Mixing Processes Utilizing Compartment Modeling P.M. Portillo, F.J. Muzzio and M.G. lerapetritou

1039

Design and Control of Homogeneous and Heterogeneous Reactive Distillation for Ethyl Acetate Process H.-Y. Lee, H.-P. Huang and I-L. Chien

1045

Contents

xvii

Decomposition Based Algorithm for the Design and Scheduling of Multipurpose Batch Plants T. Pinto, A.P.F.D. Barbosa-Povoa and A.Q. Novais

1051

A Knowledge-based Approach for Accident Information Retrieval M. Suzuki, R. Batres, T. Fuchino, Y. Shimada and P.W. Chung

1057

Feasibility Study of Batch Reactive Distillation in Hybrid Columns C Steger, E. Rev, Z. Fonyo, M Meyer and Z. Lelkes

1063

Heat Integration Between Processes: Integrated Structure Using Stage-Wise Model A. Kovac Kralj and P. Glavic

1069

Generic Model Framework for the Synthesis of Structured Reactive Separation Processes G. Sand, M. Tylko, S. Barkmann, G. Schembecker and S. Engell

1075

Knowledge Extraction During the Design of Activated Sludge Systems X. Flores, M. Poch, L Rodriguez-Roda, L. Jimenez and R. Banares-Alcantara

1083

Pharmaceutical Process Development Applying Automated Laboratory Reactors T. Chovdn, I. Markovits, B. Farkas, K. Nagy, L. Nagy, K. Nyiri and F. Szeifert

1089

Addressing the Design of Chemical Supply Chains Under Demand Uncertainty G. Guillen, F.D. Mele, A. Espuria and L Puigjaner

1095

Development of a Mixture Design Methodology for Problems with Incomplete Information. Application to PVC Heat Stabiliser Design U.G. da Cruz and G.A.C Le Roux

1101

Principles for Chemical Products Design L.A. Cisternas and E.D. Gdlvez

1107

Mathematical Development for Scaling-up of Molecular Distillators: Strategy and Test with Recovering Carotenoids from Palm Oil CB, Batistella, E.B. Moraes, R. Maciel Filho and M.R. Wolf-Maciel

1113

Case Study on Design of Regulatory Policies for Sustainable Emission Reduction A. Malcolm, L. Zhang andA.A. Linninger

1119

A Decomposition/Reconstruction Algorithmic Procedure for Computer Aided Case Based Reasoning - Implementation in Biodegradation F.ABatzias

1125

Environmentally Conscious Design of Ethanol Fed Fuel Cell System L. Hernandez and V. Kafarov

1131

Morphogenesis of Polyolefin Particles in Polymerization Reactors B. Horackova and J. Kosek

1137

Modeling and Design of a Biochemical Process for NO^ Removal CS. Bildea, M.L. Oudshoom, C. Picioreanu and A.C Dimian

1143

xviii

Contents

Multicriteria Design of Separation Sequences by Including HSE Criteria and Uncertainty K. Cziner, M. Hassim andM. Hurme

1149

Optimal Scheduling of Tests for New Product Development H.-R. Son, S.-K. Heo and I.-B. Lee

1155

PartB Topic 3 Oral State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter M Groetschy M. Mangold, M. ShengandA. Kienle

1161

Real-Time Failure Prediction for Chemical Processes: Plantwide Framework A.MeelandW.D. Seider

1167

Multiscale Analysis and Monitoring of Paper Surface M.S. Reis and P.M. Saraiva

1173

A Real Time Adaptive Dynamic Programming Approach for Planning and Scheduling N.E.Pratikakis, J.H.Lee and M.J. Realjf.

1179

Theoretical Analysis and Experimental Studies of Mixed Product Run-to-Run Control Y. Zheng, M.-F. Wu, S.-S. Jang and D.S.-H. Wang

1185

Methods of State Estimation for Particulate Processes M. Mangold, C. Steyer, B. Niemann, A. Voigt and K. Sundmacher

1191

Simultaneous Scheduling and Optimization of a Copper Plant /. Harjunkoski, H.W. Borchers andM. Fahl

1197

Coordinator MPC with Focus on Maximizing Throughput E.M.B. Aske, S. Strand and S. Skogestad

1203

Fault Diagnosis Based on Support Vector Machines and Systematic Comparison to Existing Approaches /. Yelamos, G. Escudero, M. Graells and L. Puigjaner

1209

Explicit Parametric Controller for a Batch Polymerization System M. Asteasuain, K. Kouramas, V. SakizUs and E.N. Pistikopoulos An Effective MIDO Approach for the Simultaneous Cyclic Scheduling and Control of Polymer Grade Transition Operations A. Flores-Tlacuahuac and LE. Grossmann

1215

1221

Contents

xix

On the Global Dynamic Optimization of Highly Nonlinear Systems A. Flores-Tlacuahuac and L.T. Biegler

1227

Integrating Stiction Diagnosis and Stiction Compensation in Process Control Valves R. Srinivasan and R. Rengaswamy

1233

Differential Recurrent Neural Network Based Predictive Control R. Al Seyab and Y. Cao

1239

Chance Constrained Programming Approach to Process Optimization Under Uncertainty P. Li, H. Arellano-Garcia and G. Wozny

1245

Support for Design of User Interfaces in Plant Operations X. Liu, H. Kosaka, M. Noda and H. Nishitani

1251

Online Prediction of End-of-Batch Product Quality Using Phase-Specific PLS Models X.-T. Doan andR. Srinivasan

1257

Optimal Current Distribution Control for Parallel Electrolytic Baths H. Kugemoto, K. Ozakia, Y. Kutsuwa and Y. Hashimoto

1263

Hybrid Model Predictive Control of a Sugar End Section D. Sarabia, C. de Prada, S. Cristea and R. Mazaeda

1269

Systematic Methodology for Reproducible Optimizing Batch Operation S.B. J0rgensen and D. Bonne

1275

Discriminant Analysis and Control Chart for the Fault Detection and Identification X. Pei, Y. Yamashita, M. Yoshida and S. Matsumoto

1281

Stability Analysis of Nonlinear Model Predictive Control: An Optimization Based Approach V.Dua

1287

An Optimization Framework to Computer-Aided Design of Reliable Measurement Systems R. Angelini, C.A. Mendez, E. Musulin and L. Puigjaner

1293

An Approach to Linear Control of Nonlinear Processes T. Schweickhardt and F. Allgower

1299

Control of the Synthesis Section of a Urea Plant by means of an MPC Controller OM. Agudelo Mahozca, J.J. Espinosa and J. Vandewalle

1305

Control of Thermal Runaway via Optimal Bifurcation Tailoring Aided Gain-Scheduling Feedback P. Altimari, L. Russo, E. Mancusi, M. di Bernardo and S. Crescitelli

1311

XX

Contents

Systematic Design of Logic Controllers for Processing Plants Starting from Informal Specifications S. Lohmann, O. Stursberg and S. Engell

1317

Static/Dynamic Analysis and Controllability Issues in Reactive Distillation Columns r. Lopez-Arenas, E.S. Perez-Cisneros and R. Gani

1323

Predictive Control of Polymerization Batch Reactors Using Hybrid Models /. Espinosa and W. van Brempt

1329

Dynamic Optimization of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods: An Experimental and Theoretical Investigation A. Krallis, D. Meimaroglou, V. Saliakas, C. Chatzidoukas and C. Kiparissides

1335

Multivariate Statistical Batch Process Monitoring Using Dynamic Independent Component Analysis H. Albazzaz andX.Z. Wang

1341

Model-based Optimization for Operational Policies in Seeded Cooling Crystallization A. Abbas, S.M. Nowee and J.A. Romagnoli

1347

Topic 3 Poster Controllability Analysis of Thermally Coupled Distillation Sequences for Five - Component Mixtures M. Carrera-Rodriguez, M. Ledezma-Martinez, J.G. Segovia-Hernandez and S. Hernandez

1353

Multiscale SPC in the Presence of Multiresolution Data M.S. Reis and P.M. Saraiva

1359

Nonlinear Model Predictive Control of the Wastewater Treatment Plant M.V. Cristea and S.P. Agachi

1365

Branch and Bound Methods for Control Structure Design V. Kariwala and S. Skogestad

1371

A Mathematical Programming Approach Including Flexible Recipes to Batch Operation Rescheduling S. Ferrer-Nadal, C.A. Mendez, M. Graells andL. Puigjaner

1377

Using Multi Sensor Data Fusion for Level Estimation in a Separator N.-O. Skeie, S. Mylvaganam and B. Lie

1383

Product Quality Estimation Using Multi-Rate Sampled Data B. Lin, B. Recke, T. Jensen, J. Knudsen and S.B. J0rgensen

1389

Contents

xxi

A Novel Solution Approach for Quality-based Re-trimming Optimization /. Harjunkoski and M. Fahl

1395

An Integrated Framework Based on Data Driven Techniques for Process Supervision B. Bhushan andJ.A. Romagnoli

1401

Reliable Multi-Objective Optimal Control of Batch Processes Based on Stacked Neural Network Models A. Mukherjee and J. Zhang

1407

A Thermodynamic Based Plant-Wide Control Design Procedure of the Tennessee Eastman Process L.T. Antelo, I. Otero-Muras, J.R, Banga andA.A. Alonso

1413

Stochastic Optimal Control in Batch Reactive Systems: Developments on Engineering Applications of Real Option Theory y. Rico-Ramirez, J>F. Cambero-Benitez, H. Canada-Jaime and S. Hernandez-Castro

1419

A Web Service Based Online Optimization and Monitoring System for Chemical Processing Systems Xiangyu Li, XiuxiLi and Y. Qian

1425

Using the Process Schematic in Plant-Wide Disturbance Analysis S.Y. Yim, H.G. Ananthakumar, L. Benabbas, A. Horch, R. Drath andN.F. Thornhill

1431

Constrained Control for Chemical Processes Using Reference Governor K. Kogiso, M. Noda and H. Nishitani

1437

Agent-based Diagnosis for Granulation Processes R. Lakner, E. Nemeth, KM. Hangos and LT. Cameron

1443

An Application of Metamodels for Process Optimization M.V.C Gomes, I. David, L. Bogle, D. Odloak and E.C BiscaiaJr.

1449

Time Scale Separation and the Link Between Open-Loop and Closed-Loop Dynamics A. Araujo, M. Baldea, S. Skogestad and P. Daoutidis

1455

Fault Detection and Diagnosis of Pulp Mill Process G. Lee, T. Tosukhowong andJ.H. Lee

1461

Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment R. Faber, H. Are llano-Garcia, P. Li and G. Wozny

1467

Simulation Based Engineering - From Process Engineering to Automation Engineering H. Fischer and J. C. Toebermann

1473

xxii

Contents

Large-Scale Dynamic Optimization of an Integrated Cryogenic Process M. Rodriguez and M.S. Diaz

1477

Scheduling of Storage and Transfer Tasks in Oil Refineries by Using Fuzzy Optimization L.C. Felizari andR. LUders

1483

Degrees of Freedom Analysis for Process Control M. Rodriguez and J.A. Gayoso

1489

Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction M. Baldea and P. Daoutidis

1495

Performance Assessment and Controller Design for Unknown Systems Based on Gain and Phase Margins Using Modified Relay Feedback J.-C. Jeng andH.-P. Huang

1501

Graphical Modeling for the Safety Verification of Chemical Processes /. Kim, Y. Lee and I. Moon

1509

Application of a Hybrid Control Approach to Highly Nonlinear Chemical Processes Y. Sakakura, M. Noda, H. Nishitani, Y. Yamashita, M. Yoshida and S. Matsumoto .... 1515 Dynamic Optimization of Dead-End Membrane Filtration B. Blankert, B.H.L. Betlem and B. Roffel

1521

Combined Nonlinear Model Predictive Control and Moving Horizon Estimation for a Copolymerisation Process M. Diehl P. Kuhl, H.G. Bock, J.P. Schloder, B. Mahn and J. Kallrath

1527

Automatic Adjustment of Data Compression in Process Information Management Systems F. Alsmeyer

1533

Virtual Plant, New Paradigm for Future Production Management H.A. Gabbar, K. Nishiyama, I. Shingo, T. Ooto and K. Suzuki

1539

Adaptive Monitoring Statistics Based on State Space Updating Using Canonical Variate Analysis C. Lee, S.W. Choi and L-B. Lee

1545

Scheduling of Make and Pack Plants: A Case Study C.-U. Fundeling andN. Trautmann

1551

Detection of Abnormal Alumina Feed Rate in Aluminium Electrolysis Cells Using State and Parameter Estimation K. Hestetun andM. Hovd

1557

Thermodynamic Diagram Based Estimation Structure Design for Ternary Distillation Column A. Pulis, C. Fernandez, R. Baratti and J. Alvarez

1563

Contents

xxiii

Optimal Operation of a Mixed Fluid Cascade LNG Plant J.B. Jensen and S. Skogestad

1569

Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator J.L. Fernandes, C.I.C. Pinheiro, N. Oliveira and F. Ramoa Ribeiro

1575

On-Line Data Reconciliation and Parameter Estimation for an Industrial Polypropylene Reactor D.M. Prata, J.C. Pinto and E.L. Lima

1581

Optimal Reactive Scheduling of Multipurpose, Make-to-Order Industries M.C. Gomes, A.P. Barbosa-Povoa and A.Q. Novais

1587

Optimal Steady-State Transitions Under Constrained Predictive Control D.K. Lam and C.L.E. Swartz

1593

Optimal Configuration of Artificial Neural Networks V.Dua

1599

Diagnosis of Oscillations in Process Control Loops Y. Yamashita

1605

Advances and Future Directions in Morphology Monitoring and Control of Organic Crystals Grown from Solution X.Z. Wang, K. J. Roberts and J. Calderon De Anda

1611

Molecular Weight Control in Acrylonitrile Polymerization with Neural Network Based Controllers /. Atasoy, M. Yuceer and R. Berber

1617

A New Approach to Chance Constrained Process Optimization and Control Under Time-Dependent Uncertainties H. Arellano-Garcia, T. Barz, W. Martini and G. Wozny

1623

Topic 4 Oral A Lab-on-a-Chip Simulation Framework AJ. Pfeiffer, X. He, T. Mukherjee and S. Hauan

1631

Two Level Control of the Sequence Fed Batch - Continuous Hybridoma Bioreactor LD. Ofiferu, A. Woinaroschy and V. Lavric

1637

Optimal Delivery of Chemotherapeutic Agents in Cancer P. Dua, V. Dua and E.N. Pistikopoulos

1643

Dissipative Particle Dynamics Simulation of Ibuprofen Molecules Distribution in the Matrix of Solid Lipid Microparticles (SLM)

C. Long, L. Zhang and Y. Qian

1649

xxiv

Contents

An Integrative Systems Biology Approach for Analyzing Liver Hypermetabolism E. Yang, F. Berthiaume, M.L. Yarmush and LP. Androulakis

1655

Solid Fuel Decomposition Modelling for the Design of Biomass Gasification Systems D. Brown, T. Fuchino and F. Marechal

1661

Hybrid Metabolic Flux Analysis/Data-Driven Modelling of Bioprocesses A. Teixeira, C.LM. Alves, P.M. Alves, M.J.T. Carrondo and R. Oliveira

1667

Rotavirus-Like Particle Production: Simulation of Protein Production and Particle Assembly A. Rolddo, H.L.A. Vieira, M.J.T. Carrondo, P.M. Alves andR. Oliveira

1673

Prediction of Secondary Structures of Proteins Using a Two-Stage Method M. Turkay, O. YilmazandF. Uney Yuksektepe

1679

Reconstruction of Transcriptional Regulatory Networks via Integer Linear Programming J.M.S. Natali and J.M. Pinto

1687

Systematic Design of Drug Delivery Therapies M. Xenos, L. Zhang, M.B.R. Somayaji, S. Kondapalli and A.A. Linninger

1693

Topic 4 Poster Mathematical Modelling of Three-Dimensional Cell Cultures in Perfusion Bioreactors. Part II F. Coletti, S. Macchietto and N. Elvassore

1699

Metabolic Regulatory Network Optimization Using an Information Guided Genetic Algorithm Approach Y. Zheng, C.-D. Yang, J.-W. Yeh and S.-S. Jang

1705

Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes H. Yang, M.L. Yarmush, CM. Roth and M. lerapetritou

1711

Hybrid Modular Mechanistic/ANN Modelling of a Wastewater Phosphorous Removal Process /. Peres, F. Freitas, M.A.M. Reis, S. Feyo de Azevedo andR. Oliveira

1717

Modelling Morphological Change in Endothelial Cells Induced by Shear Stress R.J. Allen, D. Bogle and A.J. Ridley

1723

Contents

xxv

Topic 5 Oral Disturbance Propagation and Rejection Models for Water Allocation Network X. Feng andR. Shen

1729

Energy Planning in Buildings Under Uncertainty in Fuel Costs: The Case of a Hospital in Greece G. Mavrotas, K. Florios and P. Georgiou

1735

Modelling an Electricity Infrastructure as a Multi-Agent System — Lessons Learnt from Manufacturing Control K.H. van Dam, M. Houwing, Z Lukszo and I. Bouwmans

1741

Global Optimization of Multiscenario Mixed Integer Nonlinear Programming Models Arising in the Synthesis of Integrated Water Networks Under Uncertainty R. Karuppiah and I.E. Grossmann

1747

Hierarchical Markov Reliability/Availability Models for Energy & Industrial Infrastructure Systems Conceptual Design A.N.Ajah, P.M. Herder, J. Grievink and M.P.C. Weijnen

1753

Agent-Enabled Dynamic Management System for Process Plants A. Kokossis, Z. Shang and E. Gao

1759

Methodology for the Design of Industrial Hydrogen Networks and the Optimal Placement of Purification Units Using Multi-Objective Optimisation Techniques L. Girardin, F. Marechal and P. Tromeur

1765

Topic 5 Poster Modelling and Simulation of a Tyre Gasification Plant for Synthesis Gas Production N.R. Mitta, S. Ferrer-Nadal, A.M. Lazovic, J.F. Perales, E. Veto and L. Puigjaner

1771

Library for Modeling and Simulating the Thermal Dynamics of Buildings /./. Videla and B. Lie

1777

A New Method for Designing Water Network Based on Variable Removal Ratio of Treatment Process L. Song, J. Du, S. CaiandP. Yao

1783

Environmental Life Cycle Impact and Cost Minimization in the Steam and Power Generation Plant P. Martinez and A.M. Eliceche

1791

xxvi

Contents

Case Study of a Regional Network for the Recovery of Hazardous Materials J. Duque, A.P.F.D. Barbosa-Pdvoa andA.Q. Novais .................................................

1797

Optimisation of a Pertraction Process for Wastewater Treatment and Copper Recovery A.M. Eliceche, M.F. Orlandi, A.M. Urtiaga and I. Ortiz .............................................

1803

Data-Centric Demand Forecasting for Utilities Z. Beran. K. MaZk and P. Stluka ................................................................................

1809

A Screening Tool for Exploring Production Chains L. Stougie, R.M. Stikkelman and M. Houwing ...............................

.........................

1815

Cost versus Network Length Criteria in Water Network Optimal Design P. Iancu, V. Plegu and V. Lavric .................................................................................

1821

Synergy Analysis of Collaboration with Biofuel use for Environmentally Conscious Energy Systems M. Turkay and A. Soylu ................................. Process Optimization and Scheduling of Parallel Furnaces Shutdown in Large-Scale Plants E.P. Schulz, J.A. Bandoni and M.S. Diaz ....................................................................

1833

Operational Optimization of the Thermoelectric System of an Oil Refinery S.R. Micheletto and J.M. Pinto .............. ..............................................................

1839

Water Reuse: A Successful Almost Zero Discharge Case R.M.B. Alves, R. Guardani, A.E. Bresciani, L. Nascimento and C.A.O. Nascimento ...............................................................................................

1845

Model Development for the Optimal Water Systems Planning E. Kondili and J. K. Kaldellis .......................................................................................

1851

Simulation of Electricity Production Systems in Autonomous Networks in Order to Maximize the Wind Energy Penetration J. K. Kaldellis and E. Kondili .......................................................................................

1857

Heat Integration in Micro-Fluidic Devices T. Zhelev and 0. Strelow .............................................................................................

863

Network Synthesis for a District Energy System: A Step Towards Sustainability C. Weber, I. Heckl, F. Friedler, F. Markchal and D. Favrat .......................................

869

Topic 6

Oral Close Loop Supply Chains: Managing Product Recovery Portfolio A.C.S. Amaro and A.P.F.D. Barbosa-Pdvoa ..............................................................

1875

Contents Simulation Based Optimization for Risk Management in Multi-Stage Capacity Expansion X. Wan, J.F. Pekny and G.V. Reklaitis A Decision Support Tool for Process Optimization of Sulphur Free Diesel Production Z Lukszo, M. Salverda and P. Bosman An Attainable Region Approach for Effective Production Planning C. Sung and C.T. Maravelias

xxvii

1881

1887 1893

A Planning Support System for Biomass-based Power Generation N. Ayoub, K. Wang, T. Kagiyama, H, SekiandY. Naka

1899

Semantic Analysis for Identification of Portfolio of R&D Projects. Example of Microencapsulation A. Kraslawski

1905

Dynamic Rule-based Genetic Algorithm for Large-Size Single-Stage Batch Scheduling Y. He and C.-W. Hui

1911

Application of Multi-Stage Scheduling P.M.M. Bongers and B.H. Bakker

1917

Slot-based vs. Global Event-based vs. Unit-Specific Event-based Models in Scheduling of Batch Plants M.A. Shaik, S.L. Janak and C.A. Floudas

1923

A Unified Approach for Knowledge Modeling in Pharmaceutical Product Development C. Zhao, A. Jain, L. Hailemariam, G. Joglekar, V. Venkatasubramanian, K. Morris and G. Reklaitis

1929

A Multistage Stochastic Programming Approach with Strategies for Uncertainty Reduction in the Synthesis of Process Networks with Uncertain Yields B. Tarhan and I.E. Grossmann

1937

A Framework for Capturing the Impact of Resource Allocation Policies in the Selection of a New Product Portfolio J.C. Zapata, V.A. VarmaandG.V. Reklaitis

1943

Multi-Period Capacitated Lot Sizing with Variable Batch Sizes Y.C. See-Toh, S.P.K. Walsh andN. Shah

1949

Integration of Discrete-Event Simulation and Optimization for the Design of Value Networks M. Schlegel, G. Brosig, A. Eckert, K. Engelke, M. Jung, A. Polt, M. Sonnenschein and C. Vogt

1955

xxviii

Contents

A CP Method for the ScheduHng of Multiproduct Continuous Plants with Resource Constraints L.J. Zeballos and G.P. Henning

1961

Stochastic Integer Programming in Chemical Batch Scheduling: Evolution Strategies vs. Exact Algorithms /. Till, G. Sand and S. Engell

1967

Scheduling and Planning with Timed Automata S. Panek, S. Engell and O. Stursberg

1973

Novel Continuous-Time Formulations for Scheduling Multi-Stage Multi-Product Batch Plants with Identical Parallel Units Y. Liu and LA. Karimi

1979

Routing and Cargo Allocation Planning of a Parcel Tanker K.-H. Neo, H.-C. Oh and LA. Karimi

1985

An Approximate Framework for Large Multistage Batch Scheduling Problems Focusing on Bottleneck Resources P.A. Marchetti and J. Cerdd

1991

On the Dynamic Management of Chemical Engineering Knowledge Using an Ontology-based Approach A. Kokossis, E. GaoandA. Kourakis

1997

Lagrangean-based Techniques for the Supply Chain Management of Flexible Process Networks P. Chen and J.M.Pinto

2003

Restructuring Methodology in Process Engineering for Sustainable Development /. Koshijima, A. Shindo, Y. Hashimoto and T. Umeda

2009

Development of a Multiobjective Scheduler for Semiconductor Manufacturing O. Baez Senties, C. Azzaro-Pantel, L. Pibouleau and S. Domenech

2015

Ontology-based Information Management in Design Processes S.C. Brandt, J. Morbach, M. Miatidis, M. Theifien, M. Jarke and W. Marquardt

2021

Workflow Support for Inter-Organizational Design Processes R. Hai, M. Heller, W. Marquardt, M. NaglandR. Worzberger

2027

Rigorous Scheduling Resolution of Complex Multipurpose Batch Plants: S-Graph vs. MILP S. Ferrer-Nadal, T. Holczinger, C.A. Mendez, F. Friedler and L. Puigjaner

2033

Contents

xxix

Topic 6 Poster Integrative Optimization of Refining and Petrochemical Plants C. Li X. He, B. Chen, B. Chen, Z Gong and L. Quan

2039

Scheduling of Identical and Parallel on/off Production Units Under Uncertainty in Cost and Demand Prediction P. Pulkkinen and R. Ritala

2045

A Flexible Design of Logistic Network Against Uncertain Demands Through Hybrid Meta-Heuristic Method Y. Shimizu, S. Matsuda and T. Wada

2051

A Flexible Framework for Optimal Biorefinery Product Allocation A^. Sammons, M. Eden, H. Cullinan, L. Ferine and E. Connor

2057

Systems for Decisions Support in Industrial use K. Coboken, G. Mogk, T. Mrziglod and U. Telle

2063

Medium Term Planning of Biopharmaceutical Manufacture Under Uncertainty K. Lakhdar, S.S. Farid, J. Savery, N.J. Titchener-Hooker andL.G. Papageorgiou.... 2069 A Multi-Criteria Optimization Model for Planning of a Supply Chain Network Under Demand Uncertainty C.L. Chen, T.Y. Yuan, C.Y. Chang, W.C Lee and Y.C Ciou

2075

A View-based Information Model for Enterprise Integration in Process Industries P. Li, M.L. Lu, Y.S. Peng and B. Hua

2081

Strategic Planning and Design Using MILP: An Industrial Application from the Tissue Manufacturing Industry J. Westerlund, P. Castro and S. Forssell

2087

Multiple Time Grid Continuous-Time Formulation for the Short Term Scheduling of Multiproduct Batch Plants P. Castro and L Grossmann

2093

An Inventory Control Scheme for Simultaneous Production Planning and Scheduling Under Demand Uncertainty T. Nishi, H. Tominaga and M. Konishi

2099

Implementation of an Integrated Platform of Process System Operations for Education and Research X. Li, Y. Qian and Y. Jiang

2105

Integration of Multi-Scale Planning and Scheduling Problems H. Stefansson, P. Jensson andN. Shah

2111

XXX Plant-Wide Planning and Marginal Value Analysis for a Refinery Complex W. Li, X. Liang and C.-W. Hui

Contents

2117

Refinery Planning Under Correlated and Truncated Price and Demand Uncertainties W. Li, LA. Karimi and R. Srinivasan

2123

An Integrated Model for the Design and Planning of Supply Chains with Product Return M.LG. Salema, A.P. Barbosa-Povoa andA.Q. Novais

2129

Pipeline Scheduling and Distribution Centre Management - A Real-World Scenario at CLC S. Relvas, A.P.F.D. Barbosa-Povoa, H.A. Matos, J. Fialho and AS. Pinheiro

2135

Scheduling Under Demand Uncertainty Using a New Multiparametric Programming Approach Z Jia andM.G. lerapetritou

2141

Information Modeling for Pharmaceutical Product Development C. Zhao, L. Hailemariam, A. Jain, G. Joglekar, V. Venkatasubramanian, K. Morris and G. Reklaitis

2147

Decentralized Supply Chain Dynamics and the Quantity Flexibility Contract V. Subramanian, J.F. Pekny and G.V. Reklaitis

2153

Planning and Scheduling of Multipurpose Continuous Plants C. Schwindt, S. Herrmann and N. Trautmann Priority-Rule Based Scheduling of Chemical Batch Processes A^. Trautmann and C. Schwindt

2159 2165

A Rigorous Approach to Coordinate Production and Transport Scheduling in a Multi-Site System C. Mendez, A. Bonfill, A. Espufia and L. Puigjaner

2171

Multi-Criteria Evaluation for the Chemical Industrial Parks J. Xiaoping, T. Zhang and L. Shi

2177

Scheduling with High Accuracy at Low Maintenance Costs: An Approach Using Discrete Event Simulation M Jung and C. Vogt

2183

An Integrated Model for Planning in Global Chemical Supply Chains A. Sundaramoorthy, S. Xianming, LA. Karimi and R. Srinivasan

2189

Information Sharing in a Distributed Enterprise: Impact on Supply Chain Performance and Decision-Making LB. Owusu and S. Hauan

2195

Contents

xxxi

Integration of Supply Chain Management and Logistics: Development of an Electronic Data Interchange for SAP Servers L. Jimenez and R. Muhoz

2201

A Novel Combined Approach for Supply Chain Modeling and Analysis F.D. Mele, C.A. Mendez, A. Espuha and L. Puigjaner

2207

A Hybrid Approach Using CLP and MILP Applied to Tank Farm Operation Scheduling S.L. Stebel, F. Neves Jr. and L.V.R. Arruda

2213

PRoduct ONTOlogy. Defining Product-Related Concepts for Production Planning Activities D. Gimenez, M. Vegetti, G. Henning and H. Leone

2219

Recipe Informatics to Shorten the Lead Time from Product Development to Production in Batch Processes T. Fuchino, T. Kitajima, Y. Shimada, K. Takeda, S. Hashizume, T. Hamaguchi, R. Batres, A. Yamada, K. Kawano and Y. Hashimoto

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Efficient MILP-based Solution Strategies for Large-Scale Industrial Batch Scheduling Problems P. Castro, C. Mendez, I. Grossmann, L Harjunkoski and M. Fahl

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Innovation and Knowledge Management: Using the Combined Approach TRIZ-CBR in Process System Engineering G.C. Robles, S. Negny and J.M. he Lann

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Decision-Making Tool for ScheduHng of Batch Processes: The Dynamic Hybrid Simulation Kernel A^. Olivier, R. Thery, G. Hetreux and J.-M. Le Lann

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Multiobjective Multiproduct Batch Plant Design Under Uncertainty A. Dietz, A. Aguilar-Lasserre, C. Azzaro-Pantel, L. Pibouleau and S. Domenech

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter Markus Groetsch ,^ Michael Mangold ,^ Min Sheng ,^ Achim Kienle , ^'^ ""Max-Planck-Institut fur Dynamik komplexer technischer Systeme, Sandtorstrafie 1, 39106 Magdeburg, Germany ^Otto-von-Guericke- Universitdt Magdeburg, Lehrstuhlfur Automatisierungstechnik / Modellbildung, Universitdtsplatz 2, 39106 Magdeburg, Germany Abstract Industrial fuel cell stacks only provide very limited measurement information. To overcome this deficit, a state estimator for a molten carbonate fuel cell system is developed in this contribution. The starting point of the work is a rigorous spatially distributed model of the system. From this model a reduced model is derived by using a Galerkin method and the Karhunen Loeve decomposition technique. An extended Kalman filter with a continuous time simulator part and a discrete time corrector part is designed on the basis of the reduced model. The filter is tested in simulations and experimentally. Keywords: Molten carbonate fuel cell, state estimation. Extended Kalman filter 1. Introduction The molten carbonate fuel cell (MCFC) is a new promising technology for decentralized stationary power generation. Due to a working temperature of 600 - 650 °C, the MCFC is capable of supplying a H i I •^^^iii consumer both with electricity and heat. Because of the direct generation of electrical energy from chemical energy, without the use of mechanical intermediate steps, the electrical efficiency of the MCFC is with approximately 50% considerably Figure 1: HotModule higher than that of conventional processes for electrical power generation, like gas engines and gas turbines [Bischof (2002)]. Further advantages are the great flexibility with respect to the fuel, the low maintenance efforts and the good ecological compatibility. In this work we consider the MCFC stack HotModule from MTU CFC Solutions. A prototype of this industrial cell stack has been installed at the Magdeburg university hospital (Figure 1). However, the operation of industrial MCFC stacks is complicated by the high degree of process integration and the small amount of measurement information available online. In this work a model-based measurement system is developed using an extended Kalman filter in order to estimate the states of the MCFC. The obtained information can then be used to improve the process operation and the efficiency of the MCFC.

I

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2. Modelling of the MCFC A detailed dynamic, spatially distributed model of a MCFC had been derived from first principles [Heidebrecht (2005)]. Figure 2 shows the structure of the modelled system and the working principle of one MCFC-cell.

CH4 + H2O ^ CO + H2O ^

3H2 + CO H2 + CO2

Figure 2: At the left: Scheme of the modelled system, consisting of one representative MCFC-cell a pre-reformer and a burner, at the right: the working principle of a MCFC-cell. The detailed model consists of 17 hyperbolic and parabolic partial differential equations and ftirther ordinary differential equations and additional algebraic equations. As this detailed model is too complex for most process control purposes and demands too much computation time, a reduced model is used for the state estimation. 0.5r

h^^/^

0.4 5 h

—

0.3 5 |i 0.3

1.8

( gl.4

ri

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solid L anode gas [ cathode gas f

^'"'''''^^^r^^

/ \

^f^"

"X.

^ " ^ ^ ^

time [-]

Figure 3: Comparison of the reduced and the detailed MCFC-model; top diagram: dimensionless cell current, randomly varied, used as input for both models; middle diagram: maximum deviation in % of the reduced from the detailed model with respect

State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter

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to the solid, the anode gas and the cathode gas temperature; bottom diagram: deviation in % of the reduced from the detailed model with respect to the cell voltage; the time is dimensionless, one time unit corresponds to about 60 seconds of real time. The reduced model was derived from the spatially distributed reference model of the cell by applying the Karhunen-Loeve-Galerkin procedure [Mangold (2004)]. The reduced model is a DAE-system and is of considerably lower order than the original one and requires less computational time, but still describes the MCFC behaviour with good accuracy (Figure 3). 3. Design of the extended Kalman filter The design of the extended Kalman filter starts with a short introduction in model-based measurement systems in Section 3.1. Section 3.2 deals with the observability analysis of the reduced MCFC-model. In Section 3.3 the design of the extended Kalman filter is described and its test using simulation data. In Section 3.4 some experimental estimation results are shown. 3.1. Principle of model-based measurement systems The basic idea of a model-based measurement system is to estimate unknown states of a given process (MCFC) using simulations of the process model (MCFC-model). The model-based measurement system (Figure 4) uses input and measurement values of the MCFC for the estimation. The task of the corrector is to compensate model errors and to suppress disturbances acting upon the MCFC in order to improve the estimation results. In this work an extended Kalman filter is used for the model-based measurement system. distur Dances no se

disturbances

input

^

1 MCFC

+ states

sensors

' measurement ~^ variables real process

Figure 4: Structure of the model-based measurement system 3.2. Observability analysis of the reduced MCFC-model The observability of the states of the MCFC is a necessary precondition for their estimation. It depends mainly on the choice of the measurement variables. At the HotModule a certain amount of measurement information is available: temperature and cell voltage measurements as well as concentration measurements using a gas chromatograph. During an observability analysis various measurement configurations are examined in order to find a configuration that shows good observability of the MCFC-states and demands only a small measurement effort. This analysis is performed upon the reduced MCFC-model with local observability criteria at different operating points [Mangold (2005)].

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As a result of this analysis a measurement configuration is found that consists only of two temperature and the cell voltage measurement. Using this set-up it is possible to stay without expensive concentration measurements at all. The found measurement configuration is then used together with the reduced MCFC-model for the design of the extended Kalman filter. 3.3. Design and Test of an extended Kalman filter An extended time-continuous Kalman filter with time-discrete measurement update is designed and implemented as a state estimator for the MCFC. The extended Kalman filter is tested with simulated measurement values generated by the reduced MCFCmodel perturbed with Gaussian noise (Figure 5). The standard deviations of the used measurement noise are chosen according to the measurement noise at the HotModule. The received estimation results are quite promising.

200 400 600 800 1000 time [-]

780

400 600 800 1000

time [-]

0

200 400 600 800 1000

time [-]

750

200 400 600 800 1000

time [-]

Figure 5: Convergence of the extended Kalman filter, if a noise-perturbed steady state of the reduced MCFC-model is used as a reference; in the first row: deviations of the filter statesfi*omthe reference states without a correction at measurement times (dotted lines) and with a correction at measurement times (solid lines); first row fi-om left to right: maximum relative error of the estimated solid temperature, the estimated cathode gas temperature and the estimated anode gas temperature; in the second row: measurement variables without a correction at measurement times (dotted lines), with a correction at measurement times (solid lines) and the measurement data taken as filter reference (dashed lines); second row from left to right: average anode gas output temperature, average cathode gas output temperature and the cell voltage. 3.4. Experimental results The extended Kalman filter is evaluated with measurement data taken from the HotModule in Magdeburg. The filter is validated by additional concentration measurements that are not used for the filter correction. The concentration

State Estimation of a Molten Carbonate Fuel Cell by an Extended Kalman Filter

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measurements are obtained from a gas chromatograph at the outlet of the anode gas channels in time intervals of about one hour. In Figure 6 and 7 are estimation results for two different load cases of the HotModule depicted. The filter shows good convergence behaviour towards the real measurement variables with only a small stationary error. The estimation results for the molar fractions are acceptable. The remaining small deviations are in the order of magnitude of direct concentration measurements. They can be explained by inevitable model errors. 0.31

0.68 r

40 60 80 time [min]

0.11 I

40 60 80 time [min]

40 60 80 time [min]

Figure 6: Convergence of the extened Kalman filter if a steady state of the HotModule is used as a reference; in the first row: measured (dashed lines) and estimated (solid lines: with a correction at measurement times, dotted lines: without a correction at measurement times) molar fractions of carbon dioxide, hydrogen and carbon monoxide at the output of the anode gas channel; in the second row: measured (dotted lines) and estimated (solid lines: with a correction at measurement times, dotted lines: without a correction at measurement times) measurement variables; from left to right: average anode gas output temperature, the average cathode gas output temperature and the cell voltage. In conclusion, the filter shows good convergence behaviour and predicts the internal states of the cell stack with reasonable accuracy. The developed state estimator can replace complicated temperature measurements inside the stack as well as slow and expensive concentration measurements. It is expected to be a usefiil tool for process operation. Of course, the approaches used here are not restricted to the considered HotModule, but can be applied directly to other high and low temperature fuel cell systems.

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20

40 60 80 100 time [min]

20

40 60 80 100 time [min]

20

40 60 80 100 time [min]

810

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^CO

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40 60 80 time [min]

v^

v~^

\j/ 20

40 60 80 100 time [min]

40 60 80 time [min]

Figure 7: Convergence of the extened Kalman filter if a steady state of the HotModule is used as a reference; in the first row: measured (dashed lines) and estimated (solid lines: with a correction at measurement times, dotted lines: without a correction at measurement times) molar fractions of carbon dioxide, hydrogen and carbon monoxide at the output of the anode gas channel; in the second row: measured (dotted lines) and estimated (solid lines: with a correction at measurement times, dotted lines: without a correction at measurement times) measurement variables; from left to right: average anode gas output temperature, the average cathode gas output temperature and the cell voltage. 4. Acknowledgement The authors thank Mykhaylo Krasnyk, IVl.Sc. for his assistance in implementing the Kalman filter. The financial support of the German Ministry of Education and Research under contract no. 03C0345B is gratefully acknowledged. References M. Bischof, G. Huppmann, 2002, Operating experience with a 250 IcW molten carbonate fuel cell (IVICFC) power plant, Journal of Power Sources, 105, 216-221 P. Heidebrecht, 2005, Analysis and optimisation of a molten carbonate fuel cell with direct internal reforming (DIR-IVICFC), Ph.D. thesis, Otto-von-Guericke-Universitat IS/Lagdeburg, Fortschritt-Berichte VDI, Reihe 3, Nr. 826, VDI Verlag Dusseldorf M. Mangold, M.Sheng, 2004, Nonlinear model reduction for a two-dimensional MCFC model with internal reforming. Fuel Cells, 4, 68-77 M. Mangold, M. Groetsch, M. Sheng, A.Kienle, 2005, State Estimation of a molten carbonate fuel cell by an extended Kalmanfilter,Proceedings of SYNCOD"2005, Springer-Verlag

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Real-Time Failure Prediction for Chemical Processes: Plantwide Framework Anjana Meer, Warren D. Seider"" ^ Department of Chemical and Biomolecular Engineering, University ofPennsylvania Philadelphia, PA 19104

Abstract Methods for plant-specific, real-time risk assessment are presented for an ethyl-benzene plant using Bayesian theory and copula to predict: (i) the frequencies of occurrence of abnormal events, (ii) failure probabilities of safety systems involving equipment and human actions, (iii) probabilities of consequences, and (iv) fuzzy memberships to various critical zones to indicate the proximity of a current plant state to a failure or disaster, utilizing accident precursor data. These methods, which involve repetitive risk analysis after abnormal events occur, are especially beneficial for operations involving complex nonlinearities and multi-component interactions, helping to achieve inherently safer operations. Furthermore, these methods demonstrate the importance of using plant-specific estimates of failure probabilities rather than generic values in the risk assessment of chemical plants. Keywords: Real-time risk assessment, Bayesian theory, copula, chemical process, abnormal event

1. Introduction Abnormal events of varying magnitudes result in incipient faults, near-misses, incidents, and accidents in chemical plants. Several efforts have been initiated to report incidents and accidents in chemical and petrochemical industries, including the CCPS process safety incident database (PSID), the EPA's RMP*Info database, and the NRC database. Efforts to increase the internal reporting of near-misses have been initiated by the Wharton Risk Management Center (Risk Center), Wharton School, University of Pennsylvania [Phimister et al., 2003]. Detection and diagnosis of abnormal events has been an active area of research [Venkatasubramanian et al., 2003]. However, estimation of the failure probabilities of safety systems to predict various consequences (end-states: continued operation, shutdown, release, and explosion), has received little attention in the CPI. In this context, recently, a method for dynamic risk assessment of an exothermic reactor was introduced [Meel and Seider, 2005a, 2005b]. Herein, these methods are extended for plant-wide, real-time risk assessment to predict: (1) the frequencies of occurrence of abnormal events of plant units, (2) the propagation of abnormal events through their safety systems, and (3) their subsequent consequences utilizing accident precursor data, helping to achieve inherently safer operations. The real-time, risk-assessment technique is applied to a reactor-separator-recycle system; more specifically, an ethyl-benzene plant having two reactors (Rl, R2) and two distillation columns (Dl, D2) [Luyben, 2002].

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A Bayesian method is introduced, based on the work of Yi and Bier [Yi and Bier, 1998], to update the performance of all of the safety systems, involving equipment and human actions, given any abnormal event in the plant. Copula [Nelson, 1999, Yi and Bier, 1998] are used to model the joint distribution of the failure probabilities of: (i) a safety system under different accident conditions, (ii) individual safety systems of a process unit to learn from the failures of the other safety systems, and (iii) similar safety systems of different plant units to learn from the failures of safety systems in other units. This type of joint distribution is effective, especially given the rare occurrence of the abnormal events. By modeling the failure probabilities of human actions under different scenarios, the proposed technique helps to learn the response of a human to an incident associated with a reactor, given an incident related to a distillation column, using the joint distribution. In this work, finite probabilities of accidents (releases and explosions) are predicted even though the accident precursor data consists of near-misses and incidents only. Also, frizzy memberships to various critical zones are determined to indicate the proximity of a current plant state to a failure or disaster using the probabilities of the consequences. This method demonstrates the importance of using plant-specific estimates of failure probabilities rather than generic values in the risk assessment The paper is organized as follows. A brief description of the case study, with focus on reactor Rl, its safety system, and event-tree, is given in Section 2. The Bayesian model for the frequency of occurrence of abnormal events is formulated in Section 3. Then, the estimation of the failure probabilities of the safety systems, and the propagation of an abnormal event across safety systems to end-states, is described in Section 4. Finally, conclusions are presented in Section 5. 2. Consequence Formulation with Event-trees Figure 1 shows an event-tree for a high-temperature abnormal event associated with reactor Rl, involving an exothermic reaction, of the ethyl-benzene process. This results in 37 consequences (Ci,i-Ci,37) including continued-operation (CO), shut-down (SD), release (REL), and explosion (EXP) based on the performance of the eight safety systems, where Cj^rn is consequence m for plant unit /, with the units Rl, R2, Dl, and D2 having / = 1,2,3,4, and m = \,.,.,NCi, with Ncj the number of consequences. The safety systems are divided into two categories: (i) equipment related (e.g., alarm systems) and (ii) human related (e.g., operator failure to perform an action). The failure probability of safety system y at branch point k of plant unit / is Xy^k. Note that similar event-trees are created for the other three plant units having consequences €2^^^^, Cg^^^, and C^j^^^ for reactor R2, and distillation columns, Dl and D2. Table 1 provides the hypothetical accident sequence precursor (ASP) data, for selected consequences, over the first five of the twenty time intervals. These data, which were used for the calculations in this paper, are reported cumulatively, showing the total number of incidents through the end of each time interval; that is, Mj^mfr>rconsequence Cj^rn of plant component /, where / = {1, 2, 3, 4} for Rl, R2, Dl, and D2, respectively. Note that a fixed time interval was chosen to demonstrate the technique, however, the analysis can be repeated on observation of every abnormal event enabling real-time risk assessment.

Real-Time Failure Prediction for Chemical Processes: Plantwide Framework

1169

1 C^ratcMT 1 1 C^raleir 1 1 Qp0m» 1 1 t'£^EI^Hi{^ tpEfip lAutam(^«Nit-| lll^i»d»iHil>| l 0 t S 1 4omi^ 1 1 «iRrm ( 8 ^ H

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Figure 1: Event-tree for the high temperature in reactor. (*ERS: Emergency Rehef System) Table 1: Accident sequence precursor data Time

Mi1,1

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EXP 0 0 0 0 0

3. Abnormal Event Frequencies A Bayesian model is introduced to estimate the frequency of abnormal events in a time interval for each of the plant units and the overall plant. It involves Poisson likelihood (K^ \y))- Gamma prior {p{X)) distributions having the posterior distribution: p{X I y) « /(A I y)p{X) - (X'e-'"){X"-'e-^') A<«^'>-'e-<^^"'\5 = £ j ,

(1)

where A is the average number of abnormal events in the time intervals and yi is the number of abnormal events in time interval /, n is the number of time intervals, and a and p are the parameters of the Gamma prior. The predictive distribution {PoissonGammd) is formulated to estimate the number of abnormal events in the next time interval. Note that an introduction to the use of prior and posterior distributions in the Bayesian analysis of abnormal event data is provided [Meel and Seider, 2005b]. A predictive z-score is used to measure the proximity of 3//(number of abnormal events in interval /) and E^y^ \ y_i ] (predicted number of abnormal events in interval / using the data from other intervals): yi-E[yi\y_i] Zi

(2)

= -

^V[y,\y-i]

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A. Meel and W.D. Seider

In the absence of information to model the prior distribution, a and p are assumed to be 0.001, providing a relatively flat distribution in the region of interest; that is, a noninformative prior distribution. The means of the abnormal events per time interval are ~ {10.9, 5.05, 6.6, 1.4} and the standard deviations are ~ {4.14, 2.5, 3.56, 1.27} for Rl, R2, Dl, and D2, using the ASP data in [Meel and Seider, 2005c]. Using R Software [Gentleman et al., 2005], the means of the predicted z-score are 0.012, 0.014, 0.02, 0.026 and the standard deviations are 1.29, 1.15, 1.45, 1.14 for Rl, R2, Dl, and D2, indicating good accuracy with occasional offsets. In Figures 2a and 2b, the sample and theoretical quantiles are in close proximity, confirming the model prediction accuracy of abnormal events for reactor Rl and the overall plant. Also, predictive accuracy is checked for each consequence (CO, SD, REL, and EXP) for each plant unit and the overall plant. Predictive accuracy for CO and SD are good. However, as seen in Figure 2c, the predictive accuracies for REL and EXP are poor due to the rarity of these events. Information gap theory has the potential to represent these rare events better, a topic of fiirther research [Yakov, 2001].

Theoretical Quantiles

Theoretical Quantiles

Theoretical Quantiles

(b) (c) (a) Figure 2: Predictive checking of abnormal eventfrequenciesusing normal Q-Q plot of z/ for: (a) Rl, (b) all plant units (Rl, R2, Dl, D2), and (c) releases in all plant units (Rl, R2, Dl, D2) 4. Failure Probabilities of Safety Systems The failure probability of safety system 7 at branch point k of plant unit / is a random variable, Xij^k, where7 = 1,...,Nsi, k= 1,...,Nbij, i= 1,... , Np, Nsf is the number of safety systems for plant unit /, Nbjj is the number of branch points for safety system J associated with plant unit /, and Np is the number of plant units. The prior distributions of the failure probabilities are updated to obtain posterior distributions using Bayesian theory and copula applied to the ASP data. In this analysis, the safety systems of the plant units are assumed to be correlated. Note that the ASP data for the plant units are affected by the transfer of material and energy throughout the plant. The prior joint probability distribution of the failure probabilities is: Np Nsi Nbjj

fix) oc c(u) X n n n(^-,M )""•" (i -

bij,k-^

(3)

%J,k

/=i j=i

^ ~ 1^1,1,1' * * '^^l,Nsi,N\j^,^

u = {u1,1,1'

•,w. %Nsi

k=\

•>'' '^•^ij,k

->'' ''>^NpXNbj^p,\

••• X '

,Nbi,Nsi ' * " ' ^''J'^'

* * *' ^NpXNbNpA

\

(4)

^P'^SNp,NbNp,NsNp -'

' "^'^Np,Ns^p,NbNp,NsNp

-*'

(5)

where c(u) is the Cudras and Auges copula density [Nelson, 1999] which is a function of cumulative distributions of x,y,^, Uy^k, and correlation parameters among them, and Oij^k and bij^k are the parameters of the prior Beta distribution for the failure probabilities. Spearman's correlation matrices [Meel and Seider, 2005c], which give the correlation

Real-Time Failure Prediction for Chemical Processes: Plantwide Framework

nil

parameters for the copula, are defined for the failure probabilities of: (i) individual safety systems under various conditions, (ii) safety systems within a plant unit, (iii) safety systems of several units. The posterior joint failure probability distribution is: Np Nsi Nbij

fix I Data) oc n n i=l j=\

Il(^,Mf"'~'i^~^i.Mt'"~'fix) k=\

Np Nsj Nbjj

(6)

II Np NSi Nbjj ''i,j,k-

/=i j=i

/=1 y=i k=i

k=i

Nsj Nhi^q

11 Np NspNbp^q

n(i-^--,M-.-,M..)|| n n(i-^.-,M-...)

""ij^k

nnn(i-^.-,M-... >

(7)

where Kij^k and Ltj^k are the cumulative number of failures and successes, respectively, at the end of each time interval and Oij^k-p,q,r is the correlation parameter between x^^k and Xp^q^r- The marginal posterior distribution of each component of x, x^^k, is obtained from fix\Data) through Markov-Chain Monte-Carlo integration [Robert, 2001]. Again, an introduction to the use of prior and posterior distributions in the Bayesian analysis of accident precursor data is provided by Meel and Seider [Meel and Seider, 2005b]. In Figure 3a, failure probabilities for operator diagnosis for Rl at two branch points, with jci 3,2 > xi,3,i, are shown that account for their correlation with other safety systems associated with the same and other plant units. Similarly, Figure 3b shows the failure probabilities of the high-temperature alarm for Rl at five branch points with xi,5,5 > X\^s,3 > ^1,5,4 >-^1,5,2 > ^1,5,1- The probabilities of consequences, CO, SD, REL, and EXP for Rl are shown in Figure 3c. Observe that the ASP data gives higher probabilities of shutdown than continued operation. It is also important that a finite probability for EXP is estimated, although no occurrences of this end-event appear in the ASP data. Note that the variations in failure probabilities as a fiinction of time interval exceed those for the Bayesian analysis of individual plant units, yielding failure probabilities that account for plantwide interactions. Furthermore, the membership values in green, blue, orange, and red zones, associated with increasing safety threats, formulated by Meel and Seider [Meel and Seider, 2005b] for Rl are graphed in Figure 3d, with higher memberships in the blue and orange zones. Similar results were obtained for the other plant units. '->^

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Figure 3: Mean posterior probabilities of: (a) failure of operator diagnosis, (b) failure of hightemperature alarm, (c) end-states; (d) Membership values in zones.

20

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A. Meel and W.D. Seider

Also, the propagation of an abnormal event through safety systems is modeled in realtime using SIMULINK in MATLAB. The dynamics of safety systemy at branch point k of plant unit / are represented as a transfer function in the Laplace domain: G,j,,{s) = ^^ (8) ©,,.5 + 1 where xsQ^k and xd/^^A: are the response and dead times of safety system y at branch point k of plant unit z, obtained by simulating various abnormal events for the ethyl-benzene plant using ASPEN DYNAMICS and the CCPS guidelines for quantitative risk analysis. These values help in obtaining the real-time failure probabilities of all of the consequences as the abnormal event propagates through the safety systems. A statistical decision-model utilizing the real-time ACI to help make decisions about early plant-shutdown is being developed currently. 5. Conclusions A Bayesian model improves the prediction of the number of abnormal events in future time intervals, as well as failure probabilities of the safety systems in a chemical plant. The application of this model in real-time provides a plant-wide failure-prediction method that models the correlation effectively. Interactions of the failure probabilities of a safety system within a plant unit, and with those in other plant units, are effectively accounted for using copula. This permits the improved identification of near-misses and incidents, which otherwise are difficult to identify. In summary, a quantitative method is introduced to estimate human and equipment reliabilities, measures that are extremely useful when estimating risk in a plant. Acknowledgement The interactions and advice of Professors Paul Kleindorfer and Ulku Oktem of the Wharton Risk Management and Decision Center are appreciated. References R. Gentleman, R. Ihaka, D. Bates, J. Chambers, et al., 2005, The R Project for Statistical Computing, http://www.r-project.org/. W. L. Luyben, 2002, Plantwide dynamic simulators in chemical processing and control. New York: Marcel Dekker, Inc. A. Meel and W. D. Seider, 2005a, Dynamic risk assessment of chemical processes: An accident precursor approach. 20th CCPS International Conference. A. Meel and W. D. Seider, 2005b, Use of Bayesian theory for dynamic failure assessment of an exothermic reactor. Submitted to Chem. Eng. Sci. A. Meel and W. D. Seider, 2005c, Accident sequence precursor data and correlation matrices for ethyl-benzene plant. http://www.seas.upenn.edu/~anjanani/PSE2006. R. B. Nelson, 1999, An Introduction to Copulas. Lecture Notes in Statistics, Springer. J. R. Phimister, U. Oktem, P. R. Kleindorfer and H. Kunreuther, 2003, Near-miss incident management in the chemical process industry. Risk Anal. 23(3): pp. 445-459. C. P. Robert, 2001, The Bayesian Choice. Springer-Verlag, New York. v . Venkatasubramanian, R. Rengaswamy, S. N. Kavuri and K. Yin, 2003, A review of process fault detection and diagnosis. Part III: Process history based methods. Comp. Chem. Eng. 27(3): pp. 327-346. B.-H. Yakov, 2001, Information Gap Decision Theory. Acedemic Press. W. Yi and V. M. Bier, 1998, An application of copulas to accident precursor analysis. Manage. Sci. 44(12): pp. S257-S270.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Multiscale Analysis and Monitoring of Paper Surface Marco S. Reis,^ Pedro M. Saraiva^ ^ GEPSI-PSE Group, Department of Chemical Engineering, University ofCoimbra, Polo Il-Pinhal de Marrocos, 3030-290 COIMBRA, PORTUGAL Abstract Paper surface plays a key role in paper quality and exhibits a multiscale structure. In this article we present a multiscale statistical process control approach for monitoring relevant surface phenomena developing at different length-scales, namely waviness and roughness. The effectiveness of the proposed approach is demonstrated through computational simulations, as well as through a pilot study involving real paper surface profiles. The results obtained demonstrate its adequacy in tracking down the relevant surface phenomena. Keywords: Multiscale analysis; Wavelets; Statistical process control; Paper surface. 1. Introduction Paper is a very complex material, exhibiting properties that derive from a structural hierarchy of arrangements covering different elements (molecules, fibrils, fibres, network of fibres, etc.), beginning at a scale of a few nanometres and proceeding all the way up to a few dozens centimetres or even meters [1]. This complexity is also present at its boundary, the paper surface, which plays a central role in many of the relevant properties from the perspective of the end user, such as general appearance (optical properties, flatness, etc.), printability (e.g. the absorption of ink) and friction features, to name a few. Therefore, it is of crucial importance to devise methodologies that assess and characterize paper surface features at different scales, as they interfere with different aspects of paper quality, as well as monitoring procedures that quickly detect detrimental structural abnormalities. Two surface phenomena, located at different scales, play a key role in paper quality: roughness and waviness. Roughness is a fine length-scale phenomenon (scales between Ijum-lmm), regarding small, high-frequency, surface irregularities. Waviness, on the other hand, refers to those larger scale deviations from a flat shape, an example of which are the so called "piping streaks", that consist of streaks aligned along the largest dimension of paper, 1-3 cm wide, that may develop as a consequence of different fibre alignment streaks across the paper machine. Being aware of their importance, the Pulp and Paper Industry developed methods to assess and characterize them [2]. However, current practice is usually based on indirect measurements (for roughness) or in subjective assessments made by trained operators (for waviness). In this context, profilometry techniques appear as valuable alternative assessment procedures, as they consist of extracting accurate profiles from the paper surface, and its subsequent processing in order to compute summary features, with a clear geometric meaning, being therefore directly oriented to the phenomena under analysis, quantitative and accurate. In this paper, we take advantage of the availability of such accurate profiles obtained using profilometry, and devise a multiscale procedure

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for the simultaneous monitoring of roughness and waviness, based upon the wavelet multiresolution decomposition. 2. Multiscale Analysis of Paper Surface Even though an extensive knowledge base is currently available regarding the way structural elements constituting the paper network interact along the whole hierarchy of scales, when focusing the analysis on a given paper grade, produced on a particular paper mill, with a given paper machine and production practice, there is certainly an amount of specific knowledge whose validity is restricted to such a paper. The extraction of such a knowledge plays an important role in its full characterization and quality supervision. Therefore, we address here the extraction of knowledge regarding those scales where roughness and waviness phenomena are active for the particular grade of paper we are interested in, in order to zoom into such scales when implementing the monitoring task. The information content spread through the different scales was analysed using the wavelet transform [3-5], through which original profiles at the finest scale (7 = 0) can be decomposed into coarser, high scale, approximate versions (y = J^^^), plus all the details lost across the scales ranging from the finest to the coarsest (y = 1: J^^^). These component signals (approximation and details) can be adequately written as the sum of products of expansion coefficients (the wavelet coefficients or the wavelet transform) with the associated basis function (wavelet and scaling functions). Figure 1 illustrates an analysis tool based upon the wavelet transform, that consists of plotting the variance of detail coefficients at each scale as a function of the scale index, in a log-log plot. We have found out that, using such a plot, the profiles for the analysed paper are such that there is a remarkable linear behaviour in the finest scales region, relative to roughness phenomena, a property that facilitates the task of figuring out where a transition point may lie. In the present case, it is possible to detect a change of pattern occurring slightly before scale 6, indicating that roughness phenomenon seem to collapse somewhere between scales 4 and 6, rather than between scales 6 and 7 (vertical lines), as suggested in the literature.

3. Multiscale Monitoring of Paper Surface Profiles The monitoring procedure, presented in this section, was designed to address those situations where the location of a given feature in the length direction of the profiles is not critical, but only the global behaviour at the relevant scales. In other words, only the localized behaviour in the frequency domain is important, which can be effectively extracted through the wavelet transformation, using its localization properties in this domain. Furthermore, flexibility regarding the incorporation of existent background engineering knowledge was also allowed for. The main steps that compose the proposed approach for conducting Multiscale SPC of profiles are the following: 1. Acquisition of profile; 2. Multiscale decomposition of the de-trended profile (i.e., the profile with the linear trend removed), obtaining wavelet coefficients at each scale ( 7 = 1 ^ J^ec -> where J^^^ is the decomposition depth); 3. Selection of relevant scales for monitoring relevant profile's phenomena (e.g., using plots such as the one presented in Figure 1);

Multiscale Analysis and Monitoring of Paper Surface

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10'

10=

10^ ^ s

> 10^

10^

10 Scale index (7)

Figure 1. Log-log plot of the variance of detail coefficients at each scale (/), for 90 surface profiles. Vertical lines indicate a transition region for the roughness phenomena, according to the literature [2].

4. Using only those scales whose indices are relative to phenomena under analysis, calculate parameters that summarize relevant information for product quality control purposes (this may require the separate reconstruction of profiles relative to each phenomenon back into the original domain, by applying the inverse v^avelet transform to a set of processed coefficients, where the only non-zero elements are those corresponding to the selected scales for each phenomenon); 5. Implementation of SPC for monitoring the parameters calculated in step 3; 6. If an alarm is produced, check its validity and look for root causes when appropriate. Otherwise, return to Step 1, and repeat the whole procedure for the next profile acquired. Step 3 allows for the incorporation of external knowledge in the process of selecting those scales that are relevant for monitoring, as well as on defining the relevant monitoring statistics. The overall procedure essentially consists of applying a bank of quadrature mirror filters [4] to the profiles, that basically works as bandpass filters, dividing the frequency domain in octave bands, i.e., bands whose ranges increase proportionally to the mean frequency covered (this organization of the frequency domain turns out to be very adequate for data analysis, since what is usually of interest is the relative band width [6]). Then, those bands relative to relevant surface phenomena are extracted and analyzed.

4. Results In this section we present the results obtained with our multiscale monitoring approach under simulated scenarios and with real world paper surface profiles. In the simulation approach, realistic paper surface profiles were generated, representing a variety of situations that go fi*om typical normal operating conditions to several degrees of

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abnormal situations (moderate and high), in order to evaluate the sensitivity of the proposed methodology to detect shifts, and therefore its potential adequacy for real world implementations. Then, using real paper surface profiles, we tested how the methodology performs in practice through a set of approximately one hundred cross direction paper surface profiles, representing mainly different levels of waviness magnitude, but where some abnormal roughness behaviour can also be found. In both situations, we have used an orthogonal Symmlet-8 wavelet filter [3], with Jj^^ = 1 1 . From the previous multiscale analysis of the paper surface structure, the scale indices, necessary for Step 4, that are associated with the roughness phenomenon were set as J^^^ ={l,2,...,6], while those associated with the waviness phenomenon are given by J^^ = {10,11} . Another task to be addressed in Step 4 regards the calculation of parameters that summarise relevant characteristics of the two phenomena, to be employed for statistical quality control purposes. Based on extended simulations and analysing real paper profiles, we found out that often a single, adequately chosen, parameter is good enough to detect magnitude changes in the roughness and waviness phenomena. The parameters selected to summarize the relevant information for each phenomena are the empirical variance of the reconstructed roughness profile (for roughness) and the maximum deviation from the mean value, D ^ (for waviness), defined as follows: Variance of roughness profiles = — ^

(2) N-l

D

=max(c,c)

(3)

\

^ ^

max

where

P

R={R^]^^^

V /

is the roughness profile, R its sample average. C

and C^

represent, respectively, the largest peak height and the largest valley depth of the profile centred at its mean value. 4.1. Simulation Results The simulated scenarios (Table 1) involve the generation of realistic profiles from typical normal operating conditions, as well as several degrees of abnormal situations regarding roughness and waviness (moderate and high). Figure 2 refers to control charts for waviness (a) and roughness (b), respectively, with 99% control limits established after a preliminary Gaussian kernel density estimation step. From these charts, we can verify that all the shifts simulated under conditions 2-5 are clearly detected in the appropriate control chart, even the one for moderate "piping streaks" irregularity. Table 1. Scenarios covered in the simulation study. Scenario 1. Normal operation 2. "Piping streaks", moderate magnitude waviness 3. "Piping streaks", high magnitude waviness 4. "Cockling", high magnitude waviness 5. Roughness, high magnitude

Multiscale Analysis and Monitoring of Paper Surface

1

200

300

nil

; 2

; 3

200

; 4

i 5

300

Obser\^ion

b) a) Figure 2. Control charts for the monitoring statistics regarding roughness (a) and waviness (b), both with 99% upper control limits. The five sectors indicated in these plots refer to the different simulation scenarios indicated in Table 1. Observation

4.2. Results for Pilot Study To further test the multiscale profile monitoring approach under conditions even closer to those found in real industrial practice, a pilot study was run in the context of a collaboration between our research group and Portucel (a major Portuguese pulp and paper company). Approximately one hundred profiles were gathered, using a MahrSurf mechanical stylus profilometer set, with a Perthometer S2 data processing unit, a drive unit PGK 120, and a MFW - skidless pick-up set. Samples were included that fall within the normal operation quality standards, as well as others corresponding to several types of abnormal situations. Table 2 presents a general description of the samples whose profiles were used in this study. Table 2. Description of surface phenomena exhibited by real surface profiles. Description Reference set No waviness Moderate waviness High waviness Upward trend on Bendtsen roughness

# Observation 1-40 41-61 62-82 83-88 89-98

Control limits were set based on the variability exhibited by the samples from the reference set, following the same approach used with simulated data. The test set contains samples with low, moderate and high waviness, as well as samples that correspond to an upward trend in roughness magnitude, as measured by the Bendtsen tester [2], an instrument based on the air-leakage principle that measures the volume of air flowing between a ring and the paper surface. The results obtained, shown in Figure 3, allow us to verify that the SPC chart for monitoring waviness does indeed follow the magnitude trends of the samples described in Table 2. As for the chart relative to roughness, it is also possible to see that it captures the upward trend in the last 10 samples, besides other significant events scattered over other samples in the test set.

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5. Conclusions In this paper we have presented an approach for conducting multiscale statistical process control of paper surface profiles. The methodology was tested under simulated and real world conditions, and the results demonstrate its adequacy in tracking down the relevant phenomena occurring at different scales. This approach can also be extended to other contexts, outside the paper industry sector, where one also needs to monitor profiles with localized features in the frequency domain.

a)

b)

Figure 3. Control charts for the monitoring statistics regarding roughness (a) and waviness (b). Thefirstpart of the data sets (1) regards reference data, the second (2) is relative to waviness phenomena with different magnitudes (see Table 2 for details) and the third (3) regards an upward trend in roughness, as measured by the Bendtsen tester. References 1. M.T. Kortschot. The Role of the Fibre in the Structural Hierarchy of Paper. The Fundamentals of Papermaking Materials - Transactions of the 11th Fundamental Research Symposium. C.F. Baker (Ed.). Pira, 1997 351. 2. I. Kajanto, J. Laamanen, M. Kainulainen. Paper Bulk and Surface. In Paper Physics, (K. Niskanen, Ed.), Vol. 16. Fapet Oy, Jyvaskyla, Finland, 1998. 3. S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, San Diego, 1998. 4. M. Vetterli, J. Kovacevic. Wavelets and Subband Coding. Prentice Hall, New Jersey, 1995. 5. B.K. Alsberg, A.M. Woodward, D.B. Kell. Chemom. Intell. Lab. Syst. 37 (1997) 215. 6. P. Wagberg, P.-A. Johansson. Characterization of Paper Surfaces Using Optical Profilometry. In Handbook of Physical Testing of Paper. J. Borch et al. (Eds.). Vol. 2. Marcel Dekker, New York, 2002.

Acknowledgments The authors would like to acknowledge Portucel for its collaboration. Financial support from FCT (project POCTI/EQU/47638/2002) is gratefully acknowledged.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Pubhshed by Elsevier B.V.

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A Real Time Adaptive Dynamic Programming Approach For Planning and Scheduling Nikolaos E. Pratikakis^, Jay H. Lee""*, and Matthew J. RealfF ^ ^School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive NW, Atlanta,+ GA 30332, USA Abstract In this paper we propose a real time dynamic programming architecture for learning a high quality policy through simulation geared towards multi-stage stochastic decision problems. The method of real-time dynamic programming is made computationally tractable for practical process scheduling problems by the adoption of a notion called 'adaptive action set.' We apply the proposed approach to an exemplary manufacturing process with stochastic demand and product yield and compare its performance with an 'ideal' solution derived from a mixed integer programming formulation. We also investigate the effect of value table initialization by comparing two different initialization schemes. Keywords: Reinforcement learning, approximate dynamic programming, real-time dynamic programming, planning and scheduling 1. Introduction Optimal resource planning, scheduling, and allocation under uncertainty is one of the important challenges arising in industrial process scheduling applications. Historically, these types of problems have been approached by using mixed integer programming (MIP) methodologies. Despite the rigor of optimality math program based methods are limited in terms of number of scenarios they can consider due to computational reasons. The focus of this paper will be on an alternative strategy for solving such problems based on real-time dynamic programming. The aim of the strategy will be to learn a high quahty policy, not necessarily an optimal policy, for infinite horizon dynamic decision problems for complex stochastic systems, in a computationally tractable way. The system is to be formulated as a Markov Decision Process (MDP), a formal definition of which can be found in [1]. The proposed architecture combines mathematical programming(e.^., MIP), localized function approximators [2] {e.g., /c-Nearest Neighbors), and real time dynamic programming (RTDP)[3]. One computational method for constructing an optimal pohcy for a MDP is Dynamic Programming (DP). An essential concept in DP is the so called 'value function,' which *The authors gratefully acknowledge the financial support from the National Science Foundation (CTS#03019993)

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maps a state SQ to the expected reward starting from that state for a given poUcy TT G 11, as expressed in Eq.(l), assuming an infinite horizon formulation: oo

J^(so) = £;„.E7V(s„7r(5,),a;*)]

(1)

In particular, the focus is on finding the optimal value function, which corresponds to the optimal poUcy TT* such that: J^*(5o) = maxJ^(5o)

(2)

TTGII

The classic D P algorithms that guarantee convergence to the optimal value function include value iteration, policy iteration, and linear programming (details in [1]). These classical algorithms are rendered impractical, however, by the "curse of dimensionality", which refers to the exponential growth of the computation with respect to the dimension of the state space. For instance the conventional value iteration requires 1) the enumeration and storage of the value for each (discretized) state, and 2) repeated updates of the value function of each state until convergence. We note that each value function update calls for the evaluation of the reward under all the possible actions with respect to that state in order to find an optimal one. In real applications, the size of the state and action space may be such that it is very difficult or even impossible to store and update the values for all the states. Also for uncountable state and action space, discretization and interpolation are essential, but the resulting errors may destroy the convergence to the optimal value function. The proposed RTDP approach, on the other hand, learns the value function in a manner similar to the reinforcement learning techniques, that is, by having the agent interact with its stochastic environment, typically through simulation, to gradually build a value table for those states encountered [3]. In addition to building the value table for the encountered states only, we also try to alleviate the curse of dimensionality associated with the action space by considering a so called 'adaptive action set (AAS),' denoted as Asuh from here on, for each state. The AAS, which ensures asymptotic convergence for stochastic shortest path problems, is used to modify the RTDP approach so that it is computationally tractable for practical scheduling problems. We call the modified RTDP method real-time adaptive dynamic programm^ing (RTADP). Note that the original RTDP architecture was proposed to solve robotic MDP path planning problems, and it considered every possible control action in updating the value table as the number of possible actions was rather limited. However, this is not the case for typical process scheduling problems and complete evaluation of all possible actions may be impractical. 2. A R e a l - T i m e A d a p t i v e D y n a m i c P r o g r a m m i n g ( R T A D P ) A p p r o a c h The proposed approach is a variant of the RTDP approach as described in Barto et al [3] and combines the RTDP approach with the k-nearest neighbor (k-NN) approximation scheme outlined in Lee et al [2]. 2.1. D e s c r i p t i o n of A p p r o a c h The procedure gradually constructs a value table starting from an empty one and gradually adding more and more entries as they are encountered. Also, as same states are

A Real Time Adaptive Dynamic Programming Approach for Planning and Scheduling

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visited multiple times, its value estimates gets better and better. One requirement for this procedure is the construction of a mathematical optimization model or a heuristic policy for each application of interest so that the state space can be explored with reasonable actions. The following describes the steps involved in each update. S t e p 1 Each iteration starts with a state Si. S t e p 2 Find the set of states Ms{si)^ which contains the states within the 8 distance of Si. Here we use the Euclidean distance metric d, as proposed by [2], with a user defined design parameter 6.

Find

N5{si)^=[seS

: d=yj{s-

s^yw^s

- 5,) < j |

(3)

where VF is a feature weighting diagonal matrix. The detailed can be found in [2]. S t e p 3 Update the value functions for the k-nearest neighbors in J\fs{si) (represented from here on as Afki^i) C A/5(5^))according to Eq.(4). In order to do each update, we construct the Agub, {^sub C A, where A is the set of all possible controls that the decision maker can exercise at any time instance for a given state). Details about the notion of Agub are given in Section 2.2. Every control in Asub is evaluated with respect to the objective function in Eq.(4) and the decision -maker follows a policy that is greedy with respect to the most recent estimate of the value function (Eq.(5)). The reader can find more insight about the optimality equation Eq.(4) and the convergence property for discounted infinite horizon MDPs in [1].

J'+\si)

= max { r ( s „ a ) + 7 E n , | . . , a J ' ( s , ) } , 7 € [0,1)

a*{si) = a.rgmajc{r{si,a)

+ -fY. ^sj\si,aJ\sj)}

(4)

(5)

The evaluation of each candidate optimal control Eq. (4) requires an initial approximation of J{sj). More details about this are outlined in Section 2.3. S t e p 4 Update the value of Si via the Bellman equation (Eq.(4)). Successor states Sj are augmented to 5 , if the cardinality of neighborhood ^fs{sj), defined by (Eq.(2)), is less than k. S t e p 5 A state Sj is sampled according to the probability distribution p{sj\si,a*) as defined from the Markov model of the random variables. The sampled Sj is set as Si^i. Set i = i-\-l and we go back to Step 1.

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2.2. K e y E l e m e n t s of A A S The AAS is constructed in such a way that, despite the partial evaluation of the action space in each update, we can guarantee a worst performance bound and convergence in the long run for a certain class of applications. It is also desired that the quality of the value estimates for each state in the table increases with the iteration. The key elements of the AAS are: 1. Heuristic actions: Heuristics map states to actions via ap priori knowledge about the problem. By including the controls given by the heuristics, we ensure that this approach gives the worst-case performance that is better than the expected performance of the best among the tested heuristics. It is also possible to use the actions resulting from the deterministic math program formulations, which represents a suboptimal policy, as we do in this paper. 2. Best known action: This action summarizes the a-priori learning with respect to the value function. If the state to be updated is a state never visited before, then its best known action is simply empty. If the state has been revisited, a best known action should have been stored with respect to the prior estimate of the value function. 3. Random actions: Random controls ensure that we effectively explore the entire action space and exclude the possibility of not visiting any portion of the state space. This enables us to ensure asymptotic convergence with probability one in applications with a goal state. 4. Other candidate optimal actions can originate from the nearest neighbors of the state to be updated. 2.3. Calculation J^+H^i) The calculation of J^^^{si) involves the knowledge of the value function of all possible successive states J^{sj) for each action in Agub- During this calculation we will encounter one of these three possible scenarios. • Scenario 1: All s/s belong to S. In this case all states have a registered value function J^{sj) in the value table. We use these values for the update of J^^^{si). • Scenario 2: Some of the Sj's do not belong to S. In order to approximate the value functions for these states, we utilize a local /c-nearest neighbor approximator [3]. The mathematical expression for approximating the value function for each Sj is as follows:

J'('i) = l

E

J'i^)

(6)

• Scenario 3: W h a t if Msisj) < k ? One approach that is intuitive and computationally the cheapest is to initialize every successive state with a low value function bound, assuming a maximization problem. An alternative tested is, for a given Sj,

A Real Time Adaptive Dynamic Programming Approach for Planning and Scheduling

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we sample M future scenarios ^i, ^ , . . •, ^M from the underlying probability distribution and construct the following sample average:

(7) 1=1

where F{sj,(i)

is the optimal objective function for each scenario for state Sj.

3. P r o b l e m D e s c r i p t i o n of a Manufacturing J o b S h o p U n d e r U n c e r t a i n D e m a n d and P r o d u c t Y i e l d We concentrate on an example of internal capacity adjustment and resource allocation of a single agent in a single product setting, in order to illustrate the RTADP approach. The specific structure of the manufacturing process considered is shown in Fig 1. The formulation of this problem as an MDP can be found in Pratikakis ^13 et al [4]. Briefly the manufacturing process is divided into three interdependent stages with physical queues to represent the in-process inventory and final product inventory The demand for the final product is stochastic and is represented by a Markov chain. A further complicating factor in this example is the uncertainty in the product yield. The rate of product failure is also a ranFigure 1: Infrastructure of an dom process characterized by a two state Markov chain agent. (with high and low failure rate states). The decisions are made on a discrete time grid. At each grid point, the number of machines (representing the production capacity) can be increased or decreased by a limited amount and for some cost. In addition, the the decision maker has the flexibility of idling a percentage of the available machines on a temporary basis.

ia

t-

4. R e s u l t s The value function of every successive state to be augmented to the value table is initialized with a constant lower value bound. We also assign the lower values, which act as barriers in the maximization problem, when exploring the states corresponding to the queues at the second, third and stock level greater than a threshold queue of 600. The exact form of the stage-wise reward r{s,a), for the results (Fig. 2) can be found in [4]. Good performance is achieved, if we meet demand D at each time period, control the stock level near the desired value of Sdes = 500 5 and minimize the queue lengths at station 2,3 {w2^Ws) by adjusting the production resources: We compared the RTADP against a MIP formulation, which uses knowledge of the future realizations of the random variables. Hence the MIP solution represents an idealistic policy that is not implementable in practice. The results show the performance gap of ~ 1 3 % in favor of the MIP. These results are very satisfactory, since the performance by

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A^.^". Pratikakis et al 500 r MIP full information

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c 200 o

CO

o 400

150

0) Avg. Performanc;egap per 30 Time Per ods RTADP |59.88

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Figure 2. Cummulative results comparing an MIP with full information and RTADP.

this ideal policy is not even attainable in practice, and prove the viability of this method for scheduling and planning applications under Markovian uncertainty. Moreover we can observe that RTADP controls St bias to Sdes and minimizes W2^w^. On the other hand the MIP controls St perfectly to Sdes but operates the shop, on average, with more resources at all stations. Gradually, RTADP "learned" good estimates for all entries in the table and operated the system smoothly with less resources and without facing any "stock out" event throughout the simulations. The simulation baseline parameters are not displayed in this paper due to the space constraint but you may contact the authors for the simulation details (email:[email protected]). REFERENCES M.L. Putterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming , New York: John Wiley and Sons, 1994. J.M. Lee, N.S. Kaisare and J.H. Lee, Jourmal Process Control, Vol.16, Issue 2, Feb.2006,pp. 135-156. 3. A. Barto, S. Bradtke and, S.Singh, Artificial Intelligence, Vol.72, (1995), pp.81-138. 4. N.E. Pratikakis, M.J. Realff and J.H. Lee, Submitted to Special Issue of J. Naval Research Logistics.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Theoretical Analysis and Experimental Studies of Mixed Product Run-To-Run Control Ying Zheng,^ Ming-Feng Wu,^ Shi-Shang J a n g , S n d David Shan-Hill Wang^ ^Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R.China ^Chemical Engineering Department,National Tsing-Hua University, Sec. 2, Kuang-Fu Rd No.l01,Hsin Chu,300, Taiwan Abstract In this study, a model two-product plant was developed to investigate the "toolbased" and "product-based" approaches. The closed-loop responses are derived analytically and control performances are evaluated. The experimental study of the preparation of nano scale SiOi particles is investigated to verify the validity of the analysis. Keywords: run-to-run control, tool-based control, product-based control, experimental study, SiOa particle size control. 1. Introduction Semiconductor manufacturing has always been an industry of high capital investment. In the last decade, run-to-run (RtR) control has been widely accepted to assure the product qualities by deleting shift, drift and patterned variation of production systems(Moyne, et al, 2001; Del Castillo, et al., 2002). Two intuitive ways of defining a control scheme for a mixed production mode are (i) each run of different products is used to estimate a common tool disturbance parameter, i.e. a "tool-based" approach, (ii) only a single disturbance parameter that describe the combined effect of both tool and product is estimated by results of runs of a particular product on a specific tool. In this work, a model of two-product plant was developed to investigate two different "tool-based" and "product-based" control approaches. The closed-loop responses are derived analytically and control performances are evaluated. In addition to the above theoretical analysis, the experimental study of the preparation of nano scale Si02 particles is investigated to verify the validity of the approach. The control objective is to produce the particles in different scale under the effect of the composition decay of ammonia solution. Note that this study is similar to a tool drift existed in a semi-conductor batch process. It is shown that the experimental study is consistent to the theoretical analysis.

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2. Theoretical Development 2.1. System model In this section, we consider a simple case that two products are manufactured on a single tool. The production schedule consists of cycles of i runs, in which j runs are used to produce product 1 and (i-j) runs are used to produce product 2. Assume that the input-output relations for the two products on the given tool are linear with different slopes O), a2 and intercept yffy, p2 and share the same tool disturbance //: y,n^ = «i+A^,v.i + '7..i

= «,+M,»;+

Yu.j =

Y it+j+l

™

fl,nj

= a,+fi2X,t+j+l

"*" ^it+j+l

where );^^(« = l,2,---,y), i;^„(«=7+l,7+2,---,/)are the outputs of product 1 and 2, and X.^^^[n=\2,"',{) are the control actions at run it+n. To model the change in tool condition, IMA(1,1) disturbance with drift is used:

//^ ~Vt-\ ~ ^~^^t ~^t-\ ->

where S is the deterministic drift and £; e A^(o,a^). In a "tool-based" control, a single "tool-noise" //.^^^ ( « = 1,2,--,/) is estimated from input-output data and two linear models for the production of the two products. This observed "tool noise" can be filtered by a EWMA filter: %.„=^%.«+(l-^)^,Y.„-i (2) whereO
In a "product-based" control approach, the EWMA filter action is performed with respect to the last run that the same product is processed. So, ^{Xt.i -h^ini -^)+(i-^)^(/-w

«=i

(4)

The deadbeat control actions for product 1 can be formulated as: ^-^l-%-l)t^J

b, ^it+n

-

n=l

(5)

Theoretical Analysis and Experimental Studies of Mixed Product Run-to-Run Control 1187 The performance of a controller can be evaluated by the asymptotic mean square error (AMSE) as AMSE{Y^) = lim[7/J t—>°°

2.2. Instability of Tool Based Control Assume ^^ =j3^/b^, q^=l-

A^j,

n=l,2 being the modeling errors in process gain, and

q2=l-

X^2 -> ^ ~ ^i~^^\

» we get the following theorem.

Theorem 1: The process is unstable by "tool-based" control if (2) {)
and the disturbance is non-stationary, i.e., contains a deterministic

or a stochastic integrating part {6 ^\)\ (3) 0 < g < 1 and there is a difference in error of the process gain estimates between products, (^^ ^^2 )• Since the plant-model mismatches for different products are unlikely to be the same in an actual plant, a "tool-based" control can be applied only when if the process disturbance contains neither a deterministic drift nor any integrating stochastic components. 2.3. Performance ofProduct Based Control Theorem 2: Given a process disturbance that is IMA(1,1) with drift, the AMSE for the response at the n*** position of the production cycle using a product-based EWMA control is given by: AMSE{Y„) = S^in)d' + '^^-^"^~^^^.'' ^ a "

with SJn) =

\
(6)

•+ -

1-q/

1-q,

1 qj

Theorem 3: The average of AMSE over all different positions of the campaign cycle of product 1 is given by

AMSE, {U) = {S^}{ij)S^ ^{S^.){ij)-2qr0^

"•—

^-*

1-.

•

[l-ql)\l-q^] J

(7)

r-

(/-*r J

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Y. Zheng etal

The argument (ij) is used to denote that this is the closed performance obtained for a particular schedule. Since a finite value for AMSE is obtained, the control scheme is stable. Some simulation studies can be found in our previous work (Zheng et al., 2005). 3. Experimental Study 3.1. The Development of the Model In planar and three-dimensional space, material refractive index causes periodical changing structure. This structure generates band gap, i.e. Photonic Band Gap, on the frequency spectrum, and this is the basis of photonic crystal. It is our objective to implement RtR control to the preparation of silica particle diameter which affects the photonic band gap wavelength of photonic crystal. Assume that particles in different scales are prepared in a single batch system, and the system is subjected to the effect of decay of the composition of ammonia solution. Stober et al.(1968) gave the total chemical reactions of the hydrolyzing reaction and the condensation reaction as following: Si{OC^H,), Si{OH),

+ 4H^0

"""' ) Si(OH), + AC^H.OH

-> SiO^ i +2H^0

In this the model derived by Bogush et al.(1988) is implemented to predict the diameters of Si02 particles given the concentrations of TEOS solution and NH3 solution as following: I d = A [Hfif Qxp(-B [H^OY) A = [TE0S]2{S2-I5l [NH,] + 1200[ NH,f -366

[NH,f)

5 = 1.05 + 0.523 [NH,]-0.12S [NH,f

where d is silica particle diameter (nm), [H2O] is water condensation (M), [TEOS] is TEOS condensation (M), [NH3J is ammonia condensation (M). Hence the particle diameter is affected by ammonia condensation, water condensation, TEOS condensation, reactive temperature, reactive time, and water/TEOS condensation proportion. The above model is linearized and the linear model is, in turn, implemented to control the particle sizes of the SiOi particles.

Theoretical Analysis and Experimental Studies of Mixed Product Run-to-Run Control 1189 3.2. The Experimental Setup TEOS*Ethanol

V

NH3*H20*Ethan«l

/

A,B measuring

Two open flask

flask

J ^ Q ^pgn

fl^sk

A,B measuring f

OO (b)

Fig.l Experimental steps Set [TEOS]=QAlM,

[H20J=5M

Temperature=35 °C . V= 100ml. The experimental steps are shown in Fig.l. After paricles preparing experiment, SEM is used to observe 3 drop liquid and 2 pictures are taken each drop. The diameter of 100 particles as shown in Fig.2 can be analyzed using software OPTIMAS. The average and the standard deviation of the particle diameters can be observed.

Fig.2 The particle analysis graph 3.3. Experimental results In this work, two different sizes particles (d=200nm and d=250nm) are prepared at one experimental setup as depicted the previous section. Fig.3 and Fig.4 show control results of the particle diameters of two products using the "tool based" and the "product-based" approach respectively. As shown in these two figures, it is found that "product-based" approach is superior to "tool based" approach. Fig.5 shows the error of the output using the two approaches. It is clear that the product output is convergent using "product-based" approach, while divergent using the "tool based" approach, and this is consistent to our theoretical derivation in the previous section.

Y. Zheng et al.

1190

290-

290-

280-

280-

11 11 ll II 1

270260•g" 240-

270260-

230-

5 230-

1

220-

6

1

1' t 11t ' I

180170160150-

T

220-

190180170-

i 1

160150-

Fig.3 Two products controlled by "tool based" approach

Fig.4 Two products controlled by "product based" approach m

J * 1

•

-

."

1

*

•

J

•

* •

\

*

J

*

1 _{

I ill t i

•§• 240-

S

190-

I

• ,—^

,__^.

,

,

,_..j_™.,

,

,

•

« • • •

Tool (T=250nm) Too! (T=200nm) Product {T=250nm) Product (T=200nm)

^—

, — _ j _ — , —

Fig.5 Errors of the products using two different approaches

4. Conclusion In this work, a rigorous analysis of two different runto-run control strategies in a high mixed production is performed. We found that a tool-based approach is some times unstable while a product-based control is stable. The experimental study of the preparation of

nano scale Si02 particles verifies the theoretical analysis. References Bogush G..H., Tracy M.A., Zukoski IV C.F. (1988). Preparation of monodisperse silica particles: control of size and mass fraction, Journal ofNon-Crystalline Solids , 104 , 95-106 Del Castillo, E (2002). Statistical Process Adjustment for Quality Control. John-Wiley and Sons, New York. Moyne, J., E.D.Castillo, and A.M.Hurwitz (2001). Run-to-Run Control in Semiconductor Manufacturing. CRC Press, Florida. Stober W., Fink A., Bohn E. (1968). Controlled Growth of Monodisperse Silica Spheres in the Micro Size Range, Journal of Colloid and Interface Science, 26, 62-69 Zheng, Y., Lin, A.H., Wang,D.S.H., Jang, S.S. and Hui,K.(2005). Stability and Performance Analysis of Mixed Product Run-To-Run Control. J. ofProcess Control, In Press.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Methods of State Estimation for Particulate Processes M. Mangold,^ C. Steyer,^ B. Niemann,^ A. Voigt,^ K. Sundmacher ^'^ ^Max-Planck-Institutfur Dynamik komplexer technischer Systeme, Sandtorstr. 1, 39106 Magdeburg, Germany ^Otto-von-Guericke- Universitdt, Lehrstuhlfur Systemverfahrenstechnik, Universitdtsplatz 2, 39106 Magdeburg, Germany Abstract Determining property distributions of particles online by measurement is difficult in many cases, especially if the particles are in the nanometre range. An alternative may be state estimation techniques, which use information from process simulations in addition to the measurement signals. Two examples of state estimators for particulate processes are presented in this contribution. The first one is an extended Kalman filter based on a population balance model. The second one is a bootstrap filter based on a Monte Carlo simulation. Keywords: particulate processes; state estimation; population balance; extended Kalman filter; bootstrap filter 1. Introduction A large amount of chemical products are generated in the form of particles. Very often, property distributions have a strong influence on the product quality. Therefore, online information on a property distribution like the particle size is desirable. However, it may be difficult to obtain such information directly from measurements, especially if the particles are very small, i.e. in the nanometre range. State estimator techniques may be a solution to this problem. A state estimator consists of a simulator part containing a process model and of a corrector part, which uses the measurement information to let the estimated state converge towards the actual state. As an example, a state estimator using a population balance model of a bulk precipitation process is presented in Section 2. After discussing the question of observability for this system, an extended Kalman filter is designed based on the population balance model. As an alternative to population balance models, Monte Carlo simulations are gaining increasing attention in the field of particulate processes. In order to incorporate a Monte Carlo simulation model into a state estimation framework, suitable correction techniques are required. Such techniques have been developed over the last years (Doucet et al., 2001). An example is the state estimator for a micro emulsion process discussed in Section 3. 2. Population Balance Model of a Bulk Precipitation Process The precipitation of nanoparticles in a batch process is considered in the following. Only the key features of the model are presented here. For details on the derivation of a population balance model, see e.g. (Oncul et al., 2006) and references therein. In the liquid phase, two components A and B react to a component C that forms solid particles. In the model, the liquid phase is assumed to be perfectly mixed. The

1191

M Mangold et al

1192

nucleation and growth of the particles is described by a one-dimensional population balance model, where the particle size is used as the property coordinate. Further model assumptions are isothermal conditions and negligible agglomeration and breakage. 2.1. Model Equations The mass balances of the liquid phase read dc

-

k^C^Cg

(1)

dt dCr dt

= ^VCACB - CG

\(^IG{CC,dp

)fddp - C^^^c^,

(2)

dp=dp.

where C/ denotes the molar concentration of component /, r{cA,CB) is the rate of the liquid phase reaction; / is the number density function of the particles; dpo and dp^rnax are the minimum and the maximum particle size, respectively; G is the particle growth rate, and CG , Cnuc J K are constant factors. A population balance for the particles of component C leads to the following equation for the number density function/

f =- ^ ^ ^ ^ ^ ^ ;

G{c,,dpMd..,t)

= B{c^

(3)

The following expressions are used for the growth rate G and the nucleation rate B: ( G{cc,dp) = k^ ^c-^^expl -11 (4) ydp J

B{cc) = k„

f r ^^ ^C

^C,c

exp

\^dpQ J J

(5)

(^G, l^nuc, b, C2, and c^^^ are constant parameters). It is assumed that the total number of particles is measurable, i.e. the following relation holds for the measurement variable y: **r,max

y =

\fddp

(6)

dp=dp

2.2. Investigation of Observability A system is observable, if it is possible to reconstruct its complete initial state from measurements taken over a finite time interval; see e.g. (Levine, 1996). Observability is a necessary condition for successful state estimation. A strict proof of observability is very difficult for a nonlinear distributed system. In the following, some preliminary results on the observability of the bulk precipitation model Eq. (l)-(6) will be given. If the system is observable, it must be possible to solve the equations for y, dy/dt, (fy/d^,... uniquely for the system states (Zeitz, 1977; RothfuB and Zeitz, 1995). From Eq. (6) and(3) one obtains for f\dp^^^ J)=0:

f=.fc)

(7)

Methods of State Estimation for Particulate

1193

Processes

Eq. (7) can be solved uniquely for cc, i.e. CQ is observable. Assuming for the moment that the integral in Eq. (2), which is denoted as I(t) in the following, is known, algebraic equations for CA and CB can be derived from Eq. (1) and (2):

do c, = Cgk^

dt

- + I + C^

(8)

f^^C n

dcc + I + C„^c, - c , dt

( A^

d'cr

dl

dr

dt

+ C„

dt (9)

Jdc, V dt

+ J + C„^^c^ = 0

Eq. (8) and (9) can be solved at least locally for Q and CB . This means that CA and CB are observable if it is possible to reconstruct / and / from measurements. In order to prove observability of the number density function / one has to show that / can be determined uniquely from the equations for j^, cfy/dt^, (fy/dt^,..., which have not been used yet in the observability analysis. By deriving Eq. (6) with respect to time, inserting the balance equations (l)-(3), and dissmmng f\dp ^^^j)=

0 , one obtains expressions

of the form \Pifddp =F.{y,dy/dt,...,dy/dt\c^,Cs,Cc\

with PQ =l;P2 = d^Gip.

i = 0,2,3,...

(10)

= 3p._i IddpG (/ > 3).

In order to be able to reconstruct/from the projections Eq. (10), the functions/>/ should form a complete set of basis functions. By developing G into a Taylor polynomial in dp and evaluating the expressions for pi, it is possible to see that the left-hand side of (10) contains arbitrary powers of dp, if G depends at least quadratically on dp, as it is the case for the model used here. This is a necessary condition that the model of the bulk precipitation process is observable. A further hint that the system is observable is the (numerically evaluated) full rank of the observability matrix of the linearised version of the discretised system. 2.3. Extended Kalman Filter For the numerical solution of the bulk precipitation model, the population balance (3) is discretised along the property coordinate dp, using a finite volume method. Based on the discretised model, a continuous-discrete extended Kalman filter with a continuous simulator part and time discrete filter update part is developed. The design equations for such a filter are well-known, see e.g. (Gelb, 1974), and are not listed here. The spectral density matrix of the process noise Q and the covariance matrix of the measurement noise R are chosen as diagonal matrices with entries on the main diagonal only. The filter is tested in simulations. Virtual measurement data are used, which are generated by adding normal distributed artificial noise to the simulation results of the bulk precipitation model. A time interval of 1 s between two measurements is assumed. A simulation result is shown in Fig. 1. The initial concentrations of ^ and B are chosen intentionally 5 % too high in the filter model. If the filter update is deactivated

1194

M Mangold et al.

(diagrams in the left column), then this small error in the initial conditions causes the estimated number density function to deviate considerably from the number density ftmction of the real system. If the filter update is active (diagrams in the right column), then the estimated particle size distribution converges rapidly towards the particle size distribution of the real system.

without filter correction •

with Kalman filter

estimate L measurement |.

^ 0.8

0)

E 3 c o ^

E 0.6 ^0.4

CO

_cg o

20

40 timet/s

60

^0

20

40 timet/s

60

with Kalman filter

without filter correction

— —

o 4

real value estimate |.

^-—__18s

a 2 F. 1- E

10"" 10" particle size d / m

10

So E10

_-36s 54s K Nv/\rs70s 18s^ 10"° 10"' particle size d / m

10"'

Figure 1. Test of the extended Kalman filter for the bulk precipitation process in simulations; lefthand side column: comparison of reference states and estimated states, if no filter update is used (pure simulation); right-hand side column: estimates of the extended Kalman filter and comparison with reference states; top row: estimated and reference measurement value; bottom row: estimated and reference particle size distribution.

3. Monte Carlo Simulation of a Micro Emulsion Process Models of particulate processes become computationally expensive, if several property coordinates have to be taken into account. In such a case, the use of stochastic Monte Carlo simulation approaches may be advantageous. Therefore, it seems worthwhile studying the application of Monte Carlo simulations to state estimation problems. As an example, the BaS04 precipitation in micro emulsion is considered in the following, where two reactants BaCb and K2SO4 dissolved in aqueous droplets react to BaS04. A Monte Carlo model of the process was presented in (Adityawarman et al. 2005). Its main features are summarized in Section 3.1. In order to use such a model in a state estimation framework, suitable filter correction techniques are required. Various

Methods of State Estimation for Particulate Processes

1195

recursive Bayesian state estimation techniques have been developed over the last decade (Doucet et al., 2001), but -with few exceptions (Goffaux and Vande Wouver, 2005)have hardly been used in chemical engineering applications. The concept of these methods is illustrated by a bootstrap filter for the micro emulsion process in Section 3.2. 3.L Model Description The Monte Carlo simulation is initialised with a given number of droplets. The droplets are filled with one of the two reactants BaCl2 (component A) or K2SO4 (component E). The droplet coalescence and redispersion are the basic steps of the Monte Carlo simulation. For a coalescence event, a pair of droplets is randomly chosen and mixed. A and B in the droplets are assumed to react instantaneously to BaS04. The number of BaS04 molecules present in the droplet is compared to critical value X^uci related to supersaturation. If the number of BaS04 molecules exceeds Xnuch a particle nucleation can occur, depending on a rate coefficient for such an event. In the redispersion step, the remaining liquid reactants are distributed randomly back into two identical droplets. The particle is put randomly into just one of the droplets. If a particle is already present in one of the droplets then all dissolved BaS04 will be used to let that particle grow. As in the previous example, it is assumed that only the total number of particles is measurable. 3.2. Bootstrap Filter The basic idea of recursive Bayesian estimation is as follows (Doucet et al. 2001, Gordon et al., 1993). One starts from the model description X, = f ( x , _ l , ^ , _ l )

(11)

y = h(x,,vj,

(12)

where x is the state vector, y is the measurement vector, co is the process noise, and v is the measurement noise (in the example of the micro emulsion process, f is given implicitly and evaluated by Monte Carlo simulation). The initial probability density distribution (PDF) of the state vector /?(xo) is assumed to be known. The PDF at a later time t can then be predicted from the previous estimate of the state vector Xt.i and past measurements according to:

p{^, I y,-i) = \p{^, I \-x )p{^,-i I y 1.-1 )^r-i

(13)

When a new measurement j ^ becomes available, the predicted PDF can be updated using the Bayes rule:

/

p\^t

.

N

p{yt\\)p{^t\yv,-x)

I y 1:^) - T ^ — \ — ^ 7 — ]

r ^

(14)

lp{yt\^t)p{^t\yi.t-i)d^t The evaluation of the high-dimensional integrals in Eq. (8) and (9) is very difficult in practice. However, Gordon et al. (1993) proposed a simple sequential Monte Carlo algorithm for an approximate solution, the so-called bootstrap filter. To test the bootstrap filter for the micro emulsion example, it is assumed that the initial ratio between the amount of ^ and the amount of B is non-stoichiometric and unknown. The filter is tested with virtual measurement data obtained by adding noise to the result of a previous Monte Carlo simulation. The initial conditions of the N samples used by the filter differ in the A/B ratio that goes from 0 to 0.5. Figure 2 shows the state of the filter after 10 measurement intervals and update step. From Figure 2 (a) it can be seen that after a few time steps most of the samples group rapidly around the reference A-to-B

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M Mangold et al

ratio of 0.4. Figure 2 (b) shows the estimated particle size distribution at time step 20 which agrees well with the reference size distribution. S 0.2 40 estimate reference

.-^0.15

30 Q. CD

(D •D

S20 E

n

0

E

!^B

10

I 0.05 "co E

^0

0.1 0.2 0.3 0.4 initial A-to-B ratio in droplets

0.1

0.5

A

L

1 2 particle diameter / nm

Figure 2. Bootstrap filter for a microemulsion process; (a) distribution of the samples at different time steps; (b) comparison between reference size distribution and estimate at time step 20.

4. Conclusions State estimation techniques or model based measurements may be a useful tool for the process operation of particulate processes. They can serve as software sensors in cases where a direct measurement of property distributions is difficult or not possible, e.g. due to a small particle size. If population balance models of moderate complexity are available, classical observer design methods can be used to set up a state estimator, as was illustrated in Section 2. However, the solution of population balance models in real time may be difficult, especially for models with several property coordinates. In such a case, Monte Carlo simulations may be used as a viable alternative. The use of Monte Carlo simulations for state estimation requires appropriate filter correction techniques. The results of Section 3 indicate that recursive Bayesian filter techniques are a promising approach to this problem.

References D. Adityawarman, A. Voigt, P. Veit, K. Sundmacher, 2005, Precipitation of BaS04 nanoparticles in a non-ionic microemulsion: Identification of suitable control parameters, Chem. Engng. Sci., 60, 3371-3383. A. Doucet et al. (eds.), 2001, Sequential Monte Carlo Methods in Practice, Springer. A. Gelb (ed.), 1974, Applied Optimal Estimation, MIT Press, Cambridge. G. Goffaux, A. Vande Wouver, 2005, Bioprocess State Estimation: Some Classical and Less Classical Approaches, In: T. Meurer, K. Graichen, E.D. Gilles (eds.). Control and Observer Design for Finite and Infinite Dimensional Systems, Springer, Berlin, 111-130. N.J. Gordon, D. J. Salmon, A.F.M. Smith, 1993, Novel approach to nonlinear / non-Gaussian Bayesian state estimation, lEE Proceedings-F, 140:107-113. W.S. Levine (ed.), 1996, The Control Handbook, CRC Press, Boca Raton. A. Onciil, K. Sundmacher, A. Seidel-Morgenstem, D. Thevenin, 2006, Numerical and analytical investigation of barium sulphate crystallization, Chem. Eng. Sci., 61, 652-664. R. RothfuB and M. Zeitz, 1995, Einfuhnmg in die Analyse nichtlinearer Systeme, In: S. Engell (ed.), Entwurf nichtlinearer Regelungen, Oldenbourg, 3-22. M. Zeitz, 1977, Nichtlineare Beobachter fur chemische Reaktoren, VDI-Verlag, Dtisseldorf

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Simultaneous Scheduling and Optimization of a Copper Plant liro Harjunkoski, Hans Werner Borchers and Marco Fahl ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Abstract This work presents a novel batch scheduling solution that has been developed for a copper plant. The major characteristics of the copper production process with some specific interdependencies are briefly summarized, the main modeling challenges and some key elements of the resulting mixed-integer optimization problem are presented, followed by a concrete example problem highlighting the benefits of the solution. Keywords: Copper production, planning, scheduling, recipe optimization 1. Introduction The copper production process is difficult to plan and schedule due to the lack of sufficient and exact measurement data, complex logistics, high raw-material variability, frequently occurring disturbances and maintenance operations that heavily affect the ideal process cycles. Normally, the scheduling is done manually and a "global" overview of the overall production process is usually missing. Thus, typically each unit ends up running at "full speed", i.e. trying to produce as much as possible as this is the intuitive local optimum from an individual equipment perspective. This results in productivity losses since the overall process efficiency may be far from optimal as many batches end up unnecessarily waiting for equipment in the next production stage. Only very few contributions on copper plant scheduling problems can be found in the literature, see e.g. Pradenas et al. (2003), where a heuristic algorithm is used to maximize the copper production at the Chuquicamata Copper Smelter in Chile. The method first generates combinations of three possible production cycles and then tests the resulting schedule against the best existing solution. The objective is to maximize the troughput. The main constraints are operational, metallurgical, mass balance related, environmental or related to the timing of loading and unloading operations. The scheduling and optimization approach that is presented here considers simultaneously, and in a rigorous way, the most important aspects affecting the production process. It enables a more efficient production, better overall coordination and visibility of the process, faster recovery from disturbances and supports optimal maintenance planning. The resulting batch scheduling model is solved as an MILP and covers the main processing steps, making it possible to optimally relate input material properties to corresponding processing times. By taking into account main chemical reactions, recipe optimization is performed simultaneously ensuring that the schedule is always optimal with regard to varying raw-material quality. In the following, we briefly summarize the major characteristics of the copper production process.

/. Harjunkoski et ah

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2. The copper production process The copper production process discussed is a batch process that covers two parallel production lines, with the typical requirement to synchronize the lines such that they serve the caster in an alternating way with anode copper, see Fig. 1. The processing is done under strictly controlled temperatures where several chemical reactions take place to enable the separation of copper from other material. The processing in each stage consists of multiple phases, e.g. the four major converter operation phases are: Charging, slag blowing, copper blowing and discharging. Typically, it takes 14-18 hours for one batch to go through the whole process.

Converter Primary Furnace

pranes

Converter

Anode Furnace

O:

Caster

Anode Furnace

Figure 1. Parallel processing equipment A special aspect on the production process is that there is only one unique end product copper anodes with the final copper content of 99.6%. The major production challenge originates from highly varying raw-material properties, which affect the optimal production recipe. An example of variations is the copper content of the matte (melt copper from the primary furnace), which may vary between 60% and 70%, depending on the chemical composition of the raw-material (copper concentrate) used. A high copper amount reduces significantly the required processing time at the converter and furthermore also results in less slag that needs to be collected along the processing. Low copper content, on the contrary leads to longer processing times, large slag amounts and a need for more so-called cooling material, i.e. recycling material with high Cu-content. The raw-material quality variations have a high impact on the scheduling decisions. For this reason, the scheduling model must also comprise main mass-balances, most important dependencies between the material amounts and purities, and the processing times. Consequently, an appropriate scheduling model also needs to encapsulate a simultaneous recipe optimization for each batch. As in most copper plants, the material is transported in ladles using cranes. Thus, the process must also be synchronized from the production logistics perspective such that parallel activities do neither overload nor block the cranes. This poses additional constraints to the scheduling problem and highlights the requirement of having a complete overview on the plant when doing the detailed production scheduling. 3. Scheduling problem According to the process description above, the scheduling problem at hand can be classified as a single-product multi-stage batch-scheduling problem with parallel processing equipment and the requirement of simultaneous recipe optimization. The main goal of the targeted solution has been to generate a detailed and optimized schedule for the production process that (from the model perspective) explicitly considers:

Simultaneous Scheduling and Optimization of a Copper Plant

1199

• two parallel converters, • two parallel anode furnaces, and, • one continuous caster that is directly fed by the anode furnaces. The primary objective is to maximize the throughput. From a practical perspective and in the light of re-scheduling requirements, the computing time for generating a valid optimized schedule must not exceed a few seconds. Therefore the scheduling model is not allowed to be too complex but only to capture the essentials, i.e. major issues that affect the production timings. Extreme operating conditions create regular maintenance needs and cause unexpected equipment breakdowns. Therefore, optimal maintenance scheduling is needed in order to minimize the impact of maintenance actions and corresponding process interruptions with respect to throughput reduction. Summarizing the above, the main decisions to be taken are: • Exact timings for the processing steps (parallel lines, maintenance) • Material amounts (melt copper, scrap material, slag) • Production cycles (converter and anode furnace) • Utilization of limited resources (timing critical) • Reserving sufficient time for necessary logistics (e.g. cranes) Since we are dealing with a single-product problem, the sequence of batches is not relevant and can therefore be given a priori. However, this refers purely to the batch numbers. The integrated recipe optimization will define individual characteristics (size, composition) for each batch. The batch routes may vary as indicated in Fig. 1. and the continuous-casting stage is treated as a batch operation. The solution that has been developed builds on a valid and robust process model that captures the main chemical reactions and is able to link the variable material amounts with predicted processing times. 4. Solution approach A mathematical programming solution approach has been selected to ensure that an optimal solution can be obtained. This poses two main challenges: • The resulting process model is in its pure form intractable, mainly due to the chemical reaction kinetics and • Logic decisions that need to be made in scheduling (commonly precedence and equipment assignment) must be represented by discrete or binary variables. In order to make the problem solvable, a linearized process model has been derived. This enables the use of standard Mixed Integer Linear Programming (MILP) techniques, for which several robust commercial solvers are available. For ensuring the validity of the linearization approach, the process model has been verified with a significant amount of plant data. A standard continuous-time job-shop scheduling formulation (Pinto and Grossmann, 1998) is used to provide the basic aspects of the production decisions, such as sequencing and assignment of jobs. The key of the mathematical solution is to build an optimal schedule by capturing the durations of each processing step and to relate it to the material amounts. Therefore, a top-down approach will be presented here to illustrate some main principles of the model only.

1200

/. Harjunkoski et al

First, some constraints related to the converter operations are shown. In the following, we consider the four major converter operation phases. The index s is used to denote a certain phase and p refers to a batch. The corresponding upper case letters refer to the respective sets, e.g. *S^ refers to the phases that are valid for the converting. Two different timings must be included. The time when a batch, p, actually is in the converter is defined by the start and end times, f^p and f^p. These are relevant for keeping track on the equipment availability. This aggregated timing is then subdivided into more detailed timings, which define the start and end time of each converter phase, t^^^p,s and f^^p,s' These, again are necessary for the synchronization of parallel tasks. Here, we consider primarily the converting phases.

C.=C

V;,eP,V.e5^|.<2

(2) (3)

.CB .CSB C =C V;, €/>,. = !

.CE

.CSE

^p =^p,s

W^ ^

(4)

D „

I cC I

/c\

ypeP,s=\S

I

(5)

Constraint (1) specifies the relation between the start and end times of a phase, where Tp^s is the processing time needed. Constraint (2) specifies that the two first phases must be followed immediately by the next one. Some flexibility is allowed for the rest of the phases, as defined by the delta-variable in Eq. (3). The relationship between first and last phases and the converter beginning and ending times are defined in Eqs. (4-5). They specify exactly the time when a converter is assigned to each batch. Similar relations are given for each phase of all equipment operations. Below, an example is shown where a phase combines two equipment. This is e.g. the case when a converter is emptied to an anode furnace (requires that both equipment are available simultaneously) or during the continuous casting, which actually is done by discharging an anode furnace. The former case is illustrated by Eqs. (6-7). fCSB p,s

_

\/peP,s=\S^ \,s' mm{S'}

(6)

T

=T , \/peP,s=^S^ \,s' mm{S'} p,s

(7)

p,s

.ASB p,s'

The start time of the last converter phase (emptying) is equal to the start time of the first anode furnace phase (filling), as expressed by constraint (6), where s' denotes the first phase of the anode fumace. The durations are equal, as shown in Eq. (7). A minor time shift, due to crane movements and initial filling at the converter, could be included but since this is typically in the order of a few minutes, it has been omitted here. Another important issue is to avoid the overlapping of crane operations. Avoiding simultaneous filling/emptying operations can be modeled by the constraints that are shown below. We assume that batch/? is immediately followed by batchp+7. Here, the major converter phases are enumerated by 1-4.

Simultaneous Scheduling and Optimization of a Copper Plant

tlfu^^?,i^K

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yp^p\p<\p\

tf^^>tfl,+^^^-^,,

(8)

^p^p\p<\p\

(9)

Constraint (8) states that the next batch should not be filled before the slag blowing has been ended plus a selected "safety" buffer, A^r- On the other hand, according to Eq. (9) the previous batch should not be emptied before the next batch has ended its slag blowing. Here, a penalty coefficient, 5^^, can be added to the objective function, if a soft constraint is preferrable. These cross-relationships between the batches make it nontrivial to find feasible sequences and also limit the flexibility of the modeling. This underlines that, the integration of recipe optimization plays an essential role in finding an optimal production schedule. Material balances are crucial for a successful recipe optimization. In its basic form the model assumes unlimited material availability, but material constraints can be included if needed. It is also important to ensure that the total amount of material does not exceed the converter capacity. A copper mass balance equation for the components is used to calculate the amounts of slag and a higher degree slag type that is recycled between the batches: [ricuj -ricu,x)'M, +[ricuj -ricua)'M2 A^cuj

-VCU,,\M,

-^[TJCUJ

-ricuA)'M, =

^^^^

Cu,f

T,.s=T;:- + n^,^,^-T,^, + S;-S;

(11)

In Eq. (10) the incoming components are just numbered for simplicity and their respective copper contents are given by rjcw The final copper content of the slag blowing phase is denoted as r/cuf- The inputs on the left hand side are matte, solid scrap material and rich slag. In the slag-blowing phase, the only outputs are slag and sulfur in form of sulfur dioxide. The mass of each component is expressed by M. A number of additional equations is needed to determine the individual masses. The processing times for the various production steps (Tp^s) are calculated using equations that are e.g. based on reaction kinetics (Eq. 11), where the first RHS-term depends on material amounts and properties. Generally speaking, the definition of it is quite different for each phase. The simplest ones are the filling and emptying phases, where only an average crane operation time per ladle is multiplied by the number of ladles. The last type of constraints that is needed is a standard scheduling constraint that relates two batches to each other and hinders possible resource conflicts.

r^^f,>/f-M-(2-X^,,-Z^,,J

\^peP\p<\P\yeeE

(12)

Constraint (12) avoids two batches from occupying the same resources simultaneously. It can also be expressed for each production stage but here we restrict ourselves to using the already existing variables only. Constraint (12) must, of course, be complemented by an additional constraint that enforces the use of one of the alternative resources only. There are a number of objective function components, such as the various slack variables and their penalty factors for violating the respective constraints, as well as, the caster throughput. Since it is very difficult to maximize the throughput explicitly.

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instead here the final batch sizes at the caster are maximized along with the minimization of the makespan.

v[\2ix{throughput] = maxjci • final batch size - C2 • makespan}

(13)

With a proper balancing of C; and C2 this turned out to be a valid strategy, which of course is very sensitive to changes in the other objective function component weighing factors. 5. Results A typical scheduling run is done for 6 batches, which covers the production of the next 20-30 hours. A longer scheduHng horizon would not make sense as it becomes almost impossible to predict the copper content e.g. in matte. On the other hand, giving a good estimate on this information as parameters is very important for the recipe generation. In Table 1, a typical problem size and optimization time is shown. Artificial result data is shown in Table 2. The resulting optimization problem is solved using ILOG CPLEX. Table 1. Problem size and solution example Constraints

Variables

0-1 variables

CPU-time

984

750

84


Table 2. Example of scheduling results Batch

1

2

3

4

5

6

Size (tons) Total production time

230 14:54

231 15:56

236 13:30

247 14:31

284 15:25

247 14:31

The solution developed is able to solve the scheduling problem very efficiently, resulting into good and realistic schedules. The solution quality depends heavily on how well the parameter estimation matches with the production process. More results and illustrations on the solution can be found in Harjunkoski et al. (2005). 6. Conclusions The overall end customer's main benefits after commissioning of the above described solution can be summarized as follows: An optimal production schedule with optimal batch recipes is available on demand within seconds. The solution allows a better overall process coordination and visibility of the process and faster recovery from disturbances through efficient rescheduling. This leads to a significant (estimated 1-2%) increase in plant throughput and revenues. References Harjunkoski I., Beykirch G., Zuber M. and Weidemann H-J. (2005). The process "copper" Copper plant scheduling and optimization. ABB Review 4/2005, pp. 51-54 Pinto, J. M., & Grossmann, I. E. (1998). Assignment and sequencing models for the scheduling of process systems. Annals of Operations Research, 81, pp. 433-466 Pradenas L., Fernandez R., Parada V., Caballero C, Zuniga J. (2003). "A Solution to the Copper Smelter Scheduling Problem", The Hermann Schwarze Symposium on Copper Pyrometallurgy, Vol YV-Pyrometallurgy of Copper 2003, Santiago de Chile, (Eds. Diaz C, Kapusta J. and Newman C), November, 2003, pp. 351-357

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 PubUshed by Elsevier B.V.

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C o o r d i n a t o r M P C w i t h focus o n m a x i m i z i n g t h r o u g h p u t E.M.B. Aske^*^, S. Strand^ and S. Skogestad^* ^Department of Chemical Engineering, NTNU, N-7491 Trondheim, Norway •^Statoil R&D, Process control, N-7005 Trondheim, Norway In this paper we suggest a "coordinator MPC" to perform dynamic real-time optimization (DRTO) on a plant. We consider the case where the plant economic criteria can be simplified to maximize the throughput in the plant. A measure for the distance to the bottleneck is formulated for a distillation column and the responses from feed to remaining capacity are expressed by experimental step-response models. The coordinator is demonstrated on a dynamic simulator and performs well for the simulated challenges. Keywords: MPC, coordinator, dynamic optimization

1. I N T R O D U C T I O N Real-time optimization (RTO) offers a direct method of maximizing an economic objective function. Typically, RTO systems are model-based and part of a closed-loop process control system which objective is to maintain the process operation as close as possible to the optimum plant operation [1]. When disturbances of economic importance have a dynamic character and especially if they occur frequently compared to the controlled plant responses, steady state RTO will be inadequate to follow the optimal operation point in periods. In such cases DRTO is more suitable, and different authors have discussed this subject. Steady state assumption in conventional RTO severely limits the frequency of optimization in integrated plants, leading to suboptimal economic performances [2]. One argument for this is that integrated plants have very long transient dynamics, so once a change occurs, it may take a very long time for the plant to reach the new steady state, thus limiting the execution frequency of the RTO. Second, optimal operating conditions calculated from a steady state model may be suboptimal or even infeasible at the local units due to the transient dynamics, unit interactions, model errors and disturbances. Third, dynamic degrees of freedom may be left unexplored leading to suboptimal dynamic solutions. Many researches have suggested that DRTO is performed at the same rate as the local unit controllers. [3] presents an approach where DRTO and MPC into a single layer by adding an economic objective function to the control objective function of MPC in a weighted average manner. Determining weighting factors and robust tunings is, however, a time consuming task. A systematic approach for integrating DRTO and MPC for large scale-industrial processes is presented by [4]. The two tasks, economic trajectory optimization and control, are decomposed into an upper layer dynamic trajectory (re)-optimization, and a lower level (nonlinear) MPC that controls the process along the current optimal trajectory determined by the upper level. Re-optimization is not necessary at each optimization sample time, instead it can be based on the disturbance dynamics. The wide use of MPC establishes a solid foundation for large-scale optimization. Dynamic coordination among MPC controllers is a key to tight integration between advanced process control and plant wide optimization [5]. In this paper we suggest to use a "coordinator MPC" with simple experimental models to solve a dynamic optimization problem for a plant. In this case, the structure of the optimization problem makes MPC an applicable tool for solving the DRTO problem. * Corresponding author: [email protected]

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2. C O O R D I N A T O R M P C The coordinator MPC task is to maximize an objective function and make decisions involving several local MPC applications, which typically handle product specifications and stability issues of smaller process units. The coordinator MPC is placed above the local MPCs in the control hierarchy and coordinates the underlying MPCs. The coordinator can perform as a DRTO if the plant economic criterion is of such kind that it can be formulated as a controlled variable (CV) in the MPC. It operates with feedback on minutes basis, compared to the typical RTO execution time of several hours. This leads to a faster correction of disturbances, model errors and transient dynamics. Most plants today operate without an RTO layer. Implementing a coordinator MPC requires less modeling effort compared to implementing RTO, but can still give a large part of the earnings. Even if RTO is implemented on the plant at a later stage, it can be combined with the coordinator MPC. Some of the calculated set points from the RTO are in that case sent to the coordinator MPC instead of the local MPC. The coordinator MPC controls the dynamic transfer to the steady-state optimum if that is feasible. It is possible to avoid the coordinator MPC layer by gathering all the local MPCs in one large application. However, for a complete plant the application will be over-complex leading to challenging modeling and maintenance. Introducing an extra layer in the hierarchy with smart decomposition reduces this complexity. 3. B O T T L E N E C K For several plants, it is optimal to maximize the throughput (production rate), subject to achieving feasible operation. Feasible operation is defined as satisfying given constraints on product qualities, flow rates, capacity, temperatures, pressures, etc. We have not defined "throughput", but for cases with a single feed it is equivalent to maximizing the feed rate integrated over a given time of operation. For cases with multiple feeds, the throughput may correspond to a (weighted) sum of the individual feeds, but the details will depend on the process and economics. When the feed rate to a process is increased, one will eventually reach a (maximum) value where feasible operation is no longer possible. This occurs as we reach a bottleneck somewhere in the plant. More precisely, we have the following definition: A bottleneck is as a constrained extensive variable, where the maximum value prevents an increase of the plant throughput (capacity, production rate, overall feed), subject to achieving feasible operation. Examples of bottlenecks are the neck of a bottle, and a 1-lane bridge on a 2-lane road. The location of the bottleneck for a given plant may change depending on external factors (disturbances) such as feed composition. If the objective is to maximize throughput, then it is generally optimal to set the throughput at the bottleneck [6]. The reason for this is dynamic. As an example, consider the 2-lane road with a 1lane bridge. At steady-state the number of cars in (feed) and out (product) are the same. However, dynamically they may vary, and it is clear that in order to maximize the throughput of cars, we must make sure that we always have the maximizing fiow at the bottleneck, that is, on the 1-lane bridge. If we for some time do not have the maximum flow through the bottleneck, then this loss in throughput can not be recovered later. The same applies in a chemical plant. However, one problem with this is that if the location of the bottleneck changes, then this generally requires reconfiguration of the basic control loops for inventory (level) control, which is undesirable. Therefore, in practice, and as discussed in this paper, we may need to adjust the throughput at some other location, for example at the feed, while trying to keep the flow at the bottleneck as close to its maximum as possible. 4. R E M A I N I N G C A P A C I T Y A N D T H E OPTIMIZATION P R O B L E M Assume the objective is to maximize the overall throughput. One approach is to set up a large optimization problem (RTO) and find the optimal solution. However, it is desirable that the coordinator does not have to duplicate the models and constraints in the local MPCs, so a more decoupled solution strategy is wanted. Since the location of the bottleneck may change, it requires an overview of the available capacity in each part of the plant. With MPC installed on each unit we may use this tool to obtain the available unit capacity.

Coordinator MPC with Focus on Maximizing Throughput

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We suggest that the steady state part of the local MPC on unit k is extended to obtain an estimate of the remaining capacity in the unit ^k

^^ ^k,max

^k

\^)

where Jk is the current throughput (feed) in unit k. The maximum feed rate Jk,max niay be solved by solving a simple LP optimization. The constraints in the LP problem are the same as in the steady state solver in the local MPC, only the objective function is different. The LP calculation in each local MPC returns to the coordinator the remaining capacity Rk in each unit. The optimization at the coordinator level is to maximize the weighted overall feed rate: max J = Y^ WiFi MV

subject to

Rk > 0; Rk = G ^ MV

(2)

^—^

where J is a weighted sum of the feed rates to the plant. The manipulated variables (MVs) at the coordinator level are typically the external feed rates and crossovers in the plant. G represents the dynamic influence from each MV to Rk. The coordinator MPC should operate such that it is possible to keep each unit specification. However, unmeasured disturbances and slow responses may require some back-off in the unit when the disturbances occur. The constraint back-off should be set according to the controller performance and the acceptable constraints violations. The magnitude of the back-off depends on the expected size of the disturbances and how strict the product specifications are. If the product is mixed on tanks before sale, violating the product specifications for a shorter period may be acceptable. The use of back-off reduces the value of J, but makes the coordinator more robust. Control target/range changes in the local MPC, like MV and CV limit changes, have a direct infiuence on the remaining capacity measure and must also be handled by feedback with the given coordinator design. Nonlinear effects in the process causes modeling error in the coordinator and must also be handled by feedback. All these effects argue for a fast feedback sampling in the coordinator MPC. The process dynamics seen by the coordinator MPC includes the local MPCs. Local MV saturation should be avoided so the local MPCs are more robust to handle disturbances and to linearize the process seen by the coordinator. To avoid local MV saturation, some back-off on each MV is included in the calculation of remaining capacity.

5. K A R S T 0 GAS PROCESSING CASE STUDY The coordinator MPC approach has been tested with good results using the Karst0 Whole Plant simulator. This is a dynamic simulator built in the software D - S P I C E ® L 5.1. The case To demonstrate the applicability of the coordinator MPC, we use a detailed simulation model of parts of the Karst0 plant. The two fractionation trains, T-lOO and T-300, both have a deethanizer, depropanizer, debutanizer and a butane splitter. In addition T-300 has two stabilizers in parallel. The simulated parts of the plant are shown in Figure 1. There are two separate train feeds, a liquid stream from a dew point control unit (DPCU) that is divided between the two trains, and a crossover. The five streams are MVs in the coordinator MPC and indicated by valves in Figure 1. The local MPCs and the coordinator are implemented in SEPTIC^ MPC software [7]. For description of the local MPCs, see [8]. 5.2. Coordinator M P C The coordinator task is to maximize the plant throughput within feasible operation. Maximizing flow rate can be realized with our standard quadratic objective function by a total plant feed as a CV with a high (not reachable) set point with lower priority than the capacity constraints. The inputs and outputs of the coordinator MPC are as follows: • CVs: Remaining capacity in each column, 10 in total (ETIOO, PTIOO, BTIOO, BSlOO, STABl, STAB2, ET300, PT300, BT300, BS300), T-lOO deethanizer sump level controller output (LC OUTLET) and total plant feed (PLANT FEED) ^Statoil E s t i m a t i o n a n d Prediction Tool for Identification a n d Control

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21FC5288VWA

Figure 1. The simulated parts of the Karst0 plant

• MVs: Feed flow from DPCU to T-lOO and T-300 (21FC5334VWA, 21FC5288VWA), train feed flow T-lOO (21FR1005VWA), train feed flow T-300 (FEEDT300VWA) and crossover flow from T-lOO to T-300 (24FC5074VWA) The total plant feed is defined as the sum of the train feeds and DPCU feeds. The level controller output as a CV follows to avoid emptying or filling up the sump level in the deethanizer T-lOO when manipulating the crossover. The remaining column capacity is calculated in each local MFC as an LP problem considering CV and MV constraints around the column. The column feed is a free variable in the LP formulation and the fiooding point is represented by a high limit on the column differential pressure. The execution is set to a slower rate than the local MFCs to ensure robustness in the feedback loop and is here chosen to be 3 minutes. The column capacity depends both on the column feed fiow and the feed composition. At the Karst0 plant, only the feed flow is manipulative. The composition is measured with gas chromatograph (GC) at the plant feed inlets and in top of the distillation columns. However, the dead time in the GC sampling makes the measurements unsuitable for control. The feed composition changes are therefore characterized as unmeasured disturbances. The coordinator models are experimental step-response models, and are found in the same way as in the local MFCs. The models were obtained at 80-95% of the maximum throughput which is a common operation area for the real plant. The coordinator MFC tuning is a trade-off between MV variation and CV constraint violation. Some constraint violation cannot be avoided due to the process response times, unmeasured disturbances and model errors. The tuning should not be so aggressive that model errors are amplified, which means that some constraint back-off will be necessary. 5.3. Results from the simulator case study The coordinator performance is illustrated with tree different cases, and the CVs in the coordinator MFC are shown in Figure 2 whereas the MVs are displayed in Figure 3. Move the plant to maximum throughput. The coordinator is turned on at t = 0 minutes to move the plant operation from a non-optimal to an optimal operation point. Figure 2 shows that the deeethanizer in T-lOO and the stabilizers are bottlenecks at the optimal operation point. The butane splitter in T300 reaches its capacity limit, however, there is available capacity in the depropanizer and downstream columns of T-lOO and the coordinator uses the crossover to reroute, removing the T-300 butane splitter bottleneck. Change in feed composition. A momentary feed composition change is introduced in the T-lOO feed at t = 360 minutes. The feed composition change increases the remaining capacity of the T-lOO deethanizer and makes it possible for the coordinator to increase the train feed. The disturbance reduces the remaining capacity in the butane splitter so the coordinator uses the crossover to keep the column within its capacity. However, the butane splitter in T-lOO is not a plant bottleneck yet, since there is still capacity for rerouting to T-300. Change in a local MPC CV limit. With the butane splitter in T-lOO operating at its capacity limit, the

Coordinator MPC with Focus on Maximizing Throughput 50

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• rs 850

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Figure 2. CVs in the coordinator MPC, remaining capacity and plant feed in t / h

g 540

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^ 480 1000

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operator reduces the bottom quality high limit in the local MPC at t = 600 minutes. The coordinator increases the crossover since there is available capacity in the T-300 string, but must also reduce the T-lOO feed some. Both the butane splitters are now bottlenecks in the plant, together with the stabilizers. 6. D I S C U S S I O N The coordinator MPC is demonstrated on a dynamic simulator and performs well for the simulated challenges. Back-off reduces the optimal function value, but is necessary due to unmeasured disturbances and the long process response times. An improvement of the coordinator MPC is to include some feed forward especially from feed composition changes. Split fraction in the column can be used, then both feed composition changes and upstream processing units operation changes will be detected. Including feed forward information, the back-off in the coordinator could then be reduced leading to a larger plant throughput. The coordinator MPC uses linear models while the process is nonlinear. In cases where the nonlinearities mostly are reflected in model gains, gain scheduling of the model improve the performance. Gain scheduling is possible to include in the current model form. Significantly model changes including dynamics, other model types in the coordinator MPC should be evaluated. Another weakness with the coordinator MPC is the simplified maximum throughput objective function. This is a special case, and if the feed turns to be limited for a period, the economic optimum will be different since energy costs and product prices should be included in the objective function. In such a case the coordinator will not lead the plant to optimal operation. Further, the buffer capacity in the distillation columns are not exploited. Faster responses can be obtained by active use of the buffer volumes, leading to a smaller loss in the economic objective function. In this simulated case the buffer volumes are limited, however, in other plants with larger buffer volumes this should be considered. Other linking variables between the units can also be considered, like increasing impurity in a column to decrease the load to the downstream unit. At last, the simplified coordinator MPC will be inadequate for longer-term planning purposes, where a more traditional RTO model will give valuable information. 7. C O N C L U S I O N In this paper we suggest to use a coordinator MPC with experimental step response models to optimize a plant with disturbances of dynamic character. The plant economic object function is simplified to maximize throughput in this case with a gas processing plant as case study. The coordinator MPC is designed and set up in a simulator together with unit MPC applications and performs well for the simulated challenges. REFERENCES [1] Y. Zhang and J.F. Forbes. Extended design cost: a performance criterion for real-time optimization systems. Comp. Chem. Eng., 24:1829-1841, 2000. [2] T. Tosukhowong, J.M. Lee, J.H Lee, and J. Lu. An introduction to a dynamic plant-wide optimization strategy for an integrated plant. Comp. Chem. Eng., 29:199-208, 2004. [3] M. Tvrzska de Gouvea and D. Odloak. One-layer real time optimization of L P G production in the F C C unit; procedure, advantages and disadvantages. Comp. Chem. Eng., 22:S191-S198, 1998. [4] J.V. Kadam, M. Schlegel, and W. Marquardt. A two-level strategy of integrated dynamic optimization and control of industrial processes - a case study. ESCAPE-12, The Hague, The Netherlands, pages 511-516, 2002. [5] J.Z. Lu. Challenging control problems and emerging technologies in enterprise optimization. Control Engineering Practice, 11:847-858, 2003. [6] S. Skogestad. Control structure design for complete chemical plants. Comp. Chem. Eng., 2S:219-234, 2004. [7] S. Strand and J.R. Sagfi. M P C in Statoil - Advantages with in-house technology. ADCHEM 2003,Hong Kong, pages 97-103, 2003. [8] E.M.B Aske, S. Strand, and S Skogestad. Implementation of M P C on a deethanizer at Karst0 gas plant. 16th IFAC World Congress, Prague, Czech Republic, 2005.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Fault diagnosis based on support vector machines and systematic comparison to existing approaches Ignacio Yelamos,^^ Gerard Escudero,^^ Moises Graells,^^ Luis Puigjaner^^ ""Chemical Engineering Department-CEPIMA, ^Software Department, ^ETSEIB Diagonal 647, 08028-Barcelona (Universitat Politecnica de Catalunya) ^EUETIB Comte d'Urgell 187, 08036-Barcelona (Universitat Politecnica de Catalunya) Abstract An innovative data based fault diagnosis system (FDS) using Support Vector Machines (SVM) is applied on a standard chemical process. Besides its simpler design and implementation, this technique allows dealing better with complex and large data sets. For that reason, it was expected to improve usual pattern classifiers performance reported in chemical engineering literature, such as artificial neural networks or PCA modeling techniques. In order to compare results with previously reported works, a standard case study such as the Tennessee Eastman (TE) process benchmark was considered and SVM achieved consistent and promising results. Besides, the difficulties encountered when comparing the results reported are discussed and a FDS comparison methodology is proposed based on reliability and accuracy of each FDS. In that sense, this study establishes a reference framework for future comparisons. Keywords: Fault diagnosis, support vector machines. 1. Introduction Parallel to the growing computational power, diverse statistical and machine learning techniques have been developed for fault detection and diagnosis (FDD). Approaches based on pattern recognition techniques (neural networks, fuzzy systems, etc) or qualitative reasoning (signed directed graph) have been used as diagnosis tools. Yet, this issue is still an open field requiring further work. Automatic data interpretation and diagnosis required by process industries must primarily prove enough confidence when classifying different plant states. SVM is a kernel based method from the Statistical Learning Theory (V. N. Vapnik, 1998) that has succeeded in many pattern recognition problems. Recently, it has been applied to fault diagnosis in chemical plants and the TE benchmark proposed by Downs and Vogel (1992) was used for studying the data sets overlapping problem (L. H. Chiang et al., 2004). However, in that work only three of the 20 TE faults were considered, so that the global diagnosis problem was not analyzed. 2. On the comparison of Support Vector Machines performance According to the review by Venkatasubramanian et al. (2003), three main methodologies have been followed in fault diagnosis of chemical processes: I) Quantitative model-based methods, II) Qualitative model-based methods and III) Process history based methods. The third approach uses historic data bases for generating models regardless of any previous expertise or first-principles knowledge

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requirements. Furthermore, these models allow dealing with the intrinsic non-linearities of most chemical processes with no additional cost. Several approaches based on the so called pattern recognition techniques (e.g. neural networks or fuzzy systems) have been implemented to overcome the limitations of the two first FDS approaches. Although data-based models are also limited by difficulties on class generalization, multiple fault diagnosis and data uncertainties, they are popular techniques in industry because of their simplicity and non-linearities processing. SVM falls in this wide group, having proved more successful than previously reported pattern classifiers. The TE case is used for comparative purposes because of its complexity and because it has become a benchmark in this topic as shown by the number of works where it is reported. The most complete fault diagnosis results on the TE process were given by Raich and ^inar, who used angular relations between data clusters (Raich and Qinar, 1997) and techniques for PCA model overlapping quantification (Raich and ^inar, 1996). The success of their fault diagnosis systems were reported fault by fault (Table 1). However, results on the TE diagnosis do not globally consider the problem and partial results reported do not allow a rigorous comparison between different methodologies. For instance, M. R. Maurya et al. (2004), propose a hierarchical SDG (signed directed graph) based on qualitative knowledge requiring further quantitative information to reduce the inaccurate fault candidate sets diagnosed; Chiang et al. (2000), show that Fisher Discriminant Analysis (FDA) gives a good lower dimensional representation in terms of maximizing the separation between different data clusters, but the results on misclassification are not easily comparable. Table 1 also includes the results by Chen and Liao (2002), who addressed the detection partial problem, but without extracting the abnormal situations cause. 2.1. Accuracy and Reliability of the FDS Accuracy and reliability are useful and complementary measures of diagnosis success and misclassification that allow rigorous FDS comparison. Accuracy: It is a measure of FDS correctness when diagnosing each state. It is estimated independently for each faulty state considered. Most studies on process fault diagnosis report results just using this index. Reliability: It is a confidence measure on the FDS. It is estimated globally for all faulty states but reliability for each fault may be also given. This crucial aspect has been hardly considered in quantitative terms. Next, the required equations to calculate both measures for each fault are shown. First, the number of FDS correct responses and the misclassification errors must be defined, Send

Cr = I,R, i = Sini

JL.

(1)

^

^f^XXEj, / =!

(2)

' = 'ini

where SM and Send correspond to the initial and final samples considered for each fault, N is the total faults considered, Rt is the FDS response diagnosing fault/on sample / {Ri = 1 if there is a correct diagnosis in sample i and Ri = 0 otherwise), Cf is the total correct responses for all the studied samples when fault/is happening, Ejt is the FDS response

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when occurring faulty in sample / (E,/= 1 if there is a /diagnosis and Ey/= 0 otherwise) and so that Vfis the total number of/diagnosis when the rest of considered states have occurred. Then the accuracy and reliability corresponding to each fault can be also defined by,

where 7/^ is the total number of samples (Sgnd) when fault/is occurring and Afand 7?/are the FDS accuracy and reliability diagnosing fault/ C/+]^ represents the FDS "votes" to the/class when all the considered faults are tested. 3. Support Vector Machines Introduced in 1992 (B. Boser et. al., 1992), SVM have been gaining popularity among the learning community since 1996. Based on the Structural Risk Minimization principle by the Statistical Learning Theory, they learn a linear hyperplane that separates a set of positive examples from a set of negative examples with maximum margin (the margin is defined by the distance of the hyperplane to the nearest of the positive and negative examples). This learning bias has proved to have good properties in terms of generalization bounds for the induced classifiers. Learning the maximal margin hyperplane is a convex quadratic optimization problem with a unique solution. When examples are not linearly separable or, simply, a perfect hyperplane is not needed, it is preferable to allow some errors in the training set so as to maintain a "better" solution hyperplane. This is achieved by a variant of the optimization problem, referred to as soft margin, in which the contribution to the objective function of the margin maximization and the training errors can be balanced through the use of a parameter called C. Despite the linearity of the basic algorithm, SVM can be converted into a dual form, allowing the use of kernel functions to produce non-linear classifiers. Kernel functions make SVM work more efficiently in high dimensional spaces, where new features can be expressed as combinations of the basic ones (N. Christianini and Shawe-Taylor, 2000). In this work, the SVM^^^* software (T. Joachims, 1999) was used. We have worked only with linear kernels, performing a tuning of the C parameter. It has to be noted that no significant improvements were achieved by using non-linear kernels. 4. Results To compare the SVM performance against other diagnosis techniques, works offering most complete quantitative results on the TE case were considered, showing the lack of comparable data. A first work [1] (Chen and Liao, 2002) only presents detection results for the 1 to 15 TE faults. A second one [2] (Raich and (^inar, 1996), includes accuracy and reliability as defined previously, but the values can not be compared as they are referred to previously detected faults instead of actual faults happening. Not taking into account any of the non-detected samples, overall reliability and accuracy are increased by removing possible wrong diagnosis. The last work analyzed [3] (Raich and ^inar, 1997), also refers results to previously detected faults and, despite the better accuracy value obtained, offers less information.

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A similar comparative analysis is used in this work, but including complete information. Results are obtained after SVM learning with treated data sets fi'om TE process simulations. Data are collected at second intervals during 24-hour runs. Aimed at enhancing performance, the exponential weighted moving average (EWMA) is used to remove useless noise and principal component analysis (PCA) is applied as a fault detection technique, taking advantage of its calculated statistics (latent scores and contributions to SPE) in fault diagnosis. Reduced training and testing sets from these treated data are randomly and independently selected to ensure rigorous results. In that way, 300 representative treated samples define each faulty data set used for training, whereas each faulty testing data set consists of 100 different observations. For comparative purposes between SVM and referred works, the total number of samples (S), the number of detected samples (D), the times the system votes to each class (V, C/+ FJr described in section 2.1) and the accuracy (A) and reliability (R) for each fault are shown in Table 1 columns. In rows, mean values for accuracy and reliability are introduced as comparative calculated measures. These measures are: the average (M, ftinction of A or R), the average weighted by the number of detected samples (Ml, that is, the rate of the total FDS correct responses and the total detected samples) and the average weighted by the number of actual samples for each case (M2, that is, the rate of the total FDS correct responses and the total samples tested). For comparative purpose also, two different computational experiments were carried out, SVM-1 and SVM-2. SVM-1 addresses the 21 faults implementing a fault detection previous SVM classification (Figure 1) in order to be able of comparing results with [2] and [3]. Besides, an extra calculation regarding all input data was done to show the actual diagnosis accuracy (M2). SVM-2 does not include information on detected faults, since this work proposes to assess performance just regarding FDS input and output (Figure 1). It takes advantage of statistics from PCA in the fault classification and only uses three columns for the complete analysis of the diagnosis performance. Moreover, the 21st TE fault is not analyzed in SVM-2, since it is a simultaneous fault that should not be treated as another single one. This work limits to the single fault problem, although the study of the multilabel problem given by simultaneous faults is underway. Results analysis shows that SVM-1 not only produces better accuracy (M), but the detection weighted average (Ml), which is much more significant, widely improves the accuracy and reliability of other works. Besides, it offers a general measure for global accuracy (M2) that is not known in [2] and [3]. SVM-2 gives consistent and promising results, and it is the only one that is able to give a global diagnosis reliability of the FDS, as it is considering all the input data in the classifier module. \

y '

Input : Data 1

DETECTION

FDS Detected

' "N

CLASSIFIER

1 Figure 1. FDS approaches regarding Input data or only detected data

! Output

1 1

Fault Diagnosis

1213

5. Conclusion and future work Support Vector Machines (SVM), a novel computational technique for data based modeling has been applied for fault diagnosis in chemical processes. The Tennessee Eastman benchmark has been used as the case study for obtaining comparative results. The results are very promising, but comparing these results has proved tough despite the number of works reporting data for this benchmark, which are incomplete or referred to previously detected faults. This work also proposes measuring and comparing the FDS performance by referring the output to the very input, which is the raw data supplied to the FDS. Discarding the intermediate step given by fault detection, a weighted average has been proposed and calculated regarding each single fault happening. Simultaneous faults and the multilabel related problem are being investigated. Further work also includes meta-leaming, combining different kernels related to similar variable sets and the use and comparison of other machine learning algorithms. These techniques are expected to improve the current SVM results. Acknowledgements Financial support from "Spanish Ministerio de Educacion y Ciencia" through the FPI program and the European community through contracts: No MRTN-CT-2004-512233, RFC-CR-04006, INCA-CT-2005-013359, is thankfully appreciated. References B. Boser, I. Guy on, V. Vapnik, 1992, A Training Algorithm for Optimal Margin. Classifiers. In Proceedings of the Workshop on Computational Learning Theory, COLT. J. Chen, Chien-Mao Liao, 2002, Dynamic process fault monitoring based on neural network and PCA. Journal of Process Control, 12, 277-289 L. H. Chiang, E. L. Russell, R. D. Braatz, 2000, Fault diagnosis in chemical processes using Fisher discriminant analysis, discriminant partial least squares, and principal component analysis,, Chemometrics and Intelligent Laboratory Systems, 50, 243-252 L. H. Chiang, M. E. Kotancheck, A. K. Kordon, 2004, Fault diagnosis based on Fisher discriminant analysis and support vector machines. Comput. Chem. Engng. 28, 1389-1401. N. Christianini, J. Shawe-Taylor, 2000, An Introduction to Support Vector Machines. Cambridge University Press. J. Downs, Vogel, E, 1992, A plant-wide industrial process control problem. Comput. Chem. Engng. 17 (3), 245-255 T. Joachims, 1999, Making large-Scale SVM Leaming Practical. Advances in Kemel Methods Support Vector Leaming, B. Scholkopf, C. Burges and A. Smola (ed.), MIT-Press. M. R. Maurya, R. Rengaswamy, V. Venkatasubramaian. 2004, Application of signed digraphs based analysis for fault diagnosis of chemical flowsheets. Engng. Aplications of Artificial Intelligence. Vol 17, 501-518. A. Raich and A. ^inar, 1996, Statistical process monitoring and disturbance diagnosis in multivariable continuous processes. Aiche J., 42, 995-1009. A. Raich, A Cinar, 1997, Diagnosis of process disturbances by statistical distance and angle measures. Computers chem. Engng, Vol. 21, No. 6, pp. 661 - 673. V. N. Vapnik, 1998, Statistical Leaming Theory, John Wiley. V. Venkatasubramanian, R. Rengaswamy, K.Y, Surya N. Kavury, 2003, A review of process fault detection and diagnosis. Comput. Chem. Engng. 27, 293-311, 313-326, 327-346.

5

2 Tablel. V: Votes to the class; D: Detected samples; S: Total number of samples; A: Accuracy; R: Rehbility; M:Average; M1: Wuglhed average by D; M2: Wdgthed average by S.*:Considerated state. **: Variable

!3-

2

e

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Explicit Parametric Controller for a Batch Polymerization System Mariano Asteasuain'^, Konstantinos Kouramas^, Vassilis Sakizlis"^ and Efstratios N. Pistikopoulos^ ""PLAPIQUI (UNS-CONICET), C. Carrindanga km 7, (8000) Bahia Blanca, Argentina. ^CPSE, Imperial College London, London SW7 2AZ, UK ^Bechtel, Hammersmith Road, London W6 8DP, UK

Abstract This work is focused on the design and evaluation of an explicit parametric controller for a batch polymerization process. A parametric controller is a novel control method that has been recently applied to a number of continuous processes but never to a batch system. In this work we aim at exploiting the properties of explicit parametric controllers and demonstrate the potential benefits of this control method in batch polymerization processes. Keywords: styrene polymerization, batch reactor, explicit parametric control 1. Introduction Control of batch polymerization reactors is a challenging task due to the non-linear and complex dynamics (heat transfer and fluid dynamics) and the varying operating conditions of these processes (Rho et al., 1998). Usually, the control objective in this type of systems is to ensure the tight tracking of a desired reactor temperature profile despite the possible perturbations of the nominal operating conditions. The problem has already been treated with the use of classical PID control or via dynamic optimization methods (see Bonvin et al, 2005 and references within); however, the research on process systems is steadily focusing on the exploitation of advanced control methods such as Model Predictive Control (MPC) that can guarantee optimal performance, constraint satisfaction and robustness (Morari and Lee, 1999). There are few works of MPC implementation for batch (polymerization) processes. Recently, Christofides and co-workers developed nonlinear MPCs for batch crystallization systems (Shi et al, 2005, 2006), based on appropriate reduced order models to achieve relatively fast calculation of the control action. However, the repeated computation of the MPC optimization problem is still a critical issue. Explicit parametric or ParOS controllers is a novel, wellestablished control method which has the ability to perform all the MPC computations off-line, thus reducing control action computations to simple function evaluations. At the same time, all other benefits of MPC are preserved. ParOS controllers have been recently applied to a number of control problems involving continuous processes (Dua et al., 2004; Grancharova et al., 2003), but never to a batch system. The aim of this work is to design an explicit parametric controller for a batch polymerization process and demonstrate the potential benefits of this control method in such process control problem. This is of particular interest since there are currently no reported case studies or implementations of the explicit parametric controller for batch

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M Asteasuain et ah

polymerization processes. First the non-linear mathematical process model is obtained and the optimal temperature profile is designed for which final product qualities are achieved. A PI controller is also designed whose behavior is compared with the one of the explicit parametric controller. A linear input output ARX approximation of the nonlinear model is then obtained and the explicit parametric controller is designed. Finally, the performances of both the PI and parametric controller are presented with an evaluation of the behavior of the two controllers in nominal and perturbed operating conditions. 2. Process description and mathematical model The system under consideration is a styrene batch polymerization reactor. The system consists of a jacketed reactor, in which the monomer (styrene) and a solvent (toluene) are charged with a proportion 70% to 30% in volume, respectively. The mixture is preheated to an initial reactor temperature, obtained for proper initiator performance. The initiator dibenzoyl peroxide (BPO) is added next to start the reaction. The kinetic mechanism considered in the polymerization model involves the following reactions: initiation by peroxide decomposition, monomer thermal initiation, propagation and termination by combination. Constant reaction volume and negligible gel effect are assumed. The kinetic parameters for the monomer thermal initiation constant were taken from Asteasuain et al. (2004), and all other kinetic parameters, physical properties and correlation parameter values were taken from Erdogan et al. (2002). The differential algebraic model obtained from the mass and energy balances is shown in Table 1. Table 1. Differential algebraic model of the polymerization system clI/dt = -KJ

(1)

Mn = Mw„„„y,/)'„

dM/dt = -K^MA, - 2fKJ - 3K^M'

(2)

Mw =

dA,/dt = 2fKJ - KX + 2K^M'

(3)

Pd = Mw/Mn

(4)

dT^ _(-AH)K^J{,M dt ~ pCp

dX,ldt = 2fKJ + K^M\

-KX^

dX,ldt = 2fKJ + K^M\ + 2K^M\ (5)

(9) (10)

Mw^rJri

dT, F.{T^-T)

(11) UA(T,-T,) pVCp

UA{T^-T.)

(12)

(13)

djjdt = Q.5KX

(6)

x = iM,-M)/M,

(14)

n = M,x

(7)

U = U,-ax

(15)

drJdt = KXA,Z,+A,')

(8)

In Table 1, / is the initiator, Mthe monomer, Aa and y^, a = 0, 1, 2, are the ath order moments of the radical and polymer molecular weight distributions. K^, K^, K^, and K^ are the kinetic constants, a n d / i s the initiator efficiency. Mn and Mw are the polymer number and weight average molecular weights, and Pd is the polydispersity index. Mwnom is the monomer molecular weight and x is the monomer conversion. Fj and T^^ are the coolant flow rate and inlet temperature. L^o and a are constant parameters.

Explicit Parametric Controller for a Batch Polymerization System

1217

2.1. Optimal temperature profile It is a common practice in most polymerization problems to control a batch reactor by pre-programmed recipe implementation and temperature control based on heat balances. Therefore, this work proceeds with the design of a temperature profile that minimizes the batch process time and ensures satisfaction of a number of constraints on the reactor and jacket temperature, process yield and final product (polymer) qualities. The polymer quaHties are expressed as its desired number average molecular weight Mn and polydispersity Pd. The following open-loop optimization problem is solved mmF = tf Differential algebraic equations

7 ; ( 0 < i i o ' c , 25^c
(16)

to obtain the optimal temperature profile T^,rc^ and the initiator charge /Q. The objective of the above optimization problem is to minimize the batch process time and the constraints are both the physical and product quality constraints. This optimal temperature profile is used as a set point for the PI and ParOS controllers whose design and properties are discussed later on this paper. 2.2. PI controller design The PI controller design follows the optimal temperature profile design. The reactor temperature Tr is the controlled variable (output) of the system and the coolant flow rate Fj is the manipulated variable. The optimal parameters of the PI controller are obtained by minimization of the ISE criterion for the nominal values of the process parameters. The PI controller equation and the resulting optimal values of its tuning parameters are

fl-

F,{t) = K{T^,At)-TM + J

I {T,,At)-nt))dt + hias

(17)

K = -46.5L/min/''C , tj = 49.95min , bias = 15L/min The PI controller will be used in the following analysis for the evaluation of the explicit parametric controller performance. 3. Explicit Parametric Controller Explicit parametric or ParOS controllers are obtained when the on-line optimization problem involved in the MPC scheme is solved by parametric optimization techniques. In an optimization framework such as model predictive control, where the objective is to minimize a performance criterion subject to a set of constraints and where some parameters vary between specified bounds, parametric optimization is a method to obtain the objective function and the optimization variables as functions of these parameters and the regions of the parameters space where these fiinctions are valid. In general any optimization framework in engineering problems can be described mathematically as z{0) =

minf{x)-\-d'^y x,y

s.t.g{x) + Ey;G7c{0,ir , ^eOcSt^ where 7 is a vector of binary variables, x a vector of continuous variables, / a scalar, continuously differentiable function of x, g a vector of continuously differentiable functions of x, 9 is a vector of parameters, b and d are constant vectors. Parametric

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M. Asteasuain et al

optimization obtains z and x as explicit functions z{6), x{9) of 6 and the regions CR{6) where these functions are valid. By treating the process operating variables as parameters, the optimization problem of the MPC is solved off-line by parametric optimization to obtain the optimal solution as an optimal mapping of the operating variables. The on-line implementation of MPC is then reduced to simple function evaluations. The main advantage of using explicit parametric controllers is that as the operating conditions of the plant fluctuate there is no need to re-optimize since the optimal solution is already given as a function of the set of operating conditions. 3.1. Model identification In order to obtain a simpler model description of the batch polymerization reactor but also maintain the dynamic properties of the system, a linear ARX input/output model is obtained by performing model identification on the batch reactor. The input/output data are obtained from simulation with Simulink of the nonlinear system along the optimal temperature trajectory, and the ARX model parameters are obtained with Matlab Identification Toolbox. The sampling time for the data acquisition is 1 min. The mathematical description of the resulting ARX model is >;,^i =3.00987, -3.3144>;,_i +1.7633>;,_3 -0.624>;,_3 +0.16555j;,_4 - 0.0085766W, + 0.0086875w,_i where y is the reactor temperature in °C and Wt is the coolant flow rate in L/min. In Figure 1 the approximation error between the nonlinear model and the ARX model output is illustrated. This error represents the model uncertainty that is incorporated in the ARX model. 3.2. Explicit parametric controller design The following MPC controller formulation is considered for the batch polymerization process

y= min y

Q{y,-yref,tf^pY

^^^'

s,t, y,^, =3M9Sy, -3.3144>;,_i +1.7633>;,_3 -0.624;;,_3 +0.165557,_4 -0.0085766W,+0.0086875w,_i

.^0)

Aw, =w,-w,_i, 0<w, <300 , 50<7, <120, 2 = 100 , R = l, p = lO~\^=5 where y is the controlled variable (reactor temperature in °C) and u the manipulated variable (coolant flow rate in L/min). The MPC takes into account the operational limitations on the manipulated (0 to 300 L/min) and the controlled variable (50 to 120°C). The problem involves five optimization variables Ut+u.,.,Ut+5 and 8 parameters ^t, Ut.\,yh yt-u yt-2, yt-3, yt-4, yref, where y^ef is the optimal temperature profile. This is a multi-parametric Quadratic Programming problem and a method for solving this problem with parametric optimization is given in Pistikopoulos et al. (2002) and Bemporad et al. (2002). The explicit parametric controller was derived with the Parametric Optimization (POP) software (property of ParOS Ltd) (ParOS, 2003). In Figure 2 one can see a projection of the regions of the controller on the yrUt sub-space. The performance evaluation of the explicit parametric controller in comparison with the performance of PI follows.

Explicit Parametric Controller for a Batch Polymerization System

100

1219

150

Time (min)

Figure 1 Process-ARX model mismatch

Figure 2 Projection of the controller regions in Vt-Ut snace

3.3. Simulation results In this section the performances of the ParOS and the PI controllers are compared. The performance of both PI and explicit parametric controller for the nominal operating conditions is illustrated in Figure 3. It is noticeable that both controllers manage to achieve efficient tracking of the optimal temperature that will guarantee the final product quality. The performance of both controllers is then tested for the case when perturbations of the nominal conditions occur. In order to provide a general view of the types of uncertainty and disturbances that could cause the perturbation of the nominal conditions, we consider three type of perturbations: perturbation of the initial reactor temperature Arr(0)=-4 "^C, of the heat transfer coefficient U= 0.97 (/nom where [/nom is the nominal value, and of the initiator charge A/o=+9.6% /o,nom where /o,nom is the nominal value. The results are shown in Figures 4 - 6 . The sensitivity of the PI controller to perturbations is obvious and temperature runway is unavoidable in all cases. On the other hand the explicit controllers behaviour is more robust. It succeeds to keep a tight tracking of the temperature trajectory. This is easy to understand if one recalls the properties of both controllers that were mentioned in the previous sections. The PI controller is designed to operate for the nominal operating conditions. However, when the perturbations of the nominal conditions occur, reoptimization of the PI parameters is required to cope with the new scenarios. On the contrary the explicit parametric controller does not need re-optimization since the optimal solution is already a function of the set of operating conditions. Thus, for any change in the operating conditions the explicit parametric controller can succesfiilly produce a control action to counteract for these changes. It should be noticed the design of the parametric controller does not take into account any robustness requirements. However, since this work is the first attempt to design an explicit controller for a batch polymerization system, the design of a robust explicit controller (Sakizlis et al., 2004) will follow as future work. 4. Conclusions In this work we delivered for the first time the design of an explicit parametric controller for a batch polymerization reactor problem. The aim is to demonstrate the benefits of the implementation of this novel controller method to batch processes by providing a study of the behavior of the controller in comparison with the classical PI control. We managed to show that although the controller is comparable to PI for

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M Asteasuain et al

105

1

Set Point — PI Controller Explicit Parametric Controller

• Set Point PI Controller Explicit Parametric Controller

fi fi

•

o 100

/'

3

IS

1 b

o "o IQ

J 90

y^

85

Time (min)

Time (min)

Figure 3. Temperature control for nominal operating conditions

Figure 4. Perturbation of the initial reactor temperature AT^iO) = -4 °C 1

- Set Point PI Controller

- Explicit Parametric Controller

140

o

1

Set Point - — PI Controller Explicit Parametric Controller

\

130

22 3 to

1 E o re

120

\ 110

100

A

/ .<-crr^

90

'__^,....^=^^=^^ Time (min)

Figure 5. Perturbation of the heat transfer coefficient U = 0.91 L^nom

^

Time (min)

Figure 6. Perturbation of the initiator charge A/Q = +9.6% /o,nom

nominal operating conditions, it successfully manages perturbations and uncertainty in the operating conditions which proves the potential of this controller for difficult processes such as batch polymerization processes.

References M. Asteasuain, A. Brandolin, C. Sarmoria, A. Bandoni, 2004, Ind. Eng. Chem. Res., 43, 5233. A. Bemporad, M. Morari, V. Dua, E.N. Pistikopoulos, 2002, Automatica, 38, 3. D. Bonvin, L. Bodizs, and B. Srinivasan, 2005, Chem. Eng. Reas. Des., 83(A6), 692. P. Dua, V. Sakizlis, S. Kershenbaum, E.N. Pistikopoulos, 2004, Proc. European Symposium on Computer Aided Process Engineering, 637. S. Erdogan, S.M. Alpbaz, A.R. Karagoz, 2002, Chem. Eng. J., 86, 259. A. Grancharova, A. Johansen, J. Kocijan, 2003, J. Europ. Contr. Conf M. Morari, J. Lee, 1999, J. Comput. Chem. Eng., 23, 667. E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad, M. Morari, 2002, Comput. Chem. Eng., 26, 175. ParOS, 2003, Parametric Optimization Solutions Ltd (http://www.parostech.com) D. Shi, P. Mhaskar, N.H. El-Farra, P.D. Christofides, 2005.Nanotechnology, 16, S562. D. Shi, N.H El-Farra, M. Li, P. Mhaskar, P.D. Christofides, 2006, Chem. Eng. Sci., 61, 268. H.J. Rho, Y-J. Huh, H-K. Rhee, 1998, Chem. Eng. Sci., 53, 3729. V. SakizHs, N.M.P. Kakahs, V. Dua, J.D. Perkins, E.N. Pistikopoulos, 2004, Automatica, 40, 189

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1221

An Effective MIDO Approach for the Simultaneous CycUc Scheduhng and Control of Polymer Grade Transition Operations A. Flores-Tlacuahuac^ and I. E. Grossmann ^ ^Departamento de Ingenieria y Ciencias Quimicas, Universidad Iberoamericana Prol. Paseo de la Reforma 880, Col. Lomas de Santa Fe, 01210, Mexico, D.F., Mexico ^Department of Chemical Engineering, Carnegie-Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213 In this work we propose a scheduhng and control formulation for simultaneously addressing scheduhng and control problems by explicitly incorporating process dynamics in the form of system constraints that ought to be met. The formulation takes into account the interactions between such problems and is able to cope with nonlinearities embedded into the processing system. The simultaneous scheduling and control problems is cast as a Mixed-Integer Dynamic Optimization (MIDO) problem where the simultaneous approach, based on orthogonal collocation on finite elements, is used to transform it into a Mixed-Integer Nonlinear Programming (MINLP) problem. The proposed simultaneous scheduling and control formulation is tested using a Methyl-Methacrylate CSTR where four different types of grades are manufactured. It is shown that the proposed methodology provides the best grade scheduling policy and optimal transition trajectories leading to maximum process profit. 1. Introduction Traditionally, scheduhng and control problems in chemical processes have been addressed in a separate way. From a scheduling point of view, the interest lies in determining optimal production sequences, production times for each product, switching times, leading to designs featuring maximum profit. Commonly, during this task, features related to the dynamic behavior of the underlying process are not taken into account. Similarly, when computing optimal transition trajectories (i.e. optimal values of the manipulated and controlled variables) between different set of products, one of the major objectives lies in determining the transition trajectory featuring minimum transition time. When addressing optimal control problems, it is normally assumed that the best production sequence has been determined in some way. Hence, normally scheduling features are neglected in optimal control formulations. In this work we propose a simultaneous approach to address scheduling and control problems, particularly those emerging during grade transition operations in polymerization reactors. In the proposed formulation, integer variables are used to determine the best production sequence and continuous variables take into account production times, cycle

1222

A. Flores-Tlacuahuac

and I.E. Grossmann

time etc. Because, dynamic profiles of both manipulated and controlled variables are also decision variables, the resulting problem is cast as a Mixed-Integer Dynamic Optimization (MIDO) problem. To solve the MIDO problem we use a methodology which consists in transforming the MIDO problem into a MINLP that can be solved using standard methods such are the Outer-Approximation method. Roughly speaking, the strategy for solving the MIDO problem consists in using a simultaneous equation-oriented approach in which the optimal control problem is solved by transforming the set of ordinary differential equations into a set of algebraic equations. The simultaneous scheduling and control formulation is appHed to a Methyl-Methacrylate CSTR where four diflPerent types of grades are manufactured. 2. P r o b l e m definition Given are a number of products that are to be manufactured in a single continuous multiproduct CSTR. Steady-state operating conditions for manufacturing each product are also specified, as well as the demand rate and price of each product and the inventory and raw materials costs. The problem to be tackled consists in the simultaneous determination of the best production wheel (i.e. cyclic time and the sequence in which the products will be manufactured) as well as the transition times, production rates, length of processing times, amounts manufactured of each product, such that the profit is maximized subject to a set of scheduling and dynamic state constraints. 3. S c h e d u l i n g a n d Control Formulation In the following simultaneous scheduling and control (SSC) formulation, we assume t h a t all products are manufactured in a single CSTR and that the products follow a production wheel meaning that all the required products are manufactured, in an optimal sequence to be determined, and that the sequence is repeated cyclically (see Pinto and Grossmann [1]). • O b j e c t i v e function.

ma^

\ 22 -TF- - E

l^E^

yi=\ ^c j=i + . . . + {x%, - xlf

'^^i-i-c fc=i/=i c=i -'c + {u),, - ulf + ... + {u%, - uTf)

E E '^A E

T^

(^^/-^fc " ""^^ }

(1)

1. Scheduling part.

Y^Vik = 1, Vz; Y^Vik = 1, VA:; y[^ = yi^k-i^ yi,k^ k=l

l;y-^ = yi^Nsi Vi

(2)

i=l

W, > D{r,, Vz; W, - G , e , , Vi; G, = F\l

- X,), Vi

(3)

An Effective MDO Approach

1223

Oik < e^'^^Vik, Vi, fc; e , = ^ Oik, Vi; Pk = Y. ^ik^ Vfc k=l

(4)

i=l

Zipk > y'pk + 2/z/c - 1, V i , p , k

(5)

Np Np

Np Np

^1 = EEtli^'Pk^ Vfc; tl = 0;tl = tl + Pfc + EEtU^ipk, Vfc i = l p—1

n = tU,

(6)

i=\ p—1

Vfc ^ 1; i^, < n, \fk; tf^u = (/ - 1 ) ^ + ^ 7 c , V/, c,

fc

(7)

2. Dynamic Optimization part. To address the optimal control part, the so-called simultaneous approach [2] for solving dynamic optimization problems was used. In this approach the dynamic mathematical model representing system behavior is discretized using the method of orthogonal collocation on finite elements. ^%k = <,fk + Olhfu i2 ^icxjik, Vn, /, c, k

(8)

1=1 Ncp


+ Olhf-i,k E ^i,N..i7-i,i,k, Vn, / > 2,

fc

(9)

1=1 ^fck

= P\^fck->

• • • 5 ^/cfc5 '^/C/C' • • • '^%k)^

^ ^ : / ^ C, k

(10)

Np

^l^,k = Jl^l,^y^.k,

^U^k

(11)

i=l Np

Np

^k = J2^l,^y^MU

Vn, k^Ns]

Xl = Y.^l,^y^^^ ^^' ^ = ^s

i=l

(12)

i=l Np

Np

uTn,k = E <s,y^,k. Vm, ^; < = E <,2/z,fc+i, Vm, A: ^ TV, - 1 ^=l

(13)

1=1

Np ^k

= ^uZ,iyi,i,\/m,k

= N,

(14)

i=l

^Nfe,Ncp,k — ^k-> Vn, A;; ttii/. — '^in,/;;' "^Tn^k '^Nfe^Ncp^k—

^lii.k^

^'^,k\

^o,l,fc — ^rn,A;5 V n , A:


(15) (16)

(17)

1224

A. Flores-Tlacuahuac and I.E. Grossmann

4. Application example To analyze the simultaneous scheduling and control problem we consider grade transitions in a bulk free-radical isothermal Methyl-Methacrylate CSTR previously described by Congalidis et al. [3] and used by Mahadevan et al. [4] to address grade transition problems from a control point of view. The mathematical model was cast in dimensionless form by dividing each state and flow rates of monomer and initiator by their maximum expected values. dxi dt

2f*kiCiX2 V {kp + kfm)xi^ Q

. p. (

X

(18)

(19) ^^ dX3 dt dX4

"dt

Qm

Qm^i

F(Q.5/c,e + hd) 2 / QmDo VMmikp

kiGiX2

-\

ktd + ktc

-h kfm)Crr

^=^

QrnDo

\

2j*kiCiX2 •Xi^

QmDi

2f*kiCiX2

Xi

\

ktd + ktc

^/m^45

Cruj

hd-^ktc

- QmXs]

(20)

(21)

where xi .^2,. ,^2 ,^3 = Crr Do' X4 = 7^,^l.n = Table 1 information regarding steady-state design and reactor scahng is shown. For testing our simultaneous scheduling and control formulation four polymer grades {A, B, C, D) were defined which correspond to molecular weight distributions of 15000, 25000, 35000 and 45000. The initiator flow rate [Qi) was selected as the manipulated variable to achieve grade transition. Table 2 contains the steady-state values of the states and the manipulated variable leading to manufacture each one of the A, B,C and D grades. Also shown in Table 2 are the demand rate and cost of each grade. The simultaneous scheduling and control formulation, as represented by Equations (1)(17), was solved using GAMS/SBB, a MINLP solver embedded in GAMS which uses a branch and bound techniques for solving MINLP's. The problem size consisted of 2897 constraints, 3138 continuous variables and 96 integer variables. The problem was started solving flrst a relaxed version of the MINLP (4 s) and used to initialize the MINLP which was solved in 430 s cpu time. Table 3 shows the results of solving the scheduling and control problem. As can be seen, the optimizer selected the cyclic sequence A -^ D ^^ C -^ B iov the production wheel with a cycle time of 400 hrs. The formulation clearly assigns one of the larger processing times (100 h) to the more valuable grade D and the shortest processing time (80 h) to the less valuable grade A. As a way to understand the role of process dynamics in the grade transition sequence t h a t was selected, the MMA polymerization CSTR was linearized around each one of the steady-state design conditions shown in Table 2. In all the cases, the dominant eigenvalue was always - 1 , which is an indication that all the grades have similar open-loop constant times. This means that, the transitions are dominated by cost and are slightly influenced by process dynamics.

An Effective MIDO Approach

1225

Table 1 Steady-state design and scaling information. Monomer feed stream [m3/h] 1 Wm Reactor volume [m3] V 0.1 Feedstream initiator concentration [kmol/m3] 8 CUn Monomer molecular weight [kg/kmol] 100.12 Mm Feedstream monomer concentration [kmol/m3] 6 ^rriin Initiator efficiency 0.58 Termination by coupling rate constant [m3/(kmol-h)] 1.3281x10^0 ktc Termination by disproportionation rate constant [m3/(kmol-h)] 1.093x10^1 ktd Initiation rate constant [1/h] 1.0255x10-1 h

r

kfm Cm Ci

2.4952x10^ 2.4522x10^

Propagation rate constant [m3/(kmol-h)] Chain transfer to monomer rate constant [m3/(kmol-h)]

5.7768 0.41534

Do

5.4794x10-3

Maximum value of monomer concentration [kmol/m3] Maximum value of initiator concentration [kmol/m3] Maximum value of zeroth moment

Di

82.219 0.05245 1

Maximum value of first moment Maximum value of initiator flow rate [m3/h] Maximum value of monomer flow rate [m3/h]

Kp

Q, ^m

(b)

Figure 1. Dynamic behavior of the MMA polymerization system during grade transition. (a) The initiator flow rate (Qi) is the manipulated variable, (b) The average molecular weight distribution (MWD) was one of the variables to be tracked during grade transition.

1226

A. Flores-Tlacuahuac

and I.E.

Grossmann

Table 2 Grade design information. The demand rate is in [kg/h] and the price in [$/kg]. Grade A B C D

Qi

Cm

0.05245 0.01673 0.006863 0.003114

5.1788 5.5068 5.6745 5.7768

Q 0.4153 0.1325 0.0543 0.0247

Do 0.0055 0.0020 0.0009 0.0005

Di

82.2185 49.3761 32.5877 22.3467

MWD 15000 25000 35000 45000

Demand rate 0.8 0.7 1 0.8

Price 10 12 13 15

Table 3 Simultaneous scheduling and control results for grade transtion in a MMA polymerization CSTR. The objective function value is $ 54 and 400 h of total cycle time. w Slot Product Process time Production rate Transition Time T start T e n d [Kg/h] [Kg] [h] [h] [h] 4 85 A 80 320 5 0 1 190 2 100 5 500 5 85 D 4 300 100 400 10 190 3 C 400 4 B 95 5 475 5 300

w

5. Conclusions In this work we addressed the simultaneous scheduling and control problem for grade transition in a MMA polymerization CSTR. Rather than assuming constant transition times and neglecting process dynamics, a mathematical model, able to describe dynamic process behavior during product transition, was embedded into the optimization formulation. Solving the scheduling and control problem taking into account process dynamics is the rigorous way to address scheduling problems. There is always a risk of getting suboptimal solutions when process dynamics are neglected. REFERENCES J.M. Pinto and I.E. Grossmann, Computers.Chem.Engng. 18 (1994) 797. Kameswaran, S. and Lorenz T. Biegler, "Simultaneous Dynamic Optimization Strategies: Recent Advances and Challenges," Proceedings of CPC7, accepted for publication (2005) J.R Congalidis, J.R. Richards and W.H. Ray, AICHE J, 35 (1989) 891. R. Mahadevan, F. Doyle, A.C. Allcock, AICHE J, 48 (2002) 1754.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

On The Global Dynamic Optimization of Highly Nonlinear Systems A. Flores-Tlacuahuac'' and L. T. Biegler ^ ^Departamento de Ingenieria y Ciencias Quimicas, Universidad Iberoamericana Prol. Paseo de la Reforma 880, Col. Lomas de Santa Fe, 01210, Mexico, D.F., Mexico ^Department of Chemical Engineering, Carnegie-Mellon University, 5000 Forbes Ave., Pittsburgh, PA 15213 We address the relationship between nonlinear behavior and the determination of globally optimal solutions in dynamic systems. Specifically, in determination of optimal transition trajectories between steady state operating points, we analyze the impact of output multiplicities in obtaining global optima around regions featuring severe nonHnearities. In order to quantify the presence or multiple local optima we propose an empirical indicator able to predict the potential for the emergence of unique global solutions. Although the presence of output multiplicities does not seem to be strictly necessary to detect multiple optima, we find t h a t unique global solutions are likely to emerge for transitions around nonlinear regions. Moreover, for dynamic optimization, we found t h a t current global optimization solvers tend to demand large computational effort when compared against local optimization solvers. Therefore, with appropriate variable bounds, good initialization strategies and, if necessary, multiple restart strategies, local optimization solvers seem to be competitive, in terms of C P U time, when faced with the decision of finding global optimal solutions. 1. I n t r o d u c t i o n Dynamic optimization tools have been proposed for improving the operation of continuous and batch chemical processes. In this framework, the optimal time-dependent values of a set of manipulated variables are computed which minimize/maximize a given objective function. Commonly, the objective function is a function of the controlled variables or is related to t h e m in some way. To address dynamic optimization problems featuring small scale systems, it is been customary to apply the Pontryagin maximum principle [1]. However, this approach becomes difficult to handle for larger systems or with p a t h constraints. To address these problems, methodologies based on sequential [2] and simultaneous [3] approaches have been widely used. Here we focus on the simultaneous approach, where the set of differential algebraic equations (DAEs), comprising the mathematical model which is part of the dynamic optimization problem, is discretized in time and transformed into a set of algebraic equations. Hence, the dynamic optimization problem is transformed into a NonHnear Program (NLP) which can be solved by standard techniques.

1227

1228

A. Flores-Tlacuahuac

and L. T. Biegler

Because dynamic optimization problems lead to nonconvex NLPs that may be difficult to solve, it is likely that an optimal solution determined with a local solver suffices for many practical purposes. However, in some cases, it is worth ensuring the global dynamic solution. This is especially true in the optimization of dynamic systems that exhibit multiplicities. In this work we consider global optimization strategies for an important class of dynamic optimization problems, the determination of optimal transitions from one operating point to another. Global optimization strategies for dynamic systems have been developed recently with sequential optimization strategies. The aBB algorithm was applied in [4]. Global underestimators and algorithms were developed for linear dynamic systems in [5] and generalized to mixed integer dynamic optimization problems in [6]. However, using either simultaneous or sequential strategies, optimal transition problems can lead to large NLPs with many degrees of freedom which can exceed the capabilities of global solvers. Also, highly nonlinear steady-state and dynamic behavior with output multiplicities, oscillatory behavior and high parameter sensitivity may be present and operating regions could exhibit both stable and unstable open-loop steady-states. As a result of these features, we opt for the simultaneous approach; unlike sequential approaches, simultaneous approaches have no difficulties in finding optimal solutions at open-loop unstable points [3]. One of our aims in this study is to evaluate the advantages and disadvantages of state of the art global optimization solvers (BARON, OQNLP) compared to well known local NLP solvers (CONOPT and MINOS) when faced with the dynamic optimization of nonlinear systems. This comparison was performed within the GAMS [7] modeling system using default options for all of the solvers. The Branch-And-Reduce Optimization Navigator (BARON) [8] finds the global solution of nonlinear (NLP) and mixed-integer nonlinear programs (MINLP). It is based on a spatial branch and bound strategy with a variety of constraint propagation and duality techniques that reduce variable ranges over the course of the solution process. In GAMS, BARON uses the MINOS local solver. Also, it is the only solver that provides both upper and lower bounds on the objective function value for the global optimum. OQNLP [9] is a multistart heuristic algorithm designed to find the global optimum of constrained NLPs. It calls a local NLP solver from multiple starting points, keeps track of all feasible solutions found by that solver, and reports back the best one as its final solution. The starting points are computed by a scatter search approach. In this study, the CONOPT local solver was used in OQNLP. While it does not guarantee global solutions nor determine lower bounds, it is claimed to perform well in finding the global solution. A more general case study comparison of global methods on a large number of test problems was recently published in [10]. In this study, a continuous stirred tank reactor model is used to demonstrate the difficulty in computing global dynamic solutions using global optimization solvers. It is shown that superior local solvers and a carefully stated optimization formulation (i.e. using a good initialization strategy, adequate variable bounds, etc.) are essential even if global solvers are applied.

On the Global Dynamic Optimization of Highly Nonlinear

1229

Systems

2. CSTR with Output Multiplicities The example, and its corresponding embedded transitions, was selected to cover a highly nonlinear operating region where unique global solutions, according to the empirical indicator detailed below, would be expected. In order to compute transition trajectories, we used the CSTR model [11]. Because the original parameter set in [11] does not lead to multiple steady-states, some of the values were modified in order to end up with a multiphcity map. In dimensionless form the model is given by: dyi

dt dy2

dt

l-yi 0 yf-y2 0

/c,oe-^/^^ yi

+ hoe-N/y2

(1) (2)

^{y2 - yc)

Vi

where yi stands for dimensionless concentration (c/cf), 2/2 is the dimensionless temperature {T/Jcf), yc is the dimensionless coolant temperature (Tc/Jcf), yf is the dimensionless feed temperature (Tf/Jcf), and u is the cooHng flowrate. Table 1 contains the numerical values of the parameters used in this work; this set of parameter values leads to the multiplicity region shown in Figure 1, along with associated operating points. Note from Figure 1 that products A and B are manufactured around open-loop stable steady-states. Product C is located at a stability transition which is also a Hopf bifurcation point. Finally, product D is manufactured within an unstable open-loop operating region. The objective function for the transition optimization is to minimize: $ = JQ^ \\y{t) — yW^dt, where y is the desired final steady state.

0 y1 =0.0944, y2=0.7766, u=340 (s) V yi =0.1367, y2=0.7293, u=390 (s) O yi =0.1926, y2=0.6881, u=430 (u) * yi =0.2632, y2=0.6519, u=455 (tv)

"50

100

150

200

250 300 350 Cooling flowrate

400

450

500

Figure 1. Multiplicity map (— stable solution, — unstable solution).

1230

A. Flores-Tlacuahuac

Table 1 Parameter values. 20 e 100 J 7.6 cf a 1.95x10-^

Residence time

Tf

(-AH)/{pC,)

ho

Feed concentration Dimensionless heat transfer area

Tc

N

300 300 290 5

andL.T.

Feed temperature Preexponential factor Coolant temperature Ei/{RJcf)

Table 2 Summary of results for the CSTR with output multiplicities with (—^ ppjT ) solved on a 1.6 GHz Pentium M laptop. In all the cases, 20 finite elements and three internal collocation points were used leading to 401 variables and 341 constraints. OQNLP CONOPT MINOS BARON Transition

B-^A A-^C C^

A

A^D D^

0.082

39.807

171.78

293.283 0.102

292.4109 91.601

346.1666 45.61

2245.9108 0.168

2255.8604

2245.9104 104.46

2389.9938

2013.5326

2019.2966 0.07

2013.5325

2050.4504 41.62

8952.4359 0.148

8697.2679 0.07

8952.4357 93.184

7068.1908 0.18

7097.5597 0.059

7068.1905 101.065

0.129

A

3. Computation and characterization of global solutions Although the relationship between nonlinear behavior and multiple optimal solutions is still not fully clear, the emergence of nonlinear behavior seems to be a prerequisite for a system to require the determination of global optimal dynamic solutions. Moreover, if a given system displays unique globally optimal dynamic trajectories, we believe trajectories must be embedded around nonlinear operating regions. However, if we operate the system around nonlinear regions, this is not a guarantee that such a system would necessarily feature unique optimal dynamic solutions. For transition optimization, there seems to be a second requirement, in addition to the presence of nonlinearities, that a system should fulfill to exhibit multiple solutions. This requirement has to do with the absolute value of the ratio (%) of the order of magnitude of the relative change of the controlled variable (A^/) related to the relative change in the manipulated variable (Aw). Here we propose the heuristic when considering the transition from steady-state i to steady-state j . A^

%•

Aii

where

A?/ •

Au =

hi-

mw

Biegler

(3)

On the Global Dynamic Optimization of Highly Nonlinear

Systems

1231

where r]ij is in fact a gain-like variable. It simply measures the size of the system response for a unit change in the system input. Such an indicator provides a quantitative measure of the system sensitivity to changes in the manipulated variable. In Eqn 3, we use norms because the presence of global optimal dynamic trajectories seems not to depend upon the direction of the response.

0.14

0

0.12

— - •

^^^X 1

1

1

1

1

1

1

1

Baron 1 Conopt Minos OqNLPl 1

Figure 2. (a) A ^^ 5 and ih) B ^ A transitions.

Prom the steady-state values of the desired transitions, depicted in Figure 1, the % indicator for the A ^ B^A <=^ C and A ^ D transitions was computed with r}AB = 2.42, r/Ac = 3.93 and TJAD = 5.29. Although it is difficult to postulate the % value from which unique global solutions emerge, a value greater than around 2 seems to be a good heuristic indicator to be able to detect these solutions. Of course no proof of this statement can be offered yet. According to the indicator values, we should expect unique global dynamic optimal solutions in all of the above transitions, as is the case from the information contained in Table 2. Here we observe that the CONOPT local solver is always able to find an optimal solution as good as OQNLP and better than the best upper bound found by the BARON global optimal solver. We also note that a different local solver in BARON should be able to produce better upper bounds. In addition, the CPU times for MINOS and CONOPT were about 2-3 orders of magnitude lower for the

1232

A. Flores- Tlacuahuac and L. T. Biegler

corresponding OQNLP and BARON solvers. Due to space limitations, only a small set of optimal dynamic trajectories are displayed in Figure 2; the rest of trajectories displays a similar shape. 4. Conclusions Even though the relationship between nonlinear behavior and multiple optimal dynamic solutions is not fully clear, we observe that steady-state transitions around operating regions with large parameter sensitivity seem to display unique globally optimal dynamic solutions. However, we believe the intended steady-state transition should feature a large enough transition (measured by the rfij empirical indicator) in order to exhibit such solutions. The case study discussed in this paper supports this statement, as do several other case studies not presented here due to space hmitations. Thus, the tendencies found in this work prevail, although we recognize the difficulty of getting general conclusions from a small set of examples. We also found that current global dynamic solvers seem to demand large computational loads and that, therefore, more work is needed if we expect to use them on a routine basis. For transition optimization, it appears that, at least for the time being, local optimal solvers, with good bounds, initialization strategies and possibly multiple restarts, seem to be a very inexpensive and reasonable alternative to global solvers. REFERENCES 1. A.E. Bryson and Y.C. Ho, Applied Optimal Control, Taylor and Francis, 1975. 2. V.S. Vassiliadis, R.W.H. Sargent and C. Pantelidis, Ind.Eng.Chem.Res., 33 (1994) 2111. 3. Kameswaran, S. and Lorenz T. Biegler, "Simultaneous Dynamic Optimization Strategies: Recent Advances and Challenges," Proceedings of CPC7, accepted for publication (2005) 4. Papamichail I., Adjiman C. S., Computers & Chemical Engineering, 28 (2004) 403 5. Singer, A. B. and P. I. Barton, Journal of Optimization Theory and Applications, 121, 2004, 613 6. Chachuat, B., A. B. Singer, P. I. Barton, AIChE Journal 51 (2005), 2235 7. Brooke, A., Kendrick, D., Meeraus, A. Raman, R., "GAMS: A User's Guide," GAMS Development Corporation, 1998 8. Tawarmalani, M. and N. V. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, 504 pages, Kluwer Academic Publishers, Dordrecht, Vol. 65 in Nonconvex Optimization And Its AppHcations series, 2002. 9. Lasdon, L., J. Plummer, Z. Ugray, and M. Bussieck, working paper, MSIS Department, McCombs College of Business, May 2004. Submitted to Mathematical Programming. 10. Neumaier, A., O. Shcherbina, W. Huyer and T. Vinko, Mathematical Programming, B103 (2005) 335. 11. G. A. Hicks and W. H. Ray, Can. J. Chem. Eng., 40 (1971) 522.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubHshed by Elsevier B.V.

Integrating Stiction diagnosis and Stiction Compensation in Process Control Valves R. Srinivasan^* and R. Rengaswamy^^ ^Honeywell Inc, ACS Labs, Phoenix, USA. ^Dept of Chemical Engineering, Clarkson University, NY 13676, USA Limit cycles caused due to valve non-linearity such as stiction can be eliminated with proper valve maintenance. Since valve maintenance is usually scheduled during production stops and with production stops scheduled typically once every six months to three years, the loss of product quality and energy loss during this intermediate period can be quite high. Stiction compensation algorithms can mitigate this problem to a large extent. In this paper, an optimization based approach for stiction compensation much in the sprit of predictive control strategies is proposed. 1. Background It has been reported that 20% to 30% of all control loops oscillate due to valve problems caused by static friction [1]. Oscillations lead to poor end product quality and wastage of energy. Hence, reliable diagnosis of valve stiction by itself will have a large economic impact. Valve maintenance, usually taken up during production stops, can reduce the static friction forces and lead to an improvement in loop performance. However, normal production stops are typically between every six months to three years. The loss of product quality and energy loss during this intermediate period can be quite high. Stiction compensation algorithms can mitigate this problem to a large extent. Two basic approaches to stiction compensation, namely dithering and impulsive control have been reported in the literature [2]. A good overview on stiction compensation in electromechanical systems can be found in Armstrong et al. (1994). Unfortunately, pneumatic valves that constitute 90% of actuators employed in a control loop filter such high frequency dither signal, making dithering technique ineffective. Similar drawback exists with impulsive control technique. Kayihan and Doyle [3] and Hagglund [4] have addressed stiction compensation algorithms for pneumatic control valves. The approach of Kayihan and Doyle [3] assumes that all the valve parameters such as mass of stem, stem position, stem velocity etc, are known a priori. Such detailed parameter values may not be available for all valves . Hagglund [4] proposed a novel *This work was done when the first author was a graduate student at Clarkson University ^Corresponding author: [email protected].

1233

1234

R. Srinivasan and R.

Rengaswamy

K Figure 1. Knocker Pulse

approach where a knocker signal is added to the control signal for stiction compensation. This is probably the best available current approach for stiction compensation. 1.1. Hagglund's Technique In Hagglund[4] technique (termed as 'knocker') short pulses are added to the control signal in the direction of the rate of change of the control signal. However, there is a need to tune three parameters that characterize the short pulses (see Figure 1): amphtude (a), pulse width (T) and time between each pulse (/z^). It was shown in our earlier work [5] that the 'knocker' performance was influenced by the pulse parameters. A brief summary of our past work is given in the next section. 1.2. Previous work[5] In our earlier work[5], we proposed a framework and a design procedure to automate these knocker parameters based on a stiction severity measure. This framework which integrates stiction diagnosis and stiction compensation tasks was shown to reduce the output variability by at least 6-7 times. However, it was pointed out that the reduction at the output variability was achieved at the cost of an aggressive stem movement. The 'knocker' in order to eliminate the output variability, shifted the variability to the stem, causing the stem to move aggressively. Such an aggressive stem movement is not preferred as it may wear the valve quickly. Hence, a design strategy for compensating stiction should ideally meet the following requirements: • Less aggressive stem movement (or valve movement) • Reduced Output variability • Less energy in the signal that is added to the control signal In this work, a novel design strategy that attempts to meet these requirements is proposed. The proposed method is devised based on the physical understanding of the stiction phenomenon and uses the stiction severity. 2. Proposed approach for stiction compensation 2.1. Problem setup Figure 2 shows the block diagram of a control loop in the presence of stiction. Valve dynamics are observed only after the start of the stem movement; stiction phenomenon, if present, will precede the valve dynamics. Also, it was discussed in [6,5] that a simple stiction model given by Equation 1 will be adequate under closed loop conditions. The simple stiction model is given

Integrating Stiction Diagnosis and Stiction

Compensation

1235

Identified Linear dvnaiiiics SP V

Controller

OP

Valve

Sliction V

Controller

Non-lineai' element

(+W

Figure 2. Process control loop in presence of sticky control valve.

Stiction Non Linearity

Plant

Figure 3. Process control loop with a compensator.

below^: X{t):

x[t-\) u{t)

if\u{t)-x{t-l)\ otherwise


(1)

Here x{t) and x{t — 1) are present and past stem movements, u{t) is the present controller output, and W is the valve stiction band. The stiction model given by Equation 1 coincides with the industrial procedure used for measuring stiction and the readers are referred to [6] for a detailed discussion on the applicability of this simple model for modeling stiction. For processes other than flow loops, the identification of the plant's linear dynamics includes the valve dynamics (that occur due to stem and plug movement, see Gp in Figure 2). This is because the flow through the valve, which is an intermediate variable, is usually not measured. In view of this assumption and combined with the fact that the stiction phenomenon precedes the stem movement, the stiction nonlinearity and the plant dynamics can be represented in separate blocks. This is depicted in Figure 3. Here ysp is process set-point (SP), y is process output (PV), e is the error (SP - PV), m is the controller output, f^ is the compensator output, u is the additive signal (m + fk) that is the input to the valve and x represents the predicted stem position obtained using the stiction model. 2.2. Problem formulation The requirements of an ideal stiction compensator can be translated into a meaningful objective function and realized using an optimizer. A possible objective function is: min J - XIISEY + X2Var{X) + X3<^{X)

(2)

Fk

Here, an optimization formulation that minimizes the objective function J over a defined prediction horizon is posed for designing the compensator signal (Fk). The first term in the objective function J is the integral square error (ISEY) of PV (with respect to set point), the second term addresses the valve stem variability Var{X), and the third term includes valve aggressiveness ((|)(X)). F and X are vectors obtained over a defined prediction horizon (p). A,i, A,2 and A-3 represents either the cost associated for maintaining the product quality or the penalty for each term

1236

R. Srinivasan and R.

Rengaswamy

in J. A compensating signal that minimizes J is the best possible signal that will meet the requirements of an ideal compensator. However, formulating such an objective function requires several assumptions to be made. These assumptions are listed below: 1. The plant model Gp is available. 2. The controller (G^) structure and its parameters are available. 3. Stiction is already detected. 4. Stiction severity measure d is ascertained a priori. With these assumptions, the optimization formulation is formally stated as: minJ

-

XxISEY + X2Var{X) +

'k2><Sf{X)

(3)

Uk-i = GcCk-i+fk-i

(4)

ek = ysp-Jk x{t) = ^{u{t),x{t-l),d)

(5) (6)

{ypred)k

=

GpXk-l

(7)

{ypred)k+p

=

GpXk+p-l k+p Y. {{ypred)i-ysp? i=k+l

(8)

ISEY

Var{X)

=

= variance{xi),i = k-{-l....{k-^p-l)

^{X) = 100* ( 1 - - ^ ) , p-l X

=

{Xk,Xk-\-l,....:Xk+p-l)

Fk

=

{fk,fk+\i'-"ifk+m-\)

(9)

(10) (11)

where, p is the prediction horizon, m is the control horizon, x denotes the predicted stem movement using the nonlinear transformation 9{^ (simple stiction model given by Equation 1) and Ut is the total number of switches in the signal between Xt and Xt^p. The stem aggressiveness denoted as (|)(X), is computed as a signal friendliness factor (see [7] for definition). The friendliness factor for the stem denotes how fast the stem changes, a value of (|) ?^ 0 indicates that the stem changes its position at every time instant and a value (j) ?^ 1 indicates that the stem changes it position only at few time instances over the time duration considered. At every time instant, a set of 'm' compensator moves are computed and similar to the MPC concept, only the first compensator move is added to the PID controller output. Since, the stiction model fA^is discontinuous and the stem friendliness factor ((|)(X)) does not have a closed form, an approximate numerical approach is used to solve the above nonlinear optimization problem. 3. Simulation results 3.1. Simulation example Consider a closed loop process with ^ , , 0.009U-i-h0.0082z-2 ^ ^ ^ 16-27.53^-1 + 11.81^-2 ^^(^^^l-1.724z-i+0.748z-i ^^^^^ ^ T^F^

^^'^

Integrating Stiction Diagnosis and Stiction

Compensation

^/WVVVV---=Tr^ Compensation started at time = 120 seconds

jlj

1237

""•

Compensation started at tinfe t = 120 S(

set-npint Runi Run 2

Umm^ 9 (sacs)'w

toJ ^"rimo (secsy^

^^^

I ^"^imefsecsy^

Figure 4. Results for optimization approach for compensation, (a) Process output and (b) ControUer output. The compensation started at time 120 seconds

Figure 5. Results for optimization approach for compensation, (a) Compensator signal, (b) Valve input obtained after addition of compensating signal to controller output and (c) Stem position.

sampled at every second. The loop oscillates due to the presence of stiction. The stiction is simulated using the simple stiction model given by Equation 1, with a stiction band d = 0.5. 3.2. Results Two simulations were performed to highlight the usefulness of the optimization approach. The optimization approach used the 'fmincon' algorithm of the MATLAB optimization toolbox used. The parameters for both the simulations are given in Table 1. Figures 4 and 5 show the comparison of the results obtained for the two simulations. It is seen that, based on the penalty imposed for each term in the objective function, the duration for the process output to reach its set-point varies. Also, the amount of variability and stem friendliness factor varies based on the weights of X.

Table 1 Parameters used in the simulation for the Optimization approach. Parameters Prediction horizon (p) Control horizon (m) ^1

y^i

h

Runl 40 2 100 0 0

Run 2 40 2 0 100 100

3.3. Discussion It was seen that the optimization approach has parameters that need to be tuned to attain efficient stiction compensation. Two main drawbacks are seen with the optimization approach:

1238

R. Srinivasan and R.

Rengaswamy

1. As the objective function (Equation 3) is non-smooth, the optimizer was not able to attain the global minimum; instead a local solution was obtained. This is evident from Figure 4, where for both runs, the process output failed to reach the set-point. This is because the stem position did not move to the correct steady state value, instead moved the stem close to it with an offset. Also the objective function values obtained for the various compensating signals (simulated as a grid of values for the next two moves) showed that the objective function is generally non-smooth but convex. 2. When the optimization approach was tried on an experimental level system at Clarkson University, the Simulink interface could not solve the optimization formulation between each iteration, due to real-time issues. Alternate non-gradient based optimization techniques that use function evaluation such as DIRECT (Divide RECTangle method). Implicit filtering can be studied to overcome the real-time issues. 4. Summary In this paper, a model based optimization approach for stiction compensation was presented. The optimization approach was observed to provide more trade-off than the 'knocker' approach. Preliminary studies suggest that the model based compensation method can be an useful strategy for stiction compensation. Further analysis of the effect of model plant mismatch, incorrect stiction measure, real time issues on the proposed stiction compensation approach needs to be done before these methods can be implemented online. REFERENCES 1. L. D. Desborough and R. M. Miller. Increasing customer value of industrial control performance monitoring - honey well's experience. Arizona, USA, 2001. CPC-VI. 2. B. Armstrong-Helouvry, R Dupont, and C. C. De Wit. A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica., 30(7): 1083-1138, 1994. 3. A. Kayihan and F. J. Doyle. Friction compensation for a process control valve. Control Engg. Practice, 8:799-812, 2000. 4. T. Hagglund. A friction compensator for pneumatic control valves. Journal of Process Control, 12:897-904, 2002. 5. R. Srinivasan and R. Rengaswamy. Stiction compensation in process control loops: A framework for integrating stiction measure and compensation. Accepted for publication in Industrial and Engg. Chemistry Research, 2005d. 6. R. Srinivasan, R. Rengaswamy, S. Narasimhan, and R. M. Miller. Control loop performance assessment 2: Hammerstein model approach for stiction diagnosis. Industrial and Engg. Chemistry Research, 44:6719-28, 2005b. 7. S. Narasimhan, R. Srinivasan, and R. Rengaswamy. Multi-objective signal design for plant friendly identification. 13th IFAC Symposium on System Identification, 2003.

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Differential Recurrent Neural Network based Predictive Control Rihab Al Seyab, Yi Cao Cranfield University, Bedford, MK43 OAL, UK

Abstract An efficient algorithm to train general differential recurrent neural networks is proposed. The trained network can be directly used as the internal model of a predictive controller. The efficiency and effectiveness of the approach are demonstrated through a two-CSTR case study, where a multi-layer perceptron differential recurrent network is adopted. Keywords: Predictive control, Recurrent neural networks. Nonlinear identification. Nonlinear control

system

1. Introduction Model predictive control (MPC) strategies have been well received by industry because they are intuitive and can explicitly handle MIMO systems with input and output constraints. Until recently, industrial applications of MPC have relied on linear dynamic models even though most processes are nonlinear. MPC based on linear models is acceptable when the process operates at a single setpoint and the primary use of the controller is the rejection of small disturbances. Operating points of modem chemical processes vary over large regions and cannot be modelled adequately using linear models. These conditions are observed in many situations such as, change over in continuous processes, tracking problems in start-up and batch processes and the control of nonlinear reactors. To properly control these processes a nonlinear dynamic process model should be used. Predictive control with a nonlinear model is referred as nonlinear model predictive control (NMPC). One of main difficulties to use NMPC is to develop a reliable nonlinear dynamic model. Most existing techniques for nonlinear dynamic model identification are only for discrete-time models although most process systems are continuous in time. A discretetime model can only work for a particular sampling frequency. If the sampling frequency is to change, the model has to be re-built. Hence, it is not convenient to use discrete-time model for multi-rate control. In contrast, once a continuous-time model has been created, it can be used for any sampling frequency, even for continuous-time NMPC. Although a continuous-time model has clear advantages, it has rare been used in NMPC except those based on first principles. The main reason is due to the difficulty to solve the differential parameter optimization problem associated with the continuoustime nonlinear model identification problem. In recent work, we have developed an efficient NMPC formulation using automatic differentiation (AD)\ This formulation has been successfully extended to train

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differential recurrent neural networks (DRNN). The developed DRNN can be directly used for NMPC within the above formulation. The training and control algorithms of DRNN are suitable for most process systems. A two-CSTR case study is presented to demonstrate the usage and benefit of the algorithms proposed. With the DRNN model developed, we can freely change the sampHng frequency to improve control performance. The paper is organized as follows. In section 2, a training algorithm is proposed using AD techniques. Section 3 presents formulations for predictive control to use a DRNN as internal model with AD techniques. These algorithms are applied to the case study in section 4 and in section 5 some conclusions of the work are provided. 2. DRNN training Assume a model-unknown continuous-time nonlinear dynamic system has riu inputs and Hy outputs. Npoints of input, u{k), ^ 0 , ,..,N-\ and output, y^{k), ^ 0 , ...,N data are collected at sampling rate h. These data are used to train a recurrent neural network (RNN) so that the RNN trained can predict the dynamic behaviour of the system in a reasonable accuracy. According to the universal approximation theory of artificial neural networks, there are many types of neural networksfi*ommulti-layer perceptrons (MLP) to radial basis functions (RBF), which can be constructed as recurrent networks to approximate the nonlinear system. The training algorithm to be discussed is suitable for any kind of networks. Hence, the DRNN to be considered is represented in the following general form. dx — = /(x,w,^), y = Cx dt

(1)

where XG R"' is the hidden state of the DRNN, we R"" the input, J^G R"' the output, ^G 7?"^ the parameters of DRNN to be trained, and C = [/ 0], i.e. the outputs are equal to the first Uy states. The collected input data are directly applied to the DRNN by assuming constant input between two sampling instants. The initial state of the DRNN is x(0) = [yp(0) 0]. Then, for a given set of parameters, 0, output can be predicted fi-om (1). The training algorithm to be proposed aims to minimize the total prediction error: (2) ^

k=l

^

To solve this optimization problem efficiently, the gradient, (p^=^is

to be calculated

using the AD techniques described as follows. Let the state of (1) be a Taylor series up to d terms, i.e. ^(0 = X/=o^['i^' ' where ^[i]~'n^' techniques^

Then, the Taylor expansion o f / c a n be directly obtained using AD as

/ = X/=0'^']^'^^^^^ Taylor coefficients,

coefficients, Xy^, with j
y[,]are functions

of

Differential Recurrent Neural Network Based Predictive Control Since dx/dt = / ,

1241

x^^+ij =xTr^/t](^[o]'--'^[/t],"»^) > ^•^- ^^^ coefficients, x^j^ can be

iteratively

obtained

from

x^^j = x(0).

Then,

at

next

sampling

point,

x{h) = ^

x^^h and it will be used as the initial value for integration of next interval.

For output, the Taylor coefficients are y^^^^ = Cx^^j and y(h)=Cx(h). More importantly, AD can be used to calculate sensitivities of (3)^.

''''"dx,,,~

dx,.,

''''"do

^^

The total derivatives are iteratively accumulated from (4) as follows.

De,^^^

de = CBe,n

(6)

For each sampling interval, the following iterative relation can be derived.

B,(„,(0) = 0, B,t„,((^+ !)/») = J 5,„(tt)/z-

k^Q,...,N-\

(7)

/=0

Jacobians of (2) are calculated from Taylor coefficients:

'

ae^^^^|, de de t'o

J.JE de

'

"^

(8)

Based on Levenberge-Marquardt algorithm^, the parameters can be iteratively updated as follows. 0,,,=d,-(j'j

+ Air'j'E

(9)

where A is determined by the algorithm to make sure the prediction error is reduced at each iterative step.

3. Predictive control The trained DRNN can be directly used as the internal model for a predictive controller. The control algorithm is based on the general formulation of nonlinear model predictive control using A D \ More specifically, at each sampling instance, a nonlinear least square optimization^ is performed to minimize the performance index represented as the integral of square error and control effort. The Jacobian matrix required by the nonlinear least square optimization is calculated using the algorithm similar to the

R. Al Seyab and Y. Cao

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training algorithm except that the independent variable is control input, u rather than network parameter. To save space, details of the control algorithm are omitted here. 4. Case study Training and predictive control algorithms proposed are applied to a two-CSTR process. The process is schematically shown in Figure 1. A full description of the system and a six-state model is available elsewhere"^. The control problem is to maintain both tank temperatures, y^=[Toi T02], at desired values in the presence of cooling-water temperature fluctuations within ±10 [K], i.e. ^=[Tcwi Tcyv2]- The manipulated variables (MV) are the cooling water flow-rates of two tanks, i.e. u=[Qcwi Qcw2], which corresponds to the second control scheme discussed by Cao and Yang (2004)\ Both MVs are subject to constraints, 0.05 < Qcwi, Q cw2 ^ 0.8 [mVs]. Two sets of 600-second input and output data are collected by applying random input signals to the 6-state nonlinear simulation model. The sampling rate of the training data is 0.1 [s], whist another set for validation is sampled at every 0.02 [s]. ^Feed 2

^Feed1

Input Cooling •"Water Out Mixer

Cooling Water Iri CSTR1

bP

Cooling l-H/Vater Out

Cooling Water! In CSTR2J

^

Pre

0=^ Figure 1. Two-CSTR process

Figure 2. DRNN Structure

The nonlinear dynamic system is approximated by a MLP DRNN shown in Figure 2 and represented as follows: dx dt

(10)

Cx

where W^ E i?"^"""^, W^ G i?"^"""" , and W^ G i?"^"""^ are connection weights, b^ G i?"^ and Z>2 G i?""" are bias vectors, whilst each element of the vector ( J ( » ) G 7?"^ represents the sigmoid-tanh function as the neural activation fiinctions, i.e.

a(n) =

l + e-

--1

The parameter vector is 6> = [vec(^^ f

(11) vec(^„ f

Z?f

vec(^2 f

^l ]^ ^ ^"'» where

n0=n^x (n^ +1) + «^ x (n^ + «^ +1). For the two-CSTR process, n^=6 and n^=6 are selected for the DRNN model and «„ = 2. Therefore, «^ = 96. For one epoch of

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training, n^xn^xN = 3456000 sensitivity variables have to be calculated. This causes an enormous computation load to DRNN training. In a typical situation, sensitivity calculation will take more than 90% of the total training time. Hence, it is really a computation bottleneck to DRNN training. The new algorithm proposed can significantly improve training efficiency as shown in Table 1 with a comparison to sensitivity calculation using ODE23 function in MATLAB. In Table 1, the computation time is for one training epoch, whilst the error is the maximum absolute error with respect to a reference solution obtained using ODE23 with a tolerance of 10"^^. AD results are obtained using ADOL-C^ with a mex wrap in MATLAB. The results clearly show that the AD based algorithm can not only reduce computation time by one to two orders of magnitude, but also significantly improve accuracy. Table 1. Computation time and accuracy comparison tolerance 10-^ 10-^ 10-^ 10-^^

ODE23 time, ms error 1.5899 40.61 162.281 0.0738 272.657 4.551x10-^ 316.375 1.864x10-^

AD time, ms 2.437 4.078 5.391 6.937

order 3 6 8 10

error 0.001 1.562x10-^ 2.606x10-^^ 1.0729x10-^^

The trained network is able to predict dynamic behaviour of the plant with a reasonable accuracy. The process data and DRNN predictions are compared in Figure 3. 380

370 f

370

365

o

°-° JM 350

CVJ

o

200 400 Time, sec

355

600

(c)

0

380

390 380

200 400 Time, sec

600

(d)

370

-^ 370

CM

, O

O

36ofeAjWlU 350

360

200 400 Time, sec

600

360 350

200 400 Time, sec

600

Figure 3. Plant data (solid) and prediction (dashed) comparison, (a) and (b) are trainging data, (c) and (d) are validation data.

R. Al Seyab and Y. Cao

1244

/

, ^^

(3

^ 50

370

^ 150

100

U'" '

2(

t-

3bb

10 100 E _- 0.5

!•—^1

in

20

30

10

ii^z^j^if

1

10 20 Time, sec

30

0.6

'^

04

t

J^L^I 100 time, sec

30

0.8

1 1 IL

1-,|_

20

150

150

Figure 4. Setpoint tracking simulation.

200

u

0.2

\

Figure 5 Disturbance rejection simulation. -10 K change in Tcwi and rcw2 at 2 sec.

A predictive controller is designed to control the two-CSTR process using the DRNN trained as the internal model. The controller uses /z=0.1 [s], prediction horizon, P=20 [s], and control horizon, M=3 [s] with the cost function weighted by output weights, [150, 100] and input weights, [1, 0.5] respectively. Simulation results shown in Figure 4 and 5 clearly indicate that the predictive controller with the DRNN model successfully achieves satisfactory performance without violating input constraints.

5. Conclusions A general DRNN training algorithm is proposed using AD techniques. The efficiency of the new algorithm in terms of computation time and solution accuracy and the effectiveness of the DRNN trained as the internal model of a predictive controller are demonstrated through the two-CSTR case study.

References 1. Cao, Y, "A Formulation of Nonlinear Model Predictive Control using Automatic Differentiation", Journal ofProcess Control, vol. 15, pp. 851—858, 2005 2. Christianson, B., "Reverse Accumulation and Accurate Rounding Error Estimatres for Taylor Series", Optimization Methods and Software, vol. 1, pp.81—94, 1992. 3. Marquardt, D., "An Algorithm for least-Square Estimation of Nonlinear Parameters", SIAMJ. Appl Math., vol. 11 , pp. 431^^41, 1963. 4. Cao, Y. and Yang, Z.J., "Muilt-Objective Process Controllability Analysis", Computers and Chemical Engineering, vol. 28, pp. 83—90, 2004. 5. Griewank, A., David, J. and Jean, U., "ADOL-C: A Package for the Automatic Differentiation of Algorithms written in C/C+-I-", ACM Transactions on Mathematical Software, vol. 22, pp. 131—167, 1996.

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Chance Constrained Programming Approach to Process Optimization under Uncertainty Pu Li,^ Harvey Arellano-Garcia,"^ Gtinter Wozny ^ '^Department of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. Des 17. Juni 135, Berlin 10623, Germany ^Simulation und Optimal Processes Department, Technical University ofllmenau, P. O. Box 100565, 98684 Ilmenau, Germany Abstract Deterministic optimization approaches have been well developed and widely used in the process industry to accomplish off-line and on-line process optimization. The challenging task for the academic research currently is to address large-scale, complex optimization problems under various uncertainties. Therefore, investigations on the development of stochastic optimization approaches are necessitated. In the last few years we proposed and utilized a new solution concept to deal with optimization problems under uncertain operating conditions as well as uncertain model parameters. Stochastic optimization problems are solved with the methodology of chance constrained programming. The optimization problem is relaxed into an equivalent nonlinear optimization problem such that it can be solved by a nonlinear programming (NLP) solver. The major challenge towards solving chance constrained optimization problems lies in the computation of the probability and its derivatives of satisfying inequality constraints. Approaches to addressing linear, nonlinear, steady-state as well as dynamic optimization problems under uncertainty have been developed and applied to various optimization tasks with uncertainties such as optimal design and operation, optimal production planning as well as optimal control of industrial processes under uncertainty. An overview on the theoretical development and practical applications in our recent work will be presented. Keywords: uncertainty, chance constraints, linear, nonlinear, optimization, control. 1. Introduction The research in the area of process optimization under uncertainty has been initiated by Grossmann and his coworkers (see Bansal et al., 2002). The emphasis in previous studies has mainly been on process design under uncertainty. Most studies follow the definition of feasibility test (Halemane and Grossmann, 1983) and flexibility index (Swaney and Grossmann, 1985a,b), in which uncertain variables over certain intervals are considered. A nested two-stage approach was proposed to solve the optimization problem (Grossmann and Floudas, 1987; Pistikopoulos and lerapetritou, 1995; Rooney and Biegler, 2003). Using these approaches, the optimality and reliability will be ensured by a conservative decision. The iteration between the first and second stage to solve the problem demands more expensive computational efforts. The violation of constraints is compensated by some penalty functions and leads to additional costs for the second stage decision. This approach is appropriate to solve planning problems under demand uncertainty (Petkov and Maranas, 1998). Furthermore, the solution with the compensation method provides no obvious information on the relation between reliability and probability which is crucial in decision making.

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P. Li et al

1 uncertainty

process Linear

\ /

Steady state

constraint

1

i

\

|

ly^

\

1

Single ^

dependent ^ound constrained output objective function improvement "^

Fig.l. a) Conservative and aggressive operation; b) Classification of chance constrained problems In the industrial practice, an overestimation of uncertainties, w^hich is a widespread practice in the chemical industry, leads to a conservative decision resulting in an unnecessary deterioration of the objective function. The main reason for these intuitive decisions in planning chemical process operations is due to the lack of systematic reliability analysis. In other cases, an aggressive decision may be preferred due to profit expectations. This strategy v^ill probably lead to constraint violations as a consequence. Accordingly, a systematic w^ay is required to evaluate the trade-off betw^een profitability and reliability. The implication of conservative and aggressive operations is illustrated in Fig. la, show^ing the distribution of a constrained output variable. Chance constrained programming is a competitive tool for solving optimization problems under uncertainty. Its main feature is that the resulting decision ensures the probability of complying w^ith constraints, i.e., the confidence level of being feasible. Thus, using chance constrained programming the relationship betw^een the profitability and reliability can be quantified. In other v^ords, the solution of the problem provides comprehensive information on the economical achievement as a function of the desired confidence level of satisfying process constraints. In the last few years, we have developed approaches of chance constrained programming to linear, nonlinear and dynamic problems and applied to different process engineering problems. 2. Optimization under uncertainty Uncertainties can be generally divided into external uncertainties like feed rate and/or its composition recycle flows, temperature and pressure of the coupled operating units, supply of raw material and utilities, customer demand, prices, market conditions. Internal uncertainties represent the unavailability of process knowledge such as model parameters. Model parameters are often regressed from a limited number of experimental data. While internal uncertainties have been well studied, external uncertainties have not been much emphasized. Moreover, they can be constant or timedependent. Usually the distribution of an uncertain variable can be gained through statistical regression from previous available data or through interpolation or extrapolation. There has been an explosive grov^h of computer-based process monitoring systems, which makes it relatively easy to acquire process data to be utilized for distribution analysis. Uncertain variables may be correlated or uncorrelated and their stochastic distribution may also have different forms. Often normal (Gaussian) distribution is considered as an adequate assumption for many uncertain variables in the engineering practice. The values of mean and variance are usually available. These uncertain input variables will propagate through the process to the output variables and the outputs will also be uncertain. For a nonlinear process it is very difficult to analytically describe the distribution of the outputs. The basic idea of optimization under uncertainty is to integrate the available stochastic information into the optimization problem formulation.

Chance Constrained Programming Approach to Process Optimization

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3. Chance constrained optimization A general chance constrained optimization problem can be formulated as follows: min E[/(x,u,^)] + 6; D [ / ( X , U , ^ ) ]

s.t.

g(x,x,u,^) = 0 Pr{h(x,x,u,^)>0}>a

(1) x(to) = Xo

where / is the objective function, E und D are its expected value and variance, respectively. The vectors g and h represents the equality (model equations) and inequality constraints, x, u and ^ are the vectors of state, control and uncertain parameters. Pr {h(x,u, ^)>0} represents the reliability of complying with the inequality constraints h, while a is the user-predefmed confidence level (0
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P.Lietal

chance constraints is also to adopt a relaxation to transform it into a deterministic NLP problem. The essential challenge here lies in the computation of probabilities of holding constraints as well as their gradients. A computation method for one single probabilistic constraint is proposed by Wendt et al. (2001). The basic idea of this method is to map the uncertain output distribution to that of the uncertain input variables. For this purpose, an uncertain variable ^^, which has a monotonic relationship with the output constrained variable yf, is to be found. Due to the monotony, the boundary of the constrained value y^^''""'^ in the output region corresponds to a limit value ^ / for ^ within the input region. This limit value can be computed by solving the model equations with given decision variables u and the boundary value j;^*^""^. Then, the probability computation of the output constraints can be transformed to a multivariate integration in the limited area of uncertain inputs:

p{/
(2)

where p(^) is the unified distribution function of ^. For the multivariate integration, we use collocation on finite elements to discretize the bounded region of the uncertain inputs. The input boundary ^ / is computed by the Newton-Raphson method based on the output value of y^*''"'^ . Since this boundary depends on the realization of the uncertain variables (^i ,...,^s-i)y it has to be computed on each collocation point of these variables. In this way, the equality constraints (model equations) are eliminated by expressing the state variables in terms of decision and uncertain variables. This solution strategy is not dependent on the distribution of the uncertain variables, if the multivariate integration in (2) can be carried out. This procedure of probability computation can be straightforwardly extended to the multiple single probabilistic constraints. However, the extension of the probability computation procedure to a joint probabilistic constraint is not a trivial task, since it is difficult to find an uncertain variable which is monotone to all outputs. A crucial issue in dealing with probabilistic constraints is the convexity analysis of the problem for nonlinear systems. An analytical result concerning this issue is given by Prekopa (1995). It states that if the constraints form a convex set and if the density function is logarithmically concave, the corresponding problem may be convex. Many but not all of the usually existing distributions share the property of a log-concave density, e.g. multivariate normal distribution or uniform distribution on bounded convex sets (Henrion et al., 2001). 3.1.3. Nonlinear dynamic systems The chance constrained approach has been extended to solve problems of nonlinear dynamic optimization under uncertainty (Arellano-Garcia et al., 2003). The approach involves efficient algorithms for realizing the required reverse mapping from the uncertain constrained output to the uncertain input. In the probability and gradient computation an optimal number of collocation points is chosen so that the original idea is applicable for dynamic optimization problems with larger scale. Furthermore, the analysis of the impact of the probability limits of the chance constraints on optimal process operation policies, particularly with regard to the optimized value of the objective function, is an important issue. Thus, an efficient search algorithm of the lower and upper bounds of the uncertain input with regard to the integration of the probability density function, and an efficient approach to gradient computation have been developed to deal with both single and joint probabilistically constraints (ArellanoGarcia et al., 2003). The joint probabilistical constraint is generally more severe than

Chance Constrained Programming Approach to Process Optimization

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single constraints with the same limit since it describes the satisfaction of all restrictions simultaneously. 3.2. Applications Probabilistic programming has been applied in many disciplines like finance and management. However, very few applications have been made in chemical process design and operations. In the past years, we have implemented our chance constrained solution approaches to address different process engineering problems. We deal with production planning problems under uncertain market conditions (Li et al., 2003, 2004) as well as refinery planning under uncertain demand (Li et al., 2004). The problems can be formulated as a dynamic mixed-integer chance constrained optimization problem which can be relaxed to an equivalent deterministic MILP formulation. An optimal decision with a desirable trade-off can be made for the future purchase, sales and operation. The stochastic dynamic optimization approach has been successfully implemented for a reactive batch distillation process with single and joint constraints (Arellano-Garcia et al., 2003). 4. Process control under Uncertainty Feedback control systems are used in unit operations to follow the decided operating point. However, many important process variables are not closed-loop controlled. This fact leads to an alternative concept of control based on chance constraints. We proposed a robust control strategy, model predictive control under chance constraints, to deal with multivariable constrained control problems. Both model and disturbance uncertainties are considered and assumed to be correlated multivariate stochastic variables. The output constraints are to be held with a predefined probability with respect to the entire horizon. The problem formulated is a stochastic program under joint probabilistic constraints. Using an efficient sampling method this problem is relaxed to a nonlinear programming problem which can be solved by SQP. Simulation results of a distillation column control show the performances of the proposed strategy (Li et al., 2000). Under the assumption of a linear system, the formulated problem is convex. Thus, it can be solved with a nonlinear programming solver. Probability values and gradients of satisfying constraints, composed of disturbance sequences with multivariate normal distribution, are computed with an efficient framework (Li et al., 2002, 2004). Unlike the linear case in which controls only affect the mean of the outputs, for nonlinear processes they have impact on both mean and covariance of the outputs. A mixing process with both uncertain feed flow rate and uncertain feed composition was used to illustrate the effectiveness of chance constrained nonlinear MPC (Xie et al., 2005). To realize the feedback control, the operating policy is adjusted from period to period based on the moving horizon approach. 5. Concluding remarks An overview on the theoretical development and practical applications of chance constrained programming is presented. Our contribution to process optimization under uncertainty is that the solution results from the approaches developed can provide both optimal and reliable decisions for process design and operation. Moreover, analysis of the results makes it possible to identify the critical constraint which cuts off the largest part of the feasible region. This is a piece of important information for the decisionmaker to relax the constraint, if necessary, so as to gain a meaningful decision. It is foreseeable that probabilistic programming will be a promising technique in solving optimization problems under uncertainty in the process industry.

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References Arellano-Garcia H., Martini W., Wendt M., Li P., Wozny G., 2003, Chance Constrained Batch Distillation Process Optimization under Uncertainty. In: I. E. Grossmann, C. M. McDonald (Eds.): FOCAPO, Proceedings pp. 609-612. Arellano-Garcia H., Wendt M., Li P., Wozny G., 2003, An Evaluation Strategy for Optimal Operation of Batch Processes under Uncertainties by Probabilistic Programming. In: B. Chen, A.W. Westerberg (Eds.), PSE8 Proceedings pp. 148-153. Arellano-Garcia H., Wendt M., Li P., Wozny G., Robust Optimization of a Reactive Semibatch Distillation Process under Uncertainty. In: A. Kraslawski, I. Turunen (Eds.): Computer Aided Process Engineering, 13, Elsevier, 551-556. Arellano-Garcia, H.; Martini, W.; Wendt, M.; Li, P.; Wozny, G., 2003, Nonlinear Stochastic Optimization under Uncertainty, Chemie-Ingenieur-Technik, 75, 814-822. Bansal, V.; Perkins, J. D.; Pistikopoulos, E. N., 2002, Flexibility Analysis and Design Using a Parametric Programming Framework, AIChE Journal, 48, 2851-2868. Diwekar, U.M., Kalagnanam, J.R., 1997, Efficient Sampling Technique for Optimization under Uncertainty. AIChE J. 43, 440. Grossmann, I. E.; Floudas, C , 1987, Active Constraint Strategy for Flexibility Analysis in Chemical Processes, Comp. Chem. Eng., 11, 675-693. Halemane, K. P.; Grossmann, I. E., 1983, Optimal Process Design under Uncertainty, AIChE Journal, 29, 425-433. Henrion, R.; Li, P.; Moller, A.; Steinbach, M.; Wendt, M.; Wozny, G., 2001, Stochastic Optimization for Chemical Processes under Uncertainty, Online Optimization of Large Scale Systems, Grotschel et al. eds.. Springer, 455-476. Henrion, R.; Li, P.; Moller, A.; Wendt, M.; Wozny, G., 2001, Optimal Control of a Continuous Distillation Process under Probabilistic Constraints, Online Optimization of Large Scale Systems, Grotschel et al. eds.. Springer, 497-518. Kail, P., Wallace, S.W., 1994, Stochastic Programming. New York: Wiley. Li, P.; Wendt, M.; Arellano-Garcia, H.; Wozny, G., 2002, Optimal Operation of Distillation Processes under Uncertain Inflow Streams Accumulated in a Feed Tank, AIChE Journal, 48, 1198-1211. Li, P.; Wendt, M.; Wozny, G., 2004, Optimal Planning for Chemical Engineering Processes under Uncertain Market Conditions, Chemical Engineering & Technology, 27, 641-651. Li, P.; Wendt, M.; Wozny, G., 2002, A Probabilistically Constrained Model Predictive Controller, Automatica, 38, 1171-1176. Li, P.; Wendt, M.; Wozny, G., 2003, Optimal Operations Planning under Uncertainty by Using Probabilistic Programming, FOCAPO, Proceedings pp. 289-292. Li, P.; Hai, R.; Wozny, G., 2004, Feasibility Analysis and Optimal Design Using a Chance Constrained Programming Framework, FOCAPD, Proceedings pp. 514-518. Li, W. K.; Hui, C. W.; Li, P.; Li, A. X., 2004, Refinery Planning under Uncertainty, I&ECR, 43, 6742-6755. Petkov, S.B., Maranas, C , 1997, Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty. Ind. Eng. Chem. Res., 36, 4864. Pistikopoulos, E. N.; lerapetritou, M. G., 1995, Novel Approach for Optimal Process Design under Uncertainty, Comp. Chem. Eng., 19, 1089-1110. Prekopa, A., 1995, Stochastic Programming, Dordrecht: Kluwer. Rooney, W. C ; Biegler, L. T., 2003, Optimal Process Design with Model Parameter Uncertainty and Process Variability, AIChE Journal, 49, 438-449. Swaney, R. E.; Grossmann, I. E., 1985a, An Index for Operational Flexibility in Chemical Process Design. Part I: Formulation and Theory, AIChE Journal, 31, 621-630. Wendt, M.; Li, P.; Wozny, G., 2002, Nonlinear Chance Constrained Process Optimization under Uncertainty, Industrial & Engineering Chemistry Research, 41, 3621-3629. Xie, L., Li, P., Wozny, G., 2005, Chance Constrained Nonlinear MPC, International Workshop on Assessment and Future Directions of Nonlinear Model Predictive Control, Lauterbad, Germany, August 26-30, 2005.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Support for design of user interfaces in plant operations Xiwei Liu, Hiroaki Kosaka, Masaru Noda, and Hirokazu Nishitani Graduate School of Information Science, Nara Institute of Science and Technology, 8916-5, Takayama, Ikoma, Nara 630-0192, Japan Abstract A human model incorporating physical and mental workload estimation is proposed, into which knowledge bases and procedures for monitoring and fault diagnosis have been built. As a case study, a user interface (UI) system, including an overview panel and alarm settings for plant operations, was evaluated from the viewpoints of shortening fault detection and identification (FDI) track and decreasing workloads. Evaluation results showed that the UI system was improved by the proposed model. Keywords: plant operations, user interface, model-based evaluation, fault detection and identification 1. Introduction It is important to construct an effective, efficient, and satisfactory user interface (UI) for a supervisory plant control system. Some human model-based evaluation approaches have been proposed to decrease human error in plant operations [1, 2]. In this study, we represent a human model to evaluate a UI system from the aspects of its operational and cognitive obstacles and physical and mental workload during the course of fault detection and identification (FDI) in an emergency. 2. Model-based Framework Based on a model human processor (MHP) [3], we built a model of a humanmachine system, as shown in Fig. 1, which includes a human machine interface (HMI) model (part a), a human model (part b), and an attention resource model (part c). The HMI model contains static and dynamic information of all user panels such as numbers, icons, layouts of user panels, and responses to key presses and mouse clicks. It communicates with a plant simulator in real time through which a human operator can access process variables and send commands to the simulator. The HMI model is adjusted according to design schemes whose usability is revealed and compared through UI evaluation. We proposed an evaluation approach for the static information of user panels by using static HMI and perceptual models [4]. 2.1. Human model The human model is a metaphor of a human operator as an information processing system (IPS), which commonly consists of perception, memory, cognition, and execution elements. When plant status is normal, human perception is in passive mode without definite intention. In an emergency, however, operators mainly concentrate on a certain few items or areas. We assume that active perception directly transfers detected items into the short-term memory (STM). Three knowledge bases (KBs) for variable information (VI), failure-symptom relation (FS), and alarm management (AM) are built in the long-term memory (LTM). VI-KB includes color, position, and the normal range of every process or control

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• 1 Part a • Partb ^ Physical Sc Mental Workloads mPartc 1 L r

Attention Allocation

1 1 1

Long-term Memory i

J

i

ii r

^M r t Cognitive -^->- Short-term -^— Perceptual Icons, Data | Pre)C essor Memory Processor Alarms 1

^'i

Motor Processor

Click, Keypress Eye Movement

Fig. 1 Model of human-machine system FL2=6

SLi=2

SL2=2

SL3=1

SL4=2

SL5=3

SLn(l^n^N)

SLN=3

Fig. 2 Failure-symptom links variable on the user panels. FS-KB contains all the known failures with these symptoms. AM-KB includes responses to alarm messages. Two procedures for normal state monitoring (NSM) and abnormal state supervising (ASS) are built in as well as knowledge bases. To simplify UI evaluation, we assume the human model never errs due to lack of knowledge. The KBs and procedures are created according to process risk analyses, operational experiences, and expert reviews. In the LTM, VI-KB, FS-KB, and AM-KB are implemented by a table, a matrix, and if-then rules, respectively. The cognitive processor sends commands to a motor processor to move gaze point or to push a button to confirm the status of the associated variables. The motor processor executes the commands from the cognitive processor. 2.2. Fault detection and identification (FDI) Once a malfunction occurs, several variables change outside of their normal ranges. These abnormal changes are symptoms of failure cause. The relationships between failure causes and symptoms can be represented by a bipartite graph, as shown in Fig. 2, where F^ is wth failure cause and Sn is n\h symptom; FLm is the number of all symptoms for the wth failure cause, and SLn is the number of all causes related to the wth symptom. FL and SL values reflect the complexity of cause and effect, respectively. The association strength {AS) of an FS link between Fm and Sn is defined as follows: 1 ^*^m,«

for

-

y{m,n)

(1)

SLnY. SL^ S„&A

where Am is a set that includes all symptoms connected with Fm- For any failure cause. the total AS of all links is normalized to 1.0, and in set A^ AS^ „ indicates the K^An

occurrence probability of failure cause FmFor simplification we assume that the FDI process starts after an alarm. The following ASS procedure is activated after detecting the alarm: 1. Obtain symptom Sn according to the AM-KB.

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2. Calculate ^ 5 values for all possible failure causes. 3. Select a possible failure cause linked to Sn in descending order oiAS, 4. Confirm symptoms connected to the selected failure cause in the descending order of AS, and totalize the AS value for the symptoms if the occurring abnormalities agree with the symptoms. 5. If confirmation fails, return to step 3 until a failure cause is identified whose total AS value exceeds a specified threshold. 23. Task decomposition and workload estimation Human behavior in the FDI process is classified into physical and mental subtasks. The latter includes perception, cognition, STM, and LTM activities: for example, reading an alarm message, remembering a former alarm, searching in a KB, and rejecting a failure cause are perception, STM, LTM, and cognition subtasks, respectively. Physical subtasks include gaze point movement and finger movement. The physical workload for these subtasks is assessed by the magnitude of motion: for example, keyboard input workload is assessed by the number of key presses. On the other hand, the mental workload for perception, cognition, STM, and LTM subtasks are scaled to several levels according to the required information of the subtasks, human subject experiments, and expert questionnaires by referring to VACP workload method [5]. 3. User Interface Evaluation and Improvement We propose the following procedure to evaluate and improve UIs: 1. Build an HMI model that includes all of the UI information. 2. Build VI-KB, FS-KB, AM-KB, and NSM and ASS procedures based on the target process, expert reviews, operational experiences, and process risk analyses. 3. Through FDI simulations, we can obtain the track of human behavior with changes in physical and mental workloads, and the required time to identify a failure cause. 4. Based on FDI performance with mental workload, evaluate the UIs including the alarm system and improve them if necessary. 5. Repeat steps 1-4 until an acceptable result is obtained. 4. Case Study 4.1. Behavior of a virtual subject We evaluated the overview panel of a boiler plant simulator, as shown in Fig. 3. The human model judged the failure cause based on KBs and procedures after a malfunction occurred. In the simulation, we employed an oil heater malfunction where a highpressure fuel oil strainer clogged. Table 1 shows a list of alarm messages after the malfunction. Figure 4 shows part of the FS links for the FDI from the A201 (O2 concentration) alarm. Five failure causes link to an abnormally low A201: induce draft fan (IDF) trips, fuel pumps (FOPs) trip, forced draft fan (FDF) degradation, oil heater failure, and turbine degradation. AS values for the five failure causes (AS26,7, AS26,h ^^26,6, ^^25,9, AS26,8) were calculated as 0.097, 0.072, 0.068, 0.038, and 0.021, respectively. In the simulation, the human model automatically recorded all FDI subtasks until the failure cause was identified. Figure 5 shows a resultant FDI track, where some subtasks are omitted for space limitations. The human model searched through links in descending order of AS and tried to confirm a failure cause with a maximum AS. IDF trip was therefore the first suspect, but it was rejected because the iconic IDF color showed that

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X. Liu et al. Table 1 List of alarm messages for original alarm settings

No.

Time

^. ^ . Item time (sec.)

Item description

Alarm Alarm description

1

12:31:44

317

F204

Air flow

LL

PV low-low alarm

2

12: 29: 58

211

F204

Air flow

LO

PV low alarm

3

12: 29: 48

201

P203

Furnace pressure

LO

PV low alarm

4

12:28:11

104

A201

Flue 02 concentration

LL

PV low-low alarm

5

12:28:01

94

A201

Flue 02 concentration

LO

PV low alarm

Oil heater failure occurred at 12: 26: 27. ^^^M

K iiff S !»• Hi m» m m m • 1A

\tkp!®

Fig. 3 Original overview panel

^F203.MVA

. , . ^ .J y

^ V

. /

rsL=2)W

^

^

Fig. 4 A part of FS links for FDIfromA201 alarm the fan v^as normal. Sequentially, FOPs trip and FDF degradation were also rejected. While the FOP-trip malfunction v^as being confirmed, the A201 low-lov^ alarm appeared. The corresponding alarm confirmation interrupted the FDI process without supplying much information except emergency level, so the FDI was continued. Finally, the human model confirmed most of the related symptoms of oil heater failure and the total AS value exceeded the predefined threshold (0.8). Figure 6 shows changes in workloads during the FDI process.

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Perception subtask Cognition subtask Physical subtask STM subtask LTM subtask — • C o n t i n u o u s Discontinuous

fheater ( A S = 0 . 0 ^ ^ ) \ ^ & first symptom ^ ^ - ^ " check F203.MV &F202.MV

check T204.PV, R034.CPV,andP204.PV

Switch to fueT supply panel

check F204.MV andF202.PV

Fig. 5 FDI track for original UI system —#— M ental workload —»™- Phy sical workload 2^ 25 r

20

26

30

37

41

Time (S) Fig. 6 Workload graph for original UI system

4.2. Improvements In the case study, some alarm messages were also related to the furnace pressure control loop. This fact misguided the FDI and confused the virtual subject, which resulted in a long FDI track. It also caused a high workload, especially in the later FDI stage because of excessive information for rejecting and confirming several possible failure causes and needless alarms. We modified the overview panel and alarm setting by taking the following modifications: 1. Add important process variables, such as A201 and A202 (CO concentration). 2. Supply shortcut buttons to relieve STM workload. 3. Adjust alarm limitations for timely guidance. Figure 7 shows the modified overview panel in which clicking anywhere inside the large rectangle can switch to the corresponding operational panel. Alarm limitations of three process variables—fuel oil temperature, fuel oil flow rate, and burner-head pressure—^were adjusted based on statistics under oil heater failure. Table 2 is a list of alarm messages after oil heater malfunction. New settings did not cause any nuisance alarms. By using the modified UI system, FDI was accomplished more quickly with decreased workloads as well. 5. Conclusion We proposed a human-machine system framework for evaluating a UI system when a plant experiences an emergency and showed a case study. Through this model-based

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approach, the obstacles and pitfalls of the UI system were found along with their causes and subsequently improved. The presented model-based approach can be used as a support tool for UI system design. In the future, the KBs and procedures in LTM and mental workload estimation will be adjusted based on human subject experiments. References [1] T. Nakagawa, Y. Nakatani, H. Yoshikawa, M. Takahashi, T. Furuta, and A. Hasegawa: Simulation based evaluation of man-machine interface in power plants, in E. Hollnagel and H. Yoshikawa (eds.), Cognitive Systems Engineering in Process Control' 96, ESSCS, Groningen, pp. 115-121(1998). [2] J. Tanji, K. Monta, J. Kawai, T. Masui, and I. Ezaki: Evaluation method of human-system interface for nuclear power plants, Proceedings of CSEPC 2000, Taejon, Korea, pp. 233-240 (2000). [3] S. K. Card, T. P. Moran, and A. Newell: The Psychology of Human-Computer Interaction, Lawrence Erlbaum Associates, London (1983). [4] X. Liu, H. Kosaka, M. Noda, and H. Nishitani, Model-based static evaluation of graphic panels for plant operations, Joumal of the Society for Industrial Plant Human Factors of Japan,Vol. 10, No. 2, pp. 103-113 (2006). [5] J. H. McCracken and T. B. Aldrich, Analyses of selected LHX mission functions: Implications for operator workload and system automation goals. Technical Note ASI479-024-84, Army Research Institute Aviation Research and Development Activity, Fort Rucker, AL (1984).

Fig. 7 Modified overview panel

rx.. Time

No. 1 2 3 4 5 6

8 8 8 8 8 8

47:51 46: 14 46:04 45:22 44:42 44:36

Table 2 List of alarm messages after modification Elapsed , . , , Item Item description Alarm Alarm description time (sec.) 202 105 95 53 13 7

P203 Fumace pressure A201 Flue 0 2 concentration A201 Flue 02 concentration F202 Fuel flow T204 Fuel temperature P204 Burner head pressure

Oil heater failure occurred at 8: 44: 29.

LO LL LO LO LO HI

PV low alarm PV low-low alarm PV low alarm PV low alarm PV low alarm PV high alarm

16th European Symposium on Computer Aided Process Engineering and 9tli International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Online Prediction of End-of-Batch Product Quality using Phase-specific PLS Models Xuan-Tien Doan^ and Rajagopalan Srinivasan^'^* ^Institute of Chemical and Engineering Sciences 1 PesekRoad, Jurong Island, Singapore 627833 ^Department of Chemical and Biomolecular Engineering National University of Singapore 10 Kent Ridge Crescent, Singapore 119260 Abstract Batch processes are commonly operated in multiple phases; phase-based approaches are intuitively well suited for batch process monitoring and quality prediction. In this paper, we therefore propose a phase-based approach to online prediction of end-of-batch quality. Our framework integrates dynamic feature synchronization and dynamic time warping (DTW) with Dynamic Partial Least Square (DPLS). The Singular Point concept [1] is applied to divide process trajectories into phases with similar dynamics. The challenge of unequal phase lengths is solved by using DTW to equalize the phase trajectories from different runs. A DPLS model [2] is subsequently built between each phase and the final quality variables. Using a well-known simulation, PenSim, we show that this phase-based DPLS (PDPLS) outperforms original DPLS in both monitoring as well as online prediction of end-of-batch quality. Keywords: phase-based monitoring, online quality prediction, dynamic time warping (DTW), singular points (SP), dynamic partial least square (DPLS). 1. Introduction As the workhorse for manufacture of low volume, high value-added products, the batch mode of processing has been the preferred choice of the specialty chemical and pharmaceutical industries. Batch manufacturing offers numerous control and supervision challenges including significant nonlinear dynamics, multi-phase processes, and batch-to-batch variations. This is further compounded by the availability of limited online measurements; usually product qualities can seldom be measured directly and infrequent lab-measurements have to be used for close-loop control and disturbance rejection. In this paper, we describe a statistical approach for process monitoring as well as online predicting end-of-batch product quality. More often than not, batch processes have multiple operating phases. Each phase in a batch trajectory has its own underlying dynamics that is significantly different from that of others. Each phase can have an independent effect on the end-of-batch product quality variables. Further the duration of phases in different runs could be different. Therefore the phase-specific or stage-based framework, which has been popular for control and monitoring of batch operations [3-6], should be adopted for quality prediction as well. Surprisingly, even though numerous methods have been proposed for online end-of-batch quality prediction, few have taken a phase-based approach. In [4, 7, Corresponding author. E-mail: [email protected].

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8], a phase-based statistical approach was proposed batch process monitoring and quality prediction. However, a number of issues remain unsolved. For example, [7] suggested that the mean trajectories and standard deviations of phases with unequal lengths can be calculated from the actual data available. However, such an estimation method effectively matches a phase-end point in one trajectory to a non-end point in another trajectory and hence the issue of mismatch in process trajectory landmarks remains. In [8], a scheme was proposed to divide a batch process into a number of phases based on clustering the correlation matrices of time-slice matrix Xk (I x J) and quality matrix Y where I and J are the number of batches and the number of variables respectively; k is the time. However, it is believed that the proposed clustering effectively imposes that one time slice Xk can only belong to one phase, and hence assumes equal lengths for trajectories in each phase, which is rarely found in practice due to batch-to-batch variation. Another approach based on multiway partial least square (MPLS) combined with batch-wise unfolding technique (MPLSB) and variablewise unfolding technique (MPLSV) was proposed for process monitoring and quality prediction in [9]. In this approach, an indicator variable (IV) is used to align trajectories in a process phase and hence its applicability is dependent upon the availability of such an indicator. In addition, its quality prediction is limited to predefined increments of batch completions; hence it is not truly online prediction of end-of-batch quality. In this paper, we further develop the phase-based approach proposed in [3] for process monitoring as well as online prediction of end-of-batch product quality. We incorporate the Dynamic PLS approach as proposed in [2] into our integrated framework previously developed and propose the phase-based Dynamic PLS (PDPLS). A well-known simulation of a batch process is used to demonstrate the superiority of PDPLS over ordinary DPLS. 2. Phase-based dynamic PLS (PDPLS) 2.1. Dynamic Feature Synchronization In [1] it is noted that ^^Information content is not homogenously distributed throughout a signar. In other words, some points, termed singular points, in a trajectory contain more information about process dynamic behaviors than other points. Examples of singular points (SP) include points of discontinuities, trend changes, and extrema. Because of their high information content, SP can be used for signal synchronization and comparison as described in [1]. In our phase-based approach, SP are used to decompose batch trajectories into multiplQphases with similar process dynamics. For SP identification, it is observed that a SP breaks normal correlation of residuals from the best fit of a window of data. The identification procedure is as follows: 1. Maintain a moving window of size x. 2. Obtain the residuals from fitting the moving window with a straight line and evaluate their standard deviation. 3. Compare current standard deviation with the corresponding past values. A SP is identified if there is significant difference. 2.2. Dynamic time warping (DTW) DTW originates from speech recognition and is capable of translating, compressing, and expanding a pair of trajectories in such a way that similar events are aligned and a minimum distance between them is obtained [6]. In this paper, asymmetric

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DTW with slope 1/2 and band global constraint B=30 is used. Details of DTW implementation in our phase-based approach have been described in [3] and references therein. Only important features are summarized here. DTW always needs a set of endpoint, global and local constraints. Global constraints define and restrict the search space while the local ones define the localized feature of the optimal path [10]. Usually endpoint constraints, which require that the endpoints of the signal pairs must match, are used in off-line analysis. For online application of DTW, the challenge is that as a batch evolves, its online data is not complete and hence its endpoints are unknown. Consequently, endpoint constraints cannot be imposed. The DTW algorithm in this paper replaces that constraint by minimizing the distance between the current trajectory with a possible match in the reference trajectory.

/*=argmin{Z)(S,R.)}

(i)

where: Ri is the reference trajectory from time t = 1 to time t = i and S is the evolving sample trajectory. A similar idea for online DTW implementation was also proposed (but not verified) in [10]. The difference is that online DTW algorithm in this paper stops at obtaining i* because our phase-based approach only needs it for scaling purpose. 2.3. Phase-based DPLS for process monitoring and quality prediction 2.3.1. DPLS In [2], Chen and Liu introduced the dynamic PCA/PLS (DPCA/DPLS) concept as applied to batch processes. Even though the concept dates back to 1998, it was not until [2] that it was first applied to monitoring batch processes. Mathematically, DPLS starts with forming a time-lagged window for each of the batches in the reference database:

^x'id+iy x'id+iy

x'idY x'(d+iy

• •• • ••

x'(ir

^

x'(2/ (2)

^ >

x!{K-\y

V ^'(KY where x(A^ j ~ L-^i A:

^ik

'"

• •• -^j ki

x!{K-d)%

^^ ^^^ J-dimensional observation vector at

time k; K is the batch length; d is the time lag; and i indicates the i^^ batch. Quality data Y^ required

for

DPLS

approach

is

F^ = [ y '

y^

•••

y]

where

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y—\yx

X.'T Doan et al

yi

"'

yNy J ^^ ^^^ Ny-dimenslonal quality vector at end of batch /.

The idea is to form a quality data matrix Y* with proper size for modelling purpose [2]. The covariance matrix S' between the time-lagged data X'a and corresponding quality data Y^ is

K-d-l and the average covariance matrix S^^^ for I batches

(^-j-iEs' S^^=

'•^^—

(4)

I(K-d) From the average covariance matrix S^^^, PLS algorithm could be applied to obtain a PLS model, as outlined in [2, 11]. 2.3.2. Phase-basedDPLS (PDPLS) In [3], we proposed an augmented DPCA approach for online monitoring of batch processes and demonstrated its superiority over ordinary DPCA. In this paper, we extend the phase-based approach to online quality prediction. The proposed method is as follows: Model building 1. SP identification for all batches in the reference database. Synchronize the identified SP and decompose batch trajectories into corresponding/j/za^^^'. 2. Group corresponding trajectories in each phase (possibly with different lengths) in all the reference batches. 3. Apply DTW to trajectories in Qdichphase to equalize their lengths. 4. Apply DPLS algorithm to obtain a DPLS model for Qdioh phase. These models can then be applied for online use as follows: Online monitoring and end-of-batch quality prediction 1. Obtain the new measurement x. Check if it is a SP. If it is, retrieve the PLS model that corresponds to the n^xX phase and go back to Step 1. Otherwise, add to time-lagged data matrix and proceed to Step 2. 2. Retrieve the reference and batch mean trajectories from database of the current phase. Identify the match x^f of x in the mean trajectory using online DTW. Scale x using Xref. 3. Evaluate T^ statistic using the scaled measurement x and current DPLS model. Compare the obtained T^ statistic with its control limit. If the limit is exceeded, announce a fault. 4. Evaluate end-of-batch quality vector y using the scaled measurement x and the current DPLS model. Go back to Step 1 for the next measurement.

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3. Case study A fed-batch penicillin cultivation process simulation, known as PenSim, is used as a case study. PenSim (available at http://www.ctiee.iit.edu/-cinar), was developed at the Illinois Institute of Technology to serve as a testbed for several applications including batch process monitoring [12]. The flow chart of PenSim simulation is shown in [13] and its variables which form the training and test data sets are listed in Table 4 in [12]. There are two main phases in penicillin cultivation process. The first phase corresponds to biomass growth and is carried out in batch mode for about 40-45 hours. The second phase, where glucose is continuously fed, occurs for the rest of the batch [13]. In this study, 41 batches of data are generated using PenSim with sampling time of 0.5 hour; initial conditions and setpoints were randomly chosen in the range specified in Table 2 in [14]. 31 of the batches have duration of 400 hours. The 32''^- 41'^ batches have their duration specified randomly between 380 and 420 hours. These 10 batches together with four other batches (28*^ - 3r^) form the training set for the proposed phase-based DPLS and MPLS approaches. For test data, 13 scenarios including 1 normal and 12 combinations of 3 faults with different locations and durations were simulated. Fig. 1 and Fig. 2 are the monitoring and prediction results for PDPLS and ordinary DPLS in Case 4 where the simulated fault corresponds to a 15\% step increase in aeration rate introduced at sample 120^^ and terminated at 160*^. As can be seen from the figures, PDPLS clearly outperforms DPLS in detecting the fault as well as predicting the final quality of the batch. Not only does DPLS fail to detect the simulated fault, it also has a few false alarms at initial stage of the batch. In addition, DPLS significantly underestimates the final quality while PDPLS is much closer to the actual values. While the fault is detected by PDPLS using T^ statistic in Fig. 1, its effect is also visible in Fig. 2. Although PDPLS shows no clear trend of eventually stabilizing on the actual values, most of the time it stays close enough (averaging 1.2% for biomass concentration and 0.7% for penicillin concentration) to be a good estimate.

Case 4 - Aeration rate fault

50

100 samples

200

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Figure 1. Monitoring results for our phase-based DPLS (PDPLS) and ordinary DPLS

Case 4: Biomass cone. 14 13.5| faillt location

Case 4: Penicillin cone.

• actual cone. DPLS predicted ' PDPLS predicted fault location

0

200

400 600 sample

800

"0

200

400 600 sample

800

Figure 2. Online end-of-batch quality prediction using PDPLS and ordinary DPLS

References I] Srinivasan, R. and M. S. Qian, Ind. Eng. Chem. Res., 44 (2005) 4697. 2] Chen J. and K. C. Liu, Chem. Eng. Sci.,57 (2002) 63. 3] Doan X. T. and R. Srinivasan, presented at AIChE annual meeting, Cincinnati, US, 2005. 4] Lu N., F. Gao and F. Wang, AIChE J., 50 (2004) 255 5] Dong D. and T. J. McAvoy, AIChE J., 42 (1996) 2199 6] Undey C. and A. Cinar, IEEE Control systems magazine, 22 (2002) 40 7] Lu N., F. Gao, Y. Yang, and F. Wang, Ind. Eng. Chem. Res., 43 (2004) 3343 8] Lu N., and F. Gao, Ind. Eng. Chem. Res., 44 (2005) 3547 9] Undey C , S. Ertunc and A. Cinar, Ind. Eng. Chem. Res., 42 (2003) 4645 10] Kassidas A., J. F. MacGregor and P. A. Taylor, AIChE J., 44 (1998) 864 II] Hoskuldsson A., J. Chemometrics, 17 (2003) 630 12] Birol G., C. Undey and A. Cinar, Computers and Chemical Engineering, 26 (2002) 1553 13] Undey C , E. Tatara and A. Cinar, J. Biotech., 108 (2004) 61 14] Lee J. M., C. K. Yoo and I. B. Lee, J. Biotech., 110 (2004) 119

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Optimal current distribution control for parallel electrolytic baths Hidekazu Kugemoto"', Keita Ozakia^ Yoshinori Kutsuwa" and Yoshihiro Hashimoto^ ^Sumitomo Chemical Co. Ltd., Niihama 792-8521, Japan bDept. of Computer Science, Nagoya Institute of Technology, Nagoya 466-8555, Japan Abstract A simple optimal balance control scheme for electrolyzation plants is proposed. It can be realized through the use of a commercially available model predictive control (MPC) package. In most cases, in order to calculate the optimal operating conditions for energy conservation, much property data or information regarding disturbances, such as the deterioration of electrodes and ion exchange membranes, is necessary. However, the proposed controller can achieve the optimal conditions without them. In this paper, it is proven that the resultant converged operating conditions of the proposed controller are identical to the actual optimal conditions. Keywords: Multi-loop control. Distribution control, Process control. Detection algorithm. Saturation control

1. Introduction Electrolysis plants are a very popular choice for chlorine chemistry, but they are typically energy intensive. Consequently, electricity cost conservation is an important issue. It is, therefore, common to operate the electrolytic baths in such a way as to take advantage of the price difference between day and night. However, in our case, we don't need to consider this because our electricity isn't received from a commercial power plant. Instead, the method used to conserve electricity power for our plant is the optimization of the balance of electricity currents for the electrolytic baths. While the total current is determined by the demand of the chlorine, the total electricity power can be changed by altering the resistance of the electrolytic baths. The resistance can be tuned through its operating temperature. Each bath is cooled to remove the heat of electrolysis. The higher the temperature, the lower the resistance, so, high temperature operation is more economical. However, we must set upper limits for the electrolytic bath's temperatures in order to protect the equipment. Upper and lower limits for the electric current also exist. The problem of optimization is how to search for the bath's optimal temperatures to minimize the total electric power. The relationship between electric current and temperature in electrolytic baths is nonlinear and it is affected by the conditions of the ion exchange membranes and electrodes, which are not monitored. Therefore, the calculation of the optimal values of the bath's temperature is impossible without measuring the resistance or other such things. However, it has been proven in this paper that the optimal solution of this nonlinear problem is always at a point on the edge of the feasible region. The region is a combination of temperature upper limits and electric current constraints. The

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temperature and electric currents are measured with a process control system. Instead of the optimal value calculation of nonlinear equations, a simple scheme using feedback control behaviour has been contrived. By continuing the feedback compensation, the operating conditions approach the optimal point. This feedback control is realized by using a commercial model predictive control package. Although the original performance index was the electric power consumption, the performance index of the MPC is the deviation of the electrolytic bath's temperatures from their upper limits. This control scheme has been working very successfully in a real industrial plant for more than two years. Energy conservation has been improved since this controller was applied.

2. Electrolytic Baths Fig. 1 shows the structure of the electrolysis process. There are nine electrolytic baths and they have ion exchange membranes in the tanks. The sodium chloride solution is decomposed into chlorine and hydrogen and the solution on the cathode side is circulated. The chlorine gas product is removed from it and fresh sodium chloride solution is added. The chlorine is the main product of this system and it is sent to the chlorine chemical plants. The solution on the anode side is also circulated and hydrogen is removed. The circulating water is cooled down by a constant temperature coolant. The bath temperatures are controlled by circulating water through heat exchangers, with the controllable ranges being about 10 degrees Celsius. I Current Control

I Current Generator

o Bath B X Bath F + Bath h

rT

^Tk

•jjTemp Control Electrolytic Heat -S—I

j Coolant Temp Control J

(Small Valve),

Circulation Water

y = - 0 . 1 2 x + 16.50

li^i^

y = -0.12 x + 16.00

Coolant

70

75

80

85

1 90

95

Temperature (°C)

Fig.2. The Relationship between temperature and resistance

Fig. 1. Electrolyzation system

The total electric current of the electrolytic baths, / is controlled by the current controller. Its set-point is determined by the demand of chlorine. The total current is the sum of the current of each electrolytic bath. n

(1) /=1

The electric power is the product of the voltage and the current. The voltage is common for each electrolytic bath and is composed of the practical electrolysis starting voltage and the load voltage. The total voltage of an electrolytic bath can be expressed as follows.

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E = E^=Eo,+RrI, (2) The resistance of the electrolytic bath depends on the temperature as shown in Fig. 2. Fig. 2 shows the relations between E11- and bath temperature, T.. The resistance can be expressed by eq. (3). R,=a,T,^b, (3) Here, a. < 0 and R^ > 0 Even if the total current, /

is fixed by the demand, the total electric power

consumption, W in eq. (4) can be reduced by adjusting the operating temperature, T.. W = E'I

= {EO.-^R.I.)I

(4)

The values of the parameters Eo-, a- and b- in eqs. (2) and (3) change as a result of the deterioration of the ion exchange membranes and electrodes.

3. Formulation of an Optimization Problem By introducing a variable ^ = J _ , the optimisation problem can be expressed as follows. n I +

W=

0<

^EOjXj

'^

/ ^min

i

< X,. <

^^^

^-

(6)

where a^ < 0 is assumed.

7-1

7=1

7=1

4. Characteristics of this Optimization Problem The boundary of the constraints in eq. (7) can be expressed as shown in eqs. (8) and (9). ^^fir^ni?^>I^i +Zl^^7 -E0,)XjX,

-I^^^Xj

7=1

7=1

Eo, ^Eoj

-z

(EOJ-EO,)\XJ

min

I '

Eoj -Eo^

i_ 1 V

+

I-L Eoj -EOf n-\ min /

min

EOj -Eoi

n-\

Eoj=EOj

+ £ {^(^-^»i„k-/.i„^,}

(8)

H. Kugemoto et al

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0 > / ; _ W - / ^ , + Z K -Eo)x.x, -I^,^±Xj 7=1

(9)

7=1

The boundaries are hyper-planes, which are composed of hyperbolic curves and lines between the two variables X. and x •. If the starting voltages of the electrolysis, Eo. and EOj , are different from each other, the relationship between X- and x • is a hyperbolic line. All of the curves and lines go through the origin of the coordinate of X-. The hyperbolic line shows a monotonic increase in the first quadrant of the space of x.. The feasible region, which satisfies all constraints in eqs. (8) and (9), is conical and its upper vertex is the origin of the coordinate of X.. The performance index shows a monotonic decrease with respect to each value of X-. I+ dW dx,

t,{Eoj-Eo,)xj 7=1

(10) V7=l

—/<0

The optimal values of X. may be an edge point of the feasible region. Unless the electric current of any bath exceeds its limits, the temperature of every bath is at its upper limit value under the optimal operating conditions. The feasible regions can be illustrated as the shaded areas in Fig. 3. The optimal points are shown by circles. Unless an electric current exceeds its constraints, the temperatures of the optimal operating points are at their upper limits as shown in Fig. 3(a). In some cases, the electric current constraints are included into the conditions of the optimal point instead of temperature constraints. When a bath condition is much better than others and its current exceeds the upper limit, its temperature should be suppressed. This is shown in Fig. 3(b). Conversely, if the electrode deteriorates, its current might be too low. Even if the temperature is at the upper limit, the electric current might be lower than the lower limit. In this case, the temperature of one of the other baths must be decreased. Fig. 3(c) shows this case.

T,=T,

Ti=Tj^,

T,==T,

T,=T,

-n=r.,

(b) Fig. 3. Optimal conditions

(c)

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5. Optimization by the MPC Since both the temperature and the electric current of each bath are measured, it is possible to judge whether the present operating condition is optimal or not. The variables manipulated in order to carry out the optimization are the set-points of the bath's temperature controllers. There are time lags between the actions to the bath's temperatures and their affects on the electric currents. Strong interactions between the baths also exist because they are connected in parallel. Therefore, optimization can be realized by using a multivariable model predictive controller. The controlled variables of the MPC are the electric currents of each bath, and their predicted variables are used in optimization. The zone control mode of the MPC is applied to electric current control. An objective function on optimizer in the MPC is to set the summation of bath temperature set-points, which are MVs of MPC, to their upper limits. In keeping the predicted electric currents within the permitted regions, MVs are increased to their upper limits. Although the system dynamics are nonlinear, a linear model is utilized. The predictive model has a nine by nine transfer function matrix. Each transfer function is of the first order and is a dead time lag system. All of the diagonal elements are the same transfer functions, whose gains are 0.2, time constants are 70min and whose dead time is ISmin. Non-diagonal elements are the same functions but the gains are -0.025 (the gain divided by eight), time constants are TOmin and the dead time is 15min. This approximation can be done easily without a lot of field tests. Due to the robustness of model predictive control, this feedback controller can achieve optimal operation, even though the predictive model has model errors. The behavior of the electrolytic bath's temperature set-points are illustrated the arrows in Fig. 3.

6. Simulation of an Optimal Balance Controller The dynamic behavior of the optimal current balance controller can be illustrated by simulation. In this simulation, the plant has three baths and the initial electric currents of all the baths are identical. r

1 Manipulated variable

a §

A,B,C

-^c

A Manipulated variable • ^

Time Upper limit

(tppff

A3,C

a

I,owcr

Controlled variable

limit

(a)

Lower limit

(b) Fig. 4 Behavior in the same resistance

(c)

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Fig. 4(a) shows the behavior in the case that the resistance values of the electrolytic baths A, B and C are identical. The electric currents are kept constant, even though all the temperature set-points are at their upper limits. The total power consumption is decreased as a result of the increase of bath temperatures. Fig. 4(b) shows the behavior of the balance controller in the case that the degrees of deterioration of the three electrolytic baths are different. A> B> C is the order of the degrees of deterioration. In this example, all the electric currents are within their constrained regions, even when all bath temperatures are at the values of their upper limit. The optimal condition corresponds to the one in Fig. 3(b). Fig. 4(c) shows some other results under the same conditions, where A deteriorates significantly more than B or C (A»B>C). In this example, the electric current reaches its lower limit. The temperature of the bath of the best condition is suppressed to increase the electric current of the worst. Its optimal condition corresponds to the one in Fig. 3(c).

7. Effects in a Real Plant This controller has been working in a real industrial plant for more than two years very successfully. Fig.5 shows the effects of the proposed balance control. The horizontal axis is the total current, which is determined by the chlorine demand. The vertical axis is the voltage. The product of the values of both axes is, therefore, the total power consumption. It is shown in Fig. 5 that the voltages after balance control are smaller than the ones before balance control for the same total current. The balance control has been successftil in reducing power consumption.

• • Before balance control •

After balance control

Total current(kA)

Fig. 5. Effects in real plant

8. Conclusion A simple optimal balance control scheme for an electrolysis plant has been proposed. Optimization under constraints can be carried out with the use of a commercial MFC package. It has been proven in this paper that it is possible to achieve the optimal operating conditions for any degrees of deterioration of electrolytic baths. Although this optimal distribution problem is nonlinear when it is solved directly, it can be realized easily using zone control and optimization schemes on the MFC. Furthermore, this MFC is robust and very simple, and it isn't necessary to measure the degrees of deterioration. It is expected that this balance control scheme will be effective for the balance control of other kinds of plants too.

References Kugemoto, Hidekazu and Keita Ozaki (2004) Optimization of electrolytic baths operation using advanced control.: Soda & Chlorine, 55, pp. 149-154 Yokogawa Electric Corporation (2004) Multi-Variable Model Predictive Controller, Exasmoc:http://www.yokogawa. com/shs/APC/Exasmoc/shs-Exasmoc-outlineOl. htm

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Hybrid Model Predictive Control of a Sugar End Section Daniel Sarabia^, Cesar de Prada, Smaranda Cristea, Rogelio Mazaeda ^Dpt. Systems Engineering and Automatic Control Faculty ofSciencies, c/ Real de Burgos s/n, 47011, Valladolid, Spain, [email protected] Abstract This paper deals with the MPC control of an industrial hybrid process where continuous and batch units operate jointly: the crystallization section of a sugar factory. The paper describes a plant-wide predictive controller that takes into account, both, the continuous objectives and manipulated variables, as well as the ones related to the scheduling of the batch units. The MPC is formulated avoiding the use of integer variables, so that a NLP optimization technique could be applied. Simulation results of the controller operation are provided*. Keywords: Hybrid predictive control, Mixed continuous-batch processes, Plant-wide control. Hierarchical control, Sugar production. 1. Introduction Plant-wide control is attracting considerable interest, both as a challenging research field and because of its practical importance. It is a topic [1] characterized by complexity in terms of the number and type of equipments involved, diversity of aims, and lack of adequate models and control policies. In this paper, the control of the final part of a beet sugar factory, the so-called sugar house or sugar end, where sugar crystals are made, is presented. Perhaps the most characteristic aspect of its operation is that batch and continuous units operate jointly, which introduce the need for combining online scheduling with continuous control. As such, it is a hybrid process that requires non-conventional control techniques. Many approaches have appeared in the literature in recent years for hybrid predictive control. A natural approach integrates in a single mathematical formulation the different elements of a hybrid process by using integer variables for representing on/off decisions and integer equations for the logic relations between variables [2] besides the continuous equations. The fact that the internal model of the MPC controller includes continuous and integer variables lead to a mix-integer optimization problem [3], which in many cases is difficult and time consuming to solve. A natural way of approaching complex systems is using a hierarchical point of view, separating the problems that can be solved locally at a lower level from the ones that require a global consideration. This paper focus on these overall decisions, and describes a controller that takes into account both continuous control of key process variables as well as the scheduling involved in

The authors wish to express their gratitude to the Spanish Ministry of Education and Science (former MYCT) for its support through project DPI2003-0013, as well as the European Commission for its support through VIFP NoE HYCON.

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the operation of the crystalHzers, which operate in batch mode. The controller follows the MPC paradigm, solving a non-linear model-based optimization problem on-line every sampling time. Moreover, the problem is re-formulates in terms of prescribed patterns of the batch units variables and time of occurrence of the events, instead of using integer variables, which allows to solve the optimization problem as a NLP one, saving computation time. The interest of this contribution comes not only from the fact that it is a challenging control problem, but because problems with a similar structure are present in many industrial process. 2. The Sugar End Section and Control Architecture Sugar factories produce commercial sugar in a set of vacuum pans or "tachas" from an intermediate solution called feed syrup. Each tacha operates in a semi batch mode following a predefined sequence. A scheme of the sugar end section of the particular case that has been considered can be seen in Fig. 1. From the three vacuum pans A, the massecuite or cooked mass, which is the mix of sugar crystals and non-crystallized syrup (mother liquor), is unloaded into an agitated heated vessel named "malaxador". From this one, the mix of mother liquor and crystals is separated by means of a set of eight centrifrigals. Massecuite from tachas type A, gives way to commercial white sugar and two kinds of syrup: the so-called lower purity syrup and the higher purity syrup. The later has a small percentage of dissolved crystals and, so, a higher purity, and its is recycled to the feeding tank (melter) of tachas A. On the contrary, the lower purity syrup is sent to another storage vessel (tank B) and processed again in one tacha named B. The proportion between both kinds of syrup can be adjusted using a timer in the local centrifrigals control. In tacha B the whole process is repeated, this time with longer operation times due to the lower purity of the syrup, but with an important difference: the sugar produced in the three centrifixgal separators B, sugar B, is not commercialised but recycled to a melter due to its color and impurities, while the lower purity syrup is discharged as a by-product called molasses. beets

water

J_L Continuous part of the factory

feed syrup

Vacuum Vacuum Vacuum pan A3 panA2 pan Al cooked mass

Vacuum panBl

' higher porily syrup

Malaxador A

Fig. 1: A simpHfied scheme of the sugar end section. The overall control objectives of the sugar end section are summarised next: 1. Processing the flow of feed syrup coming from the previous continuous sections of the factory avoiding bottlenecks in production. That is to say to maintain the levels in the malaxadors A and B and feeding tanks (melter and tank B) between certain upper and lower limits. This objective implies an adequate scheduling of the vacuum pans operation, that is, decide when every unit must load and unload its cooked mass. 2. Maintaining the quality of the crystals in terms of size and size distribution. This objective is solved locally in every tacha, where the operation of the crystallization is managed in order to obtain proper conditions for sugar crystal growth.

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3. Maintaining brix and purity in each feeding tank as close as possible to given sepoints in order to maximize the amount of sugar A produced, because the processing time and capacity of the tachas depend on both variables. This has also an influence in the first objective. From control architecture point of view, a common strategy in complex systems, is to decompose the problem in several levels or time scales, so that what can be solved locally, involving a limited set of resources or decision variables is separated from those decisions that involve variables having an effect on the dynamics of the whole system. In our case, this hierarchical decomposition recognisees at least three layers: Local SISO controllers: Such as the temperature control in the malaxadors, flow controls, etc. These are managed by the DCS of the plant and have fast dynamics compared with the ones of the sugar end. They are supposed to operate well using standard controllers. Sequence control of each batch unit: They are implemented also in the DCS or local PLCs and operate according to predefined parameters and external orders, such as load and unload. These local controllers are assumed to do its best; according to it's tuning, in order to complete its tasks each cycle. Plant wide-control: This layer is responsible for aims 1 and 3 of the previous list, in spite of the changes in the flow and quality of the incoming syrup. For this purpose, besides the scheduling of the tachas, the controller can manipulate the proportion between lower and higher purity syrup in the centrifugals and its operating frequency, which is equivalent to establishing its total processing flow. Thus, it is a hybrid control problem involving continuous and discrete variables. This task is performed very often manually by the person in charge of the section. Finally, from the point of view of this layer, sequence and local SISO controls can be considered as included in the process, operating in cascade over them. 3. Hybrid MFC Control A natural approach to many decision problems is Model Predictive Control (MPC): A model of the process is used to predict its future behavior as a function of the present and future control actions, which are selected in order to minimize some performance index. The optimal control signals corresponding to the present time are applied to the process and the whole procedure is repeated in the next sampling period. In MPC of complex systems, it is very important that the internal model being as simple as possible while still being a good representation of the process. On the other hand, it must correspond to the view and purpose of the plant-wide control, including only those variables and phenomena relevant. Internal model of sugar end section combines dynamic mass balances of total mass, solid content and sacharose in the continuous units (feeding tanks and malaxadors) with abstractions and simplifications of the other parts of the process, tachas and centrifugals. Notice that, being a semi-batch unit, when started, the model of a tacha must provide the flow demand to its feeding tank and the flows of crystals and honey to the downstream malaxador, all as a function of brix and purity of feed, assuming a fix sequence control layer. A full first principles model implementing mass and energy balances, as well as crystal growth and local control functions can perform this task, but this approach will lead to a huge model, useless for MPC. Instead, an abstract view of the tacha is employed that makes explicit use of the special patterns that its input and output flows must follow as well as of results obtained off-line from the first principles model. Input (^/„) and output {qout)flowsof a vacuum pan in a real production, where

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two batches are produced, can be seen in Fig. 2 a) and b). For simplicity in the graphic, we have named only three stages: loading, cooking and unloading. Fig. 2 c) and d) also shows, the shape approximation of ^/„ and qout used in the internal model of the tacha. batch 1 •—»*=

, ! batch 2

cooking

ru _rLjL.^^jL

UUU— \

tloadl tloadl

tload2

C) )

tloadl

tloadl ry t operationl

unloaiiiing operation

11

tunlokdl

-L unloadl •--•

tutiloadl

^)

Fig. 2: Input and outputflowsof a real evolution of vacuum pan and of simplified model. In Fig. 2 a), flow ^/„ is different from zero in several situations: The first one is when a loading order arrives at time tioadi, ^m having a known value for the also known loading period. Another one is when the semi-batch cooking stage starts. The duration of the loading stage Tioadx can be estimated fairly well, but for the value of the flow and the duration of the cooking stage Tcookx and for the rest of stages, another procedure must be used. This is a key point in obtaining a reduced model of the vacuum pan, and the approach followed has been to use tables like the one in Fig. 3 a) and b) relating the main variables of the vacuum pan with the properties of its feed, purity (P) and brix (B). Purity 90 92 94 96 98

a) 68 6213 6050 5920 5801 5718

Brix 70 5576 5522 5388 5275 5189

72 74 5172 4695 5017 4545 4885 4414 4766 4304 4689 4217

76 4252 4095 3978 3863 3771

Time duration Tcook of cooking stage (sec.)

b) 68 6.84 7.03 7.20 7.40 7.50

Brix 70 7.24 7.45 7.64 7.84 8.00

72 7.70 7.93 8.15 8.40 8.56

c) 74 8.20 8.46 8.73 8.96 9.19

76 8.71 9.10 9.35 9.64 9.92

Inflow qm in cooking stage (Kg/sec.)

Brix

68 80.39 83.49 87.07 90.93 95.24

70 80.39 83.49 87.07 90.94 95.24

72 80.39 83.49 87.07 90.94 95.24

74 80.39 83.49 87.07 90.95 95.24

76 80.39 83.49 87.07 90.95 95.24

Purity of mother liquor

Fig. 3: Typical table obtained off-line from first principles dynamic of a tacha. The value of these variables can be computed by interpolation from the tables. These have been obtained off-line integrating a frill first principles model of the vacuum pan starting from a syrup with different values of purity and brix. The variability of operation of every tacha, depends only on the state of the feed, making it possible to obtain the above mentioned tables for a range of reasonable operating conditions. The other signal in Fig. 2 b) corresponds to the outflow qout, which is zero except for the unloading period Tunhadi- Obviously the logic of operation implies that the unloading time tunioadi must be placed after the operation has finished, which can be translated into a constraint such as tunhadx > Uoadx + Tioadx + Toperationx, the latest being the intermediate operation period for the current feeding conditions and it is formed by sum of different stages duration, included Tcookx- These periods can be computed as before from interpolation in a table rj{B,P) (i=every stage) that has also been obtained off-line. In order to complete the vacuum pan model, other constraints must be added reflecting its logic

of

operation,

such

as

tioadl

>

hoadX +

TioadX +

ToperattonX +

tunloadX +

TunloadX t h a t

indicates that the next batch 2 must start after the previous batch 1 has been unloaded. Obviously, these two constraints are necessary for each batch predicted and for each vacuum pan. Also additional tables are needed, see Fig. 3 c), such as the ones relating brix and purity in the feeding tank with: the total cooked mass obtained Mcm(B,P), the brix Bnii(B,P) and purity Pmi(B,P) of mother liquor, and percentage Acin(B,P) of crystals of cooked mass in the unload stage of a tacha.

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In relation with the subjacent time model and the scheduling policy, the classical approach considers the time axis divided in sampling periods, where each sampling time j has an associated integer variable indicating if unit i starts or not it operation in period j . The scheduler solves a MIP problem to determine the optimal starting and ending times of each batch units. In this paper we have applied an alternative approach that is coherent with the use of the temporal patterns shown in Fig. 2 c) and d). It assumes as unknowns the time of occurrence of the events, tioadx and tunhadu which are real variables, instead of using integer variables in every sampling period [4]. In this way, all the decision variables of the internal model are continuous. Notice that this approach means that the scheduling problem is not computed separately but it is integrated into the overall predictive control and the need for solving mix integer optimization problems is avoided, being substituted by an NLP one that includes among its decision variables the time instants in which every tacha must load and unload along the prediction horizon. 3.1. NMPC controller Before the non-linear model predictive control problem can be solved, it is necessary to adapt some concepts used in standard continuous MPC to the context of mix continuous-batch processes. The first one is the prediction horizon (N2) that will be translated into Np minimum number of full cycles performed for all batch units (a cycle is the evolution of each batch unit from the state measured, the instant of called to controller, to the next future same state). The concept of control horizon (Nu) is split into batch control horizon (Nbi) and continuous control horizon (Nc). The first refers to the number of cycles of each batch unit i in which the decision variables tioad and tunhad will be computed. From Nbi until the end of the prediction horizon (Np), these values will be equal to the ones of the last cycle. Notice that this implies the assumption that a stable cyclic pattern will be reached at the end of the prediction horizon, in a similar way to how the future control signal is treated in continuous MPC. Each Nbi will fix the number of unknown time instants tioad and tunhad, two per cycle and per unit. Finally the Nc horizon has the classical meaning for the classical continuous manipulated variables. The control decisions are computed solving an NLP optimization problem where the aim is to minimize a quadratic cost function J:

J=f"{^a,(y,(t)-yfr

+^^jAuj(tf}dt

(1)

with the usual constraints yi"'^^ < yi(t) < yi"'"^ and Uj"'^ < Uj(t) < Uj"'^^ where the yi's extend to brixes and purities in the feeding tanks as well as the levels in these tanks and the two malaxadors, and tstop is the total time of prediction. Respect to the future manipulated variables, Uj are times of load and unload vacuum pans plus total flow and proportion of higher and lower purity syrup in the centrifugal separators of section A and B. The optimization is subjected to the internal model of the process and additional constraints imposed by the range and operation of the vacuum pans and other units. 4. Simulation results and conclusions The control strategy described in the previous sections was tested in simulation using the state-of-the-art EcosimPro environment. The process was represented by a detailed simulation built using validated models of the Sugar Processes Library [5] including sequential and local controls of all units. The MPC controller was programmed in C++ and contained the SQP algorithm, which, in turn, was able to call another EcosimPro simulation with the internal model for computing the cost function J (integration of

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dynamical internal model) each time it was needed. The sampling period was chosen as 15 min. We present an experiment of 69.4 hours (250000 sec), with an inflow of feed syrup of 6 Kg/sec. with 94.4 of purity and 72 of brix. All control horizons (Ncbi) were fixed in 2 and prediction horizon (Np) was fixed in 3 (25 hours of prediction). Fig. 4 a) and b) shows levels of the melter and the malaxador A, and its minimum and maximum values allowed, Fig. 4 c) shows purity and brix in the melter. Fig. 4 d), e) and f) shows the same variables but for the section B. The sequence of stages of vacuum pans Al, A2, A3 and B can be seen in Fig. 4 g). Time of stage 1 is the manipulated variable for load syrup and time of stage 9 is the manipulated variable to unload cooked mass, cooking stage correspond with stage 8 and load and unload stages with 3 and 11. g) Sequence of stages of tachas Al, A2, A3 and B a) Level of melter (in %)

120000

d) Level of tank B (in %)

160000

200000

240000 0

b) Level of malaxador A (in %)

Wfii^^ 0

40000

80000

80000

120000

160000

200000

240000 0

40000

80000

120000

160000

^Purity [-•'"^

80000

200000

240000 lo

200000

240000

5

3

Brix

Purity

40000

160000

f) Purity and Brix in Tank B

/ 0

120000

NNNNNNN\ N^

c) Purity and Brix in melter

]

40000

e) Level of malaxador B (in %)

120000

160000

200000

240000 0

Brix

/

]_ 40000

80000

120000

160000

200000

240000

"

40000

80000

120000

160000

200000

240000

Fig. 4: a), b) and c) Controlled variables for section A. d), e) and f) The same variables but for section B. g) Sequence of stages of tachas Al, A2, A3 and B. In this paper a plant-wide control strategy for the crystallization section of a beet sugar factory has been presented. It is based in a hierarchical view of the problem and, in the use of MPC with a simplified model that combines material balances of the continuous units and an abstract model of the batch ones. This is described in terms of tables computed off-line and prescribed patterns of the batch units variables and time of occurrence of the events, instead of using integer variables, which allows to use NLP algorithms instead of MIP ones. The strategy has proved to perform well in a simulated environment and opens the door to practical implementations at industrial scale. References A. Bemporad, M. Morari, 1999, Control of systems integrating logic, dynamics, and constraints, Automatica, 35:,407-427 K. T. Erickson, J. L. Hedrick, 1999, Plant-Wide Process Control. Wiley Publishers C. A. Floudas, 1995, Non-linear and Mix-Integer Optimization. Oxford Univ. Press C. de Prada, A. Merino, S. Pelayo, F. Acebes, R. Alves, 2003, A simulator of sugar factories for operators training, AFoT 2003, ISBN: 84-95999-46-3, 97-100 C. de Prada, S. Cristea, D. Sarabia, W. Colmenares, 2004, Hybrid control of a mixed continuousbatch process, ESCAPE14, ISBN: 0-444-51694-8, 739-744

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Systematic Methodology for Reproducible Optimizing Batch Operation S. Bay J0rgensen^* and D. Bonne^ ^CAPEC, Department of Chemical Engineering, Technical University of Denmark, S0ltofts Plads, 2800 Kgs. Lyngby, Denmark This contribution presents a systematic methodology for rapid acquirement of discretetime state space model representations of batch processes based on their historical operation data. These state space models are parsimoniously parameterized as a set of local, interdependent models. The present contribution furthermore presents how the asymptotic convergence of Iterative Learning Control is combined with the closed-loop performance of Model Predictive Control to form a robust and asymptotically stable optimal controller for ensuring reliable and reproducible operation of batch processes. This controller may also be used for Optimizing control. The modeling and control performance is demonstrated on a fed-batch protein cultivation example. The presented methodologies lend themselves directly for application as Process Analytical Technologies (PAT). 1. I n t r o d u c t i o n The batch processing types covered in this paper includes Batch, Fed-batch and periodic operation which all have the common traits of a repeated operation which start from nearly the same initial conditions. Thus the time within the bacth and the batch number are the two characteristic independent variables. Batch processing is subject to variations in raw material properties, in start-up initialization and other disturbances during execution. These different disturbances introduce variations in the final product quality. Compensating for these disturbances have been difficult in the past due to the nonlinear and time-varying behavior of batch processing and to the fact that reliable on- or in-line sensors for monitoring final product quality rarely are available. Consequently development of a systematic methodology which can ensure reliable reproducible operation may provide significant bennefits for batch processing. Each batch operation may be defined as a series of operational tasks, i.e. mixing, reaction and separation. Within each task a set of subtasks, e.g. heating/cooling, (dis)charging is handled. There may be more than one feasible set of operational tasks that can produce the specified product (-s). Consequently an optimal sequence of tasks and subtasks with respect to. a defined objective needs to be identified. This set of operational tasks is labeled the optimal batch operations model Thus the Batch Operations Model combines the batch processing tasks normally specified in a generic recipe with the batch equipment under availability and other resource constraints. *[email protected]

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Several research groups have used general empirical model knowledge to develop methods for control of batch processes including an experimental adaptation (optimization) of the Batch Operations Model, labelled "the solution model" through tracking the Necessary Conditions of Optimality (NCO) Srinivasan and Bonvin (2004). Another approach Akesson et al. (2001) exploits the knowledge that optimal batch operation consists of a sequence of operations to determine the presently (most) constrained variable. A data driven approach develops prediction of end of run properties (Flores-Cerrillo and MacGregor, 2003), while asimmilar inspiration lead to reconstruction of approximate time series models from available process data (Gregersen and J0rgensen, 2001; Bonne and J0rgensen, 2003). This paper presents methodologies based upon a model for batch or periodic operation which can ensure reliable reproducible operation and which may enable optimizing operation. The contribution comprises data driven time series modeling of batch processes and a learning model predictive control methodology. The modelling methodology produces both a Linear Time-Invariant state space model representation for inter-batch prediction and a Linear Time-Varying state space model representation for intra-batch prediction. The modelling approximates the non-stationary and nonlinear behavior of batch processes with a set of local but interdependent linear regression models parameterized as AutoRegressive Moving Average models with eXogenous inputs (ARMAX). Tikhonov Regularization is apphed to estimate the parameters of this model set. Learning Model Predictive Control is presented for control of repeated operation of stochastic Linear Time-Varying systems with finite time horizons together with tuning requirements for ensuring guaranteed convergence and hence closed-loop stability. The methodologies have been implemented as a Matlab toolbox Grid of Linear Models (GoLM). 2. M e t h o d s Batch processes are modeled with the toolbox GoLM as a sets of N LTI models. Such a set of LTI models could also be referred to as one LTV batch model. These LTI models can be parameterized in a number of ways, but in the present contribution an ARM AX parameterization was chosen. This choice of parameterization offers a simple multivariable system description with a moderate number of model parameters. The objective of the model set is to quantify the causal correlations between the process outputs y^^i G M^^, inputs Uk^i-i G W^"", and distrubances Vi G W^y, for i = 1 , . . . , t, at times t = 1 , . . . , A/" in batch k. To simphfy notation, define the input Uk, output y^, shifted output y°, and disturbance Vk profiles in batch k as Uk= [ < 0 yk=

<1

•••

K,N-1 ]'

[ VkA < 2

•••

yk,N y

(1)

Vk — [ Vk.O Vk,! • • • Vk^N-l J ^/c= [ <,l

V'k,2 •••


The GoLM toolbox models the differences between two successive batches with the ARMAX model A y , = y , - y , _ i = - A A y ^ + B/\uk

- Cv^

(2)

Systematic Methodology for Reproducible

Optimizing Batch Operation

Mil

where Awfc = Uk — Uk-i and Ay^ = y^ — y\_i- The batch ARMAX model (2) may be converted into different representations dependent on the particular application task. If the task at hand is to predict (or simulate) the behavior of a batch before it is started, the following form is convenient Ay^ = HAyk^Q - GAuk

+ Fvk

(3)

where Ayk^o — yk,o — 2//c-i,o- The change in the initial conditions Ayk^o can be considered as either an input/control variable or a disturbance. Form (3) above is also convenient for the task of classification (e.g. normal or not) of a batch after it has been completed. Furthermore, the form (3) can be used to determine open-loop optimal recipes in the sense of optimizing an objective for the batch. If the objective is to minimize the deviations e^ = [ej. p e'^ 25 • • • 5 ^k N] ^ ^k,t ^ ^"^^^ from a desired Batch Operations Model y, then (3) can be modified into ek = y-yk

= e^-i - HAyk^o + GAuk

- Fvk

(4)

The two forms (3) and (4) of the batch ARMAX model above are applicable to off-line or inter-batch type applications. For on-line estimation, monitoring, feedback control, and optimization however, it is convenient to use a state space realization of the batch ARMAX model. In an observer canonical form, which is structurally a minimal realization, the state space realization is given as Xk,t = AtXk,t-i + BtAuk,t-i

+ StVk,t

. .

^yk,t = yk,t - yk~i,t = Cxk,t with Auk^t-i = Uk^t-i — Uk-i,t-i and the initial condition 0:^,0 = CAyk^Q- Just as (3), the state space model form (5) is convenient for prediction, monitoring, and optimization type applications, and it facilitates on-line implementation of such applications. Furthermore, the state space model form (5) is particularly well suited for closed-loop or feedback control applications. For tracking control applications the state space model form (5) can be modified to Xk,t = AtXk,t-i + 13tAuk,t-i + StVk,t ek,t = yt-

yk,t = ek-i,t

-

,^. (6)

Cxk,t

In order to use (6) for tracking control, it is necessary to estimate the states based on noisy observations of the outputs. Assume that during a batch {k), observations Zk^t of the outputs yk^t are collected at times t = 0 , 1 , . . . , A^ and let the optimal estimate of the state Xk^ti i^ batch k at time ti given data up to and including time ^2 be given as the conditional mean (e.g. obtainable with a Kalman filter) Xk,ti\t2 — E{xk,ti \ ^^,^2} where the information Xk,t'' ^k,t = {Zk,t,^Uk,t,Xk,t-l},

^k,-l =^A;-l,Ar,

^0,-1 = {y-l,U-i,Z-i}

(7)

Then the tracking error ek^ti in batch k at time ti given data up to and including time ^2 is estimated as ek^ti\t2 — ^k-i,ti\N " Cxk,ti\t2 where the smoothened estimate of the error profile in batch A: — 1 is given as ^k-l\k-l

= [ ^k-l,l\N

^k-l,2\N

••'

^k-l,N\N J = y ~ yk-l\k-l

(^)

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and y/(._i|^_i is the smoothened output profile estimate from batch A: — 1 (e.g. obtainable with a Kernel Smoother). To utilize available information during a batch to obtain the best possible tracking performance, the following learning Model Predictive Control formulation {AiXfc,/, J^^^ ^ = arg

min

N-l

s.t.

Xk,i\t = AiXk^i-i\t + 6iAuk,i-i ^k,i\t = ^k-l,i\N

'^mini-1 ^ AUk^i-l Vmini SI Vi

/g\

— CXk,i\t + Uk-l^i-l

^k,i\t S

< '^maxz-l

Vmaxi

is solved at times t = 0 , l , . . . , 7 V — l i n batch k and the control sequence A<=[A<,o,o

M,i,i

•••

M,iv-i,;v-i ]'

(10)

then approximates a closed-loop optimal control sequence. I.e., at time t in batch k, (9) is solved based on updated state estimates, and the input Uk,t = Auk^t + Uk-i,t is implemented the process. 3. C a s e S t u d y : P r o t e i n p r o d u c t i o n The case study considers modehng, control and optimization of the production of a secreted protein in a fed-batch reactor (Banga et a/., 1998) as a simple illustrative example of a nonlinear batch process. The cultivation is subject to changing initial conditions, hence the specification and yield of the final product varies significantly, which clearly is undesirable a.o. for downstream processing. P r o c e s s I d e n t i f i c a t i o n : To serve as an efficient alternative to the costly development of phenomena-based models of batch processes, this modeling methodology is data-based and a model can thus be relatively readily obtained from (designed) process operation data with the use of the modeling software GoLM. Once the modeling purpose has been defined, the process of identifying a batch process model is fully autonomous. Figures 1 and 2 illustrate the variablity resulting from random variations in initial conditions versus those obtained from PRBS variations in feeding profiles. Both data sets illustrate a large but realistic variability. The nominal model set used for control design is developed from the identification data. C o n t r o l for R e p r o d u c i b i l i t y : Reproducible operation aims at reahzing the same product quality, batch time and/or production yield from batch to batch. Reproducible operation is attractive because it simplifies forecasting final batch time, scheduling of batch units, forecasting load on up- and down-stream processes and/or down-stream processing. The reproducibility of a batch process can be expressed in terms of the deviations between the desired (optimal) operation trajectories and the trajectories realized during operation of the batch process. Optimal reproducibility can thus be formulated as a problem of tracking the optimal batch operations model. The learning MFC algorithm significantly improves the reproducibility the batch process illustrated in figure 3, since it utilizes process information as it becomes available during the batch runs.

Systematic Methodology for Reproducible

Optimizing Batch

Operation

1279

•«i ?1 .S 0.5

'i B

£

0

c ^

5

10

1

10

1

1.5

^, ^ 1L 1^ ^ w ^ 8 0.5 ^ ^ ^

0

c

a .

.

5

20 J 6 10

>

Figure 1. This figure shows the evolution of 25 batch runs on a cultivation reactor producing secreted protein. The cultivation reactor is operated in open-loop and according to the nominal batch operations model. Due to stochastic initial conditions, an undesired variability in the final secreted protein concentration is experienced.

5 10 Batch time (h)

15

5 10 Batch time (h)

15

Figure 2. This figure shows the outcome of 50 designed experiments run on a cultivation reactor. The process excitation is generated by a Pseudo Random Binary Sequence (PRBS) dither signal. The cost in terms of lost productivity associated with these experiments is a modest 3% drop in the mean final secreted protein mass.

O p t i m i z i n g Control: Optimizing control is realizing the best possible product quality, batch time and/or production yield in every single batch. Optimizing control is attractive because it autonomously optimizes the product yield and/or production rate and thus the profitability of every single batch unit. Since optimizing control and reproducible operation are implemented with the same process control software, a tractable combination of the two approaches is easily implemented. The obtained result in Figure 4 achieves an increased productivity, however the variabilty is increased compared to the reproducible operation. This increase is mainly due to only using one actuator variable, i.e. substrate feed fiow rate. An extra degree of freedom available is substrate feed concentration which would enable reducing the variability further.

4. Conclusions A systematic procedure for development of pieceewise linear model sets for batch operation from operating data have been presented together with a learning model predictive control methodology which may be tuned to achieve reproducible or optimizing operation or some combined objective. The methodlogies have been implemented in a Matlab toolbox labelled GoLM. The presented methodologies hold significant promise to enable data driven predictive control to harvest even more significant benefits in batch processing compared to the success already achieved on many continuously operating processes.

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5 10 Batch time (h)

Figure 3. Closed-loop performance of a learning M F C algorithm implemented on a nonlinear batch process with stochastic initial conditions. The control objective is improved reproducibility. The mean productivity achieved is reduced by 4% compared to the nominal case.

15

5 10 Batch time (h)

15

Figure 4. Closed-loop performance of a learning MFC algorithm implemented on a nonlinear batch process with stochastic initial conditions. The control objective is maximizing the final secreted protein mass. The mean protein productivity is increased by 12% over the nominal case.

References Akesson, M.; Hagander, F. and Axelsson, J. (2001). Frobing Control of Fed-Batch Cultures: Analysis and Tuning. Control engng. practice, 9, 709-723. Banga, J. R.; Irizarry-Rivera, R. and Seider, W. D. (1998). Stochastic Optimization for Optimal and Model-Fredictive Control. Computers and Chemical Engineering, 22(4/5), 603-612. Bonne, D. and J0rgensen, S. B. (2003). Data-Driven Modeling of Nonlinear and TimeVarying Frocesses. In F. V. Hoff, editor, SYSID, pages 1655-1660. IFAC, Fergamon press. Flores-Cerrillo, J. and MacGregor, J. (2003). Within-Batch and Batch- to-Batch Inferential-Adaptive Control of Batch Reactors: A FLS Approach. Ind. Engng. Chem. Res., 42, 3334-3345. Gregersen, L. and J0rgensen, S. B. (2001). Identification of Linear Models for Batch Control and Optimisation. In G. Stephanopoulos; J. H. Lee and E. S. Yoon, editors, 6^^ IFAC Symposium on Dynamics and Control of Process Systems, pages 269-274. Srinivasan, B. and Bonvin, D. (2004). Dynamic Optimization under Uncertainty Via NCO Tracking: A Solution Model Approach. In C. Kipparisidis, editor, BatchPro Symposium on Knowledge Driven Batch Processes, pages 17-35. CEFERI, Thessaloniki.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Discrimmant analysis and control chart for the fault detection and identification Xudong Pei, Yoshiyuki Yamashita, Masatoshi Yoshida and Shigeru M atsumoto Department of Chemical Engineering, Tohoku University, 6-6-07 Aramaki Aoba, Sendai 980-8579, Japan

Abstract This paper proposes a new fault detection method using an absolute-value based Fisher discriminant analysis combined with the individuals and moving range chart. Two fault identification methods are also proposed by using the discriminant model, those are quantitative data classification method and projected trend analysis. The performance of these methods are evaluated by Tennessee Eastman benchmark problem. Keywords: fault detection, fault identification, process monitoring

1. INTRODUCTION We are all in agreement that the technique of process monitoring plays an important role in the operation of chemical processes. The work of monitoring includes fault detection, fault identification and diagnosis. Among many methods, principal component analysis (PCA) has been well studied in the last decade [1]. Although the PCA method provides optimal performance for fault detection in a sense, it is not optimal for fault identification and diagnosis. If a fault is detected, one can not do anything without correctly identifying the cause. On the other hand, Fisher discriminant analysis (FDA) is a well-known classification technique which shows optimal classification performance for fault diagnosis [2]. Based on the conventional FDA method for fault detection and identification, we proposed the absolute value-based FDA method (AFDA), which uses absolute residuals of samples and absolute differences between the means of classes in calculating the within-class and betweenclass scatter matrices [3]. In this paper, combination of the AFDA method with the individuals and moving range chart (XmR) is investigated. After reducing the dimension of the feature space by the AFDA method, the XmR chart is applied to detect faults. For the fault identification, based on the discriminant models obtained from the AFDA, two independent processes are proposed: quantitative data classification (QDC) and projected trend analysis (PTA). Fault detection and identification by using these methods are studied on the Tennessee Eastman benchmark problem.

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2. METHODS 2.1. AFDA Given rii normal observations and n2 abnormal observations of m variables ai, a2, • • • , a^^. Let Xi = [xii,Xi2, • • • , Xin^]{i — 1,2, • • , m) be the ni normal observations of variable a^, and XN = [xi,X2,- •' , Xm] ^ be the mean vector of normal observations. Let y^ = [yn, ^^2, ? yin2\ (i = 1,2, • • • , m) be the 722 abnormal observations of variable a^, and ^^^ = [^i, ^2, be the mean vector of abnormal observations. Let x*j = \xij - ^^1 be the absolute residual of xij, y*j = \yij - ^^| be the absolute residual of yij, and ^^ = jx^ — ^^| be the absolute difference between the means of normal and abnormal classes. Here, let us define the following three variables: the absolute residuals matrix X^ e 3fi"^^^i of normal observations, the absolute residuals matrix y|^ G 3f?"^^"2 of abnormal observations, and the absolute value vector z * G ^'^^^ of the differences between the means of normal and abnormal data sets. yin2 "^2711

X^ ^m2 Z

— [Zi, Z2,' ' • , Zm\

yit

^21

^22

Vml

ifm2

^2n2

'

Within-class scatter matrix S* and between-class scatter matrix St are defined as

l^X^X^^r ^on

ni

* \T . YX,{YX,) -)eW n2

(1)

St = rr' e w

(2)

JM -

(3)

From the analogy to the Fisher criterion function, a new criterion function is defined as

-''>

w^S'^w' where it; is a discriminant direction. If 5^ is nonsingular, the optimal discriminant direction w maximizing J{w) is identical to the eigenvector which corresponds to the maximal eigenvalue of the conventional eigenvalue problem {S^'^Slw = Xw. 2.2. XmR Charts The XmR charts are used when samples are individual measurements, e.g., the sample size=l. The upper control limit(UCL) and lower control limit(LCL) for monitoring are calculated from the moving range of two successive observations. Given n observations xi, X2,..., x^,..., a::^, then the moving range is defined as MRi = \xi-x^_i\.

(4)

Let X be the average of all the observations and MR be the average of all the moving ranges, then UCL and LCL are defined as MR MR LCL L128' ^ "L128' If an observation x satisfies LCL < x < UCL, the observation is judged as normal. UCL = x + 3

(5)

Discriminant Analysis and Control Chart for Fault Detection and Identification

1283

^ Fault 1 -* ^ A^

pFault 2

^ ^2

t

Fault N N ^

Fault 2 •< A

V

A • Fault N

>

Normal observations

^ ^ Fault 1-1

A

Fault i+1

A

Fault i-1 <

•Fault i+1

V

4 *" Fault i ^

(a) between normal and each fault

(b) between faults

Figure 1. Discriminant between classes

3. FAULT DETECTION AND IDENTIFICATION 3.1. Fault detection With the normal observations, the AFDA method is used to search for the optimal discriminant direction between the normal observations and the current observation. After projecting these observations on the optimal discriminant direction, one dimensional projections are obtained. By using XmR charts to these projections, faults are easily detected. 3.2. Fault identification Fault identification is to decide the classification of the current observation which has been detected as abnormal. To model the faulty situations in the system, optimal discriminant direction for each fault must be found. Two kinds of discriminant directions are considered here, those are the discriminant direction between observations in the normal and each fault (Fig. 1(a)), and the discriminant direction between observations in each fault (Fig. 1(b)). Based on these discriminant directions, the following two independent identification processes are introduced here. 3.2.1. QDC The current observation is projected onto one dimensional projection by using each learned discriminant function between any two faults. For the projection of the current observation x on each optimal direction, all the set of two faults are compared based on the following distance:

C^QDC —

(6)

where xi and Oi are the mean value and the standard deviation of the projected data for the training data of i-th fault. The fault having the smallest distance is estimated as the most similar fault.

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Table 1 Considered faults in the TE process Name Description Type IDV-1 A/C feed ratio, B composition constant Step IDV-2 B composition, A/C feed ratio constant Step IDV-3 D feed temperature Step IDV-4 Reactor cooling water inlet temperature Step IDV-5 Condenser cooling water inlet temperature Step IDV-6 A feed loss Step IDV-7 C header pressure loss Step

Table 2 Summary of fault detection Method \ Data set IDV-1 IDV-2 IDV-3 IDV-4 IDV-5 IDV-6 IDV-7 XmR with AFDA 1 2 1 1 1 1 1 1 1 1 XmR with FDA 1 1 1 3

3.2.2. PTA By using discriminant functions between normal and each fault, several successive data samples are projected to each direction. The corresponding class of the current data is identified based on the distance between the projected sample trend with each projected fault trend. Although any kind of distances can be used, in the case study examples we used the following distance measure: 7 V V•Strain a p T A ^^

»^testj

V*^train ,

3 MR/1.128

3?testj ,

^^x yl)

where a?train and attest represents a sequence of projected sample points of training and test data sets, respectively. 4. CASE STUDY The proposed methods are appUed to the Tennessee Eastman (TE) benchmark problem [4,5] for on-line fault detection and identification. In this study, seven kinds of step disturbances, which are shown in Table 1, are used with the 22 continuous measurement variables. 4.1. Discriminant models The AFDA method generates two sets of discriminant functions corresponding to Fig. 1(a) and (b) by using the training data sets, which include 100 normal samples and 100 abnormal samples. For comparison, corresponding discriminant models by using conventional FDA method are also made.

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Table 3 Identification accuracy by the QDC methods Methods \ Data Set IDV-1 IDV-2 IDV-3 IDV-4 IDV-5 IDV-6 IDV-7 QDC with AFDA 94% 90% 99% 100% 98% 100% 100% QDC with FDA 98% 100% 100% 100% 100% 100% 99%

4.2. Fault detection For the test observations of each disturbance, the results of fault detection by the proposed method are shown in Table 2. For comparison, the results by the conventional FDA method combined with XmR charts are also shown in this table. Numbers in this table represents the elapsed time of the detection from the occurrence of the fault. Both of the methods work well and detect the correct fault. Although the detection speeds of these two methods are almost comparable, the XmR with AFDA method is faster than the XmR with FDA method for the detection of IDV-2 fault. 4.3. Fault identification 4.3.1. QDC All the 100 abnormal samples for test data are classified into seven classes based on the learned AFDA discriminant functions. Accuracy of fault identification are sunmiarised in Table 3. In this table, the result by using the FDA method is also listed. Both of the methods accurately classified the data. 4.3.2. PTA Distances of the current and fault projected trend patterns, which are represented by a consecutive five observations, were compared by using learned discriminant functions. The results are summarised in Table 4. The smallest number in each row shows most similar class for the test data. The table shows that all the faults are correctly identified for all the test data by PTA on AFDA discriminant functions. PTA on conventional FDA shows similar result but IDV-4 was misclassified as IDV-3. 5. DISCUSSION PTA uses projected trend rather than a single projection, it distinguishes unknown trend patterns from the known fault. When the observed trend is not similar to any of the learned trend, the method indicates that the class is not known. This characteristic of the identification is important in finding unknown new faults. From the view point of the computation time, QDC and PTA methods do not have large differences. Suppose the number of known faults is A^, computation using discriminant functions of QDC has the order of N - 1 for each observation, and the computation using discriminant functions of PTA is the order of A^. However, the amount of discriminant models is obviously different when A^ becomes large. The amount of discriminant functions of QDC is N{N ~ l)/2, but the amount of discriminant functions of PTA is A^.

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Table 4 Distances between projected trend vectors (a) PTA onAFDA Test data Training data IDV-1 IDV-2 IDV-3 IDV-4 IDV-5 IDV-6 IDV-7 IDV-1 1.55 9.18 43.01 26.37 12.34 70.54 42.57 1.27 IDV-2 7.32 7.83 20.61 8.55 65.61 39.93 6.53 IDV-3 9.95 1.46 17.61 8.34 64.75 39.10 IDV-4 3.64 63.59 38.01 15.98 25.06 28.91 1.28 1.71 65.09 38.10 IDV-5 6.16 2.90 5.81 18.91 3.53 IDV-6 17.36 37.48 50.31 41.38 0.36 49.06 IDV-7 11.65 103.86 368.41 86.85 73.68 18.48 0.86 (b) PTA on FDA Test data IDV-1 IDV-2 IDV-3 IDV-4 IDV-5 IDV-6 IDV-7

IDV-1 IDV-2 1.71 11.63 1.63 5.95 4.21 7.69 11.30 12.13 4.72 1.47 41.59 43.15 7.23 70.65

Training data IDV-3 IDV-4 IDV-5 IDV-6 IDV-7 12.84 28.86 9.23 61.45 42.42 2.72 22.51 7.04 57.15 34.20 0.88 20.16 7.34 56.91 31.96 1.08 1.37 6.18 56.62 27.25 4.83 20.50 1.11 57.53 33.03 39.82 23.90 17.50 0.56 45.23 108.14 11.22 39.65 23.82 0.75

6. CONCLUSION On-line fault detection method combining the AFDA with XmR charts was proposed. First, dimension of the feature space was reduced by the AFDA, and then XmR chart was used to detect a fault. Two kinds of discriminant models, those are QDC and PTA, were also proposed for fault identification. The characteristics of these two discriminant functions were discussed. An application study on the TE simulator verified the feasibility of the proposed methods, and the method detected and identified the given faults fast and correctly. The application result showed that these two identification methods worked well but had their own advantages on different kinds of faults and were complementary in a certain extent. The use of these methods together could increase the accuracy of fault identification.

REFERENCES [1] V. Venkatasubramanian, R. Rengaswamy, K. Yin and S.N. Kavuri, Comput. Chem. Eng., 27 (2003) 327. [2] L.H. Chiang, E.L. Russell and R.D. Braatz, Chemometr. Intell. Lab., 50 (2000) 243. [3] X.D. Pei, Y. Yamashita, M. Yoshida and S. Matsumoto, PSE-Asia, Seoul (2005) 567. [4] J.J. Downs and E.F Vogel, Comput. Chem. Eng., 17 (1993) 245. [5] T.J. McAvoy and N. Ye, Comput. Chem. Eng., 18 (1994) 383.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Stability Analysis of Nonlinear Model Predictive Control: An Optimization Based Approach Vivek Dua Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London JVCIE 7JE, United Kingdom Abstract In this work stability of discrete time nonlinear model predictive controllers is analyzed by formulating and solving a mixed-integer nonlinear program. Corresponding to each state and control variable a set of 0-1 binary variables is introduced where a binary variable in a set represents a discrete time interval. The stability criterion is introduced by incorporating constraints such that the state and control variables reach the corresponding target values at the end of the time period where a time period is given by summation of the binary variables in a set. Keywords: Nonlinear Model Predictive Control, Stability Analysis, Mixed-Integer Program 1. Introduction Control of chemical processes is considered to be a challenging problem due to the presence of inherent nonlinearities in these systems. This problem is further complicated by the presence of constraints on control and state variables. Model predictive control provides a systematic framework that can take into account the model of the system as well as satisfy constraints on control and state variables (Pistikopoulos et al., 2002; Bemporad et al., 2002). This is achieved by solving an optimization problem that minimizes the weighted sums of the deviations, of the state and control variables from their respective target values, which are predicted over a future time horizon and the problem is solved subject to the model of the system and the constraints on state and control variables. The future time horizon over which the state of the system is predicted is known as the prediction horizon and similarly the time horizon over which the control variables are optimized is known as the control horizon. The prediction and control horizons are tuning parameters that are obtained while designing the controller for a system and they play a crucial role in the controller performance and stability of the system. An increase in the prediction horizon can improve the stability of the system, but results in slower response and increase in the size of the problem. For linear time-invariant systems by choosing an infinite prediction horizon and a finite control horizon the system can be stabilized (Rawlings and Muske, 1993). Keerthi and Gilbert (1988) and Mayne and Michalska (1990) have presented a detailed study of stability analysis of model predictive controllers. Sistu and Bequette (1995) presented verifiable expressions that can be used to find a stabilising set of tuning parameters for nonlinear systems, similar to the results by Garcia and Morari (1982) for linear systems. In this work an optimization based approach for carrying out stability analysis is presented where discrete time intervals are represented by 0-1 binary variables. The rest of the paper is organized as follows: a mixed integer mathematical formulation for stability analysis is presented in the next section and concluding remarks are given in section 3.

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2. Stability Analysis using Mixed-Integer Programming Model Predictive Control (MPC) problems can be formulated as the following optimization problem: min

J = y [x' {k)Qx{k)] + Y [u{ky Ru(k)]

s.t. x{k + \) = f{x{k\u(k)) ^min < X(A: + 1) < X < X ^ a x , k = 0 , 1 , . . . ,

Wmin ^
N,

^ "max ^ ^ = O , ! , - , A^c

where x and w are the vectors of state and control deviation variables respectively, Ny, Nu and Nc are the prediction, control and constraint horizons respectively, Q and R are weights on deviations of the state and control variables, and k denotes a time interval. The basic idea of an MPC implementation is shown in Figure 1 where at the current time interval, k, the optimization problem is solved to minimize the state and control deviationsfromthe set point by implementing the optimal vales of the control variables.

past < — State Variable 'I I

Set Point -> future I Control Variable

I I I I


—{

h

H—\—\—\—\—> k+Nv

k k+1 Prediction Horizon

Figure 1. Model Predictive Control Note that only the first control element is implemented and this sequence is repeated at the next time intervals, for the new state measurements or estimates, until the desired or

Stability Analysis of Nonlinear Model Predictive Control

1289

set point values are obtained. The key advantage of MPC is that it can take into account the constraints on state and control variables. Consider the following reactions and the corresponding model of an isothermal CSTR (Sistu and Bequette, 1995): A-^B

ii dt dx [

B-^C

2A->D

— —k^X^

—f^lXj

^3-^5 "*" (-"-y^F

Kq^X u

^A)^

X nil

dt where X; and X2 are the state variables representing the concentrations of A and B respectively, XF= \^ mol/liter is the concentration of A in the feed, u is the dilution or feed rate, ki = 50 h r \ ^^ = 100 hr^^ and k3= 10 liter/(mol hr). ite Profile with Time

Figure 2. Controller performance for horizon length 5, yi = 72 "^ 1000,73 = 1 and unconstrained u

Figure 3. Controller performance for horizon length 5, 71=73=0, 72=1000 and unconstrained u The objective is to maintain X2 at 1.0 and this can be achieved for two steady state solutions of the model equations: {xy, X2, u) = {2.5, 1.0, 25} and {6.6667, 1.0, 233.33}; the second solution is preferable since it requires a lower dilution rate. Meadows and Rawlings (1997) analyzed the performance of the following objective function: t+T

J= \rx[x^{t)-2,5f +r^[x^{t)-\,of + r,{u(t)-25fd.t

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for various values of yi, 72,73, and prediction and control horizons. In this paper the model equations are discretized by using a simple forward difference method (Euler's method) and the performance of the controller for a sampling time of 0.002 hrs, horizon length of 5, and yi = 72 = 1000 and 73 = 1 and unconstrained u is shown in Figure 2. For the case when yi = 73 = 0 the system settles to the other steady state as shown in Figure 3 and for the same penalty weights when an additional constraint, u < 50, is imposed X2 does not reach the target value of 1.0 and the performance of the controller is shown in Figure 4 and a decrease in the offset from the steady state value of X2 for increasing horizon length is shown in Figure 5.

Figure 4. Controller Performance horizon length 5, yi = 73 = 0, y2 = 1000 and w < 50

Prediction Horizon

Figure 5. Steady state of X2 versus horizon length Instead of carrying out the stability analysis for various horizon lengths, the problem can be formulated and solved a mixed-integer nonlinear program (Viswanathan and Grossmann, 1990) where time intervals are modelled by introducing 0-1 binary variables as proposed next. Let corresponding to each state variable x there is a vector of 0-1 binary variables 7;^ and similarly there is y^ for u where y^ and yu represent discrete time intervals. The summation of y^ for an x will represent the time intervals it will take x to reach the desired state and similar statement holds for u. The objective is to minimize the time intervals. Mathematically this can be achieved by formulating the following logical conditions:

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{x{k)-x,„f<M*y,ik) {u{k)-u{k + \)f < M*y^{k) y,{k)>y,{k

+ \)

y,{k)>y,{k

+ \)

where Xset is the set point value of x and M is large positive number. The first constraint implies that iiyx= 1, x(k) < Xset and xfy^ = 0, x(k) = x^eu an alternative formulation which does not involve square terms and relies on introducing slack variables can also be employed (Floudas, 1995). Since we are interested in minimizing the time period the elements oiy^ which take a value of 1 will indicate the time intervals during which x has not reached the set point value. To ensure that the time periods are arranged in a sequential order third and fourth constraints are introduced. The second constraint is similar to the first one except that it indicates that at the end of the time period for control variables the control variables do not change; an alternative formulation involving set point value of control variables, similar to first constraint for state variables, can also be used. The stability of the isothermal CSTR example presented earlier was analyzed by using the proposed formulation and the same results as reported in the open literature were obtained - for the case when the system was unstable integer infeasible solution were obtained and feasible solutions were obtained for stable systems; all the cases were solved by using GAMS/DICOPT (Brooke et al., 1998). The key advantage of the proposed formulation is that one does not have to try different horizon lengths and instead solving a mixed-integer program provides the answer regarding the stability of the nonlinear MPC. The proposed approach provides a tool for stability analysis but does not give optimal horizon lengths and this is being currently investigated.

3. Concluding Remarks A mixed-integer programming approach, where 0-1 binary variables represent discrete time intervals, for stability analysis of nonlinear model predictive controllers was presented. The issues pertaining to global optimization and guaranteeing stability for all the values of state variables given within certain lower and upper bounds will be the subject of future work. The proposed approach does not require testing different values of horizon lengths, is quite general and can be extended to other systems.

References A. Brooke, D. Kendrick, A. Meeraus and R. Raman, 1998, GAMS: a user's guide, GAMS development corporation, Washington. A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, 2002, The expHcit Unear quadratic regulator for constrained systems, Automatica, 38, 3. C.A. Floudas, 1995, Nonlinear and mixed-integer optimization, Oxford University Press, New York. C.E. Garcia and M. Morari, 1982, Internal model control . L a unifying review and some new results. Industrial & Engineering Chemistry Process Design and Development, 21, 308.

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S.S. Keerthi and E.G. Gilbert, 1988, Optimal infinite horizon feedback laws for a general class of constrained discrete-time systems: stability and moving horizon approximations. Journal of Optimization Theory and Applications, 57, 265. D.Q. Mayne and H. Michalska, 1990, Receding horizon control of nonlinear systems. IEEE Transaction on Automatic Control, 35, 814. E.S. Meadows and J.B. Rawlings, 1997, Model predictive control, in: Nonlinear process control, M.A. Henson and D.E. Seborg (Editors), Prentice Hall, New Jersey. E.N. Pistikopoulos, V. Dua, N.A. Bozinis, A. Bemporad and M. Morari, 2002, On-line optimization via off-line parametric optimization tools, Computers and Chemical Engineering, 26, 175. J.B. Rawlings and K. Muske, 1993, The stability of constrained receding horizon control, IEEE Transactions on Automatic Control, 38, 1512. P.B. Sistu and W. Bequette, 1995, Model predictive control of processes with input multiplicities. Chemical Engineering Science, 50, 921. J. Viswanathan and I.E. Grossmann, 1990, A combined penalty ftinction and outer-approximation method for MINLP optimization. Computers and Chemical Engineering, 14, 769.

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AN OPTIMIZATION FRAMEWORK TO COMPUTER-AIDED DESIGN OF RELIABLE MEASUREMENT SYSTEMS Raffaele Angelini, Carlos A. Mendez, Estanislao Musulin, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The optimal design and upgrade of sensor networks have been receiving an increase in attention for the last few years. This work presents a new methodology based on a MINLP formulation to the optimal design of reliable measurement systems with minimum cost. The applicability of the MINLP model is illustrated through the resolution of a benchmark case study taken from literature and its computational performance is compared with a genetic algorithm-based approach. Keywords: Sensor networks, design and retrofit, MINLP model. 1. Introduction A wide-ranging variety of approaches relying on exhaustive enumeration, algorithmic procedures and meta-heuristic techniques have been developed to address the complexity of the sensor network placement problem, usually assuming a steady-state system at the nominal operating conditions. Within this context, the sensor placement can be regarded as a highly combinatorial optimization problem where the main goal is to find the optimal balance between the performance indicators and the cost of the data acquisition system. The performance of the whole measurement system as well as of individual process variables is typically determined through multiple criterions such as precision, flexibility, reliability, robustness and so on, to which minimum requirements are usually to be satisfied. The concept of observability is also a necessary condition of any sensor network since that the state of the entire process must be known, which means that a numerical value needs to be determined for every process variable by using direct measurement or deduction from other process variables. On the other hand, the total cost of the measurement system can comprise the investment cost of new sensors, the installation and maintenance charges and, in the case of retrofitting, the sensor reallocation cost. The design of a sensor network that allows for the observation of all process variables was first addressed by Vaclaveck and Loucka (1976). Later, this problem was solved by Madron and Veverka (1992) regarding the minimum total cost. However, sensor failure can lead to a reduction of measurements, thereby seriously affecting the whole process performance. Thus, it is necessary to assure process observability even after sensor failures. The observation of a process variable can be expressed mathematically by making non-null the probability of estimating this variable at a given time t (reliability). The evaluation of this probability is closely related to the different ways of estimating a process variable given a sensor failure probability and a

Author to whom correspondence should be addressed: [email protected]

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specific sensor network. On the basis of these concepts, a method for optimal sensor location in a pure flow process was developed using graph-theory by Ali and Narasimhan (1993). This preliminary approach did not directly consider either the network cost or the reliability of the individual process variables. However, these key topics were later addressed by Bagajewicz and Sanchez (2000). Additionally, these authors transformed the problem presented by Ali and Narasimhan into a mathematical programming problem. Later on, Benqlilou et al. (2004) proposed an approach for evaluating the reliability of process variable estimation taking into account all the redundancies offered by the system in terms of functional as well as hardware. In this evaluation, both quantitative process knowledge and fault tree analysis are considered and combined, which leads to a more suitable and practical evaluation of reliability. The reliability of estimating each one of the key process variables is then used to determine the sensor network reliability, which, in turn, is used as a set of sensor placement constraints in the design and retrofitting procedure. Thus, a sensor placement method was initially proposed to consider the number (hardware redundancy or multiplicity) of sensors of a given type (reliability) that are to be assigned to a given process variable while satisfying the reliability requirements at the minimum total cost. This proposal was applied for network design as well as for retrofitting. Although the formulated sensor placement optimization problem was successfully solved using genetic algorithms, it showed some typical limitations of stochastic optimization methods, such as the impossibility to assure a global optimum and the high difficulty in dealing with hard-constrained problems. The optimization problem as posed involves a very large number of discrete decisions to determine the number and type of sensors to measure each process variable. The main challenge is that the problem size scales exponentially in the number of process variables and sensors. An additional complexity is the lack of direct algebraic equations to compute the reliability of process variable estimation which was typically calculated by algorithmic procedures. This work is focused on the optimal design and upgrade of reliable sensor networks. A novel MESFLP-based approach that takes into account sensor redundancy as well as multiple sensor types is presented. Different objectives functions besides cost can be easily implemented. The successful applicability of the proposed method is illustrated through a challenging case study taken from literature. 2. Reliability of process variable estimation The probability, rj{t), of estimating a process variable j at time t must simultaneously consider sensor, hardware and functional reliabilities (see Figure 1). For the sake of simplicity, the sensor reliability is assumed to be constant over time and the time reference is omitted from here on. As an example, consider a simple process unit with one feed stream 1 and two product streams 2 and 3, as shown in Figure 1. For this process unit, the three flow variables (Fi, F2 and F3) are related to each other through the pure mass balance expressed by the equality Fi = F2 + F^. Let us assume that all sensor failures occur randomly and independently and that the mass flow of all streams can be measured using similar flow-meters k = SI, SI, S3, SA, S5 and S6, all of them with a sensor reliability equal to 0.80. Therefore, using the property that if PI and Fl are the probabilities of two independent events, the logical condition P\ ox Fl is equal to P\+P2-P\.F2. The reliability of estimating variable Fi is summarized in Table 1. For these three illustrating cases (sensor reliability, hardware reliability and functional reliability), vpi > 0 and variable Fi is said to be observable. Moreover, in the case of hardware reliability, the reliability of estimating variable Fi is greater than the reliability

Optimization Framework to Computer-Aided Design of Reliable Measurement Systems 1295 of each one of the sensors measuring it, showing the existence of more than one way to estimate Fi. Thus, r^j is directly related to the degree of redundancy and can still be estimated even if one sensor fails. However, in cases of sensor reliability and functional reliability, no sensor failure is permitted. These results illustrate the direct effect of the sensor placement on the reliability of measuring or estimating the process variables. Sensor reliabilily

Hardware relrability

1

^

Funcfonai reliability

^ %

Figure 1. Reliability of process variable estimation Table 1. Calculation of reliabilities depending on sensor placement Reliability equation r'si ^S3 o r r^S4 = ^S3^ /S5^rid/s6=

^S4 - f^S3 • ^S4

f^S5-r^S6

Reliability rFj{t) Reliability type 0.80

Sensor

0.96

Hardware

0.64

Functional

The lowest reliability associated to the estimation of a process variable is used to determine the entire sensor network reliability, which in turn, is used as a criterion for the design and retrofit of the instrumentation system. Thus, a reliable sensor network design and retrofit problem can be formulated and solved through a rigorous mathematical programming model. Data, variables and constraints involved in the mathematical representation of this problem are introduced in next section. 3. The proposed MINLP-based approach 5^/5: J (process variables); Jk (process variables that can be measured through sensor k); Je (process variables involved in the estimation e); K (sensor types); Kj (sensor types that can be used to measure process variable y); N (sensor multiplicities); Njk (allowable sensor multiplicities for measuring process variable j through sensor k); E (alternative ways of process variable estimation); Ej (alternative ways for estimating process variable y) Parameters: c/""^ (acquisition cost of sensor k)\ Ck"^ (installation cost of sensor K)\ c^^'"^^ (deinstallation cost of sensor A:); n^"" (minimum number of sensors for measuring variable y); nf""^ (maximum number of sensors for measuring variable 7); nj^i""^ (number of sensors k already installed in variable 7); r^ (reliability of sensor K)\ rj'"'" (minimum reliability requirement for variable7); c"""^ (maximum total cost of the sensor network) Continuous variables: F^'^jXri (hardware failure probability for measuring process variable 7 through n sensors keKj); F^^'^^j (hardware failure probability for measuring variable 7); F^^'^'j (functional failure probability for estimating variable 7); C"'j^k (sensor installation cost); C^^'"'j^k (sensor de-installation cost); Cost (total sensor network cost); i?""'" (minimum reliability of the sensor network)

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Binary variable: Yj^n (binary variable denoting that n sensors A:GX, are used to measure process variable y) 3.1. Model constraints • Sensors already installed in the process (Only needed in case of retrofitting).

Z«M^E j

Z«^y,M

Vfc

(1)

jeJ^ riENj^^

• Minimum and maximum allowable sensor redundancy.

«f"<2

Z"h>^.x^"7'"

v/-

(2)

keKj neNj 1^

• Hardware failure probability considering a single sensor and multiplicity. FjZn -^-Yj,t,„+{i-rjYj,„

\fj,ke Kj,ne Nj^,

(3)

• Hardware failure probability combining multiple sensor types and sensor redundancy.

^f''=n u^M.

v/-

(4)

Functional failure probability considering direct measurement and estimation.

^r=u1- Yli'-F^^") e&E

Vy

(5)

J'^Je

Reliability of a process variable as a minimum requirement to be satisfied. 1 _ rfi^^ > r^'" ^ 7 - J

(6)

• Total sensor acquisition cost. jns

Cp>c^^cq

\/k

(7)

• Total sensor installation cost.

C j,kms

v^ ins ~ ^k

ll"YjXr,-4l

yj,keKj

(8)

V"^^;.*

Total sensor de-installation cost.

C j,kdeins

-^

deins ^^k

\/J,keKj neN

j,k

(9)

Optimization Framework to Computer-Aided Design of Reliable Measurement Systems 1297 • Sensor network cost including acquisition, installation and de-installation costs. Cost = ^

Cf^ + 2 ^ ^ +

^jf'

(10)

• Reliability of the whole sensor network as a problem variable to be maximized. X-pfi^'^ ^Jl^riin

Vy

(11)

4. The ammonia plant case study The dynamic case of the ammonia plant initially introduced in Benqlilou et al. (2004) is addressed in this section to illustrate the performance of the proposed MINLP model. This problem comprises eight flows and five level process variables which can be measured through a catalogue of 5 different flow-meters and 3 level meters, respectively. A schematic representation of this plant is depicted in Figure 2. The available flow-meters have reliabilities of 0.70, 0.75, 0.80, 0.85 and 0.90 and their associated costs are 1500, 1700, 2000, 2300 and 2800 euros, respectively. For the level meters, reliabilities are 0.60, 0.70 and 0.80 having a cost of 2000, 3500 and 5000 euros. A maximum hardware redundancy of 2 sensors was considered. A large spectrum of reliability requirements ranging from 0.6 to 0.99 was evaluated in order to test the performance of the proposed deterministic model under different scenarios. The design and retrofit problems of the sensor network of the ammonia plant were solved through the MINLP formulation using the optimization codes BARON (2004) and DICOPT (2004). For each scenario, maximum computational times of 1 minute, 10 minutes and 1 hour were enforced to BARON. In these cases, the reliability was gradually increased from 0.6 to 0.99 by 0.01, 0.03 and 0.13 steps, respectively. The initial point for each minimum reliability requirement was taken from the best solution of the previous step. No time restriction was needed to DICOP because of the low CPU time required to prove local optimality. Figure 3 shows the sensor network cost as a ftinction of the minimum reliability requirements. Solutions obtained throilglTttie different solvers with different time restrictions are compared in this Figure. For the sake of comparison, an enhanced version of the proposed GA-based approach introduced in Benqlilou et al. (2004) was also utilized. Figure 4 presents the results obtained by means of the determinist and the stochastic approaches. In order to perform a fair comparison, a time limit of 10 minutes was enforced both for BARON and GA. The results put clearly on evidence the better performance of the MILP approach, which was able satisfy the minimum reliability requirements with lower cost in most of the cases.

Figure 2. Schematic representation of the ammonia plant.

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Figure 3. Comparison of solutions obtained with different time restrictions, steps and solvers GA-MINLP COMPARISON AMMONIA PLANT (Design and Retrofit) _ —.*•— 10 min - 0.03 step - BARON • —•

10 min - 0.03 step - 5000 ind - GA O.03 Step - DICOFT

design

Reliability

Figure 4. Comparison of solutions obtained through MINLP and GA-based approaches 5. Conclusions An efficient MINLP-based approach for the optimal design and retrofit of reliable sensor networks with minimum cost has been presented. The formulation takes into account most of the problem features such as sensor redundancy, different sensor types and sensor network cost. The reliability of estimating individual process variables is explicitly computed through a set of constraints that considers sensor, hardware and functional reliabilities. The performance of the deterministic model was successfully compared with a GA-based approach. Results showed a more stable and robust behavior in the MIL? method which was able to find better solutions than the stochastic approach in most of the cases. Global optimality could also be proved through a global optimizer such as BARON with reasonable computational effort.

Acknowledgments Financial support received from the European Community (MRTN-CT-2004-512233; RFC-CR04006; INCO-CT-2005-013359) isfrillyappreciated. Literature AH, Y. & Narasimahan, S. (1993). AIChEJ. 39, 820 - 826. Bagajewicz, M. J.& Sanchez, M. C. (2000). Comp. & Chem. Eng., 23, 1757 - 1762. Baron 7.2. (2004). Solver Manual. Benquilou, C; Graells, M.; Musulin, E. & Puigjaner, L. (2004). Industrial & Engineering Chemistry Research, 43, 8026 - 8036. Dicopt2x-C. (2004). Solver Manual. Madron, F. & Veverka, V. (1992). AIChEJ. 38, 227 - 236. Vaclaveck, V. & Loucka, M. (1976). Chemical Engineering Science. 1199 - 1205.

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An approach to linear control of nonlinear processes T. Schweickhardt^, F. Allgower^ ^Institute for Systems Theory and Automatic Control, University of Stuttgart, 70569 Stuttgart, Germany Nonlinear controller design is a demanding task. Therefore, linear methods are favored over nonlinear ones in practice although it is not always clear a priori whether the former are appropriate in face of a nonlinear process. Nonlinearity measures are a means to quantify the nonlinearity inherent to a process. However, as yet there were no rule on how to design a controller based on the results of nonlinearity assessment. In this paper, an approach is presented that integrates nonlinearity assessment and the design of linear controllers for nonlinear systems by using methods from linear robust control. 1. I N T R O D U C T I O N Linear techniques for systems analysis and controller design are well developed. For many control-related engineering problems, methods are available that are theoretically sound as well as practically implement able. Also for mildly nonlinear systems, one can attempt to use a linear model and linear controller design methods, hoping that the nonlinear effects are too small to destabilize the closed-loop system or to deteriorate closed-loop performance. The most common approach is to use the local linearization as a linear model, to design a linear controller on the basis of this model, and to analyse (e.g. by simulation) the stability and performance of the closed loop with the nonhnear plant. However, no stability and performance guarantees are given this way and the nonlinearity of the plant is not taken into account. In order to decide whether linear controller design is adequate, nonlinearity measures can give important insight [1-3]. Some results exist on the suitabihty of linear controllers for nonlinear systems [4-7], but no practical methods are available how to use the results of nonlinearity assessment in the design of linear controllers for nonlinear processes. In this paper, an approach is presented that integrates nonhnearity assessment and the design of linear controllers for nonhnear systems in order to (i) decide for a given control problem whether linear controller design is adequate, (ii) derive a suitable linear model (not necessarily the local linearization) and (iii) describe a linear controller design procedure that guarantees stability. This paper is structured as follows. We first give a brief introduction to nonlinearity assessment in Section 2. In Section 3 an approach is presented that integrates nonlinearity assessment and the design of linear controllers for nonlinear systems based on linear robust control techniques and based on the nonlinearity measures introduced earlier. An example is given in Section 4. The paper concludes with Section 5.

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2. LINEAR MODELS OF NONLINEAR SYSTEMS The fundamental idea of the error-gain nonlinearity measure as defined in [1,2,8] is depicted in Fig. 1.

A= u

N-G

G L

+r

1 2/ = i V p

+ 1

.JN

Figure 1. A general nonhnear system N can be decomposed into a linear model G and a nonlinear uncertainty A. The best linear model is defined as the linear model for which the associated uncertainty has the least system gain.

A general nonhnear (i.e. not necessarily linear) stable dynamical system N, described by the transfer operator N : u \-^ y = Nu is modeled by a linear model G described by the linear transfer operator G \ uv-^ y = Gu. The error system A = N — G represents the plant/model mismatch and its gain is a measure of the model quality of G. We therefore define the error-gain nonlinearity measure of N onU as u A . .

7 ^ = mf sup

\\Nu-Gu\\

-—n—n—

(1) .1/2

where ||x|| = (j^ \^{t)\ dtj is the L2-signal norm, the symbol |-| denotes the Euclidean vector norm and the plant N is required to be (finite-gain) L2-stable [9]. This nonlinearity measure gives the gain of the error system A, when the worst case input signal u ^U is considered. The best linear approximation G is chosen among the set of all causal stable linear transfer operators Q such that the resulting worst case gain is minimized. The set U usually describes the region of operation in which the nonlinearity of the system N is to be assessed. In this case U contains only signals not exceeding a certain maximal amplitude. Note that the nonlinearity measure can not decrease when a larger operating regime is considered. Furthermore, the error gain nonhnearity measure is bounded by 0 < TJV < ||A^||f = sup^^^^^^o} | l ^ where the right-hand side is the system gain of N on hi. A value of 7^ close to zero indicates a negligible nonlinearity, whereas a large value suggests that also the best linear model results in a significant model/plant mismatch. In order to quickly get a lower bound, one can for instance consider the steady-state behaviour of the process [8]. Different computational schemes are available to get approximations of the dynamic nonlinearity measure [2,3,10]. We will not treat the computational details here, but we mention that the computation of exact values of the nonlinearity measures can be quite demanding.

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3. A N A P P R O A C H T O L I N E A R C O N T R O L O F N O N L I N E A R S Y S T E M S Once we have decided that Hnear controller design is adequate for the nonlinear control problem at hand, we turn the attention to the design of a suitable controller. The usual approach to the design of linear controllers for nonlinear systems is to use the linearization around the operating point as a linear model, design a hnear controller and analyze (e.g. by simulation) the stability and performance of the closed loop with the nonlinear plant. By this method, no stability and performance guarantees can be made and the degree of nonlinearity of the plant is not taken into account in the controller design step. In this Section, a novel approach is presented that integrates nonlinearity analysis and linear controller design for nonlinear systems in order to 1. decide for a given control problem whether linear controller design is adequate, 2. derive a suitable linear model (not necessarily equivalent to the local hnearization) 3. describe a linear controller design procedure that guarantees stability of the closed loop containing the nonlinear process. The suggested method in this Section will deal with stable nonlinear systems exhibiting a finite gain onU: \\N\\^ < oc. The idea is to use the small gain theorem in order to maintain stability despite a nonlinear uncertainty (for the small gain theorem, see e.g. [9]). With the help of the small gain theorem, a controller C that satisfies \\N\\^ • ||C||- < 1, where N is the plant, will achieve closed loop stability, see Fig. 2 (a). While stability can be achieved this way, it remains unclear how performance requirements can be met and how the controller should look like apart from the requirement ||C||. < ||A^||~^ In particular, this approach excludes controllers with integral action as the controller has to be stable itself. To circumvent these problems, a different approach is taken. We therefore spht the nonlinear system N into a (stable) linear part G and a (stable) nonlinear part A = N — G and use linear techniques to design a (not necessarily stable) linear controller C for the linear part G. With the help of the loop transformation theorem [9, Ch. 6], we can reorder the A^ — C—structure of the closed loop into the M — A—structure as depicted in Fig. 2 (b). Now, let the linear closed loop transfer operator be M = CS = C{I + GC)~^. The small gain theorem then states that the closed loop system is stable if the gains of M and A satisfy ||A||. • ||M||. < 1. The linear controller can then be designed such that the performance requirements are met at least for the linear model G and integral action is possible. In order to get the best results, the linear model must be chosen such that the gain of the model uncertainty|| A||. takes the least possible value. This is achieved by choosing G such that ||A||. = ||A^ — G||- is minimized. But this procedure exactly corresponds to the definition of the error gain nonlinearity measure 7 ^ - MG \\N -G\\^, where a model G* with ||iV - G*||f = 7 ^ is a best linear model. The nonlinearity assessment then consists of fi) computing 'j^ and G* and (ii) check (e.g. with linear i/oo-techniques) whether ||M||. < 1/7^ is achievable. If so, G* gives a suitable linear model and hnear controller design is applicable (as long as the constraint -/^ • ||M||f < 1 is respected). Up to now, we have neglected the fact that the nonlinearity measure is computed for a restricted region of operation (given by a range of admissible amplitudes), whereas the

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T. Schweickhardt and F. Allgower N

u,

A = A^ - Gh

+ ^ei

?

yi

^^i^nfi-

1—•

2/2

62

CH

U2

(a) Linear closed loop with nonlinear uncertainty

^l

(

2/1 - G e i

A = A^-G

J

G

C

'_ -H.y^

\t" ^2 — Gwi rO4T

(b) Reformulation for use with small gain theorem

Figure 2. Formulation of the nonlinear control problem as linear control problem with norm bounded nonhnear uncertainty.

framework of the small gain theorem allows for signals that are unbounded (in amplitude). In virtually all applications, control engineers have to deal with processes whose manipulated variables can take values in a certain range only. But for such processes, an anti-windup control strategy has to be applied anyway, and we can combine it with nonlinearity assessment in the following way. T h e o r e m [11] Consider the system M = CS = C{I + GC)~^ where the linear model G has a saturating input. Let the saturation bounds correspond to the bounds for which the error-gain nonlinearity measure 7 ^ was derived. Under these conditions, stability of the closed control loop in Fig. 2 a) is achieved if ||M||f < 1/75^. An upper bound on the gain of the linear saturated feedback subsystem M can for instance be derived using the result from [12], 4.

EXAMPLE

We will illustrate the presented approach with a small example. We therefore study the ultra-filtration membrane reactor given in [13]. A chemical reaction is conducted in loop with a filtration unit in order to keep the biocatalyst in the reactor. We make the simphfying assumptions that no enzyme deactivation occurs and we neglect any transport delays between reactor and filtration unit. We thus obtain the first-order model y

VR

+ KM

VM {CSLO -

+ {csLQ -y)

y)

+ KMpy

where the manipulated variable u is the inlet flow rate and the controlled variable y the product concentration. We consider here a steady state input of uss = 1.5 cm^ min~^ and an input range oi u = uss i 1-0 cm*^ min~^. The corresponding steady state output is y — 22.21 mmol dm~^. The nonlinearity measure is computed to be "^ — 3.82. We

An Approach to Linear Control of Nonlinear

1303

Processes

80

Figure 3. Responses of the closed loops to steps in the reference signal of different height. The solid line corresponds to the controller based on the best linear model, the dotted line to the controller based on the local linearization.

obtain for the best linear model G*{s) — ^f^q 04'5^ as opposed to the linearization at the operating point which yields Giin{s) = -^^^^ Two important differences become apparent when comparing the two models. Firstly, the linearization has no transmission zeros, while the best model has a non-minimumphase zero. And secondly, the steady-state gains are different. The steady-state gain of the best model leads to a better steady-state approximation. But the real plant does not show inverse response behaviour, and we would expect that the best model G* should be minimum-phase. But the best model is not unique, and nothing in our setup requires G* to be preferably minimum-phase. Obviously the algorithm leaves space for improvement so that the most suited model out of the pool of best models is chosen. For our simple example, however, this fact does not play an important role as will become clear. For the model G* we design a P/—controller with anti-windup and an additional first-order filter. The controller parameters are chosen such that the gain of the resulting M—system (with saturating plant) has an upper bound of 0.184. The required condition ||M||^ - 7 ^ = 0.18-3.82 = 0.70 < 1 is satisfied and closed-loop stability is guaranteed. For purposes of comparison, a similar P/—controller with anti-windup was designed for the plant model Gun so that the same bandwidth was achieved. Step responses of the closed loop with the "real" plant are shown in Fig. 3, the solid line corresponding to the first controller (based on the best model) and the dashed line corresponding to the second controller. For this simple example, the performance of the two controllers is similar, but note that considering the design process of the second controller, closed-loop stability was not guaranteed. Due to integral action, both controllers achieve vanishing steady state errors, while the anti-windup schemes prevent the control systems from going unstable.

13 04 5.

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and F. A llgower

CONCLUSIONS

Linear controller design can be implemented much more easily than nonlinear controller design. One therefore requires tools that allow to determine prior to controller design whether linear controller design is adequate or whether nonlinear techniques have to be used. In this work, we presented a novel approach that combines nonlinearity assessment and controller design by using linear robust control methods and nonlinearity measures. The key features of this method are that (i) closed-loop stability is guaranteed and (ii) conservatism is reduced by considering the nonlinearity of the plant only for the actual operating regime. This is an important advantage, as global control of nonlinear systems is usually neither desirable nor feasible. For the future, further developments are desirable to obtain more robust and more easily realizable methods that also guarantee a certain level of performance. REFERENCES 1. 2. 3. 4. 5. 6.

7. 8.

9. 10.

11. 12.

13.

C. A. Desoer and Y.-T. Wang. Foundations of feedback theory for nonlinear dynamical systems. IEEE Trans, on Circuits and System, CAS-27(2): 104-123, February 1980. F. Allgower. Definition and computation of a nonlinearity megisure. In 3rd IFAC Nonlinear Control Systems Design Symposium, pages 279-284, Lake Tahoe, CA, 1995. D. Sourlas and V. Manousiouthakis. On the computation of the nonlinearity measure. In Proc. 37th IEEE Conf. Decision Contr., pages 1434-39, 1998. A. J. Stack and F. J. Doyle III. The optimal control structure: an approach to measuring control-law nonlinearity. Comp. and Chem. Eng., 21 (9): 1009-1019, 1997. S. A. Eker and M. Nikolaou. Linear control of nonlinear systems: Interplay between nonlinearity and feedback. AIChE J., 48(9): 1957-80, 2002. T. Schweickhardt, F. Allgower, and F . J. Doyle III. Nonlinearity quantification for the optimal state feedback controller. In European Control Conference (ECC), Cambridge, U.K., September 2003. Paper #056. M. Quay, R. Dier, J. Hahn, and P. J. McLellan. Effect of process nonlinearity on linear quadratic regulator performance. J. Proc. Contr., 15(1): 113-124, 2005. T. Schweickhardt and F . Allgower. Linear modeling error and steady-state behaviour of nonlinear dynamical systems. In Proc. 44th IEEE Conf. Decision Contr., pages 8150-8155, Seville, Spain, December 2005. M. Vidyasagar. Nonlinear Systems Analysis (2nd ed.). SIAM, Philadelphia, PA, 2002. D. Kihas and H. J. Marquez. Computing the distance between a nonlinear model and its linear approximation: an I2 approach. Comp. and Chem. Eng., 28(12):2659-2666, 2004. T. Schweickhardt and F. Allgower. A robustness approach to linear control of nonlinear processes. 2006. submitted. B. G. Romanchuk and M. C. Smith. Incremental gain analysis of piecewise linear systems and application to the antiwindup problem. Automatica, 35(7): 1275-1283, 1999. A. Bodalo, J. L. Gomez, M. Fuensanta Maximo, and M. C. Montiel. Development and experimental checking of an unsteady-state model for ultrafiltration continuous stirred tank reactors. Chem. Eng. ScL, 60(15):4225-4232, 2005.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Control of the Synthesis Section of a Urea Plant by means of an MFC Controller Oscar Mauricio Agudelo Manozca ^ Jairo Jose Espinosa ^, Joos Vandewalle ^ ^ Katholieke Universiteit Leuven, Electrical Engineering Department (ESAT), Research Group SCD, Kasteelpark 10, B-3001 Leuven, Belgium. {mauricio. agudelo, joos. vandwalle} @esat. kuleuven. be ^ IPCOSN. VISMC office, Technologielaan 11/0101 B-3001 Leuven, Belgium. jairo. espinosa@ipcos. com. Abstract The control of the synthesis section of a urea plant is a challenge due to delays, recirculation flows and large residence times. This paper presents the results of applying MPC control techniches for controlling the synthesis section of a urea process. The control system stabilizes the syntehsis section while maximizing the throughput of the synthesis section by maximizing the feeding flows. This control scheme keeps the main variables (NH3/CO2 ratio, reactor level, reactor pressure and reactor temperature) on their desired values or within a defined operating range. The results obtained with this control scheme showed improved stability and increased production. Keywords: Predictive control, Process Control, Chemical Industry, Multivariable systems, PID controllers. 1. Introduction In the last decades, model predictive control (MPC) has been widely accepted by the industry, mainly because of its ability to handle constraints explicitly and the natural way in which it can be applied to multivariable processes [6]. Initially, MPC was applied to meet the specialized control needs of power plants and petroleum refineries, nowadays this control technology can be found in a wide variety of application areas including chemicals, food processing, automotive, aerospace, metallurgy, and pulp and paper. This paper presents the results of using an MPC controller to control the synthesis section of a urea plant. The control integrates the reactor, condenser and stripper units providing maximum integration of the plant via the control system. The controller acts dynamically improving the operating point according to the dynamic limits of the process and exploits the embedded optimization of the MPC to maximize the production. In order to test the control scheme, a complete and very accurate urea model has been used. Such model was developed by Protomation and it has been implemented in a software package called Process Studio, which was created by the same company. For the Implementation of the MPC strategy, an industrial MPC controller developed by IPCOS has been used. This MPC controller is called INCA Suite[l], and it was interconnected with Process Studio via OPC (OLE for Process Control) to control the urea model.

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2. Description of the Process The urea simulator developed by Protomation, models a "Stamicarbon Urea CO2 Stripping Process" [2], where we can distinguish 4 sections: synthesis, recirculation, evaporation and finishing technique and waste water treatment. In a CO2 stripping process, ammonia (NH3) and carbon dioxide (CO2) are fed directly into the synthesis section, where at high pressure (13-30 MPa) and high temperature (170-200°C) the urea formation takes place, according to the following reactions: 2NH3 + CO2 ^ NH2COONH4 + (heat) NH2COONH4 <^ NH2CONH2 + H2O - (heat)

(1) (2)

In reaction (1), ammonia and carbon dioxide are converted into ammonium carbamate (NH2COONH4). This reaction is highly exothermic and fast. In the reaction (2), ammonium carbamate dehydrates to produce urea and water. Unlike reaction (1), reaction (2) is slow and endothermic. Figure 1 shows a block diagram of the synthesis section. In order to achieve maximum urea conversion the process requires a molar ratio of 3 among ammonia and carbon dioxide (NH3/CO2) at the outlet of the urea reactor [3]. The urea formation takes place in the pool condenser and in the urea reactor. Since part of the ammonia and carbon dioxide are not converted to urea and water, the reaction mixture coming from the urea reactor is subjected to a stripping process (it occurs in the CO2 Stripper), where carbon dioxide is used to remove the unreacted NH3 and CO2. The off-gases (NH3 and CO2) from the stripping process are introduced into the pool condenser together with the carbamate solution from the High-Pressure scrubber and fresh ammonia. The gases leaving the reactor are fed into the High-Pressure Scrubber where they are washed with the carbamate solution from the recirculation stage [3]. Thus a major portion of gases is recovered and returned to the pool condenser [4]. The inert gases and some ammonia and carbon dioxide coming from the scrubber are washed in a low-pressure absorber, thereby minimizing ammonia emissions. CO2 + Air

enriched Carbamate Solution Pool Condenser Inert gases, NH3 I C02,H20 Urea Reactor

HP

Scrubber

Inert gases NH3,C02 H2O LP Absorber

Reaction Mixture

CO2 Stripper

Carbamate Solution

Recirculation Stage

Fig 1. Block diagram of the Synthesis Section In the recirculation section unconverted ammonia and carbon dioxide are removed from the urea solution. These unconverted materials are condensed, forming a carbamate solution which subsequently is recycled to the synthesis section. The urea

Control of the Synthesis Section of a Urea Plant by Means of an MFC Controller

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solution after the removal of the unreacted carbamate, ammonia and carbon dioxide is sent to the evaporation section where the water in the urea solution is removed. Afterwards, the urea melt is sent to the granulation unit (finishing technique) where the ft)rmation of the urea granules takes place. Finally, the condensated material coming from the evaporation stage is sent to the waste water treatment section, where residual ammonia and carbon dioxide present in the process condensate are removed and sent back to the recirculation stage. 3. MPC Control of the production of the Synthesis Section Figure 2 shows a control scheme designed to maximize the production of the synthesis section of a urea plant [1]. The goal is achieved by maximizing the feeding flows (NH3 and CO2). This objective is accomplished while keeping the NH3/CO2 ratio, the reactor level and the reactor pressure on their desired values and the reactor temperature within a defined operating range. In this control scheme the production of urea can be set by selecting the setpoint of the NH3 fiow rate controller (FRC106). In this way, if the setpoint of the NH3 flow rate controller is augmented, the amount of urea produced will be increased. Since the reactor level shows an integral behavior and INCA Suite can deal only with stable processes, it was necessary to stabilize the reactor level by means of a proportional controller (LIC201). The manipulated variable of this proportional controller is the position of the valve HIC201 that has a direct impact on the reactor level. This valve also influences the reactor residence time (conversion) and traditionally it has been operated manually in the urea plants; with this control scheme such valve is manipulated by the MPC through the LIC201 controller. ^

Recirculation Stage

MPC Controller

Fig 2. Control diagram for controlling the production of the synthesis section. In order to have a good disturbance rejection for the NH3/CO2 ratio, this control system has a feed-forward action in the PID Loops. The setpoint of the CO2 flow rate controller (FIC108) is determined by the NH3 flow rate multiplied by a constant K (feed-forward action) plus the output of the AIC202 proportional controller. The fiinction of the

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AIC202 controller is to stabilize and to allow setpoint changes of the NH3/CO2 ratio at the reactor outlet. For the MPC controller, the manipulated variables are the setpoint of the LIC201 controller (IRL), the setpoint of the AIC202 controller (INC), the setpoint of the PRC902 (LPS - Low pressure steam controller, which has a direct impact on the carbamate reaction), the setpoint of the PIC904 controller (HPS - High pressure steam controller, which influences the stripper efficiency), the position of the HIC202 valve (HIC202, which influences the reactor pressure) and the setpoint of the FRC106 controller (FNH3 - NH3 flow rate controller). The controlled variables are the reactor level, the NH3/CO2 ratio at the outlet of the reactor, the reactor pressure (PR201) and the reactor temperature (TI201). Observe that there are more MV's than CV's this feature provides an extra freedom for the system to search the optimal operating point. Sub-space multivariable identification (N4SID [5]) was used to obtain the mathematical model (in this case a multivariable linear model) that allows the MPC to predict the future behavior of the process. The order of the model was selected by the singular values of the Hankel Matrix. The estimated order was 9. The model was validated by comparing the simulated and the predicted outputs. The operating ranges of the manipulated variables of the MPC controller were chosen very carefully to avoid problems with the downstream sections of the urea plant. The tuning of the parameters of the MPC controller was performed in a trial-and-error procedure. For evaluating the performance of the MPC controller the ideal resting value (value where this MV should go) of the NH3 flow rate was changed from 77.2 t/h (nominal value) to 80.5 t/h. The setpoints (nominal values at full load) of the reactor level, the NH3/CO2 ratio, the reactor pressure and the reactor temperature were kept on 50%, 3, 144.6 barg and 181.96°C respectively. It is important to mention that the carbamate solution coming from the recirculation stage constitutes one of the main disturbances for the synthesis section. Figure 3 shows the increment of the flow of urea solution leaving the synthesis section (throughput) during the experiment. At initial steady state conditions, the throughput of the synthesis section was 241.82 t/h of urea solution. Once the NH3 flow rate was increased from 77.2 t/h to 80.5 t/h the new throughput of the process in steady state conditions was 253.63 t/h of urea solution. We have obtained an increment of 11.81 t/h in the throughput of the synthesis section. Of course, the achievable improvement on a real plant will depend strongly on the bottlenecks of the given plant.

5

10 15 Time (hours)

5

10 15 Time (iiours)

Fig 3. Urea solution flow rate and controlled variables during the experiment

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\ariable setpoint

5

10 15 Time (hours)

20

5

10 15 Time (hours)

20 "

10 15 Time (hours)

20

5

10 15 Time (hours)

f

variable 1 setpoint

)

0

5

10 15 Time (hours)

20

20

variable | setpoint | 5

10 15 Time (hours)

20

Fig 4. Manipulated variables during the experiment Figures 3 and 4 show the controlled variables and the manipulated variables of the MPC controller along the test. From figure 3 it is remarkable how the controlled variables are kept close to their setpoints. So, the maximum peak for the NH3/CO2 ratio is only 0.0053, for the reactor level is only 1.79 %, for the reactor pressure is 0.88 barg and for the reactor temperature is 0.3°C. In figure 4 one can observe how the ammonia flow rate is changed by the MPC controller slowly from 77.2 t/h to 80.5 t/h. During the experiment the downstream sections of the plant were operating without problems. The control scheme presented has increased the production while keeping the unit under control. Although it is not shown in this paper, this control scheme can also perform setpoint changes for the NH3/CO2 ratio, the reactor pressure, and the reactor level. Independent setpoint changes in the reactor temperature are not possible since the reactor temperature is linked to the reactor pressure. 4. Final comments and Conclusion In this paper the results of using an MPC control scheme for controlling the synthesis section of a urea process have been presented. The main goal of this control system was to maximize the throughput of the synthesis section by maximizing the feeding flows. Additionally this control scheme kept the NH3/CO2 ratio under control and also the reactor level, the reactor pressure, and the reactor temperature. The results obtained with this control scheme were fairly good from the control point of view and from the production point of view. The increment of the production was limited by the maximum feeding flow capacity usually limited by the available raw material or feeding facilities like the CO2 compressor. However every plant can have its limitation on a different unit. Acknowledgements The authors would like to acknowledge the support of IPCOS ISMC office, Rob Faessen from Protomation and Klement K. Riegman from Stamicarbon. Additionally this research was supported by: • Research Council KUL: GOA-AMBioRICS, BOF OT/03/12, several PhD/postdoc & fellow grants; • Flemish Government: - FWO: PhD/postdoc grants, G.0407.02 (support vector machines), G.0080.01 (Collective

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Behavior and Optimization), G.0211.05 (Nonlinear Systems), research communities (ICCoS, ANMMM); - IWT: PhD Grants; - Toumesol 2005; - Belgian Federal Science Policy Office lUAP P5/22 ("Dynamical Systems and Control: Computation, Identification and Modelling"). Dr. Joos Vandewalle is a full professor at the Katholieke Universiteit Leuven, Belgium. References 1. O. M. Agudelo, J. J. Espinosa, J. Vandewalle, "Control of the synthesis section of a urea Plant using MPC control techniques". Internal Report 05-157, ESAT-SISTA, K.U. Leuven, Leuven, Belgium, 2005. 2. Protomation bv, "Process Studio. Application Urea-Operating Manual", Netherlands, 2003. 3. Stamicarbon, "Stamicarbon - Pure Knowledge", http://www.stamicarbon.com 4. Uhde, "Urea Brochure", brochure available in http://www.uhde.biz/ cgi-bin/ byteserver.pl/pdf/broschueren/Duengemittel/Urea.pdf 5. P. Van Overschee and B. De Moor, "Subspace Identification of Linear Systems: Theory, Implementation, Applications", Kluwer Academic Publishers, the Netherlands, 1996. 6. D. Henriksson and J. Akesson, "Flexible Implementation of Model Predictive Control using Sub-optimal Solutions", Internal Report, Department of Automatic Control, Lund Institute of technology, Sweden, April 2004. 7. IPCOS, "INCAengine version 2.1. IPCOS User Manual", Boxtel, Netherlands, 2003.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Control of thermal runaway via optimal bifurcation tailoring aided gain-scheduling feedback Pietro Altimari,^ Lucia Russo,^ Erasmo Mancusi,'' Mario di Bernardo,'^ and Silvestro Crescitelli^ ^Dipartimento d'Ingegneria Chimica Universita ''Federico IF' Piazzale Tecchio 80, 80125 Napoli, Italy ^Dipartimento di Ingegneria Chimica Alimentare Universita di Salerno, Via Ponte Don Melillo, 84084, Fisciano (SA), Italy ^Facolta di Ingegneria, Universita del Sannio, Piazza Roma, 82100, Benevento, Italy ^Dipartimento di Informatica e Sistemistica, Universita degli Studi di Napoli 'Federico IF', Via Claudio 21, 80125 Napoli, Italy. Abstract Thermal runaway of chemical reactors has been proved to be strictly related to the experience of catastrophic bifurcation operating conditions. In this framework, biftircation control theory, dealing with the modification of the bifurcation structure of nonlinear systems, is to become a powerful tool to design control systems able to tame thermal runaway hazards. In this work we focus on a novel bifurcation control strategy: the bifurcation tailoring. Bifurcation tailoring aims to globally reshape the bifurcation diagram of a nonlinear system. In order to cope with process constraints, an optimal formulation of the bifurcation tailoring method is proposed and implemented to remove thermal runaway hazards in chemical reactors. Keywords: runaway, bifurcation control, reactor. 1. Introduction Thermal runaway of a chemical reactor describes a situation in which a small change in a operating parameter induces a large change in the temperature profile of the reactor. This situation is rather common for high exothermal reactions where a small increase in the reaction temperature can lead to a large increase in the reaction rate due to the Arrhenius temperature dependence of the rate constant. When the heat generation is faster than the heat removal the reactor temperature continuously rises with a consequent acceleration of the reaction and an explosion is possible [1]. A considerable amount of research on runaway of chemical reactors has been carried out by several authors during the last 30 years. Many studies have been focused on deriving criteria to detect runaway boundaries in the parameter space for different types of reactions and different types of reactors. These criteria allow to predict numerically, or if possible, analytically runaway operating conditions. In spite of great efforts carried out to derive effective methodologies for runaway boundaries prediction, there is little work on developing active control strategies to tame thermal runaway hazards. In many cases safe operating regions are very narrow and a control is needed.

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Recent studies relate chemical reactor runaway to the experience of catastrophic bifurcation operating conditions [2]. In this framework, bifurcation control theory [3] is to become a powerful tool to design control system suitable to prevent thermal runaway events. Bifurcation control deals with the modification of the bifurcation structure of a nonlinear dynamical system and, hence, can provide useful guidelines for thermal runaway control. Typically, bifurcation control looks for a local modification of the bifurcation structure of a nonlinear dynamical system. Departing from the idea of a local bifurcation control, Lowenberg and Richardson [4] proposed a control strategy, named bifurcation tailoring, aimed to globally reshape the bifurcation diagram of the nonlinear dynamical system. Bifurcation tailoring enables the production of feedforward control signals to change the bifurcation diagram of a nonlinear dynamical system to a desired one. In its simple feedforward implementation bifurcation tailoring does not guarantee stability and uniqueness of the desired solution regimes. However this problem can be overcame by an added feedback control [5]. Bifurcation tailoring aided feedback control has been implemented successfully to tame instability motion of aircraft occurring at high angle of attack [4]. Recently, bifurcation tailoring has been appeared effective for thermal runaway control [6]. The problem addressed in this paper is the suppression of catastrophic bifurcations induced by measurable slowly varying disturbance parameters. When implementing bifurcation tailoring with this control objective, preliminary reshaping of the bifurcation diagram of the uncontrolled system is required. The bifurcation diagram is reshaped to meet desired output behaviour as bifurcation parameters vary. Once reshaping is designed, feedforward control (i.e. scheduling) in the manipulated parameters is synthesized to force the controlled output to follow the desired bifurcation diagram. This approach fails when process constraints are imposed. Indeed, a reshaped bifurcation diagram, yet meeting desired output behaviour, may require feedforward control violating constraints on manipulated parameters and/or state variables. A modification of the bifurcation tailoring technique is proposed to take into account process constraints. The proposed approach allows the reshaping of the bifurcation diagram of the uncontrolled system and determine the feedforward control law through on-line constrained optimization. Performance of the method is discussed through thermal runaway control in a partial oxidation process. 2. S t a t e m e n t t h e p r o b l e m Partial oxidations are typical industrial processes where runaway phenomena can affect dramatically the yield and/or selectivity of the desired product. In these processes the main control objective is to maximize the production of the desired intermediate product (partially oxidized), and avoid the complete oxidation. Although the reaction mechanisms are complex, these systems can be well approximated by two consecutive or parallel reactions where the undesired reaction usually exhibits a larger activation energy than the desired one [ 1 ]. Here is considered the case of two consecutive exothermic reactions with first order reaction rates, i.e.: A—i-^B—^-^C

ri = k i C ^

r2 = ^2^B

(1)

We assume the reactions take place in a cooled continuous stirred tank reactor and not negligible cooling jacket dynamics. Details on the process can be found in [1].

Control of Thermal Runaway

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The results of the bifiircation analysis of the partial oxidation process are reported in Fig. 1. Steady state reactor temperature (Fig. la) and yield of the desired product B (Fig. lb) are shown as the inlet temperature is varied. (b)

f

? ^2

/

y^

s. § HBi

Figure 1: Solution diagram as the feed temperature varies; (a) steady state reactor temperature; (b) yield of the intermediate product. The solution diagram shows an hysteresis and two saddle node bifurcation points delimitate the range where three steady states coexist. On the upper branch of the hysteresis, a subcritical Hopf bifixrcation point, HBi, is detected and a branch of unstable limit cycles arises and then vanishes through a global homoclinic bifurcation. It is of great concern for control purposes to note that the optimal reactor performances (maximum yield of the desired product) are detected at steady state regimes close to the catastrophic saddle node bifurcation point S2. When the reactor operating conditions are close to the bifurcation point S2, a slight increase of the feed temperature can lead to reactor runaway, giving rise to a large jump of the reactor temperature and a dramatic fall of the yield of the desired product. 3. Optimal bifurcation tailoring A modification of the bifurcation tailoring control scheme [5] which we named optimal bifurcation tailoring is proposed to take into account process constraints. Consider a continuous-time dynamical system described by: x = f(x,p,q)

y = h(x)

(2)

where x G Q" and y e n " " are the vector state and vector output of the system, respectively. Let p a slowly varying measurable disturbance parameter and q the vector of the manipulated parameters. The optimal bifurcation tailoring problem is to design a control law q = q(p) and a vector state function x = g(p) such that the following statements are met as the bifurcation parameter varies: (g(pXq(P)) = mina)(g,p,q)

(3)

x,q

s.t F(g,p,q) = 0 g G X, q G Q

(4) (5)

where 0(x,p,q) is the control costs function, X is the admissibility domain of the solution of the system (2), Q is the feasibility domain of control actions. When

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implementing optimal bifurcation tailoring, static optimization is sampled in time and the evolution equations of the open-loop controlled system are the following: x = f(x,p,q(tk))

y = h(x)

s.t. (g(tk ), q(tk )) = min 0 ( x , p(tk ), q)

(6)

x,q

s.t. F(g(tk),p(tk),q(tk)) = 0 g(tk)e X,q(tk)G a

p(tk)e [Pa.Pbl. t e [tk,tk+i]

We assume the control costs function in (6) to be expressed by quadratic deviation of the operating conditions with respect to the nominal ones, i.e.: n

m

m

0(x,p,q) = X ^ ^ ( ^ i - < ) ' + Z ^ i ( ^ i - ^ i > ' + ^ ( P - P * ) ' +Z>"i(^i(^k)-qi(tk-i))'

(7)

where (x*,p*,q*) is the nominal operating point and Oh, a,. A, jUf are weight coefficients. The last sum in (7) aims to guarantee bounded and, hence, feasible variation of the scheduling at each sampling time. 4. Numerical results Prevention of thermal runaway induced by feed temperature disturbances (other disturbances can be easily considered in the same way), requires reshaping of the steady state solution diagram. In particular, hysteresis has to be suppressed. Open-loop optimal bifurcation tailoring control is implemented to achieve hysteresis suppression as the feed temperature is varied. Feed cooling flow rate and the feed flow rate are assumed as manipulated variables. Preliminary static optimization of the process costs function is carried out as the feed temperature is quasi-statically increased. Fig. 2a displays the optimized steady reactor temperature diagram. When the optimized bifurcation behavior is followed as the feed temperature is varied, reactor temperature does not experience large variations and a complete reactant oxidation is prevented. A block scheme of the control loop is reported in Fig. 2b. The feed temperature is the input to the schedule whose outputs are the feed cooling flow rate and the feed flow rate which should assure the system follows the optimized solution branch. (a)

(b)

HB,^-^''

^

f

Schedule

4 2

' " ' ^ • ^ - - ^

J ^in

,

6

Si

QcQ

0

r

^r ^ ^ w

-2

Bin

Figure 2: (a) solution diagram for the uncontrolled reactor and a optimized steady state reactor temperature behaviour (triangle points); (b) feedforward control loop.

Control of Thermal Runaway

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Feed cooling flow rate and feed flow rate predicted through static optimization as the feed temperature varies are reported in Fig. 3. When the feed temperature ranges between the values 0<^>,<2 both feed cooling flow rate and feed flow rate are manipulated to attain constrained minimization of the process costs function.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

in

0.5

1.0

1.5

2.0

2.5

3.0

^in

Figure 3: optimized cooling flow rate (a) and flow rate (b) feedforward control laws. At feed temperature values larger than ^„~2, feed cooling flow rate is unchanged attaining a saturation constraint and the feed flow rate alone is called to realize optimization. The reactor simulation response when implementing optimal bifurcation tailoring control is reported in Fig. 4a and Fig.4b. 1 (a)

[(b)

1

r"—

nfcfc-J

Figure 4: (a), (b) simulation response for the feedforward controlled reactor; (c) feed temperature disturbance; (d) steady state solution diagram of the feedforward controlled reactor. As the feed temperature varies according the ramp reported in Fig. 4c the reactor temperature does not match the expected optimized bifurcation behavior. At a feed temperature value ^>,~0 a sudden and large jump of the reactor temperature is detected and autonomous oscillations arise as the feed temperature is further increased. The reasons of the mismatching between the optimized bifurcation behavior and the reactor simulation response are well understood by bifurcation analysis of the open-loop controlled reactor. Fig. 4d displays the steady state bifurcation diagram of the open-loop controlled reactor. In a wide feed temperature range the optimized solution branch is found to coexist with undesired solution branches. When coexistence of steady state regimes is detected, the optimized solution branch is stable and close to an unstable branch of saddle points. Hence, the basins of attraction of the desired solution regimes are very narrow in this range and sudden jumps of the reactor temperature could arise when feed temperature disturbances occur (Fig. 4a). As the feed temperature is increased the optimized solution branch is unstable in the range delimitated by the hopf

P. Altimari

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bifurcation points HB2, HB3. In particular, the h o p f bifurcation point HB3 located o n an u n d e s i r e d s o l u t i o n b r a n c h g i v e s rise t o l a r g e a u t o n o m o u s o s c i l l a t i o n s . A n a d a p t i v e g a i n - s c h e d u l e d f e e d b a c k c o n t r o l is i m p l e m e n t e d t o g u a r a n t e e r o b u s t stability o f t h e d e s i r e d s o l u t i o n r e g i m e s . A b l o c k s c h e m e o f t h e f e e d f o r w a r d / f e e d b a c k c o n t r o l l o o p is r e p o r t e d in F i g . 5a. T h e c o o l i n g feed t e m p e r a t u r e is a s s u m e d a s m a n i p u l a t e d v a r i a b l e for f e e d b a c k c o n t r o l . T h e s c h e d u l e p r o v i d e s static o p t i m i z a t i o n o f t h e b i f u r c a t i o n b e h a v i o r b y f e e d a n d feed c o o l i n g f l o w r a t e s c o n t r o l w h e r e a s t h e f e e d b a c k c o n t r o l o f t h e c o o l i n g feed t e m p e r a t u r e g u a r a n t e e s r o b u s t stability o f t h e o p t i m i z e d s o l u t i o n r e g i m e s . F i g . 5 b d e s c r i b e s t h e biftircation d i a g r a m o f t h e f e e d f o r w a r d / f e e d b a c k c o n t r o l l e d r e a c t o r . T h e d e s i r e d s o l u t i o n b r a n c h is s t a b l e a n d u n d e s i r e d s o l u t i o n are n o t d e t e c t e d as t h e feed t e m p e r a t u r e is v a r i e d . (a)

^in

x= X (6>„)

Schedule ^c,Q

< ^

Feedback controller

66>

Reactor

F i g u r e 5: (a) f e e d f o r w a r d / f e e d b a c k c o n t r o l l o o p ; (b) s t e a d y state s o l u t i o n d i a g r a m o f t h e f e e d f o r w a r d / f e e d b a c k c o n t r o l l e d r e a c t o r a s t h e feed t e m p e r a t u r e v a r i e s . 5. C o n c l u s i o n s A n o p t i m a l f o r m u l a t i o n o f t h e b i f u r c a t i o n t a i l o r i n g c o n t r o l m e t h o d is p r o p o s e d t o t a k e into accoimt process constraints. Performances of the m e t h o d and implementation issues are thoroughly discussed through thermal r u n a w a y reactor control. A continuous stirred t a n k r e a c t o r for a p a r t i a l o x i d a t i o n p r o c e s s is c o n s i d e r e d . T h i s r e a c t o r e x h i b i t s a catastrophic saddle n o d e bifurcation leading to thermal r u n a w a y as the feed temperature is i n c r e a s e d . T h e o p t i m a l b i f u r c a t i o n t a i l o r i n g c o n t r o l is i m p l e m e n t e d t o s u p p r e s s t h i s b i f u r c a t i o n a n d t o g u a r a n t e e o p t i m a l b i f u r c a t i o n b e h a v i o r a s t h e feed t e m p e r a t u r e is v a r i e d . O p e n - l o o p o p t i m a l b i f u r c a t i o n t a i l o r i n g c o n t r o l d o e s n o t g u a r a n t e e stability a n d uniqueness of the desired regimes. Hence, a feedback control system based on an a d a p t i v e g a i n - s c h e d u l i n g c o n t r o l s t r a t e g y is d e v e l o p e d in o r d e r t o r o b u s t l y s t a b i l i z e t h e optimized steady state solution branch. References [1] Varma, A., Morbidelli, M., and Wu, H., 1999, Parametric sensitivity in chemical systems, Cambridge University Press, Cambridge. [2] Patil, V. K., Sudhakar, S. and Balakotaiah, V., 1997, Ind. Eng. Chem. Res., 36, 3230. [3] Chen, G., Moiola, J. L., Wang, H. O., 2000, Int. J. Bif. & Chaos, 10, 5 1 1 . [4] Lowenberg M. H. and T. S. Richardson, 1998, Proc. OfAIAA Atmospheric Fligth Mechanics Conf., paper no. AIAA-98-4267, 286 [5] Wang, X., di Bernardo, M., Stoten, D. P., Lowenberg, M. H. and Charles, G. 2003, Int. J. Bif. & Chaos, 13, 677. [6] Altimari, P., Russo, L., Mancusi, E., di Bernardo, M. and Crescitelli S., 2005, The Italian Conference on Chemical Process Engineering-!, Giardini di Naxos, Italy, May 15-18

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Systematic Design of Logic Controllers for Processing Plants starting from Informal Specifications Sven Lohmann^, Olaf Stursberg^, and Sebastian Engell^ ^Process Control Lab (BCI-AST), Department of Biochemical and Chemical Engineering, University of Dortmund, 44^21 Dortmund, Germany. Abstract In contrast to industrial practice in logic controller design, where manual, experience-based, and thus error-prone procedures are commonly used, this paper proposes a methodology that aims at systematizing the design procedure as much as possible. Given a set of informal specifications and a technical flowchart, the control program is constructed as a sequential function chart (SFC) by transforming the requirements via clearly defined intermediate formats. The resulting SFC can be translated algorithmically into timed automata, composed with an appropriate plant model, and then rigorous verification can be used in order to analyze whether all specifications are satisfied. 1. Introduction and Method Overview In processing as well as in manufacturing plants, a large part of the automation software consists of logic control functions which supervise and coordinate local controllers, establish sequential procedures, and ensure a safe operation and high productivity. However, the design of logic controllers does in most cases not follow a rigorous design pattern based on precisely defined models - rather informal requirement specifications are translated manually into commonly used programming languages (lEC 61131-3 [1]). The disadvantages of this procedure are that (a) specific design decisions are only insufficiently (or not at all) documented, (b) the correlation between a requirement and the respective piece of logic control code is often not made explicit, (c) the validation of the design is difficult, and (d) subsequent modifications are time consuming. This motivates the proposed systematic design methodology. The scheme of the procedure shown in Figure 1 provides an overview of the different representations and models involved and can be divided into three steps: (I) the actual design procedure, (II) the translation of the SFC into a formal modeling language. The target format chosen here is timed automata, since SFC may contain time-limited or delayed actions. Step (III) is the algorithmic analysis of the formal controller model together with an appropriate plant model. The starting point of the design procedure is a set of informal requirements, a piping and instrumentation (P&ID) diagram, and applicable standards and guidelines (1). Firstly, these requirements are transformed into functions^ each of which represents a control procedure that realizes a requirement. Each function comprises of a set of actions representing the different steps of the control procedure. An action itself consists of a name, the condition which must apply for its execution, and

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a list of the identifiers of the relevant sensors and actuators. The functions with their respective actions are listed in the function table. In order to establish the order and interdependencies of the functions, a so-called dependency chart (DC) is employed (3). It represents, in graphical form, which functions can (or must) be executed sequentially, alternatively, or in parallel. Together with the function table, the dependency chart provides the basis to stepwise construct the controller code as SFC. The structure represented by the DC is trans(I) (1) Informal specification, formed into a function plan (4) consisting of P&I diagram steps (representing functions), transitions (to X (2)| express the sequential order of functions), and Function Table parallel and alternative branching to formulate (3) ';i corresponding interdependencies as identified -*\ Dependency Chart (DC) in the DC. Each step contained in the func(4) tion plan is then replaced by a step-transition Step1 Function 1 Transitionchain. Such a chain implements the sequence plan Chain (STC) of actions associated with the function according to the function table. By adding the sen(5) Sequential Function Chart (SFC) sor and actuator information contained in the (11) function table to the step-transition chains, the (6)r Timed Automata (TA) complete sequential function chart is obtained (III) i (7) Process (5). To illustrate this procedure in more detail, model the application to a control task for a labora(8)r Result of Analysis tory example (Sec. 2) is described in Sec. 3. modify In order to check if the control program com•ready plies with the requirements formulated initially, formal verification can be applied as fol- Figure 1: Scheme of the method, lows (as an alternative or in addition to testing): The SFC is transformed into timed automata by an algorithmic procedure (6). Usually some requirements (operational as well as the safety-related) express desired or undesired behavior of the controlled plant, thus a plant model (7) is indispensable for verification. In the proposed procedure, the plant is modeled by timed or hybrid automata, and then composed with the controller model. A completely algorithmic verification procedure then determines whether the given requirements are fulfilled (8). In case of faults, the analysis result is used to identify in which design steps modifications have to be introduced.

'f

I I

r

2. Sequential Control of a Batch Evaporation System The design procedure is illustrated for the batch evaporation system sketched in the left part of Fig. 2. The system consists of two tanks (Tl, T2) with heating devices (HI, H2), a condenser with cooling (CI), connecting pipes equipped with valves (VI, V2, V3) and a pump (PI). In addition, the plant contains different sensors for liquid levels (LIS), temperatures (TI), and the concentration in T l (QIS). The desired operation is to evaporate liquid from a mixture in T l until a required concentration is reached, to collect three batches of the product in T2, and to empty the latter afterwards through P I . Two different disturbance scenarios are considered: (a) In case of a cooling failure (detected by FISlOl, error 1), the evaporation is continued for a short period of time and, if the concentration target

Systematic Design of Logic Controllers

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is not reached by then, the remaining content of T l is disposed through V3. (b) In case of a heating failure, T l is emptied immediately through V3 (since the production goal cannot be reached anymore). In both cases the nominal operation is resumed if the faulty devices are repaired or replaced. Two critical states can be identified: a continuing heat supply for scenario (a) may lead to overpressure in the condenser, and for scenario (b), the content of T l crystallizes when a lower temperature bound is reached. 3. Data Formats used in the Design Procedure The method proposed here is based on SFC, a graphical high-level representation of a hierarchical control structure in which any of the other common languages for logic control programs (instruction list, function block diagram, ladder diagram or structured text) can be embedded. These characteristics provide a clear structuring of sequential and parallel processes within the logic control program. As shown for an example in Fig. 2 (right part), the building blocks of an SFC are steps with an attached action block, transitions with an associated condition, parallel executions enclosed by double horizontal lines, and alternative executions marked be single horizontal lines. The vertical lines represent the flow by which the blocks are processed from top to down, with the exception of loops which point in the opposite direction. For the purpose of the proposed method, a clearly defined syntax and semantics of SFC are necessary. These definitions are not provided by the standard [1] as pointed out in [3], but were specified in a rigorous manner in [5]. In contrast to the precise SFC-model, the record of all relevant requirements (the requirement specification) is not given in a structured form but in natural language.

Ifiril^^^'^^

\ ^qualifier - | - " ^ ^ ° ^ - - ^ transition cor

-|-notlis204

i

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I]_ V

-Unotlis204

/LISN

T2

W

/TIS\ ^302^

H2:|:

^

<

T

/Lis\

i^

Figure 2. P&ID of the evaporator system (left) and a possible logic controller as SFC (right).

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In addition, it often has a varying degree of detail and is incomplete, making the direct translation into programmable control code difficult. The P&ID contains the plant components considered for the control design (including sensors and actuators) on a well-defined and appropriate level of detail, hence it forms, together with the function table, the basis for obtaining a list of uniformly structured requirements. In the function table, a function refers to the operation to be carried out in order to satisfy a particular requirement. The operation can usually be separated into a list of actions^ each of which intuitively represents the manipulation of an actuator, e.g. upon receiving a sensor signal. An excerpt of the function table (FT) for the evaporation system is shown in Fig. 3 (left part). For each action, the following information is specified: a precondition as the condition that has to be satisfied prior to the execution of the action, the information on the sensor for which the precondition is formulated, a description (in natural language) of the control action to be performed, and the control command sent to the actuator(s). All actions of a function (represented as rows in the table) are chronologically ordered (where possible) and separated into nominal operation and error handling. The function table does not include information on the relation between the different functions. In order to provide this information, the design procedure uses dependency charts (DC), as shown for the example in Fig. 3 (right part). A DC is motivated by Gantt-charts which are common in production planning. The functions are shown as rectangles over an abscissa that represents time in a qualitative fashion. Graphical symbols mark strict sequential execution (single arrows), strictly parallel execution (vertical bold lines connecting the beginning and/or ending of two or more functions), or the alternative execution of functions (multiple arrows). The time points at which a function starts or ends are denoted by the event name taken from the FT (column sensor). To each row of the chart one function is assigned. Along the ordinate all functions are listed in chronological order (where applicable) from top to bottom. The dependency chart and the function table are refined iteratively until all specifications are considered, i.e. until a level of detail is reached that corresponds to the one identified as suitable in the P&ID. Subsequently, FT and DC are transformed into the SFC via the two intermediate formats function plan (FUP) and

Function Precondition Function i/Action al Action a2

Sensor

J Function START

&mj\ dose_Tl

Precondition

tlkm^

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for the example

Sensor

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Actuator

Procedure Initial step

D^tl ^Iffit^

Actuator All valves closed, all devices disposell switched off Oi«aaVl Close Vi LIS201 error2

QIS202 LIS264 ersolved

Figure 3. Function table (left) and Dependency Chart (right) for the evaporator example.

Systematic Design of Logic Controllers

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step-transition-chains (STC). The function plan is generated from the DC and represents the coarse structure of the SFC. One step represents a complete function and the structuring elements of the DC are converted into (1) simple transitions between steps (for two sequentially executed functions), (2) alternative branching (for multiple arrows starting from one block in the DC), or parallel branching (for a bold vertical hne connecting blocks). The syntax of the FUP follows the grammar specified for SFC in [5], which ensures safe structures [3]. The STC then replace steps of the FUP by sequences of steps and transitions representing the actions listed for the corresponding function in the FT. This is shown exemplarily in Fig. 4 for two chains. Each STC consists of the SFC elements transitions^ steps and action blocks^ and an STC always starts with a transition and ends with a step. The transition conditions and contents of the action blocks follow from the fields sensor and actuator of the FT. The refinement of the FUP by inserting the STCs eventually leads to an SFC, as shown in Fig. 2 (right part) for the evaporation system. 4. Translation into Timed Automata and Verification For analysis, the SFC obtained by the above procedure is translated into timed automata (TA) [2]. Different transformation schemes were proposed in [3], and in [5]. While the procedure in [3] explicitly models the cyclic scanning mode in which SFC are executed on programmable logic controllers, the simplified scheme in [5] considers the scanning mode only if an event from the plant model is received. Abstracting from the strict PLC-cycle leads to an automaton structure of considerably reduced size. The common principle of both schemes is that the SFC is first partitioned into syntactical units, and then translated into a modular timed automaton model. The communication between the automata is modeled by synchronization. The state-transition structure of each automaton follows di-

Figure 4. Function plan (middle): intermediate format to represent the interdependencies of different functions, which are modelled by step-transition chains STCs (a).

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rectly from the step-transition sequences of the SFC. The actions associated with the steps are mapped into the corresponding automata considering that clocks and time conditions have to be used where the SFC contains time-dependent action qualifiers. The controller model can be composed with a TA model of the plant. The verification task then is to analyze whether the production goal is reached while the critical states are never encountered. The latter (over-pressure and crystallization in T l , see Sec.2) are explicitly specified as states in a TA modeling the state of aggregation in T l . The verification with Uppaal [4] led to the result that the critical states are not reachable, and that the production goal is met if no breakdown of the cooling occurs. Thus, the SFC controller shown in Fig. 2 (right part) meets the requirements. 5. Conclusions A systematic procedure for the design of logic controllers has been presented. A set of intermediate formats is used to prceed step by step from the initial informal specifications to implement able controller code. The sequence of formats forces the designer to systematically gather and refine the relevant information, and to reason about the interdependencies between the control functions. Furthermore, the collection of FT, DC, and SFC establish a documentation of the design process that makes later changes of the control code easier. Finally, the algorithmic verification reveals design flaws if an appropriate plant model is available. While the steps of converting the FT into STCs, deriving the FUP from the DC, combining the STCs and the DC to the SFC (as well as the transformation in TA and the verification) can be carried out algorithmically, the creation of the FT and the DC is obviously a creative process. Therefore an objective of current and future work is to provide tool support for the complete procedure. REFERENCES 1. Int. Electrotechnical Commission, Tech. Committee No. 65: Programmable Controllers - Programming Languages. Standard lEC 61131-3, 2003. 2. R. Alur, D.L. Dill: A theory of timed automata. Theor. Comp. Science, 126/2 (1994), 183-235. 3. N. Bauer, S. Engell, R. Huuck, S. Lohmann, B. Lukoschus, M. Remelhe, O. Stursberg: Verification of PLC Programs given as Sequential Function Charts. In: Integration of Software Specification Techniques for Applications in Eng., Springer, LNCS, Vol. 3147, 2004, 517-540. 4. G. Behrmann, A. David, and K.G. Larsen: A Tutorial on Uppaal. In: Formal Methods for the Design of Real-Time Systems, Springer, LNCS Vol. 3185, 2004, 200-236. 5. O. Stursberg and S. Lohmann: Analysis of Logic Controllers by Transformation of SFC into Timed Automata. Proc. IEEE Conference on Decision and Control, Sevilla, 2005, 7720-7725.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Static/dynamic analysis and controllability issues in reactive distillation columns Teresa Lopez-Arenas,^'^ Eduardo S. Perez-Cisneros,^ Rafiqul Ganf ^ CAP EC, Department of Chemical Engineering, Technical University of Denmark, 2800 Lyngby, Denmark ^Departamento de Ingenieria de Procesos e Hidrdulica, Universidad Autonoma Metropolitana-Iztapalapa, 09430 Mexico D.F., Mexico Abstract The design, operation and control of reactive distillation columns are more complex than those involved for either reactors or conventional distillation columns. Therefore, a good understanding of the process helps to avoid unsafe operation conditions and further facilitates various subsequent studies such as control, monitoring, data reconciliation, parameter estimation and optimization of existing plants. In this work the study of the steady state and dynamic behavior as well as the analysis of some controllability issues for reactive distillation columns, based on the applicability of the (equilibrium) element-based approach and thermodynamic issues, are presented. The MTBE process is selected as case study, so that previous reported results can be compared and verified. Results are obtained using ICAS and ASPEN simulators. Keywords: reactive distillation, steady state multiplicity, dynamic simulation, controllability, passivity. 1. Introduction The flowsheets of several important processes in the refinery (i.e. MTBE, ETBE, TAME, hydrodesulfurization of diesel, etc.) consist of a reactor followed by a sequence of distillation columns. Reactive distillation (RD) has emerged as a highly promising process because it combines the requirements of in-situ separation with reaction, bringing several advantages such as: capital and operation costs are reduced, energy integration is improved, reactant conversion is increased and product selectivity is improved. In general, RD has a complex behavior, such as high non-linearity due to complex kinetics and thermodynamic models, strong input-output interactions, steady state multiplicity, high potential to instability, sensitivity to disturbances, etc. However, the complexity of the RD process is known to create unusual responses to changes in operating variables. Many of these effects are related to the phenomenon of multiplicity of steady states (which occurs as either input or output multiplicity) and have significant implications on the effectiveness of operability and control schemes. On the other hand, from a thermodynamic point of view, as the driving force is increased, the separation becomes easier, the energy consumption is reduced and the control requirement becomes easier. Moreover from a control point of view, it is reported that passive systems are easy to control and a passive state, in chemical processes, is related to state of minimum entropy production (Coffey et al., 2000). In this work, the main objective is the study of the steady state and dynamic behavior as well as of controllability aspects based on the equilibrium (element-based) approach (Perez-Cisneros et al., 1997) and a thermodynamic analysis for RD columns. The element-based approach uses chemical elements rather than the actual components, so

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that it has the advantage of obtaining reduced order models (i.e. smaller number of states, since the number of elements are always less than the number of compounds), making easier the selection of the control structure. While from the thermodynamic analysis, controllability insights can be obtained by using the driving force and the Gibbs energy concepts: a state related to the largest driving force and to the minimum Gibbs energy will provide a passive system, and therefore a system easier to control. These conclusions are corroborated through steady state and dynamics simulations on ICAS and ASPEN for the MTBE (methyl tert-butyl ether) process as case study, so that previous reported results (Hauan et al., 1995; Sneesby et al., 1998; Eldarsi et al., 1998) can be compared and verified. The results show the implications of both input and output multiplicities on the column operation, design and control, and insights on the control configuration are stated. 2. MTBE process 2.7. Reaction scheme MTBE is produced by reaction of isobutene and methanol, where the liquid-phase reaction is catalyzed by an acidic catalyst. A typical mixed butenes feed for MTBE synthesis consists of about 40% isobutene and 60% n-butene, which is an inert component. The reaction scheme is: i-C^Hs + CH3-OH ^ C5H12O

(1)

An alternative to the transformed composition variables for representing reactive systems is the element composition concept proposed by (Perez-Cisneros et al., 1997). The main advantage of this element composition representation is that it reduces the composition space domain and simplifies the phase equilibrium with chemical reaction (equilibrium or kinetically controlled) calculations. The chemical elements are the molecule parts that remain invariant during the reaction. Thus, defming elements as A = isobutene, B = methanol and C = n-butene, then the reaction can be written as: A+B^^C

(2)

Therefore, there are Nc = 4 components and M = 3 elements (see the corresponding element matrix in Table 1), so that the 4-component RD column can be modeled as a ternary conventional distillation column. 2.2. Reactive distillation column configuration A reactive distillation column typically consists of three zones: rectification, reaction and stripping (see Fig. 1). The reactive distillation zone contains catalyst in the form of structured packing material. Separation of the inert takes place in the rectification section of the column, and the purification of the product takes place in the stripping section. In particular, for our case study, the RD column has a total number of 15 stages, plus a total condenser and a partial reboiler. It is desirable to obtain a bottom product containing high-purity of MTBE and a distillate containing high-purity of n-butene. The column design specifications and operation conditions are given in Table 2. Table 1. Element matrix (A) for MTBE reaction

Isobutene Methanol MTBE n-butene

Element A

Element B

Element C

1 0 1 0

0 1 1 0

0 0 0 1

Static/Dynamic Analysis and Controllability Issues

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Table 2. Design specifications Units Parameter Value STAGES Total number 15 Reactive stages 4-11 Holdup 250 Kg atm Pressure 11 METHANOL STREAM Feedrate 28.3366 Kmol/h K Temperature 320 Pressure 11 atm Feed stage 3 BUTENE STREAM Isobutene feedrate 25.641 Kmol/h N-butene feedrate 46.022 Kmol/h K Temperature 350 Pressure 11 atm Feed stage 9 -

Parameter Value Units BOTTOM&TOP SPECIFICATIONS K/(Kmorh) Heat duty 650203 Reflux ratio 7 COLUMN GEOMETRY Radius 0.155 m m^ Plate active area 0.366 Plate spacing 0.31 m Weir length 0.53 m Weir height 0.038 m Plate efficiency 1.0 CONDENSER SPECIFICATIONS m Area 0.366 0.2 m Liquid height REBOILER SPECIFICATIONS 0.732 Area m^ m Liquid height 0.5

2.3. Process modeling and simulation ICAS simulator (Gani et al., 1997) was used for both static and transient simulations, where the element-based approach has been implemented recently for RD columns. This package allows for a number of standard thermodynamic property methods as well as property estimation methods. All of the independent equilibrium-model equations are simultaneously solved using Newton's method for steady state simulations and the BDF (backward differential formula)-method for dynamic simulations. In particular for the MTBE process, the thermodynamical data was taken from the DIPPR databank, the Soave-Redlich-Kwong equation of state and the UNIFAC model were used to represent the vapor and liquid phases respectively. For comparison purposes, simulations on the commercial ASPEN simulator (2001) were also done using the RADFRAC module, which is an equilibrium model that uses the inside-outside algorithm. RADFRAC has been used and validated against experimental data by many other authors (Taylor and Krishna, 2000), so that ASPEN simulations will be used here also to validate the results from ICAS. 3. Thermodynamics and controllability connection Passive systems are easy to control, therefore there is a significant interest in finding coordinate representations that render a system passive (Byrnes et al., 1991). In earlier publications (Coffey et al., 2000; 2001) it was illustrated how the use of passivity and thermodynamic based analysis can be used to better understand the open-loop stability and control problems in conventional distillation columns. Typical passivity (Liapunov) theory is used for stability analysis. A system is passive (Byrnes et al., 1991) if there exists a non-negative storage function {A) that captures the dynamics of the system and satisfies the following inequality A{t)
(3)

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Moreover, the passive state will be stable if the derivative of the storage function is negative (i.e. AAl&t < 0). Specifically using the negative of the entropy as storage function, Farschman et al. (1996) found that in chemical processes, a passive state is a state of minimum entropy production. Taking as a point of departure these resuhs, our basic assumption is that a process also related to minimum Gibbs energy will provide an easier separation with reduced energy consumption, implying a passive system that will be easier to control. Thus, the implication of Gibbs energy on the column operation and control is studied in next section. The Gibbs energy (G) is calculated, rather than the well known definition in terms of chemical potentials and component compositions, using element potentials {X) and the element matrix {Aik, as given in Table 1) as follows:

G = RTYlMi^

H.=T!-JA.

(^ = 1...A^J

(4)

4. Open-loop analysis 4.1. Steady State multiplicity The steady state multiplicity is analyzed through bifurcation diagrams, which are built using a continuation method (Guckenheimer and Holmes, 1983), raking the steady state of the model when the value of a bifurcation parameter (BP) is increased or decreased. Various BP were studied finding different multiplicity regions, for instance: (/) no multiplicity was found using the methanol feedrate as BP (as reported by Eldarsi et al., 1998); (//) input multiplicity was found using the reboiler heat duty as BP at fixed reflux ratio (as reported by Wang et al., 2003); (///) output multiplicity was found using the methanol feed stage as BP (as reported by Hauan et el., 1995); and moreover (iv) a combined input-output multiplicity was also encountered using the reboiler heat duty as BP at fixed reflux flowrate (as also reported by Wang et al., 2003). Results using reboiler heat duty as BP are shown in Fig. 1. The numerical simulations from both ICAS and ASPEN wcYQ quite similar for all cases (Fig. la shows results for case /z), where the main differences were due to the RD models used in each simulator. Fig. lb shows the input-output multiplicity for case iv. The Gibbs energy analysis was done using the bifurcation diagram of Fig lb. Fig. 2a shows the Gibbs energy profiles (calculated using ICAS) when output multiplicity occurs (i.e. when the three steady states A, B and C appear, having all of them the same reboiler heat duty). Observing that in the reactive zone (stages 4-11), GA ^ GB ^ ^c> 1

S ^

;A

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^ ^ \ ^ ^^^\

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1000

1500

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2000

2500

Reboiler heat duty (KW)

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B 380

Reboiler heat duty (kW)

Fig. 1. Bifurcation diagram using rebolier heat as BP at fixed: (a) reflux ratio, (b) reflux flowrate.

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Static/Dynamic Analysis and Controllability Issues

-IxlO'

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Stage

Fig. 2. Gibbs energy profiles: (a) output multiplicity and (b) input multiplicity. indicates that the steady state A related to a high conversion will be easier to operate, with minimum Gibbs energy, and therefore this steady state will be easier to control in comparison with states B and C. While Fig. 2b show the Gibbs energy profiles when input multiplicity occurs (i.e. when two steady states B and D appear, having the same bottoms MTBE purity but at two different reboiler heat duties). The results shows that the steady state D is related to a smaller energy consumption and has lower Gibbs energy values, implying that this state will be easier to control. 4.2. Dynamic behavior Fig. 3a shows the dynamic response for the RD column. Starting at the nominal steady state, a step change in the methanol feed flow is applied from 40 Kmol/h to 28 kmol/h, reaching a second steady state. Simulation results from ICAS and ASPEN are quite similar, only there is a slight difference in the settling times (the response from ICAS is faster than the one from ASPEN). The corresponding responses of the Gibbs energies along the column are presented in Fig. 3b, where it can be seen that the time response is slower at the top of the column. The information obtained from these figures helps in controller design such as the allocation of sensors and the tuninf of controllers. 4.3. Controllability remarks In particular for conventional distillation columns, Coffey et al. (2001) proposed a control configuration where: (a) the inventories were defined as: \) = [My, ..., MNC, E^ {Mi are the mass holdups and E is the internal energy), (b) the thermodynamical potential was selected as 8(\)) = - S {S is the entropy), (c) the storage function A{\)) = .*T, (0\\) - u*) + 5' - 5'* (A > 0) fulfilled the passivity definition [Eq. (3)], and (d) the

G

8x10*6x10'4x10'2x10'-

Stage

^ / / • " " • " " • • ^ • • ~ \ . ,

~ ' ~ ' ~ " '

U '\

(KJ/Kmol) 0-2x10'1 ^1/ "••'-••--. -4xl0'_ _ l./" -6x10'[^ _ -8xlO'-Ixio'- • - • l " " " -IxlO'0.4

0.6 0.8 Time (h)

0.6 0.8 time (h)

1.0

Fig. 3. Dynamic behavior: (a) bottoms MTBE purity and temperature, and (b) Gibbs energy.

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T. Lopez-Arenas et al

intensive variables O) conjugated to the inventories were: co*^ = [dS/dv]y = y*= [-liy/T, ...,-|iN/r, 7/r]^. So that temperatures (7) were used for energy balance control and chemical potentials (ju^ were related to composition control. Following this methodology, as future work, we propose the following definitions for the RD column: (a) inventories in terms of the element compositions (i/): O) = [bi, ...BM, E]^; (b) thermodynamical potential in terms of the Gibbs energy [Eq. (4)], and (c) the intensive variables co conjugated to the inventories in terms of the element potential, element composition and temperature. So that the Gibbs energy will be related to the energy balance control and the element potentials to the composition control. The main advantages of using these definitions are that: firstly, as the number of elements is always less than the number of the components then the selection of the control structure is easier, and secondly, using the Gibbs energy as thermodynamical potential will imply a passive system and therefore a system easier to control. 5. Conclusions A general strategy for analysis of steady state multiplicity and dynamic behavior of RD columns has been presented and illustrated for MTBE synthesis. Numerical results were compared using two simulators, ASPEN that uses a conventional equilibrium model and ICAS that uses the element-based equilibrium model, obtaining similar results. From the Gibbs energy analysis some controllability insights were obtained: a process state related to minimum Gibbs energy will provide an easier separation with reduced energy consumption, implying a passive system, and therefore easier to control. Furthermore, it was shown that both modeling and control problems of a RD column can be reduced through the element-based approach since the number of states is also reduced (as the number of elements is always less than the number of components). Current and future work is testing the closed-loop controller design related to minimum Gibbs energy. References ASPEN® Version 11.1, 2001, Tutorial, Aspen Tech. Byrnes, C.I., Isidori, A., Willems, J.C., 1991, Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, IEEE Transac. Autom. Ctrl, 36, 1228-39. Coffey, D.P., Ydstie, B.E., Andersen, T.R., Jorgensen, S.B., 2001, Distillation control using passivity, Proc.ESCAPE-11, 627-632. Coffey, D.P., Ydstie, B.E, Farschman, C.A., 2000, Distillation stability using passivity and thermodynamics, Comp. Chem. Eng., 24, 317-322. Eldarsi, H.S., Doglas P.L., 1998, Methyl-tert-butyl-eter catalytic distillation column. Part I: Multiple steady states. Trans IchemE, 76, Part A, 509-516. Farschman, C.A., Nismanath, P., Ydstie, B.E., 1998, Process systems and inventory control, AIChE J., 44 (8), 1841-1857. Gani, R., G. Hytoft, C , Jaksland, Jensen A. K.,1997, An Integrated Computer-Aided System for Integrated Design of Chemical Processes, Comp. Chem. Eng., 21,1135-1146. Guckenheimer, J., Holmes, P., 1983, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag, New York. Hauan, S., Hertzberg, T., Lien, K.M., 1995, Why methyl tert-butyl ether production by reactive distillation may yield multiple solutions, Ind. Eng. Chem. Res., 34, 987-991. Perez-Cisneros, E.S., Gani, R., Michelsen, M.L., 1997, Reactive separation systems I. Computation of physical and chemical equilibrium, Chem. Eng. Sci., 52, 527-543. Taylor, R., Krishna, R., Modeling reactive distillation, 2000, Chem. Eng. Sci., 55, 5183-5229. Wang, S.J., Wong, D.S.H., Lee, E.K., 2003, Effect of interaction multiplicity on control system design for a MTBE reactive distillation column, J. Proc. Control, 13, 503-515.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Predictive Control of Polymerization Batch Reactors Using Hybrid Models

Jairo Espinosa, Wim Van Brempt IPCOSN.V., Technologielaan 11-0101, LeuvenB-3001, Belgium Universidad Nacional de Colombia- Facultad de Minas Carrera 80 # 65-223 Medellin-Colombia Abstract The use of model based control for polymerization reactors is a very attractive alternative to the classical master slave control of temperature. However the bottleneck has been always the construction of a reliable model. Classical identification techniques fail in the control of batch processes because the operating point changes along the reaction. Rigorous modeling is sometimes a very complicated task because of the lack of knowledge about the process or because the fitting of the dynamic parameters demands specific experiments. Also the complexity of the model and the lack of observability represent a major obstacle for the online operation of a full dynamic rigorous model. This paper presents the development of a gray-box model that can be fitted with minimal experimentation time and provides reliable predictions and information for the INC A (Ipcos Novel Control Architecture) model based predictive controller (MPC). The model not only provides reliable predictions of the evolution of the temperature but also generates online information about the status of the conversion. The use of such controller improves the performance of the control system providing extra room to increase the temperature of the reaction in order to increase productivity. Keywords: Predictive control, Batch, Semibatch, Hybrid Modeling. 1. Introduction The use of optimized recipes becomes a challenge for any control system. Fast recipes carry an increase of the temperatures of the reactor, such increase speeds-up the reaction mechanism (Arrhenius Law) and in general the dynamics of the process. The increase of speed of the reaction mechanism is not paired with an increase on the effectiveness of the heat exchanging installation; this mismatch must be compensated by the control system. Classical control systems (i.e. FID) are tuned to deal with the dynamics of the heat exchanger mechanism and in a limited basis with the disturbances generated by the reaction. In terms of frequency response one should say that the controller should provide a large bandwidth to reject the disturbances caused by the heat of reaction and good tracking capabilities to follow as close as possible the optimized trajectory. Normal FID controllers must make a trade-off between these two goals. On the other hand Model Predictive Control (MPC) can exploit the prior knowledge given by the knowledge of the optimized trajectory (important for tracking purposes) and the knowledge of the disturbances generated by the reaction (important for disturbance rejection).

1329

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J. Espinosa and W. V. Brempt

The use of MPC in batch processes has been motivated by multiple factors. Among those are worth to mention: 1. MPC uses the information provided by a process model (knowledge of the process dynamics) and future values of the references. 2. MPC is aware of the physical limits of the process providing good and safe disturbance rejection when the process is driven close to the limits of the installed capacity 3. MPC can provide an excellent performance on trajectory tracking no matter if the recipes are fast or not. What do you mean? An essential step in the implementation of a model predictive controller is the identification of the process model. Standard identification methodologies are however not suited for batch processes for two reasons: 1. Standard model identification tools generate linear dynamic models. These are not suited for strongly nonlinear processes such as batch processes. 2. Batch production facilities often consist of a variety of reactor units, in which a broad variety of product grades is being produced. Standard model identification technology requires several tests, which represent costly project time. In order to enable economic feasibility of batch process modeling, a dedicated modeling technology must be developed which allows for fast (non-linear) model derivation.

2. Integration of MPC into the a classical batch process The MPC controller used in this project is an extended version of the MPC controller INCA created by IPCOS N.V. Figure 1 shows the INCA-MPC controller acting as a Master control which receives the temperature profile of the recipe. It generates an optimal set of references for the slave PI controller in charge of the Jacket temperature. This way of integrating the system has various advantages, but perhaps the most important is that it helps to maintain the heat exchange dynamics constant, despite of changes in steam and water temperatures and pressures; aging and fouling of the pipelines in the heat exchange utilities; etc. Temperature Recipe

INCA Reactor Temperature

Setpoint Jacket Temperature

W^^Mi^ hy-;ry%ia^

Jacket Temperature

S/R ! Steam ^ Cooling

A

Predictive Control of Polymerization

Batch Reactors

1331

Figure 1 Integration of the INC A -MPC controller in to a batch Process

3. An MPC strategy for Batch The features of batch processes demand a different approach for the MPC controllers compared with the classical applications in continuous processes. Batch processes are in many cases temporarily unstable as well as strongly nonlinear since they do not have a unique operating point. The INCA-MPC controller can be configured to rely on nonlinear dynamic models for prediction of future process signals, as well as for derivation of subsequent linear dynamic optimization models. It solves the nonlinear MPC (NMPC) problem using a proprietary technique which resembles a time unfolded Sequential Quadratic Program (SQP). This advance feature is quite powerful, but requires an efficient process description. Such a description should exploit the distinctive features of batch processes and reflect these features in a special model representation explained in the next subsection. 3.1. Model representation The main controlled variable in a batch process is the reactor temperature. This temperature is determined by two mechanisms: the heating and cooling capacity of the heat exchangers and the heat absorbed or generated by the reaction. Models of the heat exchanger tend to be simple and very easy to obtain, either by direct physical modeling or simple identification experiments. On the other hand the chemical reactions tend to be complex and difficult to observe. In some cases the only way to evaluate the state of the reaction (conversion) is to check the evolution of the heat generated/absorbed by the reaction. The heat generation in the reactor can be seen as follows:

AHeat = AHeatReaction + AHeatExchanger

(1)

In terms of process variables a simplified representation of the system is given by:

teacor = ^0AHeatReaction + k, (7.,,,, - r_,„,) where k^ = XjcpV

and k^ =

(2)

, with c, p, F being respectively the heat capacity,

cpV the density and the volume of the reacting material and U and A are respectively the heat transfer coefficient and the area of heat transfer between the jacket an the reactor. Observe that if the reaction is not reversible the first term will vanish after the reaction is completed and the identification of the heat exchanger can be carried in two possible ways depending on the capacity of the heat exchanger. One can measure the temperature of the heat exchanger or the change in temperature of the fluid used to remove/add heat from the system.

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J. Espinosa and W. V. Brempt

Once the second term (heat exchanger) has been quantified, the contribution of the heat of reaction to the temperature of the reactor can be quantified. The idea is to define a simpHfied dynamic system to represent the reactor as follows: ^

A ~ J \ ^ A y-^ reactor )

(3) •^reactor

~ ^hJ

where X^

\ ^ A ^-^ reactor / "'" ^ 1 (.-^jacket ~ -^reactor )

is the conversion (0<X^

< 1 ) and / ( . , . ) is a nonlinear empirical

parametric function representing the dynamics of the reaction, and dependent on the temperature and the current conversion. The function f(.,.) can be estimated, since its input parameters can be either estimated or measured. 3.2. Model construction The construction of the model demands the separation of the main effects governing the dynamics of the process, namely the heat of reaction and the heat exchanger. Therefore the dynamics of the heat exchanger are identified when no reaction is taking place. In fact the parameter kj can be identified as a static constant or a simple dynamic system. reactor

'^

1V jacket

reactor)

v /

Afterwards, the effect of the heat exchanger is subtracted from the global thermal effect, producing: '^h J V ^ A ' ^ reactor ) ^ ^ reactor ~ ^\ \^ jacket ~ ^ reactor )

\^)

with this values and a proper estimation of the parameter XA the function f{X^, Treactor) ^^^ ^^ estimated using a parametric representation which combines basic process knowledge and fiizzy basis function. Figure 2 shows the representation of such a function for an industrial batch process. The parameters of the function are obtained using a nonlinear optimization procedure, where certain constraints enforce first principle relations. It generates a fimction with good generalization properties, allowing the use of different trajectories without retuning of the model. A very important fact is that this function can "refined" after each batch run. Such a refinement improves the prediction capabilities of the model improving the performance of the MPC. 3.3. Controller Implementation The NMPC is implemented using the model described in equation (3). The model is kept synchronized with the plant by using a nonlinear observer based on the model. The controller is designed to make a tight tracking of temperature. It exploits the knowledge of the prediction of the "peaks" of heat generated by the reaction; this is an important advantage which improves the performance of the NMPC. Such a tight control delivers the possibility of running the recipes at higher temperatures, which on its turn reduces batch time duration.

Predictive Control of Polymerization

Batch

1333

Reactors

Conversion Function f(X^,T)

Temperature - Kelvin

Figure 2 Parametric function to predict Conversion rate vs. Temperature-Conversion

4. Final Remarks and Conclusions This paper has presented the use of a new simple nonlinear model, capable of capturing the main dynamics of batch reactions. Such a model can be calculated with simple and "cheap" experiments, delivering an improved control system with minor disturbance to the production. It allows in general for a fast model identification phase, which is crucial in a batch production environment where often a large variety of products and reactors is present. The controller has been tested in industry providing not only a tighter control but also online conversion estimation. The methodology is currently being extended to semibatch processes and implementations are foreseen in the coming months. Acknowledgments Part of this research has been financed by the Flemish Government in Belgium via research funds of the IWT in the framework of the research project BATCH. References Abel, O., Helbig, A., Marquardt, W., Zwick, W., Daszkowski, T.. (2000). "Productivity optimization of an industrial semi-batch polymerization reactor under safety constraints." Journal ofProcess Control^ 10, 351-362. Abel, O., Marquardt, W. (2003). "Scenario-integrated on-line optimisation of batch reactors." Journal ofProcess Control, 13, 703-715. Bonvin, D., Srinivasan,B.,Ruppen,D.(2001) "Dynamic Optimization in the Batch Chemical Industry." The sixth International Chemical Process Control meeting (CPC-6), Tucson, Arizona. Crowley, T. J., Choi, Kyu Yong. (1999). "Control of copolymer hydrodynamic volume distribution in a semibatch free radical copolymerization process." Computers and Chemical Engineering, 23, 1153-1165.

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and W. V.

Espinosa, J., Vandewalle, J., Wertz, V. (2004) Fuzzy Logic Identification and Predictive Control, Ed. Springer, London. Garcia, V., Cabassud, M., Le Lann, M.V., Pibouleau, L., Casamatta, G.. (1995). "Constrained optimization for fine chemical productions in batch reactors." The Chemical Engineering Journal, 59,229-241. Lee, K. S., Bang, S.H., Yi, S., Son, J.S., Yoon, S. C.. (1996). "Iterative learning control of heatup phase for a batch polymerization reactor." Journal of Process Control, 6(4), 255-262. Ma, D. L., Braatz, Richard D . . (2003). "Robust identification and control of batch processes." Computers and Chemical Engineering, 27, 1175-1184. Machado, R. A. F., Bolzan, A.. (1998). "Control of batch suspension polymerization reactor." Chemical Engineering Journal, 70, 1-8. Ruppen, D., Benthack, C , Bonvin, D. (1995). "Optimization of batch reactor operation under parametric uncertainty- computational aspects." Journal of Process Control, 5(4), 235-240. Sampath, V., Palanki, Srinivas, Cockbum, Juan C , Corriou, Jean-Pierre. (2002). "Robust controller design for temperature tracking problems in jacketed batch reactors." Journal of Process Control, 12, 27-38. Srinivasan, B., Bonvin, D.,Visser, E.,Palanki, S. (2003). "Dynamic Optimization of Batch Processes IL Handling Uncertainty Using Measurements." Computers and Chemical Engineering, 27(1), 27-44. Ubrich, O., Srinivasan, B.,Lerena, P.,Bonvin, D.,Stoessel, F. (2001). "The Use of On-line Calorimetry in the Optimisation of Isothermal Semi-Batch Reactors." Chemical Engineering Science, 56(11), 5141-5156.

Brempt

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1335

Dynamic Optimization of Molecular Weight Distribution Using Orthogonal Collocation on Finite Elements and Fixed Pivot Methods: An Experimental and Theoretical Investigation Apostolos Krallis, Dimitris Meimaroglou, Vassilis Saliakas, Christos Chatzidoukas, Costas Kiparissides Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University ofThessaloniki, P.O. Box 472, 54006 Thessaloniki, Greece Abstract In the present work a theoretical and experimental investigation on the optimization of the molecular weight distribution (MWD) for linear free-radical polymerization systems is presented. The MWD calculations were carried out by two different computational methods, the orthogonal collocation of finite elements (OCFE) method and the fixed pivot (FP) technique. The predictive capabilities of the model were tested against experimental data on MWD. The two numerical methods were compared in terms of the accuracy and the computational effort spent for the MWD calculations. An optimal MWD control problem was additionally addressed within this study using the OCFE method for the MWD calculations. The reactor temperature, as a key process condition, was selected as the main control variable. The optimal trajectories were applied to a free-radical MMA batch polymerization pilot plant. The optimal temperature profile was easily maintained by the regulatory cascade control system applied to the reactor unit. Keywords: Numerical Methods, Optimal Reactor Control, Suspension Polymerization 1. Introduction Commonly employed polymer product quality indicators (e.g., mechanical strength, tear strength, rheological properties and so forth) are directly or indirectly linked with molecular structural properties of the polymer chains (e.g., molecular weight distribution (MWD), long-chain branching (LCD), and so forth). Hence, the ability of a model to predict accurately the molecular properties of polymers in a polymerization reactor is of profound interest to the polymer industry. In the past a number of mathematical approaches have been proposed to describe the molecular weight developments in free-radical polymerization reactions (Tobita and Ito, (1993); Crowley and Choi, (1997); Nele et. al., (1999); Kiparissides et. al., (2002)). In general, the above numerical attempts suffer from two key kinetic limitations, (e.g., the use of the quasi-steady state approximation (QSSA) for 'live' radical chains and the overlook of gel and glass effects). In the present study the orthogonal collocation on finite elemets (OCFE) and fixed pivot (FP) methods were used to calculate the MWD for linear polymers. A detailed

1336

A. Krallis et al

polymerization model is used in conjunction with the FP and the OCFE methods to determine the MWD. The impact of the QSSA for the 'live' polymer chains on the accuracy of the calculated MWD was additionally investigated. The OCFE method was used to determine the optimal sequence of reactor temperature set points which produces a specific desired MWD. The optimal temperature set point sequence was determined using sequential quadratic programming (SQP). The optimal trajectories were applied to a free-radical MMA batch polymerization reactor. 2. Free-Radical Polymerization Model The kinetic scheme for the free-radical suspension polymerization of methyl methacrylate (MMA) is shown in eqs. (l)-(6). Initiation:

/_k^2Pi?'

,

Pi?*+M—^P/

(1)

Propagation: p ; + M — ^ p „ , ;

(2)

Chain Transfer to Monomer: p ; + M — ^ D „ + i ? ;

(3)

Termination by Combination: p*^p*—^£^^Z)

(4)

Termination by Disproportionation: p*+P*—!hd_^D n

m

+D n

(5) m

v /

To account for the effect of the diffusion-controlled phenomenon on the termination and propagation rate constants, the model of Chiu et. al. (1983), was employed. Given the above detailed kinetic mechanism, the following balances for the "live" and "dead" polymer chains of chain length "n" (ne [1,660000]) are obtained: di

;^

3. Fixed Pivot Technique The inherent limitations of the numerical solutions resulting from the discretization of the chain length, can be avoided in the more general formulations (i.e., fixed and moving pivot techniques) of Kumar and Ramkrishna (1996). The last methods

Dynamic Optimization of Molecular Weight Distribution

1337

guarantee the correct calculation of any two moments of the distribution and are applicable to any type of discretization of the chain length. The fixed pivot technique is a very efficient method for the calculation of the weight chain length distribution (WCLD). It assumes that the overall polymer chain population can be assigned to selected discrete lengths. Specific reaction steps (i.e., termination and propagation) leading to the formation of polymer chains other than the representative ones, are incorporated in the set of discrete equations in such a way that selected properties of the WCLD (i.e., total number, mass, etc.) corresponding to any two moments of the WCLD, are exactly preserved. For the preservation of the zero and first moments of the WCLD, the assigned polymer chain fi*actions to the discrete lengths ni and nj+l will be given by the following equation: n, =—^^^^^

n.
, n, =

^-^n.,-1 < n < n .

W

4. Orthogonal Collocation on Finite Elements A key characteristic of the OCFE method is the treatment of the discrete polymer chain length domain as a continuous one. Hence, the concentrations of "live" and "dead" polymer chains are handled as continuous variables. Accordingly, the chain length domain is divided into a number of finite elements, NE, with element boundaries at the points: ^o=l, Ci» C2»---> CNE-I, CNE = Sf, where Sf is the final degree of polymerization [Sf = 660000]. For each element a number of n interior collocation points, [si, s2,..., sn] are specified. The concentrations of the "live" and "dead" polymer chains are then approximated by continuous low-order polynomial functions within each finite element. In the present study, Lagrange interpolation polynomials were used to approximate the concentrations of the "live" and "dead" polymer chains: R{s) = tw:j{s)R{s,,)

C-.^^
7=1,-,A^^

(1^)

P(s) = M ( s ) p ( s , 3 )

Cj-,^s
j = l,...,NE

(11)

i=l

The tilde denotes approximation variables. The functions, W-j(s) and W-j(s) are Lagrange interpolation polynomials of order n+1 and n, respectively, and are equal to zero at the collocation points Si,j, for k^i and equal to unity for k=i. To take into account the concentration of "live" polymer chains at chain length x=l, the left boundary point of the first element was included as an interpolation point. A set of residual equations was then derived, by substituting eqs. (12)-(13) into the material balance equations (7) and (8). The main requirement of the OCFE formulation forces the residual balances to vanish at the selected collocation points, Sij. 5. Dynamic Optimization of the MWD For the dynamic optimization of the molecular weight distribution of the fi*ee-radical MMA polymerization, the reactor temperature is chosen as manipulated variable, because it is a key process condition that has the greatest direct impact on the MWD. The optimal reactor temperature sequence can be used as temperature set points in order

1338

A. Krallis et al

to control the MWD. This can be based on the assumption that the reactor temperature can be controlled effectively during the batch. The objective is to minimize the time required to produce a polymer with desired properties. Thus, the objective function to be minimized, can be defined in terms of the final MWD and polymerization time: min J{T) — ^ w.

( MWD,-MWD^^^ ^^Nd^l^f

i=\

subject to single bound constraints:

\
(14)

where, MWDai are the desired values of the MWD, Nd is the number of the desired values of the MWD, tf is the final polymerization time and Wj appropriate weights. 1 and u represent the lower and upper single bound constraints. T stands for the vector of the sequence of the reactor temperature during the polymerization. Each element of the vector T represents a constant manipulated control variable during a specific time interval. The number of the elements j of the vector T is equal to the ratio of the final polymerization time over the number of time intervals selected by the user. All the time intervals have equal length. The objective function is minimized using NPSOL, which is based on a sequential quadratic programming (SQP) method. 6. Results In order to observe the development of macromolecular chains with the progress of polymerization, and to realize whether the mathematical model captures such dynamic features, the model predicted MWD was confronted against experimental data taken from the open literature (Tefera et. al., 1996). From Figure 1, it is apparent that a very good agreement exists between the model predictions and the experimental results for the.whole range of the conversion values. The model successfully predicts the change of the shape of the MWD from a narrow unimodal profile in the beginning of the polymerization to a broader one at an intermediate conversion and to final bimodal one at a higher conversion. Figure 2 shows a comparison between the MWDs calculated via the application of the FP and OCFE (with and without the QSSA) methods at different monomer conversions. Notice that all the methods are capable of predicting broad and bimodal MWDs. It can be seen that the predicted MWDs by the FP and OCFE (without the QSSA) methods perfectly agree for all the values of the selected degrees of conversion. The impact of the QSSA on the MWD calculation is traced particularly at high degrees of conversion. A model-based study of the optimization problem for the minimum-time production of a polymer with desired MWD is presented next, employing the OCFE method (without the QSSA) as a representative and reliable method for the MWD calculations. A bimodal MWD was selected as desired profile to be produced by an optimal temperature trajectory. The validity of the model was further tested by direct implementation of the model produced optimal temperature trajectory to a cascade control system of pilot scale batch polymerization reactor. The reactor system comprised an agitated 1-gal jacket vessel, a heating-cooling unit, a monomer feed unit, a sampling unit, a vacuum unit, a reaction termination unit and a supervisory computer and process interface. Heating and cooling of the reaction mixture was achieved by controlling the flows of two water streams (e.g., a hot and a cold one trough the reactor jacket). The cascade control system consists of a primary proportional-integralderivative (PID) and two secondary PI controllers. Figure 3 depicts a comparison

Dynamic Optimization of Molecular Weight Distribution

1339

between model predictions and experimental results on the reactor temperature produced from the implementation of the optimal temperature trajectory as set points to the reactor's cascade control system. The optimal MWD in comparison with the desired MWD is shown in Figure 4. As can be seen from Figures 3 and 4, initially the optimizer forces a low temperature in the reactor to restrain the polymerization, facilitating simultaneously the production of long polymer chains. At the onset of the gel effect and onwards a kick of the reactor temperature is necessary leading to the production of short polymer chains that result in the formation of a second peak in the MWD. Finally, in Figures 5 and 6, model predictions on monomer conversion and hot and cold water flowrates are compared with experimental data produced from the optimal temperature profiles depicted in Figure 5. It is apparent that a very good agreement exists, between model predictions and experimental results, proving the reliability of the proposed model to produce desired MWDs. X=6% 0.5-

• /

y^

' / o

.^

A

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X=62%

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/

7 • / iV^.^>'TY

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Figure 2: Comparison of the FP and OCFE (with and without QSSA) methods for the prediction at different monomer conversions (Io,AiBN=3.74g/kg of MMA, T=62V

Figure 1: Comparison between model predictions (OCFE method) and experimental measurements, at different monomer conversions.

o

V*^

I

10

Chain Length

Desired IVIWD Optimal MWD

310 Model 300

0

20

40

60

80

-

Set Point

°

Experimental

100 120 140 160

Time, (min)

Figure 3: Comparison between model predictions and experimental results on the reactor temperature (Io=2 g/kg of MMA).

180

10'

10'

10*

Chain Length

Figure 4: Desired and optimal bimodal MWD using the OCFE method (lo=2 g/kg of MMA).

A. Krallis et al

1340

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[-2.0

1-1.6 1.2

jkr^\r\r

0.8 0.4 0.0

100

120

140

160 180

Time (min)

Figure 5: Comparison of model predicted and experimental monomer conversion produced from the optimal temperature profile (Io=2 g/kg ofMMA).

1Q0

120 140 160

Time, (min)

Figure 6: Comparison of model predicted and experimental measurements on hot and cold water flowrates produced from the optimal temperature profile (Io=2 g/kg of MMA).

References Chiu W.Y., G.M. Carratt, D.S. Soong (1983). A Computer Model for the Gel Effect in FreeRadical Polymerization, Macromolecules, 16, 348-357. Crowley, T.J. and K.Y. Choi (1997a). Calculation of Molecular Weight Distribution from Molecular Weight Moments in Free Radical Polymerization, Industrial Engineering Chemistry Research, 36, 1419-1423. Kiparissides, C , P. Seferlis, G. Mourikas, and A.J. Morris (2002). Online Optimizing Control of Molecular Weight Properties in Batch Free-Radical Polymerization Reactors, Industrial Engineering Chemistry Research, 41,6120-6131. Kumar, S. and D. Ramkrishna (1996). On the Solution of Population Balance Equations by Discretization - 1 . A Fixed Pivot Technique. Chemical Engineering Science, 51, 1311-1332. Nele, M. C. Sayer, and J.C. Pinto (1999). Computation of Molecular Weight Distributions by Polynomial Approximation with Complete Adaptation Procedures, Macromolecular Theory and Simulations, 8, 199-213. Tobita H. and K. Ito (1993). On the calculation of molecular weight distribution from the moments using Laguerre polynomials. Polymer Reaction Engineering, 1(3), 407-414. Tefera N., Weickert G. and Westerterp K.R. (1996). Modeling of Free Radical Polymerization up to High Conversion. II. Development of a Mathematical Model. Journal of Applied Polymer Science, 63(12), 1663-1681

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Multivariate Statistical Batch Process Monitoring Using Dynamic Independent Component Analysis Hamza Albazzaz and Xue Z. Wang Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, The University of Leeds, Leeds LS2 9JT, UK Abstract Use of independent component analysis (ICA) in developing statistical monitoring charts for batch processes has been reported previously. This paper extends the previous work by introducing time lag shifts to include process dynamics in the ICA model. Comparison of the dynamic ICA based method with other batch process monitoring approaches based on static ICA, static principal component analysis (PCA) and dynamic PCA is made for an industrial batch polymerization reactor and a simulated fed-batch penicillin fermentation process. For all the faults studied, the dynamic ICA method was found to be the only approach that detected all the faults, and it detected the faults earlier with less ambiguity than other approaches. Keywords: multivariate statistical process control, independent component analysis, batch process monitoring, dynamic ICA 1. Introduction Batch operation is the preferred operational mode for the production of many lowvolume and high-value added products such as pharmaceuticals, polymers, food and biological products. A major objective of batch process operation and control is to achieve consistent product quality from batch to batch run. Multivariate statistical process control (MSPC) based on Hotelling's 7^ and squared prediction error {SPE) charts, in particularly with the use of principal component analysis (PCA) for data dimension reduction, has been a key technique for this purpose (Nomikos and MacGregor, 1994, Wise and Gallagher, 1996). Early PCA based MSPC models were based on static PCA: they implicitly assumed that the observations at one time point for a variable is statistically independent to observations at past time points of these variables, as well as that of other variables. Later, various efforts have been made to include the dynamic behavior to the PCA model (Wold, 1994, Ku et al., 1995, Srinivasan et al., 2004, Junhui and Kun-Chih, 2002). More recently, an alternative approach to PCA has attracted attention which is based on independent component analysis (ICA) for data compression. In the early work, ICA was used for the purpose of dimension reduction of process variables (Li and Wang, 2002). Investigations on ICA for process monitoring were reported by Xia and Howell (2005), Lee at al. (2003a,b), Kano et al. (2003), Yao et al. (2003), Yoo et. al (2004), and Albazzaz and Wang (2004) who found improved monitoring performance than techniques based on some other data compression methods. The purpose of the paper is to extend our previous work (Albazzaz and Wang, 2004) on applying ICA to batch processes by introducing dynamics to the ICA model and comparing the new method with other batch statistical approaches for batch process monitoring based on PCA and ICA.

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2. The method for batch process monitoring based on dynamic ICA 2.1. ICA algorithm To introduce the ICA algorithm, it assumed that there are / linear mixtures Xj, x^, ..., x/ ofm independent (source) components S], $2,..., Sm, i.e., Xi =mnsi + mi2S2 + ... + rriimSm,

i=l, ••., I (l>m)

If denote n as the number of the samples, X = [x(l), x(2), . ..., x(n)] E: R

(1) is the data

matrix of observed variables, S = [s(l), s(2) ... s(n)] G xv is the independent component matrix and M matrix with elements M=[ai,a2, UrJ G R , namely mixing matrix, the above equations can be written as X-=MS (2) The goal of ICA consists of the estimation of the mixing matrix M and/or the independent source vector s from the observed mixed matrix X. This goal is equivalent to find a separating matrix W G R S=WX

that satisfies (3)

where S is the estimation of ^S. The independence of random variables can be reflected by non-gaussianity, which can be measured by kurtosis or negentropy. The process of ICA is an optimizing process of maximizing the non-gaussianity. A few of algorithms have been developed based on this principle. In this paper, the fast Fixed-points (FastICA) algorithm developed by Hyvarinen and Oja (2000) is used. The details of the algorithm can be found from the literature therefore will not be repeated here. 2.2. Statistical batch process monitoring based on dynamic ICA The ICA method for statistical batch process monitoring (Albazzaz and Wang, 2004) can be extended to include process dynamics by augmenting each observation vector with previous h observations and stacking the data matrix. The crucial step in the dynamic ICA method is to select the number of the time lags h, and the method used here is identical to the approach adopted for dynamic PCA by Ku et al (1995). The ICA algorithm is then applied to augmented data matrix to find the independent components from auto and cross-correlated variables. In batch operations, unlike in continuous processes, the process variables do not satisfy Gaussian distribution along the time axis. This is because in batch operations there are different stages including such as charge and discharge of feed ad intermediate products, valve closing and opening, and equipment topology variations. As a result, the trajectory of an operational variable does not follow Gaussian distribution along the time axis. However, if multiple normal batches producing the same products using the same feeds and operational strategies are considered, the trajectories of a variable for all the batches should follow a Gaussian distribution, centered a round a mean, which we can call it "the mean trajectory". To simplify the discussion here, we have assumed all the batches have the same batch length. Following this line of thinking, it is also fair to assume that for a specific time point, say t, the values at t on all the trajectories of a variable corresponding to all batches also satisfy Gaussian distribution. As a result, an upper control limit (UCL) and a lower control limit (LCL), as well as a mean // can be obtained for time point t. Similarly, UCL, LCL and ju can be calculated for all other time points. Plotting all UCLs, LCLs, and all ju together, time varying curves for UCL, LCL and ju can be obtained. The proposed SPC monitoring charts using dynamic ICA is based on the above idea. In this respect, the methodology is the same as the previous wok of Albazzaz and Wang (2004). The difference to the previous work is that the proposed

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dynamic ICA method takes into account the serial correlations between sampling points * by augmenting each observation vector with the previous h observations and stacking the data matrix as well as not only is concerned with the correlation among the batches, but also with the auto-correlation on the process variables for all batch run. The normal operation of original three-way data matrix (JxJ^K), where J is the number of variables, / the number of batch runs and K the sampling points, is unfolded to a twoway matrix X (JxJK). Then time lag h of all batches together is determined gradually: first static case (h=0) should be dealt with. The number of static relations equals the difference between the number of all variables X and the number of chosen independent components (ICs) based on number of principal components (PCs). In the second step when the time lag =1 the number of process variables are increased and the number of new static and dynamic relations is determined. This could be done by subtracting from the number of all variables the number of ICs based on number of principal components and the number of discovered relations from previous steps. The procedure is stopped when no new relations are found and appropriate time lag is selected. In our proposed method, the augmenting matrix with time lagged data of whole measurements of all batches has the structure in the following manner,

xi ^[(x'(k)yix'(k-i)y

ix'(k-h)Y]

(4)

where h is the lag time, k is sampling time, i the number of batches and/ the number of the variables. The augmented data are then scaled for each variable, i.e. for each row of the matrix (JxJK), using mean centering by subtracting the mean. Then noise from data can be removed if required. ICA is applied to the augmented matrix (JxJK) to compress it to Y (L xIK) where L is the number of ICs, L < J, and to obtain the separating matrix W. Then the new matrix Y (Lx IK) is split into L matrices, each corresponding to one IC and has the dimension oiil^K) where / is the number of batch runs and K the sampling points. Each column of the matrix (I^K), y-± for k = 1,2, .. .K, is assumed to satisfy Gaussian distribution and therefore UCL, LCL and mean ju can be calculated. The UCLs and LCLs T. ui i ^7 • ui j j r ^ j i ^u „ , ,. . ^ .„ Table 1 Variables recorded for case study 1 - the over all the sampling points will penicillin fermentation process No. Process Variables form the UCL and LCL curves. 1 Aeration rate (L h"^) For monitoring purpose of a 2 Agitator power (W) new batch operation, select the 3 Glucose feed temperature (K) time lag and construct the time 4 Dissolved oxygen concentration (% 5 saturation) lagged matrix of measurements 6 Culture volume (L) Xnew. Then apply the same scale 7 Carbon dioxide concentration (mmol L"^) as used in the model 8 pH development stage, and then the 9 Temperature (K) values for the independent 10 Generated heat (Kcal) components are calculated by Cooling water flow rate (L h'^) W^Xnew where W is the separating matrix obtained in during the ICA based SPC model development stage. Finally the instant values corresponding to each independent component is plotted in the SPC diagram. 3. Application 3.1. Case study 1 Case study 1 is a simulated process for fed-batch penicillin fermentation. The simulator, PenSim v2.0, was developed by the Monitoring and Control Group of the Illinois

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Institute of Technology and is downloadable from the website http://www.chee.iit.edu/~cinar. It models the concentrations of biomass, C02, hydrogen ion, penicillin, carbon source, and oxygen, as well as heat generation, during the production of penicillin under various operating conditions. Table 1 gives a summary of the 10 variables recorded. The simulator was used to generate 15 batch runs corresponding to normal operation and another four batch runs correspond to abnormal operation (Table 2). Each batch data has duration of 400 time points and contains random noise. The noise was suppressed using wavelet before frirther analysis. Two PCs and two ICs were chosen to develop the static PCA and static ICA control charts. The time lagged chosen for all batches is 1, four PCs and four ICs were chosen. Figure 1 compares the results of the four models i.e. static PCA, dynamic PCA, static ICA and dynamic ICA for detecting fault 4, a linear decrease of glucose feed rate from 0.04 L/h to 0.03 L/h at 60^*^ hour to the end of the batch operation. It was found that the times the fault was detected, and the degrees of unambiguousness for the four approaches are different. Firstly, static PCA and dynamic PCA seemed to have detected the fault at about 100^^ hour, but it was ambiguous because the deviation from the UCLs were not obvious, and then returned to normal. It was not until the time point of a bout 230^*^ hour, both static PCA and dynamic PCA showed clear abnormal operation. In contrast both static ICA and dynamic ICA showed clear abnormal operation from the time the fault occurred, and remained indicating abnormal operation until the end of the batch run. Between static ICA and dynamic ICA, the later detected the fault at a bout 80^^ hour, earlier than static ICA at about 110^^ hour. Table 3 summarizes the results for case study 1. Neither static PCA, nor dynamic PCA or static ICA detected fault 1. Only dynamic ICA detected fault 1 and it was detected with no obvious delay from the time when the fault was occurred at 50* hr. A very similar scenario was observed for faults 2 and 3, i.e. dynamic ICA detected the subtle changes clearly with no obvious time delays, while the other three approaches, static PCA, dynamic PCA, and static ICA failed to detect the event.

100

(a) static PCA

100

T j ^ 200

300

(b) dynamic PCA

-nme 200

(c) static ICA (d) dynamic ICA Fig 1 Monitoring charts for case study 1 based on four methods, and fault 4 (black line). All charts contain upper and lower control limits (UCL and LCL, gray lines) set at 99.87%.

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Table 2 Normal and abnormal operational data for case study 1 - penicillin fermentation Cases Explanation Normal 15 batches Fault 1 Agitator power is suddenly step-decreased by 15% at 50* hr and maintained at that lower rate to the end of batch operation (400 hrs) Fault 2 Agitator power is suddenly step-decreased by 15% at 100* hr and maintained at that lower rate to the end of batch operation (400 hrs) Fault 3 Aeration rate is suddenly step-decreased by 40% at 150* hr and maintained at that lower rate to the end of batch operation (400 hrs) Fault 4 Glucose feed flowrate is linearly decreasedfrom0.04 to 0.03 1 hr"^from60* hr to the end of batch operation (400 hrs) Table 3 Performance comparison between static PCA, dynamic PCA, static ICA, and dynamic ICA for detection of the four faults of case study 1 (no - not detected, @ detected at that time) Faults Fault 1 Fault 2 Fault 3 Fault 4

Static PCA no no no @230*hr

Dynamic PCA no no no @230* hr

Static ICA no no no @110* hr

Dynamic ICA @50*hr @ 100*hr @ 150*hr @80*hr

Table 4 A summary results of comparison of different approaches for detection of the seven faults- case study 2. V- detected the fault at the same time; VV- detected the fault more clearly. Faults Static PCA Dynamic PCA Static ICA Dynamic ICA Batch 49 V V V V Batch 50-55 V V V V and VV 3.2. Case study 2 Case study 2 is an industrial batch polymerization reactor of Dupont that has been studied before by Nomikos and MacGregor (1995). The data set used for the study contains 18 batches which have been identified in literature as normal operations, and another 7 batches (named as batches 49-55) which were identified as abnormal. Each batch had duration of 100 time intervals and 10 variables. Variables 1, 2 and 3 are temperature measurements inside the reactor, variables 6 and 7 are temperature measurements in heating-cooling medium, variables 4, 8, and 9 are pressure measurements, and variables 5 and 10 are flow rates of materials added to the reactor. Three PCs and three ICs were chosen to develop the static PCA (Wold et al., 1998) and static ICA control charts (Albazzaz and Wang, 2004). To develop the control charts based on dynamic PCA and dynamic ICA, the time lagged for all batches is 3, six PCs and six ICs were chosen. For batches 50-55, it was found that all approaches predicted the faults almost at the same time, but dynamic ICA showed clearer abnormality than other approaches, as demonstrated by fault 50 in Figure 2. Although all four approaches detected the fault almost at the same time, but for static PCA, dynamic PCA and static ICA, the abnormality is more obvious in the time ranges, 0-20* hr and 80'^-100'^ hr, than in other time ranges, while dynamic ICA show abnormality in almost the whole batch period, and the abnormality is shown to be more obvious than other approaches. Similar observations can be found for 51-55 batches. No obvious differences were found between the approaches for batch 49. Table 4 gives a summary of the results. 4. Conclusions Introduction of time lag shifts to include process dynamics in the ICA model for batch process monitoring has resulted in improved monitoring performance for the two case

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studies investigated. For case study 2 about an industrial polymerization reactor, all four approaches showed that they could detect the faults in batches 50-55, but dynamic ICA approach gave clearer results than static PCA, dynamic PCA or static ICA. For case study 1 of a simulated fed-batch fermentation penicillin process, static PCA, dynamic PCA, and static ICA only detected the 4^^ fault while the dynamic ICA detected all four

faults.

Time

(b) dynamic PCA

(a) static PCA

100

(c) static ICA (d) dynamic ICA Fig 2 Monitoring charts for case study 2 for four methods for PCI and ICl, and fault 50 (black line). All charts contain upper and lower control limits (UCL and LCL, gray lines) set at 99.87%.

References H. Albazzaz, X.Z. Wang, 2004, Ind. Eng. Chem. Res., 43, 6731-6741 A. Hyvarinen, E. Oja, 2000, Neural Networks, 13, 411-430 C. Junghui, L. Kun-Chih, 2002, Chem. Eng. Sci., 57, 63-75. M. Kano, S. Tanaka, S. Hasebe, I. Hashimoto, H. Ohno, 2003, AIChE J., 49, 969-976. W. Ku, R. Storer, C. Georgakis, 1995, Chemom. Intell. Lab. Sys., 30, 179-196. J.M. Lee, C. Yoo, LB. Lee, 2003a, J. Chem. Eng. Japan, 36, 1384-1396. J.M. Lee, C. Yoo, LB. Lee, 2003b, J. Chem. Eng. Japan, 36, 563-577. M.J. Lee, C. Yoo, and I.B.Lee, 2004, Chem. Eng. Sci., 59, 2995-3006. R.F. Li, X.Z. Wang, 2002, Comput. Chem. Eng., 26, 467-473. P. Nomikos, J.F. MacGregor, 1994, AIChE J. 40, 1361-1375. R. Srinivasan, C. Wang, W.K. Ho, K.W. Lim, Ind. Eng. Chem. Res., 43, 2123-2139. B.M. Wise, N.B. Gallagher, 1996, J. Proc. Contr., 6, 329-348. S. Wold, N. Kettaneh, H. Friden, and A. Holmberg, 1998, Chemom. Intell. Lab. Sys., 44,331-340 S. Wold, 1994, Chemom. Intell. Lab. Sys., 23, 149-161. C. Xia, J. Howell, 2005, Contr. Eng. Pract., 13, 1027-1035. Z.X. Yao, Y. Qian, X.X. Li, Y.B. Jiang, in: B. Chen, A.W. Westerberg (Eds.), Computer-Aided Chem. Eng. 15 - Proc. 8th Int. SympPSE. (Elsevier, Kunming, China, 2003) 1112-1117. C.K. Yoo, J.M. Lee, P.A. Vanrolleghem, LB. Lee, 2004, Chemom. Intell. Lab. Syst, 71,151-163.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Model-based optimization for operational policies in seeded cooling crystallization Ali Abbas^, S. Mostafa Nowee^, Jose A Romagnoli'^

'^ Nanyang Technological University, Singapore ^University of Sydney, Australia ^Louisiana State University, USA In this solution crystallization study, a population balance model that predicts the crystal size distribution (CSD) is used for targeting, by optimization, the product mean size and the coefficient of variation of the final CSD. The model is robust over a wide range of conditions and predicts the effects of heating during a batch since the kinetics of dissolution have been identified and incorporated into the model. The dynamic temperature profile and the initial seed size distribution are optimized while the initial seed induction time is conferred through knowledge of saturation conditions. This is quite a highly developed approach in contrast to existing anecdotal and/or ruleof-thumb experience currently dictating industrial operational methods. Results from optimizations under different objective functions are presented and validated experimentally. Keywords: Crystallization; Optimization; Seeding 1. Introduction Crystallization has been referred to as an art rather than a science. The main objective of crystallization systems engineering is to develop advanced understanding of crystallization phenomena enabling the efficient bulk processing of high quality crystalline products. Crystallization, in many industries, is the most common way of production of high value chemicals with high purity and desired size and shape. Crystal size is typically the most important factor to be controlled because of its close relation to the characteristics of the crystalline product, such as flowability and bulk density, and also different applications require different particle size. Crude methods are still being used in industrial operations to operate/control their crystallization processes and those industries that implement any control rarely do this at optimal conditions. Nevertheless, there have been a number of successfiil research investigations as evidenced in the literature on the control of the crystallization process with most of these studies aiming to control the product crystal size or size distribution. Crystallization processes are complex in nature and are known to be highly non-linear and thus pose difficulties in their modeling and control. They also carry liquid as well as solid (particulate) phases adding to the process's complexity. Their mathematical description is based on complex multi-dimensional population balance equations, coupled with integral DAEs. In addition, the kinetic history of the process largely influences the end product quality. Any unanticipated disturbances during process operation could lead to irreversible changes to product particle characteristics. It is thus evident and based on research observations that the use of the model is crucial for optimizing the crystallization operations allowing for end-product crystal size reproducibility. Researchers have recently recognised the significant role a crystallization model can play in optimizing operations. Manipulated variables that would be candidates for optimization decision variables are traditionally the working temperature and seeding. Considerable effort has been made on particle size control in batch cooling crystallization (Mullin, 2001). Most of these works consider supersaturation as the key variable to be optimized. This strategy which is based on the determination of the optimal temperature profile was first discussed by Mullin and Nyvlt (1971). It was though based on a key understanding namely that constant nucleation rate is required in order to keep the supersaturation low and within the metastable zone. Other works dealing with an

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optimal cooling profile generation intended to keep supersaturation low in the beginning stages of the process inhibiting nucleation but promoting crystal growth. Although seeding was believed to be an important factor for controlling crystal size (and allowing for the production of mono-dispersed CSDs), the amount of seed to be added was largely based on empirical knowledge. Only in recent years some empirical research appeared considering seeding as a factor to be optimized (Doki et al., 1999, Doki et a l , 2001, Doki et al., 2002, Jagadesh et al., 1996, Jagadesh et al., 1999, Kubota et al., 2001). In these investigations the seed chart was used to determine optimal seeding policies. These works demonstrated the impact seeding has on the crystallization showing even under un-controlled profile of cooling (eg. natural cooling), the rate of nucleation could be kept low. It could also be shown via simple mass balance calculations and the assumption that only seeds consume the generated supersaturation during the batch, the predicted final crystal mean size correlates well with the experimental ones. Other finds included the ability to produce an end-product with narrow monomodal CSD and further due to secondary nucleation, the final product could have a bimodal distribution. A model-based dynamic optimization of seeded cooling crystallization was discussed by Chung et al. (1999). Seeding parameters in addition to supersaturation were optimized and the objective function used was maximisation of final mean size with minimum coefficient of variance. They concluded that the effect of seeding variables on the final product characteristics are much more pronounced than that of supersaturation. Al-Zoubi and Malamataris (2003) and De Faveri et al. (2004) incorporated factorial experimental design using a surface response model in determining initial seed concentration and seed introduction time as well as the temperature profile. Their target was a specific crystal yield, size, shape and purity. Another investigation that dealt with model-based optimization of seeded cooling crystallizations was done by Choong and Smith (2004), who determined the optimal seeding characteristics, mean size and seed mass, by maximisation of product mean size and minimisation of CSD coefficient of variance as objective functions, respectively. Recent research trends, as evident in the literature, show an increase in the use of the dynamic crystaUization model for achieving various optimal objectives. The simultaneous use of the two variables, temperature and seeding, is evident in more recent works. The most comprehensive study appears to be that of Patience et al. (2004) who solve the population balance equation (PBE) for a growth-dependent dispersive crystal system and use temperature profile as well as two seeding variables as decision variables. They also validate their optimization resuhs experimentally. In this study, we extend on a previously developed dynamic crystallization model (Abbas, 2003) to include a mathematical description of (1) both super- and under-saturation crystallization regimes and (2) seeding. In (1) dissolution and disappearance kinetic rates are introduced to the model. We report the results of a model-based parameter estimation study undertaken to determine the kinetics parameters and thus identify the crystallization model. Finally an offline model-based optimization analysis was conducted to determine the optimal cooling profile in conjunction with the optimal initial seed size distribution. The latter being a unique determination that is novel to this work and extends on model-based optimization work by previous workers. We further present results of experimental validation of such dual decision variable offline optimization. 2. The crystallization model Since the crystallization process is a particulate one, the population balance equation (PBE) as proposed by Hulburt and Katz (1964) is used accounting for the evolution of crystal particles across temporal and size domains. For a batch crystallization system such as the ammonium sulfate one we are dealing with in this study, crystal growth is assumed to be non-dispersed and independent of crystal size and agglomeration and attrition are considered negligible, the population balance simplifies to

dt

dL

Where n(L,t) is the number density of crystals, t is time, L is the characteristic crystal size, G is the growth rate of the crystals while B is the nucleation rate. In the case where crystal dissolution by heating exists such as when the process is operating under temperature control giving rise to

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Optimization for Operational Policies

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temperature increases, the PBE will change and terms denoting dissolution (Z)) and disappearance (Da) of particles will need to be incorporated (Equation 2). The structure of PBE remains the same as in Equation 1, but the dissolution and disappearance terms are introduced to replace the growth and nucleation ones respectively and with opposite signs.

M^=^M^_z)

(2)

dt dL "" The switch from supersaturation state to undersaturation state results in elimination of growth and nucleation processes, and in the occurrence of dissolution and disappearance processes. The reverse occurs when the process switches from under- to supersaturation (eg. switching from heating to cooling). In other words, where there are high supersaturation levels, one would expect to observe nucleation and growth of crystals and Equation 1 would hold true. Where the process is going in the direction of under-saturation, i.e. heating, one would observe dissolution and disappearance of crystals rendering Equation 2 to be true. Clearly, this is only true for those solute systems that have a solubility that increases with temperature. In crystallization processes some phenomena that may occur are nucleation, growth, agglomeration and breakage. These phenomena govern the process and ultimately dictate the product characteristics and particularly the shape of the product PSD. In this study, two additional kinetic rates are considered namely dissolution and disappearance. As the objective of this research is to identify optimal operational policies, the kinetic model structure is kept simple and confined to a power law structure.

R = k^C^

(3)

Where R is the vector of kinetic rates [B G D Da\ representing the rates of nucleation, growth, dissolution and disappearance, k is the vector of kinetic rate parameters [kb kg kd ka], AC is the supersaturation which is equal to the difference between the concentration of the solute C and the saturated concentration C . Finally 0 represent the vector of rate orders [b g d a] corresponding to each of the kinetic rates respectively. To complete the model, the mass and energy balances, after Rawlings et al. (1993), for the solute and crystallizer temperature, are incorporated. The model was solved by finite difference discretization method. For the supersaturated and undersaturated states, the backward and forward finite techniques were used respectively, provided stability. The initial condition for the system is the size distribution of seeds. The solution was implemented and simulated using the process modeling package gPROMS (Process Systems Enterprise Ltd, UK) which executed the solution within a timeframe of a few seconds using a computer running a Pentium 4 processor.

3. Model identification and optimization Using optimization based (maximum likelihood criteria) parameter estimation, it was possible to identify the parameter sets k and 0, that provide the highest probability of the model predicting the real data. The gEST function of gPROMS was used for parameter estimation here. The problem was split in two sections, cooling and heating. The parameters in nucleation and growth kinetics were estimated using the experimental data from the cooling phase while the dissolution and disappearance kinetics were estimated from the heating phase. The estimated parameters are log(A:) = [3.25 -6.55 -10.04 -6.83] and 0= [10.8 6.0 2.0 1.0]. Experiments for the purpose of parameter estimation were conducted in a glass jacketed crystallizer unit with a volume of 1 litre. Cooling/heating was implemented by means of a circulator (Julabo, USA) connected to the Honeywell industrial distributed control systems (DCS). The temperature profile was preprogrammed and realized using a standard PID controller using set-point tracking. The feed to the crystallizer was prepared from nano-pure water and Laboratory reagent grade ammonium sulfate.

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The ammonium sulfate solution was prepared so as to have a concentration corresponding to the saturation temperature of 50°C and a volume of 0.85 litres. We use a conductivity (Orion conductivity probe model 018020A, USA) and temperature to track the solution concentration while CSD was determined offline using laser diffraction (Malvern Mastersizer-S, UK). The experiments designed for parameter estimation consisted of two phases of linear temperature change; one cooling and one heating. During the cooling phase supersaturation develops affecting the crystallization. This is then followed by a heating phase affecting the dissolution of the crystals formed during the cooling phase. Solution concentration and CSD data under these cooling/heating regimes are recorded providing information on the kinetic rates and allows the quantification of the parameter sets. For experiments leading to offline optimization validation, the temperature profiles and seed determinations from optimization results were implemented. The subsequent step of optimization is discussed next. The optimization of the model was performed using the gOPT function in gPROMS. Two objective functions were posed; one is to maximize the mean crystal size, L , in the shortest time while keeping the distribution variance {Var.) at a minimal value (Figure la) and the other is to minimize the distribution variance while constraining L to a specific end-point value again in minimum time (Figure lb). The control variables are the temperature of the crystallization and seed loading profile. The formulation of these objective optimization problems is summarized in Figures la and lb. Ten control intervals were implemented in this dynamic optimization. Temperature being the time-variant control variable was specified as piecewise-linear while the seed distribution was specified as a piecewise-constant profile for each class of crystal size. Finding the optimal seed distribution was possible by applying anecdotal knowledge, bounding the seed distribution to a size range known to provide adequate seeding conditions. For instance the seed size must be above a critical size determined from experimental investigations otherwise the seeds will disappear before realising their effect. A number of constraints were implemented and consisted of the path rate of cooling constraint and end-point constraints reflecting the end-product characteristics.

4. Results and Discussion The solution of this non-linear optimization problem represented the optimal combination of //, T(t) and n(L,t) initial that gives the maximum or minimum value of the objective function while satisfying the given constraints. The significance of the dynamic optimization problem is that a path to optimality is drawn. In this case the optimal path is described by a temperature trajectory and a corresponding mean size trajectory (Figures 2a & 2b). The optimal temperature profiles are very similar and comprise of three distinct regions. In region one, we observe an initial decrease in temperature that can be explained by the drive to cross over the saturation temperature to generate supersaturation. Region 2 sees a phase of slow cooling in which the supersaturation is being stabilized and growth of crystals is occurring under controlled nucleation towards the maximum size. In the third and final region, a sudden decrease in temperature is observed and is the result of the optimization attempting to minimize tf and so forcing the temperature down sharply. The corresponding mean size profiles simulated under the optimal temperature profiles are shown in Figure 2b. Again similarities between the two exist thought the results of the maximization of the mean size are slightly better. No size reduction of the seeds is because the induction time of the seeds is fixed at a point very close to saturation conditions rendering the rate of dissolution to be slow so that the reduction in seed size is not appreciable. The crystals continue to grow and stabilise at about the same size for both optimizations leading to suggest that the two optimization problems provide similar end-size solutions and thus are similar in the way they are posed. As for the time of the batch, no significant difference is observed between the two optimizations. The optimal seed distributions for the two optimization problems are shown in Figure 3 a. The two optimizations observed very similar seeding optima. The end product size distribution was found to vary for the two optimization problems which are shown in Figure 3b. The height and shape of the distributions are distinctly different suggesting the possibility of shaping the CSD to a specific shape desired. These outputs from the optimizations where implemented to the bench scale crystallizer and the results show a good agreement. The temperature profiles consist of four linear profiles reasonably close to the profile given by optimizations (Figures 4a & 4b). Mean size measurements were close to what it was expected

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from simulation studies, although there is a very minor fluctuation from the mean size growing profiles presented from simulations (Figures 5a & 5b), which could be mainly due to the oscillatory behavior in the controlled measured temperature inside the crystallizer . However, as could be seen in Figure 6 the final size distribution has a very narrow shape compared with a process without seeding and with linear cooling profile, which proves the claim of controlling the size distribution by the addition of the seed. Objective function 1:

mcLX,L\tf) t,T.(^\o.t,\

Objective function 2:

min

Subject to:

Subject to:

Crystalliser model: Equations 1-3

CrystalHser model: Equations 1-3

Initial conditions: n(L,t) = 0

Initial conditions: n(L,t) = 0

C(t) = Co Batch time:

t=0

C(t) = Co

t=0

t"^'" < tj. < t'J'"

Control variables: T'"'" < T{t) < T"'"'

Batch time:

J""'"
End-point constraints:

T """ < T\ty j
cr

Figure la. Optimization formulation for maximum mean size.

t=0

t=0

t'J'" ^tf ^ tf"''

Control variables: T'"'" < T{t) < T'"'^

0
Var.ytj^j

^^^

Path constraint:

0
End-point constraints:

T'"'" < T\tj ^
cr
Figure 2(a) Optimal temperature profiles for the two optimization objectives (left) and 2(b) corresponding optimal mean size profiles (right).

Figure 3(a) Optimal seed distributions for the two optimization objectives (left) and 3(b) corresponding optimal end-product size distributions (right).

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Figure 4(a) Optimal temperature profile for objectiveftinction:maximum mean size (left) and 4(b) corresponding optimal mean size profile versus experimental (right).

Figure 5(a) Optimal temperature profile for objective function: minimum variance (left) and 5(b) corresponding optimal mean size profile versus experimental (right).

Size [micron]

Figure 6. Comparison between fmal product distribution of optimal and linear cooling profiles. References: A. Abbas, (2003), A model-based framework for advanced operation of crystallization process - A pilot plant study, PhD dissertation, The Department of Chemical Engineering, University of Sydney. N. Al-Zoubi and S. Malamataris, (2003) International Journal of Pharmaceutics, 260, 123-135. K.L. Choong and R. Smith, (2004) Chemical Engineering Science, 59, 313-327. S.H. Chung, D.L. Ma and R.D. Braatz, (1999) Canadian Journal of Chemical Engineering, 77, 590-596. D. De Faveri, P. Torre, P. Perego and A. Converti, (2004) Journal of Food Engineering, 61,407-412. N. Doki, N. Kubota, A. Sato and M. Yokota, (2001) Chemical Engineering Journal (Lausanne), 81, 313-316. N. Doki, N. Kubota, A. Sato, M. Yokota, O. Hamada and F. Masumi, (1999) AIChE Journal, 45, 2527-2533. N. Doki, N. Kubota, M. Yokota and A. Chianese, (2002) Journal of Chemical Engineering of Japan, 35, 670-676. H.M. Hulburt and S. Katz, (1964) Chemical Engineering Science, 19, 555-74. D. Jagadesh, N. Kubota, M. Yokota, N. Doki and A. Sato, (1999) Journal of Chemical Engineering of Japan, 32, 514-520. D. Jagadesh, N. Kubota, M. Yokota, A. Sato and N.S. Tavare, (1996) Journal of Chemical Engineering of Japan, 29, 865873. N. Kubota, N. Doki, M. Yokota and A. Sato, (2001) Powder Technology, 121, 31-38. J.W. Mullin, (2001) Crystallization, Butterworth-Heinemann, Oxford. J.W. Mullin and J. Nyvlt, (1971) Chemical Engineering Science, 26, 369-377. D.B. Patience, P.C. Dell'Orco and J.B. Rawlings, (2004) Organic Process Research & Development, 8, 609-615. J.B. Rawlings, S.M. Miller and W.R. Witkowski, (1993) Industrial & Engineering Chemistry Research, 32, 1275-96.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Controllability Analysis of Thermally Coupled Distillation Sequences for Five - Component Mixtures Marcelino Carrera - Rodriguez, Minerva Ledezma - Martinez, Juan Gabriel Segovia-Hernandez, Salvador Hernandez Universidad de Guanajuato, Facultadde Quimica, Noria Alta s/n, Guanajuato, Gto., 36050, Mexico, [email protected]

Abstract The control properties of thermally coupled distillation sequences for the separation of five - component mixtures of hydrocarbons were compared to those of conventional distillation sequences. Seven thermally coupled arrangements were investigated. The preliminary steady - state designs of complex schemes were obtained by starting from conventional distillation sequences and then optimizing for minimum energy consumption. The control properties of the sequences considered were obtained by using the singular value decomposition technique at zero frequency and analyzing rigorous dynamic simulations. It was found that, in general, the coupled schemes present theoretical control properties similar or better than those of conventional distillation sequences. This result establishes that the energy savings predicted for thermally coupled distillation sequences can be achieved in practice.

1. Introduction Effective screening of separation systems constitutes a critical stage, as engineers need to review and understand trade-offs ahead of detailed modeling and simulation. In the separation, it is often desired to explore the use of complex rather than simple columns, because the complex units reduce mixing losses, use available vapor and liquid more effectively, and improve the

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separation efficiency (Hernandez et al, 2003). Among all possible complex schemes for multicomponent distillation process, the thermally coupled distillation sequences are very promising for both energy and capital cost savings. However, due to the complexity, research on these complex distillation configurations is only restricted to three - component mixtures, for only a few promising flowsheets that can be constructed for ternary mixtures (Tedder and Rudd, 1978; Hernandez and Jimenez, 1996). In these studies, it has been reported that the thermally coupled configurations are capable of achieve energy savings of up to 30% in contrast to the conventional direct and indirect distillation sequences. There are few works on extensions toward the design of integrated systems for mixtures of more than three components (Christiansen et al, 1997; Blancarte-Palacios et al, 2003). Recently, Rong et al (2000; 2003) have parametrically studied some thermally coupled distillation sequences for five - component mixtures from viewpoint of economic evaluation and optimal synthesis of the multicomponent thermally coupled distillation flowsheets. The expectance that the dynamic properties of those coupled systems may cause more operational problems than the conventional sequences is one of the factors that have contributed to their lack of industrial implementation. In this work we developed a comparative study of the control properties between seven thermally coupled distillation sequences (previously studied by Calzon McConville et al, 2006) and seven conventional distillation sequences (Figures 1-7) for the separations of five - component mixtures. 2. Design and optimization strategies Strictly, the design of the thermally coupled distillation sequences could be modeled through superstructures suitable for optimization procedures with mathematical programming techniques. However, the task is complicated and is likely to fail to achieve convergence. In this work, in order to overcome the complexity of the simultaneous solution of the tray arrangement and energy consumption within a formal optimization algorithm, we have decoupled the design problem in two stages: (1) tray configuration; (2) energy-efficient design (optimal energy consumption). In the first stage, our approach begins with the development of preliminary designs for the complex systems based on the designs of the conventional distillation sequences (as proposed by Hernandez and Jimenez, 1996). For the conventional sequences, it is assumed that each column performs its respective split (i.e., the separation of the light and heavy key components) with molar recoveries of 98% respectively. Then, by using the shortcut method of Fenske-Underwood-Gilliland, the tray structures of conventional distillation schemes are obtained. After the tray arrangements for the integrated designs have been obtained, an optimization procedure is used to minimize the heat duty supplied to the reboilers of each coupled scheme; taking into account the constraints imposed by the required purities of the five

Controllability Analysis of Thermally Coupled Distillation Sequences

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products streams. The resulting sections serve as a basis for the tray arrangements of the complex scheme.

a)TCDS -1

g)TCDS -7

Figure 1. Thermally coupled distillation sequences (TCDS) for separation of five component mixtures.

The remaining degrees of freedom (the number of thermal links; see Figure 1) after design specifications and tray arrangement are used to obtain the proper values of the interconnecting vapor streams which provide minimum energy consumption. In this work, the process simulator Aspen Plus 11.1^^ is used for this purpose. If the product compositions are obtained, then the design is kept; otherwise, proper adjustments must be made. The values of the interconnecting flows are optimized for minimum energy consumption in the reboilers (more details about the procedure are in Calzon - McConville et al., 2006).

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3. Theoretical Control Properties Open loop dynamic responses around the assumed operating point (which corresponds to that with minimum energy consumption for each configuration) were obtained. The responses were obtained through the use of Aspen Dynamics 11.1^^. Transfer function matrices (G) were then collected for each case, and they were subjected to singular value decomposition (SVD): G = VSW"

(1)

where Z = diag (Oi, ,an), <5\ = singular value of G = Af^ (GG^); V = (vi, V2, ) matrix of left singular vectors, and W = (wi, W2,....) matrix of right singular vectors. Two parameters of interest are the minimum singular value, a*, and the ratio maximum to minimum singular values, or condition number (y*). The minimum singular value is a measure of the invertibility of the system and represents a measure of potential problems of the system under feedback control. The condition number reflects the sensitivity of the system under uncertainties in process parameters and modelling errors. These parameters provide a qualitative assessment of the theoretical control properties of the designs. The systems with higher minimum singular values and lower condition numbers are expected to show the best dynamic performance under feedback control. For this initial analysis, we simply estimated the SVD properties for each separation system at zero frequency. Such analysis should give some preliminary indication on the control properties. 4. Case of Study The components of the mixtures considered are n-butane, n-pentane, n-hexane, n-heptane and n-hexane for mixture 1, and n-butane, isopentane, n-pentane, nhexane and n-heptane for mixture 2. Two molar compositions Fl (A=0.35, B=0.1, C=0.1, D=0.1, E=0.35) and F2 (A=0.125, B=0.25, C=0.25, D=0.25, E=0.125) for each mixture are analyzed. Molar recoveries of 98, 94, 94, 94 and 97 are assumed for each component in the products. A feed flowrate of 45.5 kmol/h as saturated liquid is used. The operational pressure for each column is chosen in order to guarantee the use of cooling water in the condensers. As far as energy consumption is concerned, the optimized steady - state design provides energy savings of -35% with respect to the best energy - efficient sequence based on conventional distillation columns. 4. Results Table 1 presents the results for the SVD test for each sequence. The TCDS - 1 option has similar values of minimum singular value and condition number to

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those of the conventional sequence (CS - 1), which implies that the coupled arrangement has similar control properties (lower control efforts under feedback operation and better conditioned to the effect of disturbances) that the conventional design. The TCDS - 2, TCDS - 3, TCDS - 4, TCDS - 6 and TCDS - 7 systems present higher values of the minimum singular value; therefore, it can be expected that these coupled systems exhibit better control properties than the conventional sequences under feedback control. The results for the condition number show that the complex sequences offer the best values. As a result, it can be expected that thermally coupled distillation systems are better conditioned to the effect of disturbances than the conventional arrangements. Table 1 Dynamic properties of the distillation systems. Sequence

O*

Y

CS-1 TCDS-1

0.0219 0.0218

562.85 564.29

lAE (Component A) 1.4879x10"^ 1.4751x10"^

CS-2 TCDS- 2

0.0012 0.2026

14106 33.37

4.0077 X 10"^ 5.4113x10"^

CS-3 TCDS- 3

0.0005 0.0418

33318 576.8

6.8872x10"^ 3.0709x10"'

CS-4 TCDS - 4

0.0064 0.0071

59934 56650

2.0089x10"^ 1.9999x10'^

CS-5 TCDS- 5

0.0350 0.0001

562.85 14000

9.9784x10-' 5.3308 X lO-"^

CS-6 TCDS- 6

0.00220 0.00500

13588 2630.14

7.6608 X lO"* 6.9002 X lO"*

CS-7 TCDS- 7

0.00003 0.00009

19004 1973.8

8.5508 X lO"* 8.0078 X 10"^

Two details are worth highlighting, the TCDS - 1 and TCDS - 5 are thermodynamic equivalents schemes. The results (Table 1) show that TCDS - 5 have the worst control properties in comparison with the TCDS 1. This result is important because two thermodynamically equivalent schemes have different dynamic properties: the structure with side columns have better control properties than a thermodynamic scheme with retrofit. Some authors have claimed that the reduction of interconnection flows in complex arrangements might provide better operating properties. When we do a general comparison of control properties of the seven coupled scheme, the partially thermally coupled (TCDS 6 - 7 ) arrangements show worst values (minimum singular value and condition number) than the fully thermally coupled schemes (TCDS 1 - 4). This observation is interesting because the number of the thermal links affect the control properties of coupled systems. Similar results were obtained using

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dynamic simulations (Table 1). The lAE criterion shows the best dynamic behavior for the thermally coupled distillation sequences using PI controllers (Table 1). For the dynamic analysis, individual set point changes for product composition were implemented for each of the three product streams. For all cases (conventional and integrated sequences), the five control loops were assumed to operate under closed loop fashion 5. Conclusions It has been shown that the controllability properties of seven thermally coupled distillation sequences for the separation of five component mixtures are better than those of the conventional sequences. The results show that the structure has different effect to dynamic performance for a specific separation in the case of thermodynamically equivalent schemes. These results are significant because they let us establish that coupled schemes not only require energy demands lower than the conventional distillation sequences but also present control properties similar or better to those of the conventional distillation sequences used in the preliminary design of the thermally coupled distillation sequences. It is apparent that the presence of recycle streams instead of deteriorating the dynamic behavior of separation sequences, may contribute positively to their dynamic properties. 6. References Blancarte - Palacios, J.L., Bautista - Valdes, M.N., Hernandez, S., Rico Ramirez, V., and A. Jimenez, 2003, Ind. Eng. Chem. Res., 42, 5157. Calzon - McConville, C.J., Rosales - Zamora, M.B., Segovia - Hernandez, J.G., Hernandez, S., and V. Rico - Ramirez, 2006, Ind. Eng. Chem. Res., 45, 724. Hernandez, S., and A. Jimenez, 1996, Trans IChemE, 74, 357. Hernandez, S, Pereira-Pech, S, Jimenez, A, and V. Rico-Ramirez, The Can. J. Chem. Eng. 2003; 81 (5) lOST, Rong B. G., Kraslawski, A., and L. Nystrom, 2000, Comput. Chem. Eng., 24, 247 Rong B. G., and A. Kraslawski, 2003, Ind. Eng. Chem. Res., 42 (6), 1204. Tedder, D.W., and D.F. Rudd, 1978, AIChEJ, 24, 303. Acknowledgements The authors acknowledge financial support received from PROMEP and Universidad de Guanajuato, Mexico

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Multiscale SPC in the Presence of Multiresolution Data Marco S. Reis,^ Pedro M. Saraiva^ " GEPSI-PSE Group, Department of Chemical Engineering, University ofCoimbra, Polo Il-Pinhal de Marrocos, 3030-290 COIMBRA, PORTUGAL Abstract Multiscale statistical process control (MSSPC) has the ability of handling the multivariate and multiscale nature of data arising from industrial processes. However, it assumes that such data is available at a single resolution. In this paper, we present an approach for conducting MSSPC that adequately integrates data available at different resolutions (multiresolution data) and demonstrate its merits though simulated examples. The proposed approach, called MR-MSSPC (standing for multiresolution and multiscale statistical process control), proved to be able to provide a clearer definition of the regions where faults occur, when compared to alternatives based on single resolution data, without compromising detection speed. Keywords: Multiscale statistical process control; Multiresolution data; Wavelets. 1. Introduction Processes going on in modem chemical plants are typically complex, and this complexity is also reflected in collected data, which contain the cumulative effects of many underlying phenomena and disturbances, with different location and localization patterns in the time/frequency plane. For such reasons, multiscale approaches designed to handle and take advantage of information contained at different scales have been developed for addressing different tasks [1]. In this context, Bakshi [2] proposed a multiscale approach for conducting multivariate statistical process control (MSSPC), that exploits the synergy between principal component analysis' (PCA [3]) ability to extract the correlation structure of variables, and the wavelet transform's [4-6] ability to approximately decorrelate autocorrelated processes and extract deterministic features embedded in noisy signals. The combination of both approaches consists of monitoring, independently, the PCA scores for the wavelet coefficients at each scale, that result from the wavelet transformation of process data. This step allows for the effective extraction of abnormal multiscale patterns, and the final decision regarding the state of operation is always made at a second stage, using data arising from the reconstruction, back into the original timedomain (inverse wavelet transformation), of the wavelet coefficients pertaining to scales where significant activity was detected. The good detection properties of MSSPC were already addressed in the literature [2,7]. On the other hand, industrial data usually have associated a number of features that rise serious difficulties in their manipulation and in the extraction of potentially useful information. Some examples are the presence of missing data patterns and the existence of data at different resolutions. In spite of several developments that have already been proposed to address the sparsity problem created by multirate [8] and missing at random data [9], the issue of handling multiresolution (MR) process data remains, to a large

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extent, unexplored, with developments mainly centred around signal and image processing problems [10]. • () O C^ • 0

1

2

3

4

( ) ( ) ( ) • ( ) ( ) ( ) • C) ( ) () • 5

6

7

8

9

10

11

12

13

14

15

a)

16

T) • () () () • () o () • () () n •16 b) Figure 1. Time ranges over which average values are actually calculated (a) and those where the values are held constant in a conventional strategy to incorporate multiresolution data in single resolution methodologies (b). In industrial applications, multiresolution data structures usually arise when process sensor information is combined with data taken from other sources, such as the following: averages over several readings taken from process measurement devices and made by operators during their shifts; measurements from pools of raw material or products accumulated during a period of time and mixed thoroughly before testing; averages of process variables over regular periods of time, computed automatically by local DCS computers; aggregated measurements from each batch operation. Singleresolution techniques usually handle this type of data structures by holding the last computed value constant until the next average value becomes available, at the end of the averaging period, as illustrated in Figure 1 (relative to a variable whose values consist of averages taken over four successive observations). Such a strategy clearly creates a mismatch between the time supports where averages are calculated, and those attributed to the average values by such a procedure. Therefore, we found both opportune and relevant to address the development of approaches that adequately handle MR data structures, and in this paper we show how that can be achieved for MSSPC, currently relying strictly upon the availability of data at a single resolution. We call the proposed methodology MR-MS SPC.

2. Description of MR-MSSPC The MR-MSSPC methodology begins with the specification of resolutions associated with each variable. Quite often there is a finest resolution, corresponding to variables collected at higher sampling rates, which is used to set the finest grid (scale index 7 = 0). If variable X. correspond to averages computed over time supports of length 2^' times that of the finest resolution, then its scale index, or resolution level, is set to J.. A variable at resolution J. can only be decomposed in the wavelet transform to scales coarser (higher) than J., and therefore it does not contribute to the monitoring procedures implemented at finer scales (y < J.). This attribution is straightforward in situations where the low resolution variables represent averages over dyadic supports. In case this does not happen, we propose setting J. as the immediately next coarser scale, i.e. J. = [log2 {^Sy\, where AS stands for the averaging time support, and [log2 {ASy\ for the smallest integer n, such that n > log2 {AS).

Multiscale SPC in the Presence of Multiresolution Data

A

1361

(...) o o o oo

(5 6 6 6 6 6 o


Figure 2. Schematic illustration of the dyadic moving window for orthogonal wavelet transform calculations used in the implementation of MR-MS SPC. The decomposition depth to be used in the wavelet transformation phase of standard MS SPC, J^^^, is another parameter to be set before implementing the methodology. The whole procedure can be summarized in the following sequence of steps, divided into two phases: 1. Compute PCA models at each scale using reference data. 1.1. For each variable, perform the wavelet decomposition from J. +1 to J^^^; 1.2. Calculate mean vectors and covariance matrices at each scale; 1.3. Select number of PCs and calculate PCA models at each scale. 2. Implement MR-MSSPC methodology. For each observation index, k, multiple of 2^"""^^ ( J^.„ = min{ J.}.^^^ ): 2.1. Get dyadic window corresponding to current observation (length equal to 2'^max(^) y pigure 2 illustrates the type of moving window used in MR-MSSPC; 2.2. Decompose those variables X. for which J. < J^^(k), from J. +1 to J^^(k); 2.3. Implement PCA-based SPC at each scale (where coefficients are available), and select scales where significant events are detected; 2.4. Using scales where significant events are detected, reconstruct data at the resolution levels between the coarser scale where a significant event was detected, and that corresponding to the resolution for some variable. Do the same for the mean vectors and covariance matrices associated with the selected scales; 2.5. Using the reconstructed data, recombined covariance matrices and mean vectors, calculate PCA-based SPC statistics at each intermediate resolution, and look for significant events in these charts. If none detects a significant event, consider the process to be operating under normal conditions; if any of them do show an abnormal value, than trigger an alarm, and study the contribution plots for the reconstructed statistics at the scale where the signal occurs. We would like to point out that, in the above outlined procedure, the PCA models at different scales may have a different number of variables associated with them, the models for the finest scales having fewer variables than those for the coarser scales, as these also integrate lower resolution variables. For this reason, the number of components for the models at each scale must be chosen individually. Furthermore, a rule has also to be defined in order to specify the number of PCs to be used for the final PCA model regarding reconstructed data at the finest scale, when it has contributions from several scales, whose models have different numbers of components. We will use the minimum, so that the number of principal components (PCs) will always be smaller than the number of reconstructed variables. Stage 2.4 corresponds to an extension of the MSSPC's reconstruction phase. As we are now dealing with variables having different resolutions, we test the statistics derived

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from the reconstructed data at these intermediate resolutions as well, besides 7 = 0, provided that they stay below the coarser scale where significant activity was detected. These intermediate scales do always correspond to time supports that are dyadic powers of the finest one, something that results from the bare nature of the wavelet transform, being the natural extension of the confirmation test conducted at 7 = 0 in the classical MSSPC methodology. Therefore, we may end up with more than one plot for each statistic and, in order to maintain their overall level of significance {a\ the individual significance levels for reconstructed statistics were adequately adjusted (using the Bonferroni inequality) so that the combined level is the one required.

3. Results In this section, the main features of the proposed MR-MS SPC methodology are illustrated through its application to a simulated system. The good properties of MSSPC methodologies in the monitoring of systems exhibiting autocorrelation were already widely explored in the literature [2,7], and such properties are inherited by the proposed MR-MS SPC. In fact, since our methodology is based upon a dyadic discretization strategy, the decorrelation ability of the multiresolution decomposition is even higher than that obtained with the uniform discretization scheme used in [2], as there is no superposition in the windows used to compute the successive wavelet coefficients, as happens with data windows used for the uniform discretization scheme. Thus, the focus in this study will be mainly over the following stationary uncorrected system, where the main features of the method can be clearly illustrated:

'x;

"1 X, 0 X{k) = \ M= 1 X, 1 .^4.

0' 1 \'L(k) + £(k\Wi\h\ 1 -1

V , /7W

2

^j^

where I^ stands for the identity matrix with dimension m. For the purposes of illustrating our MR-MS SPC framework, we assume that variable X^ contains coarser resolution information, given as the successive averages over non-overlapping windows of length AS (to be defined for each example), while the remaining variables are all available at the finest resolution (values for variable X^ will only be available at the end of each period of time, representing the mean over it). 3.1. Example: Averaging support for X^ equal to 4 A reference set with 4096 observations was generated using model (1) with AS = A for variable X^, in order to estimate the monitoring parameters. Then, a test set containing 128 observations was created, where a shift of magnitude +1 was imposed in all variables between observations 43 and 83 (included). The fact that transition times do not fall in the dyadic grid at a boundary between two averaging windows is intentional, in order to see how the method behaves in such less favourable conditions. Figure 3 presents the results obtained regarding control charts for the square of prediction errors

Multiscale SPC in the Presence of Multiresolution Data

1363

( 0 P] at the two resolutions available in the data set, i.e. at J. ={0,2} {AS = A implies J^=2\

where we can verify that the faulty region is correctly identified.

50

100

50

Time index

100

Time index

Figure 3. Plots of one of the PCA-MSPC statistic ( 0 at the two resolutions available in the data set: J. = 0 (a) and J^ = 2 (b), using data reconstructed from significant scales. Control limits are set for a confidence level of 99% (horizontal line segments). Vertical lines delimit faulty region. Figure 4 presents the results obtained, under the same conditions, by using also the Q statistic for (a) the MS SPC approach with uniform discretization [2] and (b) the classical PCA-SPC approach [11], where, in both of them, the multiresolution structure was handled as illustrated in Figure 1. These control charts clearly present a noticeable distortion of the faulty region, due to poor handling of multiresolution data. 4.5

^

h

4

I

3.5 # 9

SS

3

1y IW n\

# 9

2.5 2

1 .5

'

1 0.5 O

20

a)

^^

"0^ # ^

^H
40

60

SO

^»3

UVAJI/!

.... :^>€a£aioe_ 100

Time index

120

140

b)

•11U l -

3

^

100

Time index

Figure 4. Plots for Q statistic in MS SPC with unif discretization (a) and classical PCA-SPC (b). 3.2. ARL characterization study As MR-MS SPC involves a certain delay in the calculation of coefficients, since it is based on an orthogonal wavelet transformation, we studied whether the benefits illustrated in the previous example come at the cost of higher detection times, through an ARL study where two situations regarding the resolution of variable X^ were contemplated, J^=2 and ^4 = 3, along with several shift magnitudes. The following methods were considered in the study: MR-MSSPC, Dyadic-MSSPC (similar to MRMSSPC, with a dyadic discretization strategy, but using data at a single resolution - the finest one), Unif-MSSPC (i.e., MSSPC with uniform discretization) and PCA-SPC. Figure 5 presents the ARL results obtained, where we can see that any time delay associated with MR-MSSPC only becomes an issue for shifts of magnitude greater than 2, after which stabilizes around 0.5, an acceptable value for most applications.

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M.S. Reis and P.M. Saraiva

-H^ ^ ^

MR-MSSPC Dyadic-MSSPC Unif.-MSSPC PCA-SPC 1.5

2

2.5

Magnitude of Shift

1.5

2

2.5

3

3.5

Magnitude of Shift

Figure 5. ARL results for the different methodologies, using shifts of variable magnitude and two levels of resolution for variable X^ (J^=2 and J^=3).

4. Conclusions In this paper we present an approach for integrating multiresolution data in MSSPC. The approach was tested using a simulated latent variable system, leading to improvements in the definition of the region where faults occur, without compromising detection speed.

References 1. R.L. Motard, B. Joseph (eds.), 1994, Wavelet Applications in Chemical Engineering, Kluwer, Boston. 2. B.R. Bakshi, 1998, Multiscale PCA with Application to Multivariate Statistical Process Control, AIChE J. 44, 7, 1596-1610. 3. J.E. Jackson, 1991, A User's Guide to Principal Components, Wiley, New York. 4. S. Mallat, 1998, A Wavelet Tour of Signal Processing. Academic Press, San Diego. 5. M. Vetterli, J. Kovacevic, 1995, Wavelets andSubband Coding. Prentice Hall, New Jersey. 6. B.K. Alsberg, A.M. Woodward, D.B. Kell, 1997, An Introduction to Wavelet Transforms for Chemometricians: A Time-Frequency Approach, Chemom. Intell. Lab. Syst. 37, 215. 7. H.B. Aradhye, B.R. Bakshi, R. Strauss, J.F. Davis, 2003, Multiscale SPC Using Wavelets: Theoretical Analysis and Properties, AIChE J., 49,4, 939-958. 8. N. Lu, Y. Yang, F. Gao, F. Wang, 2004, Multirate Dynamic Inferential Modeling for Multivariable Processes, Chem. Eng. Sci., 59, 855-864. 9. P.R.C. Nelson, P.A. Taylor, J.F. MacGregor, 1996, Missing Data Methods in PCA and PLS: Score Calculations with Incomplete Observations, Chemom. Intell. Lab. Syst., 35, 45. 10. A.S. Willsky, 2002, Multiresolution Markov Models for Signal and Image Processing, Proceedings of the IEEE, 90, 8, 1396-1458. 11. J.F. MacGregor, 1995, T. Kourti. Statistical Process Control of Multivariate Processes, Control Eng. Pract. 3, 3, 403-414.

Acknowledgments Financial support from FCT (POCTI/EQU/4763 8/2002) is gratefully acknowledged.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Nonlinear Model Predictive Control of the Wastewater Treatment Plant Mircea V. Cristea and Serban P. Agachi "Babe§-Bolyai" University ofCluj-Napoca, Faculty of Chemistry and Chemical Engineering, 11 Arany Janos, 400028 Cluj-Napoca, Romania, e-mail: mcristea@chem. ubbcluj. ro Abstract Sustainable development of the modem society implies special care of the water resources and their efficient management. The biological Wastewater Treatment (WWT) plants play a major role for preserving the health of the environment. This work presents alternatives for the design of the control system for the aerobic suspended growth wastewater treatment plant. Model Predictive Control (MPC) of the WWT plant soluble substrate and dissolved oxygen are investigated in the presence of typical disturbances. The proposed combination of feedback-feedforward MPC control design improves the control performance, compared to traditional control or simple feedback MPC, for both disturbance rejection and setpoint tracking. The control performance presents short settling time and reduced overshoot for the multivariable approach of MPC with output and manipulated variables constraints, showing benefits for industrial implementation. Keywords: Biological wastewater treatment, model predictive control. 1. Introduction The main task of the biological treatment is to convert soluble organic contaminants into insoluble organic and inorganic constituents or to CO2 and H2O. Aerobic or anaerobic treatment is based on the work of enzymes that transform hydrocarbons into food for the bacteria, breaking down the undesired hydrocarbons. The activated sludge process is widely used and consists in an effective wastewater treatment for the removal of dissolved and colloidal biodegradable organics. This is a treatment technique well suited for cases when organically contaminated wastewater exists. A wide range of municipalities and industries that treat wastewater containing organic chemicals, petroleum refining wastes, textile wastes, and municipal sewage use the activated sludge process. Wastewater treatment processes usually include: flocculation, neutralization, and biological processes. The active sludge process converts dissolved and colloidal organic contaminants into biological sludge, further removed by settling. The microorganisms usually found in the activated sludge consist of bacteria, fungi, protozoa, and rotifers. With proper detention, activated sludge systems have good performance for Biochemical Oxygen Demand (BOD) removal [1]. Success of the continuous biological treatment process depends on the way that suitable conditions are achieved for bacterial growth, death and endogenous respiration phase. These requirements may be only obtained, especially for large wastewater treatment units, by appropriate control techniques intended to keep the operation of the unit at the most efficient working regime. Model Predictive Control is a good candidate for these demanding control tasks. The way this control algorithm may be used for the control of a suspended growth aerobic system is presented in the present work.

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2. Model description and model predictive control approach The aerobic activated sludge (suspended growth) wastewater treatment system has been considered in this study. Organic wastes are introduced in a (completely mixed assumed) reactor where the bacterial culture is held in suspension. Aerobic environment is achieved by diffused or mechanical aeration. The biological solids form a floe, settle out, and separate from the treated water. A portion of the settled cells is recycled to maintain the desired concentration of organisms in the reactor. A satisfactory floe is necessary for effective separation of the biological solids. Important for settling are proper design of a settling unit and proper operation to prevent the presence of filamentous organisms and fungi. Bacteria in activated sludge are capable of performing hydrolysis and oxidation reactions. Both the oxidation of complex hydrocarbons and the hydrolysis of polysaccharides occur outside the cell. They are catalysed by exoenzymes secreted from the cell wall into the surrounding aqueous environment. Oxidation is conducted by aerobic organisms which use dissolved oxygen present in the biological system. Hydrolytic reactions are caused by aerobic organisms using water present in the biological system [1]. For this completely mixed continuous-flow aeration unit, presented in Fig. 1, the influent is fed uniformly along the entire length of the basin. AIR SUPPLY

WASTEWATER

SLUGE WASTE

Figure 1: Schematic representation of the continuous-flow aeration unit. The aeration is essentially homogenous, resulting in uniform oxygen demand throughout the basin [2]. This results in a homogeneous concentration of solids and substrates in the basin. This system is stable and is less affected by disturbances. Disturbances will be uniformly distributed in the basin and subsequently diluted, because of the relatively uniform population of organisms. The first principle model presented in the following equations has been used both for the dynamic behaviour description of the unit and for building the simulator on which the model based control strategies have been investigated [3]. The mass balance for the concentration of biomass X(t), substrate S(t) and dissolved oxygen DO(t) are described in the Eq. (1) to (3): • = ju (t)X(t) - D(t)(l + r)X(t) + rD(t)X^ (t) dt

(1)

Nonlinear Model Predictive Control of the Wastewater Treatment Plant

— = -^X(t)-D(t)(l+r)S(t) dt Y dDO = dt

+ D(t)S,Jt) + rD(t)S^(t)

Knju(t) '-^-—X(t)-D(t)(l^r)DO(t)^a Y

1367

(2)

W(DO^^ -DO(t))^D(t)Dq^ +rD(tpO^(t) (3)

Constant volume of the mixed liquor in the aeration basin has been assumed. Mass balance for the recycled biomass Xr(t), has been considered for the settling basin, assuming again constant volume, and thus yielding to the following equation: ^

dt

= ^[D(t)(l V

+ r)X(t) - D(t)(P + r)X/t)]

(4)

r

The specific bacterial growth rate // is depending on both soluble substrate and biomass concentration, according to the Monod's relationship: S(t) DO(t) 1^(0 = M max— ^Z;;7-^d K,^S(t)Kj^O^DO(t)

... ^^^

The dynamic model succeeds to capture the main dynamic features of the biological wastewater treatment unit, needed for the control investigations. Model Predictive Control, also referred as moving or receding horizon control, has become an attractive control strategy especially for linear but also for nonlinear systems subject to input, state or output constraints [4, 5]. MPC determines the control action based on the prediction of future dynamics of the system, allowing early control action to be taken in order to accomplish the control performance based on the expected future behaviour. The incentives of MPC algorithm, compared to traditional control algorithms, are associated to the need the system usually has for satisfying input, state or output constraints. These constraints put limitations on the achievable control performance, but this task is systematically managed by the MPC optimisation objective (with associated constraints), compared with the ad-hoc solutions frequently used in conventional control. The MPC algorithm used for the control of the wastewater treatment unit uses the combination of feedback and feedforward control design in order to reduce the effects of the large time constants presented by the process. 3. Results of the MPC approach The main control objective of the wastewater treatment unit has been considered the maintenance of the effluent soluble substrate (pollutant) concentration S(t) at the desired level of 43.5 mg/l. Three most important disturbances have been taken into account: the inlet soluble substrate concentration Sin(t), the inlet dissolved oxygen concentration Sin(t) and the inlet flowrate of the feed, the last disturbance being taken into account by the dilution rate D(t) variable of the previously presented model. The control performance has been first investigated for the cases when two of the aforementioned disturbances act stepwise. Namely, at the time moment of t=10 h acts the inlet soluble substrate concentration step, with an amplitude change of +12.5 %

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M V. Cristea and S.P. Agachi

(from 200 mgA to 225 mg/l), and for the second case acts the dilution rate step disturbance with an amphtude change of -\-14.2 % (from 0.07 h'^ to 0.08 h'^). The first investigated MPC control approach was to control the dissolved oxygen DO value in the aeration basin at a desired setpoint value of 3.21 mg/l, in the presence of the considered disturbances. Results of the dissolved oxygen DO(t) and soluble substrate S(t) concentration change in the WWT unit are presented in Fig. 2 and Fig. 3, when the inlet soluble substrate concentration disturbance is acting stepwise.

Figure 2: Dissolved oxygen variation for MPC Figure 3: Soluble substrate concentration control of DO in the presence of inlet soluble variation for MPC control of DO in the presence of inlet soluble substrate substrate concentration Sjn disturbance. concentration Sin disturbance. It may be noticed that both soluble substrate S and dissolved oxygen DO concentration are kept, by the MPC control of the DO concentration, at their desired values after the disturbance is acting, although DO is the only controlled variable. This result is usefiil for the control system design of WWT units where the inlet soluble substrate concentration Sjn is the most important disturbance and the dilution rate may be straightforwardly maintained at constant value. Unfortunately, for this first control approach and when the dilution rate D disturbance is acting, the DO variable is maintained at the desired setpoint but the soluble substrate S is irreversibly disturbed from its desired value. Due to this undesirable behaviour, the MPC control of both dissolved oxygen DO and soluble substrate S has been fiirther investigated. Disturbance rejection and setpoint following performance have been studied. Additionally, in order to improve the MPC performance, the combined feedback-feedforward control has been proposed and tested. For this second MPC approach the simulation scenario consists in putting into operation three disturbances: the dilution rate D step disturbance acting at time moment t=20 h (changing from the initial value of 0.07 h'^ to the final value of 0.075 h'^X followed by the inlet soluble substrate Si„ concentration step disturbance applied at time moment t=100 h (changing from the initial value of 200 mg/l to the final value of 275 mg/l) and ending by the dissolved oxygen Z)0,„ inlet concentration step disturbance applied at time moment t=250 h (changing from the initial value of 0.5 mg/l to the final value of 0.1 mg/l). The comparative results showing feedback MPC and combined feedback-feedforward MPC, both acting for the disturbances rejection, are presented in Fig. 4 and Fig. 5.

Nonlinear Model Predictive Control of the Wastewater Treatment

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1369

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Figure 4: Dissolved oxygen variation for MPC control of both DO and S, (feedback MPC dashed line and combined feedbackfeedforward MPC solid line), in the presence of the Z), Sin and DOin step disturbances.

50

100

150 •nme[h]

200

250

Figure 5: Soluble substrate concentration variation for MPC control of both DO and S, (feedback MPC dashed line and combined feedback-feedforward MPC solid line), in the presence of the D, Sin and DOin step disturbances.

Furthermore, the results of the combined feedback-feedforward MPC, for the case of two a priori known (programmed) setpoint changes (one for the dissolved oxygen concentration setpoint change of +9.3%, changing from 3.21 mg/l to 3.51 mg/l, at the time moment t=100 h, and one for the soluble substrate concentration change of +11.5%, changing from 43.47 mg/l to 48.47 mg/l, at the time moment t=225 h) are presented in Fig. 6 and Fig. 7.

I49 O 3.45

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I j

^ 0

50

100

150 Time [h]

200

250

300

Figure 7: Soluble substrate concentration variation for MPC control of both DO and S, (combined feedback-feedforward MPC solid line, setpoint dashed line), in the presence of DO and S setpoint changes.

The results presented in Fig. 4 to Fig. 5 reveal the incentives of the combined feedback-feedforward MPC approach, compared to the case of feedback MPC. The advantages of this latter MPC control approach are sustained by the shorter reset time and by the considerable smaller overshoot. The setpoint tracking performance of the MPC control approach is also very good. The combined control of DO and S variables provides stability to the WWT unit and good tolerance to the main disturbances.

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M.V. Cristea and S.P. Agachi

4. Conclusions Different model predictive control design schemes have been proposed and tested in order to maintain the effluent soluble substrate concentration at the desired value. Both setpoint tracking and disturbance rejection performance of the proposed MPC approach have been tested, considering three representative disturbances: the inlet concentration of the soluble substrate, the dilution rate and the dissolved oxygen concentration of the inlet flow. The investigations revealed that dissolved oxygen MPC control strategy is simple and successful as long as the dilution rate can be maintained at constant level. As this situation is not usual for practical applications, MPC control of both dissolved oxygen and soluble substrate concentration has to be considered for an effective control design. Although this feedback MPC control approach presents favourable performance, the combined feedback-feedforward MPC control approach succeeds to accomplish superior control performance, shown by its short settling time and reduced overshoot. As MPC control is performed in the presence of constraints, for both manipulated and control variables, the proposed control design presents improved performance compared to the traditional control approach and reveals incentives for the practical implementation. Notation D(t) = dilution rate (flowrate/basin volume) Sr= soluble substrate concentration in [h'^] the recycle flow [mg/l] DO(t) = dissolved oxygen cone, [mg/l] V/Vr = ratio between the aeration DOin = inlet dissolved oxygen cone, [mg/l] basin and settling basin volume DOmax = maximum dissolved oxygen W = ratio of air flowrate to basin concentration [mg/l] volume [K^] DOr = dissolved oxygen concentration n the X(t) = biomass concentration [mg/l] recycle flow [mg/l] X/t) = recycled biomass Kd = endogenous decay consant [mg/l] concentration [mg/l] Koo = half oxygen sat. constant [mg/l] Y= growth yield Ks = half-velocity constant [mg/l] a = oxygen transfer rate Ko = model constant fi = ratio of waste to influent flowrate r = ratio of recycled to influent flowrate ju = specific growth rate [h'^] S(t) = soluble substrate concentration [mg/l] ju^ax = max. specific growth rate [h'^] Sjn = inlet soluble substrate cone, [mg/l] References [1] N.P. Cheremisinoff, 1996, Biotechnology for Waste and Wastewater Treatment, Noyes Publications, New Jersey. [2] E. R. Alley, 2000, Water Quality Control Handbook, McGraw-Hill, New-York. [3] N. Nezam and I. Dumitrache, 2001, Intelligent Control Systems in Wastewater Treatment Plants, Control Engineering and Applied Informatics, Vol. 3, no.4, 69-76. [4] M. V. Cristea, S.P. Agachi, V. Marinoiu, 2003, Simulation and Model Predictive Control of a UOP Fluid Catalytic Cracking Unit, Chemical Engineering and Processing, No. 42, 67-91. [5] M. V. Cristea, L. Toma, S. P. Agachi, 2005, Neural Networks Used for Model Predictive Control of the Fluid Catalytic Cracking Unit, Proceedings of the 7-th World Congress of Chemical Engineering, Glasgow.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1371

Branch and Bound Methods for Control Structure Design Vinay Kariwala* and Sigurd Skogestad^ Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway We demonstrate the potential of branch and bound methods for handling the combinatorial problems arising in control structure design. The method efficiently solves large-scale benchmark problems for selection of controlled variables using minimum singular value rule and variable selection for stabilization. We also provide an overview of some other interesting problems that can be solved in the same framework using non-standard bounding strategies. Keywords: Branch and bound, Control structure design. Controlled variables. Decentralized control, Stabilization. 1. Introduction Control structure design (CSD) deals with the selection of controlled and manipulated variables, and the links interconnecting them [1]. Many of the available methods for CSD require an enumeration over the entire set of alternatives and thus suffer from computational intractability. In this paper, we show that for many problems arising in CSD, the combinatorial growth in the number of alternatives can be handled efficiently using branch and bound (B&B) method. Effective B&B methods require tight lower and upper bounds on the objective function value for all feasible solutions that can be generated by expanding upon the partial solution (node). It works by eliminating partial solutions, which cannot give a better solution than the best known lower bound (often taken as the objective function value for best known integer solution). As an example, consider that we want to select m out of n variables by maximizing ^ and ^' is the best known lower bound. Let ft be the set of feasible solutions, which can be generated by expanding upon the partial solution. Then, ft can be eliminated (pruned), if an upper bound on The use of B&B methods for CSD problems has earlier been explored in [2] and also in [3], where the minimization of RGA-number [1] was shown to be a mixed integer linear program. Here, we apply B&B method using different bounding strategies to: (a) controlled variable selection using minimum singular value (MSV) rule [1] and; (b) variable selection for stabilization. The latter problem has also been considered in [4], but the bounding strategy used here is computationally more efficient. Later in this paper, we also give an overview of some other problems that can be cast in the B&B framework. * Present Address: Division of Chemical & Biomolecular Engg., Nanyang Technological University, Singapore. +Corresponding Author: Tel: +47-735-94154; Fax: +47-735-94080; E-mail:[email protected]

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2. Controlled variable selection using MSV rule The MSV rule [1] is a simple method for selecting the set of primary controlled variables (CVs). Let J denote the scalar economic objective function and, G and Gd represent the linearized (unsealed) process and disturbance gain matrices, respectively. For finding the locally optimal set of CVs, the rule requires that giG') be maximized where q_ is the MSV and G' is the scaled gain matrix. The output and input scaling matrices are J°;f and diag(|G J~J Jud — Gd\-\-en), respectively. Here, J^^ = dJ/dx dy and e^ is the implementation error (see [1] for details). A quick observation reveals that {G J~J Jud-Ga) can be calculated beforehand, as the scaling procedure is independent of the choice of CVs. Thus the MSV rule is equivalent to squaring down a matrix such that the MSV of the squared matrix is maximum. Let Gaii be the scaled gain matrix for all candidate CVs and H = diag(/ii), hi = {0,1}, ^ ^ hi = mhQ the decision matrix such that f^ CV is selected, if hj = I. The MSV rule requires solving max a(i7 Gall)

(1)

H

2.1. Bounding strategy When the constraint hi = {0, 1} is relaxed a s O < / i ^ < l , ( l ) can be posed as a linear matrix inequality (LMI). The solution to this relaxed convex program provides the desired upper bound for pruning branches. A computationally cheaper approach, used in this paper, is based on Weyl's inequalities. Let ai be the singular values, where (7^+1 < a^. For A e R^^^, p < q, [5, Th. 4.3.6], CTip+l) ( ^ ] <(7p{A)

(2)

In words, when rows are added to a fat matrix, the least non-zero singular value is nonincreasing. Thus, the least non-zero singular value of a partially completed solution provides the required upper bound on the MSV The lower bound is obtained sequentially by adding one variable to the partially completed solution at every step such that the least non-zero singular value is maximum, until the resulting matrix is square. To illustrate these concepts, we consider an example where we want to select 3 of 4 CVs

'•all •

10 10 10 10 9 1 2 1 3 2 1 0.1

With only yi selected ai — 17.321, where as ^2 for sets {yi, ^2}, {2/1, ys} and {yi, y^} is 5.699,1.386 and 1.337, respectively. Thus, 17.321 is the upper bound for node {yi} as it bounds (72 (and as) for all solutions that are generated by expanding it. The sequential approach first adds y2 to the set {yi} and then ys to {yi, y2}- The set {yi, ^2, Vs} has as = 0.78, which is a lower bound for {yi}- Similarly, for {yi, 2/2}, 5.699 and 0.78 are the upper and lower bounds, respectively. 2.2. Application For this problem, we use a breadth-first B&B method. The initial feasible solution is obtained using the same sequential approach as used for obtaining the lower bound. Numerical

Branch and Bound Methods for Control Structure Design

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results suggest that the sequential approach usually performs better than other alternatives for generating an initial solution, e.g. non-square relative gain array (RGA) [1]. The B&B method is applied to the Hydrodealkylation of Toluene (HDA) process, which has 8 degrees of freedom and 129 candidate CVs giving 1.52 x 10^^ alternatives. The MSVs of the solutions obtained using sequential and RGA methods are 13.63 and 2, respectively, which emphasizes the superiority of the sequential approach. The solution obtained using B&B method has an MSV of 14.73, which requires evaluation of 8.1 x 10^ alternatives (5.3 x 10~^% of total alternatives). Motivated by controllability reasons, Cao et al. [2] proposed using an asymmetrical tree based B&B method for solving (1). In comparison, their method requires evaluation of 2.5 x 10^ alternatives. Though the improvement in MSV is only minor as compared to the sequential approach for this process, in general, the relative difference can be as large as 70%, as confirmed using randomly generated matrices. 3. Variable selection for stabilization For complex unstable systems, often the system is first stabilized using a subset of CVs and manipulated variables (MVs) and then another controller is designed to satisfy the performance requirements. The question remains: which CVs and MVs should be used for stabilization? These variables can be selected such that the input usage for stabilization is minimized as it reduces the likelihood of otherwise destabilizing valve saturation [1]. The 'Hoo-optimal achievable input performance is given as minx(s) ||i^5(5)|| = q_~^(U{G{s)y), where Z//(G(s))* is the mirror image of anti-stable part and a^ is the minimum Hankel singular value (HSV) [6]. Then, the optimal subset of variables is chosen by maximizing afj{U{G{s)y). For systems with single unstable pole, the optimal solution can be obtained using pole vectors [1]. This approach, however, cannot provide the optimal solution for systems with multiple unstable poles and a method for obtaining the same is discussed next. 3.1. Bounding Strategy Without loss of generality, we assume that G{s) has only unstable poles. Let the triplet {A, B, C) denote the minimal state-space realization of G{s). Then, o;^(G(s)*) > 7 if and onlyifP,Q > 0 solve [7] A * P - P A + C*C

QA'-AQ

> 0

+ BB'' > 0 ' P

jl'

> 0

(3)

When the system has distinct unstable poles, P, Q can be removed by noting [6] ny

p = X^((airc„)°M

(4)

riu

Q ^ 5^(P,*P*JoM

(5)

where Mij = l/(pi + p*), p^ are unstable poles, o is the Hadamard product and JBJ*, C^i denote the f^ row and i^^ column of B and C, respectively. Note that the products depend only on the

V. Kariwala and S. Skogestad

1374

CV and f^ MV, respectively. Then the variable selection problem requires solving 7/ 7/

>

0

(6)

where a^, Pj = {0, 1}. The i^^ CV is selected if a^ = 1 and vice versa. For selecting m variables out of (ny+Uu) CVs and MVs, we need the additional constraint ^^ • a^ + ^ f3j = m. An upper bound is obtained by relaxing the decision variables as 0 < 0;^, ^^ < 1 and a lower bound is obtained by rounding-off a, (3 that optimally solves the convex program. Note that the assumption of distinct unstable poles can be satisfied by numerically perturbing the poles. 3.2. Application There are many software packages available for solving integer semi-definite programs. In this paper, we use the B&B algorithm available with Yalmip [8], where the package Sedumi 1.05 [9] is used for solving the LMI problems resulting on relaxation of binary variables.

(a) Total number of CVs and MVs

fixed

(b) Number of CVs and MVs fixed individually

Figure 1. Selection of variables for stabilization of Tennessee Eastman process. The algorithm is applied to Tennessee Eastman process with six unstable poles. The model with 22 CVs and 12 MVs is scaled as outlined in [4]. The variation of maximum HSV with the total number of CVs and MVs fixed is shown in Figure 1(a). Cao and Saha [4] suggest using m = ^, for which maximum HSV is 926.65. The bounding strategy used in [4] is similar to the strategy used for bounding MSV Out of 1.72 x 10^° alternatives, the proposed method evaluates only 1441 alternatives cumulatively for all m, which is an order of magnitude less than reported in [4]. It is also possible to fix the number of CVs and MVs individually through the constraints Yli^i — ^y» S j A? — ^ u ^^^ the results are shown in Figure 1(b). For vrty — rriu = 3, the optimal solution is same as for m = 6, as expected. 4. Other problems Similar to the problems discussed earlier, the combinatorial nature of many other problems arising in CSD can be handled using B&B methods. Usually, the challenging task is to find a tight upper bound on the objective function. Next, we discuss some problems requiring novel manipulations of the partial solution to get monotonic relationships useful as upper bounds. The first problem consists of selecting pairings such that iJ.{E) is minimized (or —iJi{E) is maximized), where /x is the structured singular value [1] and E = {G — G^diag)^diig ^^^^ ^diag

Branch and Bound Methods for Control Structure Des ign

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consisting of diagonal elements of G. Selection of pairings with minimal IJ^{E) (equivalent to finding optimal permutation of G) puts least restrictions on decentralized controller synthesis using independent designs [1]. When there exists a pairing with /x(^) < 1, iterative-RGA provides the optimal solution explicitly [1], otherwise a B&B method is required and the bounding strategy is discussed next. For any given partial solution (node), a lower bound on —fi{E) can be obtained using the sequential approach discussed in § 2. Now, assume that G is permuted such that the (1,1)-block of permuted G correspond to gain of partially selected pairings. By partitioning E similar to G, we then have —/i(E'ii) > —iJi{E) [1], which serves as the upper bound on all the alternatives that can be generated by expanding on this node. This happens as any expansions on this node do not affect En. Another problem consists of selecting pairings such that the achievable decentralized output performance (Jdecen) is minimal. Though exact characterization of Jdecen is an open problem, sub-optimal methods providing upper bounds on Jdecen are available. Then, at any node, the lower bound on —Jdecen is obtained by completing the partial solution using any reasonable heuristic and the sub-optimal methods. For a given pairing, an upper bound on —Jdecen was recently presented in [10]. This method can be adapted for partially completed solutions by using a block decentralized controller, where the remaining (unpaired) CVs and MVs are controlled using a full multivariable controller. As only bounds on Jdecen can be computed, B&B method may end with a number of alternatives, for all of which the upper bound on —Jdecen is larger than the best available lower bound. Nevertheless, B&B method is useful for eliminating inferior pairing alternatives. 5. Conclusions B&B methods can efficiently handle the combinatorial problems arising in CSD. In comparison with standard mixed integer non-linear programs, CSD problems require novel bounding and search strategies. We demontrated the use of monotonic relationships and integer semi-definite programs for obtaining tight upper bounds on partial solutions. The algorithms scale well with problem dimensions and are shown to provide optimal solutions for large-scale benchmark problems. The future work will focus on extending these ideas to multi-objective optimization problems. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

S. Skogestad and I. Postlethwaite. John Wiley & Sons Ltd., Chichester, UK, 2005. Y. Cao, D. Rossiter and D. H. Owens, Proceedings of DYCOPS 5, Korfu, Greece, 1998. I. K. Kookos and J. D. Perkins. Ind. Eng. Chem. Res., 40, 2001. Y. Cao and P Saha, Chem. Engg. Sci., 60, 2005. R. Horn and C. Johnson. Cambridge University Press, New York, NY, USA, 1985. V. Kariwala, S. Skogestad, J. F. Forbes and E. S. Meadows. Intl. J. Control, 78, 2005. S. Boyd., L. Ghaoui, E. Feron and V. Balakrishnan. SIAM, Philadelphia, PA, USA, 1994. J. Lofberg. Proceedings of CACSD, Taipei, Taiwan, 2004. (Available from h t t p : / / c o n t r o l . ee . e t h z . c h / ~ j o l o e f / y a l m i p . p h p ) . 9. J. F. Sturm, Optimization Methods and Software, 11-12, 1999. (Available from h t t p : //sedumi .mcmaster. ca/). 10. V. Kariwala, J. F. Forbes and E. S. Meadows. Proceedings of ACC, Boston, MA, 2005.

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6. Appendix: Average performance We further evaluate the performance of B&B methods for maximization of MSV and HSV using randomly generated examples. For maximization of MSV, the method is applied to 1000 randomly generated matrices of different sizes and the results are shown in Table 1. Table 1 Performance of B&B method for maximization of MSV for randomly generated matrices n m Alternatives % of alternatives evaluated Mean CPU Time Mean Median Max Min (sec) 10 3 120 44.1 40.8 132.5 11.7 0.0042 20 3 1140 11.1 8.8 2.1 66.5 0.0067 15504 75.2 20 13.7 10.9 0.6 5 0.1188 2.8 30 5 142510 3.8 37.1 0.1 0.3409 For maximization of HSV, the efficiency of this algorithm is tested using 100 random square transfer matrices of different dimensions generated using Matlab command rss and the results are shown in Table 2. For the last two cases, the maximum number of alternatives evaluated are not reported, as they were restricted to 5000 and 20000, respectively, and there exists examples that exceed these bounds. Table 2 Performance of B&B method for maximization of HSV for randomly generated systems rip n m Alternatives % of alternatives evaluated Mean CPU Time Mean Median Max Min (sec) 3 5 3 120 43.3 32.5 155.8 0.8 4.0657 11.7 120 84.5 87.5 142.5 10.858 5 5 3 5 1140 63.3 70.5 0.6 10 3 134.6 75.631 15504 4.17 1.5 0.0 56.057 5 10 5 0.2 0.0 298.67 5 142510 1.7 15 5 Some of the salient observations are: 1. The efficacy of the method, measured in terms of the number of the alternatives evaluated, increases with problem dimensions. This is reasonable, as many more branches can be pruned, when less variables are selected. 2. There exist cases that are worse than complete enumeration. As most combinatorial problems are AfV-hard, the worst-case performance is likely to increase nonpolynomially with problem dimensions, when any ingenious B&B strategies are used. 3. The average performance is much worse than the large-scale benchmark problems considered previously. This happens as real problems often show more structure than randomly generated problems. For example, for the HDA process, the scaled gain for 23 out of 129 candidate CVs is nearly zero, which are eliminated in the first step of the B&B method resulting in substantially reduced search space. 4. As we need to solve an LMI problem iteratively for finding an upper bound for the HSV maximization problem, the mean CPU time is much larger than the MSV maximization problem. An advantage of this formulation is that there exist cases, for which the optimal integer solution is obtained through evaluations of only one alternative.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantehdes (Editors) © 2006 PubHshed by Elsevier B.V.

A mathematical programming approach including flexible recipes to batch operation rescheduling Sergio Ferrer-Nadal, Carlos A. Mendez, Moises Graells, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract The inherent dynamic nature of industrial environments often needs not only the execution of the required rescheduling actions but also the proper adjustment of the production recipe to the current process conditions. Therefore, the concept of flexible recipe becomes an important part of the rescheduling fi-amework that allows ftiU exploitation of the batch plant intrinsic flexibility. This work introduces a rigorous mathematical approach that incorporates the concept of recipe flexibility to batch operation rescheduling. Keywords: rescheduling, flexible recipe, batch operations, MILP model. 1. Introduction Batch processes have received great attention over the last years because of their higher flexibility compared to continuous processes and the increasing demand for specialty, high added-value chemical and pharmaceutical products. Within this context, the shortterm scheduling deals with the optimal allocation of a set of scarce plant resources over time to manufacture one or more products following a batch recipe. Most of the scheduling approaches assume that batch processes are operated at nominal conditions following predefined fixed production recipes (Mendez and Cerda, 2003 a). However, in many cases a flexible recipe operation may result a more suitable way of incorporating systematic recipe adaptations depending on the actual process conditions. The flexible recipe concept was originally introduced by Rijnsdorp (1991) as a set of adaptable elements that controls the process output. Afterwards, Verwater-Lukszo (1994) presented a flexible recipe approach for the adjustment of control recipes during production which has been applied to several case studies (Sel et al., 1999; Rutten and Bertrand, 1999). One of the first attempts to extent the flexible recipe approach to a plant-wide scheduling problem was carried out by Romero et al. (2001). These authors proposed to integrate a linear flexible recipe model into a multipurpose batch process scheduling model based on an S-graph algorithm. In addition to changes in nominal process conditions, frequent unexpected events can also take place during the normal batch plant operation (equipment failures, late order arrivals, order cancellations and so on). These unforeseen changes may lead the inprogress schedule to become suboptimal or even infeasible. Although rescheduling techniques have a central role in process operations, only a few developments have focused their attention on this challenging problem (Cott and Macchietto, 1988; Mendez and Cerda, 2003b). These rescheduling approaches allow performing certain corrective actions such as partial resource re-allocation, re-sequencing and re-timing assuming a Author to whom correspondence should be addressed: [email protected].

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fixed batch production recipe. This work introduces a MILP-based framework to batch operation rescheduHng including the concept of recipe flexibihty. 2. The flexible rescheduling framework The proposed rescheduhng approach is based on the MILP model for single-stage multiproduct reactive scheduling introduced in Mendez and Cerda (2003b). The original model was adapted and extended to address the rescheduling problem of multistage multipurpose batch plants involving different storage policies, non-zero transfer times and flexible recipes. This model relies on the notion of general precedence which reduces the number of binary variables and so the computational effort, as reported in Mendez and Cerda (2003a). Flexible recipe constraints are incorporated in this model to account for the possibility of changing the processing time of some tasks tweaking the rest of the parameters of the product recipe. The cost for modifying these process variables from their optimal economic conditions is taken into account to represent how plant productivity is increased regardless of the cost of altering the nominal plant conditions. Different incidences during the processing horizon can be considered such as insertion of new orders, equipment failures, due dates changes, delay in arrivals, variations in the cost of the raw materials or products, etc. Therefore, when an unexpected event arises, the following groups of tasks can be defined: • Executed tasks (T^^^^) already processed at the rescheduling point which are not included in the rescheduling formulation since they are past events with no influence in the remaining schedule. • Non-directly affected tasks (T"^^) by the unexpected event that are being processed or still have to be processed at the rescheduling point. • Directly affected tasks (T^^) by the unexpected event. They include rejected tasks , running at the rescheduling point that have to be transferred to an alternative unit in order to be reprocessed. Successive stages of these rejected tasks in the processing sequence are also included in this group. • ^New task (T"®^) from late order arrivals to be scheduled. All these different tasks are depicted in Figure 1 which represents a production scenario where two products (dark grey and light grey) are manufactured. rnda

I

Unit U l Unit U 2 Unit U 3 Rescheduling point

Time 'Cxec rpnda rpda ,

Figure 1. Basic representation of task types: r^^^ T'^, T"^ and T"'^. Rather than re-optimizing the sequence of the remaining tasks, only local changes are allowed in order to reduce the impact over the schedule in progress. Partial rescheduling actions linked to each group of tasks are summarized in Table 1.

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3. Case study The proposed rescheduling strategy under the flexible recipe framework will be illustrated by solving a modified version of the case study proposed in Romero et al. (2003). Five products are manufactured in four stages being available alternative process equipment units The flexible recipe for the production of benzyl alcohol (PI) is introduced within this production scenario. Transfer times as a 5 % of the processing time of each processing task have been also considered. Production orders comprise a single batch of one or several products with specific due dates. Batch processing times, available processing units and order due dates are reported in Table 2. Table 1. Allowable rescheduling actions (X: No ; V: Yes) ^ , Task type

.^ .... (Re)Alloc.

.^ s^ (Re)Seq.

.^ xr^. . (Re)Timing

1. Executed tasks, T""^

X

X

X

X

2. Non directly affected tasks, T"^

X

V

V

V

V V

V V

V V

V V

3.Directly affected tasks, T''* 4. New tasks, T™

Recipe adjustment ^ i / ^ ur . i N (only for flexible tasks)

The crossed-Cannizaro reaction for the batchwise production of benzylalcohol (second stage of product PI at unit U2) from the reduction of benzaldehyde has been studied by Keesman (1993). This author proposed a quadratic model to predict the yield of the reaction for a priori known disturbances in the process inputs. Equation 1 shows the linearized model required to be incorporated into the MILP formulation, assuming a small flexibility region around the current operating conditions (see Table 3). y = 4A X i + 4 JC2+95 X3+95 X4

(1)

Additional flexibility has been considered in the first stage of product PI (unit Ul). This is a preheating stage where the temperature is directly proportional to the processing duration and the temperature for the reaction in the next stage. The selected objective function to be minimized is the total cost associated to the order earliness (1 m.u./h), tardiness (5 m.u./h) as well as the corresponding cost for manipulating the process conditions (See Table 3). The proposed approach is implemented within the modelling language GAMS (Brooke et al., 1998) using CPLEX version 7.5. 4. Results This rescheduling approach is applied to the schedule in progress shown in Figure 2 which has to be updated at time 3h in order to face the breakdown of unit 7 with a repairing time of 17 hours. In addition, new batches corresponding to the arrival of late orders must be also inserted in the on-going schedule (see Table 2). Figure 3 shows the proposed reschedule plan without considering the recipe flexibility. Finally, Figure 4 depicts a flexible reschedule that, despite the recipe modification cost, results in a better solution in terms of the proposed objective function. This improvement comes not only from the recipe changes but also from the several modifications of sequencing decisions, which can be easily observed in Figures 3 and 4. It is worth nothing that the

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computational effort for updating the current schedule remains very low, even in the case that additional rescheduling actions are considered. Short reaction times are highly important in real industrial environments. Table 4 summarizes the main features of the schedules generated without and with recipe flexibility. In order to improve the customer satisfaction in the flexible production environment, the second stage of the first, third, fifth and sixth batch of product PI (flexible reaction task) reduces their processing time at the expense of increasing the amount of formaldehyde Table 2. Process data for the case study Product PI Stage

Product P5

Unit

PT,h

Unit

PT,h

Unit

PT, h

Unit

Ul

0.5

Ul

1

U7

2

U2

1.5

U8

U2

0.75

U9

2.5

U8

2

U8

2

U3

2

U3

2

U3

1

U9

2

U4

1

U9

2.5

U6

1

U7

2

U4

2

U5

1.5

1.75

(Flexible stage)

4

Product P4

PT,h

U2

3

Product P3

Unit

(Flexible stage)

2

Product P2

U3

2

U4

1.5

U9

1.75

U7

2

U4

0.5

U6

1

U7

0.75

U5

1

PT, h

U5

1

U2

1.25

Ul

1

Due dates (h) Order 1

10

10

9

10

15

Order 2

10

20*

9

22*

15

Order 3

10

15

Order 4

15

20*

Order 5

19*

Order 6

19*

(*) New ordei

Table 3. Recipe parameters, flexibility region and cost for deviation from nominal conditions Flexible process variable

Flexibility Region Lower bound

Deviation cost

Upper bound No deviation is allowed in quality

5y

reaction yield

5xi

reaction temperature

-0.7'^C

0.5 °C

3 m.u.AC

8x2

reaction duration

-0.3 h

0.1 h

2m.u./C

5x3

amount of KOH

-2.7 g

8.5 g

5 m.u./g

6x4

amount of Formaldehyde

-30 g

7.5 g

4 m.u./g

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and KOH. However, in none of the batches of PI, flexibility is exploited for the first stage. This situation arises because flexibility has always a cost and this stage does not suppose a bottleneck in the process and, consequently, no improvement can be achieved. Therefore, this example clearly reflects the high importance of recipe flexibility in the rescheduling process of critical and hard-constrained batch operations. I

Product PI ] Product P2 ] Product P3 Product P4 Product P5 111^^81 Waiting time WMMi Rejected task Figure 2. Schedule in progress at the rescheduling point. U1 U2 tJ3 LJ^

US U6 UT-

US U9

Figure 3. Optimal rescheduling considering fixed production recipe at nominal conditions.

Figure 4. Optimal rescheduling consideringflexibleproduction recipes. 5. Conclusions An efficient MILP-based reschedulingfi*ameworkthat incorporates the recipe flexibility as an additional rescheduling opportunity has been presented. The approach is based on

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a continuous-time domain representation and the generalized notion of precedence. The rescheduling strategy considers the on-going production schedule and the current process conditions in order to simultaneously adapt the production recipe to the new scenario and re-optimize the schedule of the batches still to be processed. Different objective functions can be employed to regain feasibility or optimality at minimum cost. Efficiency and applicability of the proposed strategy is demonstrated by successfully solving a complex rescheduling problem in a multipurpose batch plant with reasonable computational effort. The results reported put clearly on evidence the significant benefits of exploiting the inherent flexibility of batch plants. Table 4. Comparison between fixed and flexible rescheduling Fixed reschedule

Flexible reschedule

Tardiness cost, m.u.

131.81

116.44

Earliness cost, m.u.

7.93

11.54

Process conditions change cost, m.u.

0.00

1.95

Objective function, m.u.

139.74

Binary vars., cont. vars., constraints

5, 555, 974

CPU time, s (AMD Athlon 2600 MHz, CPLEX 7.5 )

68

129.93

283,676,1131 230

Acknowledgements Financial support received from the European Community projects (MRTN-CT-2004512233; RFC-CR-04006; INCO-CT-2005-013359) and the Generalitat de Catalunya with the European Social Fund (FI grant) is fully appreciated. References Brooke, A., Kendrick, D., Meeraus, A. & Raman, R. (1998). GAMS - A user's guide. The Scientific Press. San Francisco. Cott B.J. & Macchietto S. (1989). Minimizing the effects of batch process variability using online schedule modification. Computers & Chemical Engineering. 13, 105 - 113. Mendez C.A. & Cerda, J. (2003a). A MINLP continuous-time framework for short-term scheduling of multipurpose batch processes under different operation strategies. Optimization & Engineering. 4, 7 - 22. Mendez C.A. & Cerda, J. (2003b). Dynamic scheduling in multiproduct batch plants, Computers & Chemical Engineering. 27, 1247 - 1259. Rijnsdorp, J.E. (1991). Integrated Process Control and Automation, Elsevier, Amsterdam. Romero J., Espuna A., Friedler F. & Puigjaner L. (2003). A new framework for batch process optimization using the flexible recipe. Industrial & engineering chemistry research. 42, 370 379. Rutten, W. G. M. M. & Bertrand, J. W. M. (1998). Balancing stocks, flexible recipe costs and high service level requirements in a batch process industry: A study of a small scale model. European Journal of Operational Research. 110, 626 - 642. Sel, D., Hvala, N., Strmcnik, S., Milanic, S. & Suk-Lubej, B. (1999). Experimental testing of flexible recipe control based on a hybrid model. Control Engineering Practice. 7, 1191-1208. Verwater-Lukszo, Z. (1998). A practical approach to recipe improvement and optimization in the batch processing industry. Computers in Industry. 36, 279 - 300.

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Using multi sensor data fusion for level estimation in a separator Nils-Olav Skeie^, Saba Mylvaganam^, Bernt Lie^ ^Telemark Technological R&D Centre, Kj0lnes Ring 30, N-3918 Porsgrunn, Norway ^Telemark University College, R O . Box 203, N-3901 Porsgrunn, Norway Abstract A data driven model is developed to be used as a soft sensor to predict the hquid and interface levels in an oil / water separator. The methodology uses a set of absolute pressure sensors together with multi sensor data fusion for estimation of the levels. Experimental results are provided for model validation. Key^vords: Multi sensor data fusion, soft sensor, oil / water separator, interface level. 1. Introduction The product from oil drilling consists of a mixture of oil, gas, water, and sand. Oil / water separators are used to separate these components: gas is vented off at the top, sand is removed from the bottom, while settling is used to separate oil and water. Settling requires a certain residence time to be completed, and it is necessary with a certain oil volume in the separator. The water feed can be controlled to achieve the required oil volume and settling time, but it is necessary to have a measure of the oil / water interface level. Further complications for the measurement are foaming, and sand in the water. A number of principles for level measurement in separators exist, but today mostly sensors based on nuclear principles are used. Since operators work in the neighbourhood of separators, it is desirable to avoid using nuclear based sensors. In this work, we consider the possibility to combine information from several sensors in a multi sensor data fusion (MSDF) algorithm to develop a soft sensor for oil / water interface level. The paper is organized as follows: in section 2, sensor types are discussed, and a specific sensor type is chosen in this initial study. In section 3, a linear regression model is proposed for describing the relationship between sensor data and interface level. Then in section 4, the experimental separator rig is described. In section 5, experiments are described, and a soft sensor is developed. Finally, the results are discussed. 2. Sensor selection In this study only standard sensors will be considered, and if possible, used. Using MSDF, many sensors can be used, both of the same type and of different types. The following types of sensors have been considered [1]: • floats: in an oil/water separator two floats must be used, but due to mechanical solutions these sensors are not desirable in the separator.

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• hydrostatic devices: differential or absolute pressure devices; - differential pressure devices are not suitable due to the possibility of mud, condensation rates, and plugging in the dry-leg, — absolute pressure devices can be used, using a flush diaphragm to avoid plugging. The challenges regarding absolute pressure devices are dependency of density, and that each device must be able to measure the whole pressure range of the separator and still be accurate for density measurement, • load cells: difficult to make a reliable mechanical solution for weighing the separator, • ultrasonic transmitters: will not work due to sand in water, and foam on top of oil, • radar transmitters: will not work due to foam on top of the oil, and a low dielectrical constant of the oil, • guided wave radar: working, but not stable in real life. Very dependent on the thickness of oil layer, the dielectrical constant of the oil, and the thickness of the emulsion layer between oil and water. The separation is working due to different densities of oil and water. Normally the density of oil is 65% — 90% of the density of water. Using more than one pressure sensor, it is possible to measure the density difference between the sensors. The relationship between pressure and density is:

r pgdh

(1)

J/ho he,

where p is the value measured by the sensor, po is the gauge pressure, p is the density, g is acceleration due to gravity, and h is the height of the liquid. Using more then one pressure sensor, po will be the pressure from the sensor above, and h is the height between these sensors. In this first study, five absolute pressure sensors are used, together with a top pressure sensor measuring the gauge pressure in the separator. An overview of the sensor locations are shown in Figure 1. Due to a small distance between the pressure sensors, sensor fusion is selected instead of single sensor values and a logic system for estimation of the liquid and interface levels. Particularly the interface level is difficult to estimate. 3. Linear regression model A software application was developed using .NET technology and C # as a programming language, for communication with the sensors, logging sensor data, configuration, and user interaction. Using absolute pressure sensors, an atmosphere pressure sensor is needed to be able to display the gauge pressure. The relative pressures of the sensors are calculated

Using Multi Sensor Data Fusion for Level Estimation in a Separator

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• the gauge pressure in the separator: Vtop

Ptopsens

Patmsens

(2)

• the pressures for the sensors that can be covered by Hquid: Preli ^^ Psensori

Ptop

Patm_sens

Psensori

Ptop sens

(3)

Patm sens IS the valuc from the atmosphere pressure sensor, ptopsens the value from the top sensor, and Psenson the value from pressure sensor i that can be covered by liquid. Using the Prei values for the pressure sensors that can be covered by liquid, these values will be independent of the gauge pressure in the separator. The experiment can then be done for any value oiptopThe relative pressures will be used as inputs to a data driven model of the separator. A linear relationship between the relative pressure values and the liquid level and oil / water interface is assumed: y = bx -^ e

(4)

where y contains the estimated levels, b is the regression coefficients, x contains the relative pressures, e is the error / noise. Let matrix X be composed of x vectors from N experiments, and Y be composed of y vectors from the same experiments. Then equation 4 can be written as:

Y = XB + E

(5)

and the least squares solution of B is implicitly given by the normal equation [2]:

X^XB

- X^Y

(6)

To calculate the B matrix in Equation 6 requires that the X^X matrix is nonsingular. If the X^X matrix is singular, Principal Component Regression (PCR) will be used. In PCR, Singular Value Decomposition (SVD) may be used to factor the X matrix into the matrixes X = UEV^ [3]. Matrix E gives the singular values Ci > ... > cr^ > 0. The singular values tell us if any of the rows in X is linear dependent on other rows, in this study meaning that one or more sensors may be redundant. The soft sensor model for estimating the liquid level and the oil / water interface of the separator will then be:

Y=XB

(7)

4. Separator rig A separator rig is built as a 1:3 scale model of a first stage vertical separator rig in a real separator train. The separator rig used for the experiment is equipped with pressure sensors, valves, pumps, and buffer tanks as shown in Figure 1.

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N.-O. Skeieet al ^ - j Over pressure valve

Top pressure sensor Well input

Figure 1. Process overview of the separator rig.

5. Soft sensor from experimental data Water and olive oil are used in the experiment, and the input pumps are used to set up different filling fractions of water and oil in the separator. The sensor data x are logged using the C # based software application, while the output data y are measured manually by visual reading against a centimeter tape which is glued to the separator. MATLAB was used for further processing of the logged data. The relative pressure data were mean-centered [4], and a MATLAB script was used to generate the meancentered X matrix, and SVD of the X matrix is performed. The E matrix from the SVD gives the singular values cr^ G {594.5, 83.4, 42.4, 28.1, 11.1}. The X matrix is thus of full rank, and the least squares method can be used; no pressure sensors are redundant, as used with different heights shown in figure 1. Equations 6 and 7 are then used, and figure 2 displays the estimated levels and the manually read levels from the training set used for the model development. The levels are the liquid level (water 4- oil) and the interface (water) level. This is the first model developed in this study, and a clear relation is found between the estimated and manually read levels. For validation of the model, a validation set was acquired. The estimated and manually read values of the validation set using equation 7 are shown in Figure 3, still showing a reasonably good relationship between the values. Manually read levels and the estimated soft sensor levels relationship and residual of the validation set are shown in Figure 4.

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Manually read levels and soft sensor levels using the training set 800 r

. 650h

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Figure 2. Evaluating the soft sensors with the training set.

Manually read levels and soft sensor levels using the validation set SOOr750{700 ^650h E

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Figure 4. Manually read level and goft sensor level relationship and residual.

6. Conclusions This first study shows that it is possible t o use standard pressure sensors and multi sensor data fusion t o estimate the levels in an oil / water separator. In spite of this being the first soft sensors developed in this study, there is a reasonably good relation between the read and estimated values. The current sensor system has an absolute accuracy of approximately 45 mm. In t h e next study a more accurate sensor system will be used, and the accuracy will be improved to approximately 10mm. The level residuals for t h e interface level shown in figure 4 are better than ±50 mm. Using an improved accuracy of the sensor system, and more samples, the next soft sensor developed is expected t o give an even better result. Future work will include adding a guided radar, attempting t o make t h e soft sensor model even more accurate. The training and validation set will also be used with a neural network t o test the performance and accuracy of a neural network model for estimation of the separator levels. REFERENCES 1. 2. 3. 4.

J. Fraden, Handbook of modem sensors^ Physics, Designs, and Applications, 3rd Edition, Springer, 2004, ISBN 0387007504 D. C. Montgomery, E. A. Peck and G. G. Vining, Introduction t o linear regression analysis, 3rd Edition, Wiley, 2001, ISBN 0471315656 G. Strang, Introduction to Linear Algebra, 3rd Edition, Wellesley-Cambridge Press, 2003, ISBN 0961408898 T . Naes, T . Isaksson, T. Fearn and T. Davies, A user-friendly guide to Multivariate Calibration and Classification, NIR Publications, 2002, ISBN 0952866625

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Product quality estimation using multi-rate sampled data Bao Lin^, Bodil Recke^, Torsten Jensen^, J0rgen Knudsen^, Sten B. Jergensen^ ^CAPEC, Department of Chemical Engineering, Technical University ofDenmark ^FLSmidth Automation, Valby 2500, Denmark Abstract This paper investigates different approaches to develop soft sensors from multi-rate sampled data. The data lifting approach consists of two steps, identifying a model with a slow/lifted sampling period and extracting a fast model. Approaches based on direct extraction and linear regression are briefly reviewed, followed by reformulating the task as an unconstrained optimization problem. An illustrative example concerning design of a free lime soft sensor for cement kiln systems is presented. Using data collected from a simulation system based on first principles models, a weighted partial least squares (WPLS) approach for soft sensor development is compared with data lifting techniques. Case studies reveal the superior performance of the WPLS approach. In addition the product quality for cement kiln systems can be estimated reasonably well, demonstrating the potential to be used for effective quality control. Keywords: soft sensor, multi-rate data, regression analysis, cement kiln process 1. Introduction In most chemical processes, quality measurements have slower sampling rate than other process variables. A major obstacle for effective quality control is the lack of frequent or real-time information, due to the time delay associated with lab analysis or slowly processed quality measurements of online analyzers. Therefore, soft sensors that are able to provide a reliable real-time prediction can be essential for effective product quality control. This paper presents a systematic framework to develop soft sensors from multi-rate sampled data. Plant log systems usually provide the operating data logged with different sampling frequency: high frequency process measurements and low frequency quality measurements. Therefore it is impossible to directly identify a process quality model at the fast sampling rate using the original data. Lu and Fisher (1989) proposed a least squares output estimation approach for multi-rate datasets. A transformation is performed on the original system by multiplying a polynomial such that the intersample value of the output is no longer required for identification. Lifting technique is a widely used approach to handle data collected at multiple sampling rates (Wang et al, 2004; Srinivasaro et al., 2005). The data lifting technique transforms multi-rate sampled data into a uniform sampling rate dataset at the lower frequency by stacking the variables with a smaller sampling interval. A model in the lifted time domain is first identified, from which a model of fast sampling domain is extracted. A weighted partial least squares (WPLS) approach has been applied to develop quality estimators from multi-rate data (Lin et al., 2005). A weighting value of 1 is applied to process measurements that correspond to the time instant when a new quality

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measurement is obtained, 0 to others. A soft sensor with fast sampHng period is then directly derived using available process measurements. For a data lifting technique, the key issue is to reliably extract the fast sampling model. When the ratio of slow/fast sampling period is large, this approach might suffer from high numerical sensitivity or even numerical infeasibility. In addition, the direct extraction of the fast system might also lead to an infeasible system, i.e., with imaginary elements in the system matrix. This paper reformulates the fast model extraction within an optimization framework, in order to minimize the possibility of obtaining infeasible systems. The purpose of this paper is to investigate development of a systematic procedure for multi-rate soft sensor and to illustrate the application on cement kilns, which feature an offline quality measurement. A comparison between an estimator obtained with the lifting technique and WPLS approach is also performed. Section 2 briefly reviews a data lifting technique and techniques to extract fast models, followed by the proposed optimization framework. Section 3 presents the systematic multi-rate soft sensor development procedure. Case studies for multi-rate soft sensors for cement kiln systems are illustrated in section 4, followed by discussion and conclusions. 2. Lifting technique for multi-rate system Process measurements {w(/)| are sampled with the period T^, while the quality measurements, {j^(A:)jhave a sampling period pT^.

Data hfting technique

(Khargonekar et al., 1985) reorganize the original data set by stacking the fast sampled variables. The lifting operator is defined as:

u = LpU

["(1)" 'uip+iy =J

u{p + 2)

M(2) ?

where p>l

(1)

? ' ' * 9

lu(p)_ u(2*p) u(l)

' u{k*p + l) ~ u(k*p + 2)

u(2)

u{{k + r)*p)_ uik)

is an integer. Therefore, the lifted input sequence U has the same period

as the slowly sampled output. A process model with a slower sampling period,

pT^,

can be identified straightforwardly. In order to obtain the inter-sample value of the slow-sampled process measurements, {j^CA:)}, a fast model is commonly extracted according to the relationship between the slow and the fast system. Given a discrete linear system with a sampling period of 7^: x^+i=Ax,+Bu, (2) A new discrete linear system is obtained using lifted data:

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y^=cxt where the parameter relations can be derived:

B = [A^

B

A^'B

AB

(4)

B]

C=C It is straightforward to extract matrices B and C from the lifted system. The system matrix A can be obtained using at least one of the following two approaches:(Wang et al, 2004) The first approach starts from the equation A = A^ where A is calculated j_

directly as A = A ^ . It is necessary to obtain a real-valued matrix. Therefore, A is j_

j_

approximated with the real part of A ^ . That is, A = Re( A ^ ) . The second approach derives A

B = [A^

B

AB

through a linear regression. Given

BJ,

B, = A ' - ^ B / = 1,2,...,;7

(5)

which is the i^^ block of B . The following relationship holds: B.^i=AB. z = l,2,-..,;7-l

(6)

AB = AB^

(7)

and rewrite in a matrix form:

[AB B^

B^_,

-

BJ=A[B^

r

B^_,

B^_,

-

BJ

(8)

Y

Then,

A = (rY^)(YY^)"'

(9)

A third approach is proposed here in which the extraction of fast models is reformulated within an optimization framework. Assume the system matrix of the fast system A is expressed as: A = Re(A) + zIm(A) (1) the objective function is expressed as the deviation between A^ and A , plus the norm of the imaginary part of A :

argmin|A^-AII +||lm(A)|U where

L can be taken as 1, 2 or the oo norm.

(1)

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The objective function can also be defined as the deviation between the impulse response of the lifted and fast models:

?ir%mm^{h'{i)-h{pi)y

(12)

An unconstrained optimization problem is then formulated, and solved, e.g. with the "fininsearch'' function of MATLAB or any other unconstrained minimizer. fminsearch finds the minimum of an unconstrained nonlinear scalar function with simplex search. Since the quality measurements are sampled at a slower rate than other process measurements, the same value is inserted in the log system till a new lab analysis is obtained. Therefore, a weighting vector {w(A:)}: w(^) = 1^

,,

0

.

(13)

otherwise

is used to downweight the inter-sample value of the quality measurement as follows: M(l)

y{2) yip

"(2) uip - 1) u{p) u{p + \)

-1)

yip) yiP + 1)

f

(14)

uik)

yik) VL ' J

L

•

J y

A regression relationship is developed using the weighted matrices of regressors and the dependent variable, which is then applied to all available samples of regressors in order to obtain the inter-sample estimation. A soft sensor derived using the weighted partial least squares (WPLS) approach is compared with those developed by extracting the fast model from a lifted system. 3. Case studies The proposed approaches to design data driven soft sensors from combining fast process measurements with slowly sampled quality data are compared for a cement kiln simulation system. The product quality of a cement kiln is indicated by the amount of CaO (free lime) in clinker. In practice, the quality measurement is generally only available every 1 or 2 hours with a time delay of about 40 minutes. It is desirable to develop a soft sensor that can accurately predict the content of CaO in real time for effective quality control. Data collected fi:om a Cemulator® simulation system which was based on a first principles model (Delfter, 1982), are used in this study. Pseudo Random Binary Sequence (PRBS) signals are applied to the kiln speed, feed, tertiary air damper and IDFan power to excite the process. The free lime content of the clinker is available every hour, while other process measurements are logged every 10 minutes, which include kiln speed, kiln feed, position of the tertiary air damper, induced draft fan

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Sample Figure 1. Free lime estimation obtained with quality measurements and lifted data (IDFan) power, fuels to calciner and kiln, plus several temperature measurements within the kiln system. A data block of 2000 samples is selected: 1800 samples to derive the model (300 quality measurements) and 150 samples for validation. A state space model is identified from lifted data using the identification toolbox of MATLAB. As shown in Figure 1 {start - quality measurement; solid line state space model estimation), the free lime soft sensor obtained with lifted data is able to predict the trend of the quality measurement reasonably well. In order to estimate a model for the inter-sample behaviour the three approaches described above are investigated. The fast model obtained from the first approach is unstable, due to the omission of imaginary part of A^^. The second approach yields a solution, however, the validation shows significant sensitivity as seen in Figure 2. The PRESS (one-step-ahead prediction residual sum of squared errors) between the extracted model and quality measurement is 31.29. Since the simplex search method is a local optimization approach, the final result depends on the initial guess. 10 runs are performed for approach 3 using both objective functions (eqs. 11 and eql2). The best solution of 10 runs performs similarly to the 2"^ approach. However, large variations are observed in the results of 10 tests, which indicate the high probability of arriving at a locally optimal solution. Therefore, a more reliable algorithm is desirable to obtain a consistent solution.

60

80

Sample Figure 2. Free lime estimation with fast model extracted by approach 2 A CaO soft sensor is developed using WPLS approach. As mentioned before, weighting vectors are multiplied onto inter-sample inputs and outputs of modelling data. During the model validation phase, this regression model is applied to all input samples. Therefore, the inter-sample behaviour of the soft sensor can be calculated.

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2

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Sample Figure 3. Free lime estimation with WPLS approach during vahdation period As shown in Figure 3, the soft sensor follows the trend well, however, deviations are observed in a couple of periods. Comparing with the lifting technique based approaches (see Figure 2), WPLS approach shows improved performance for providing inter-sample estimation. The PRESS between the extracted model and quality measurement is reduced to 7.25. Therefore lifting with regularization should be developed to investigate which regularization is to be preferred. 4. Conclusions This paper investigates the approaches of developing soft sensors fi-om multi-rate sampled data sets. Since product quality is normally infrequently sampled and only available with a time delay, inter-sample information is most valuable. A two-step approach based on data-lifting techniques is investigated: the identification of a system with slow sampling rate followed by the extraction of a fast model. An optimization reformulation is provided to overcome the shortcomings associated with direct extraction methods. The proposed approach is applied to design a free lime soft sensor for product quality. Data collected from a simulator based on first principle models are used to compare the approaches based on data-lifting techniques and WPLS method. Preliminary results reveal that the soft sensor from the WPLS approach outperforms those of the investigated data-lifting techniques. One reason might be the large ratio between the frequency of standard measurements and that of the product quality. More importantly, the soft sensor developed with the WPLS approach is able to provide reasonable prediction for the free lime, demonstrating potential of using regularization combined with lifting to provide estimates for effective quality control and optimizing process operation. References Delfter, O. (1982). Simulation for Start-up and Operation of a Rotary Dryprocess Kiln,(in Danish), Thesis, Technical University of Denmark. Khargonekar, P.P., K. Poolla and A. Tannenbaum (1985). "Robust Control of Linear TimeInvariant Plants Using Periodic Compensation." IEEE Transactions on Automatic Control 30(11): 1088-1096. Lin, B., B. Recke, J. Knudsen and S.B. Jorgensen (2005). A Systematic Approach for Soft Sensor Development. European Symposium on Computer Aided Process Engineering -15, Barcelona. Lu, W. and D.G. Fisher (1989). "Least-Squares Output Estimation with Multirate Sampling." IEEE Trans. Automat. Control 34(6): 669-672. Srinivasaro, M., R.D. Gudi and S.C. Patwardhan (2005). Identification of Fast-Rate Nonlinear Output Error ModelsfromMulti-Rate Data. IF AC World Congress, Prague. Wang, J., T. Chen and B. Huang (2004). "Multirate Sampled-Data Systems: Computing Fast-Rate Models." Journal ofProcess Control 14(1): 79-88.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A Novel Solution Approach for Quality-based Retrimming Optimization liro Harjunkoski and Marco Fahl ABB Corporate Research, Wallstadter Str. 59, 68526 Ladenburg, Germany Abstract In this work, we present a novel mathematical programming formulation for solving the re-trimming problem. The proposed approach leads to optimal or close-to-optimal solutions with the effect of a significant reduction in quality loss, i.e. the economical loss based on degraded quality. The model introduced is able to consider quality profiles along the jumbo-reel as well as the requirements attached to each roll providing a geometric representation on the trim-loss problem. Keywords: Trim-loss problem, cutting stock, jumbo-reel quality, re-trimming. 1. Introduction The primary objective for cutting problems in paper production is trim loss minimization. A typical example is the cutting of a wide paper reel (jumbo reel) into smaller paper rolls, which are either end-customer rolls or intermediate products waiting for further processing, see Fig. 1. The general problem formulation tries to find a cutting strategy for producing the required roll widths using as little material as possible, i.e. by minimizing the trim loss. A secondary objective may also be to minimize the number of different patterns and to sequence them such that knife-setup actions are avoided.

Figure 1. Jumbo-reel cutting & the trim-loss problem The trim-loss problem as such is a discrete problem. Thus, the problem size explodes fast with the number of rolls considered due to the large number of alternative cutting strategies. Consequently, there exist lots of heuristic-mathematical approaches to solve the problem efficiently, without guaranteeing global optimality of a solution. These include rounding heuristics, column generation, solving problem only partially and other knapsack-type of algorithms. In modem paper mills, the trim-loss problem for the paper machine/winder is commonly solved as an integrated part of production planning already long before the jumbo-reels are actually being produced. Thus, uniform quality distribution of the paper to be produced is assumed. During the paper-making process, product quality data is collected from various on-line measurement devices. Comparing the actual quality distribution with the original trim patterns of a jumbo reel, the predetermined cutting plan may be far from optimal. For instance, the most valuable customer roll might have been assigned to the worst position with respect to product quality of the reel and must

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thus be rejected. Many providers of trim optimization packages/solutions therefore allow the user to perform a quality check on the trim solution and to manually adjust the cutting patterns to better meet the customer requirements and to react to quality deviations or defects. Today's modeling approaches of the trim optimization problem do not support qualitybased trim optimization, since the standard trim-loss (or cutting stock) problem does not take into account the exact position of each roll in a pattern, but merely focuses on total amounts. Only very few contributions into this direction are reported in the literature (e.g. Bergman et al., 2002). 2. A Novel mathematical programming approach Since adding the quality aspect to existing standard models would increase the complexity such that the problem becomes intractable (additional non-convexities, high amount of new discrete variables), an alternative modeling approach for quality-based trim optimization is needed. Here, we present a novel mathematical programming approach for automatic computation of an optimized solution of the re-trimming problem. The model covers quality profiles of a jumbo-reel through a geometric representation of the trim-loss problem. Driven by tight performance requirements, the mathematical model presented here considers only one cutting pattern at a time. A repositioning of the rolls on the jumbo reel is suggested taking into account the exact geometrical position as well as the quality information along the reel-width. The approach results in optimal or close-to-optimal solutions significantly reducing the quality loss, i.e. the economical loss based on degraded quality. In the following, we propose a two-step approach. First, a discretization-based method is used to generate a good approximation for the optimum solution, In a second step, a continuous exact approach is applied in order to ensure the feasibility of the final solution. This allows us to overcome non-convexities of the problem and to ensure that a close-to-optimal solution is obtained fast. From a conceptual point of view, a close relationship to general scheduling methodology and corresponding model formulations can be observed. Both approaches are first presented separately, before the combination of them is discussed. 2.1. The discrete model (discretization-based model) The main idea of the discretization-based approach is to divide the jumbo-reel width into fixed "slices". For each slice a respective cost coefficient for every roll is defined. The cost coefficient is based on quality classes, e.g. A-, B-, or C-quality (see Fig. 2). The final quality class for each roll is calculated by combining the quality mapping with key product-roll requirements.

AjA; A'A|A|A;Aii^A|BiBiBiB|Ai A'AiA|/J

Figure 2. Discrete approach illustration The approach leads to solving an MILP and is similar to the one proposed by Ronnqvist (1995) for cutting wooden boards. However, to meet the requirements of the particular problem at hand, some aspects of the Ronnqvist formulation (e.g. considering different cut options) can be removed. It is worth noting that the simplified Ronnqvist problem is almost equivalent to the State-Task Network problem formulation for short-term

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scheduling by Shah et al. (1993). There, the scheduling time-horizon is discretized by a fixed number of gridpoints, which are assigned through binary variables. In the retrim optimization context, the width takes the corresponding role of the time. Here, the binary variable, xdrp equals one at the jumbo-reel position, 7, in which the roll, r, starts. For the full problem formulation we also need a roll value coefficient for each potential roll position, Crp and the width of a roll, Wr, as well as a discretized roll width, Wdr. The problem formulation follows.

max^c^^. W^ 'Xd^j

r

(1)

j\{jp-Wd,+\

Y,xd,j=l

Vr

(3)

J

xd^je {0,1} The objective function (1) maximizes the quality yield, i.e. ensures that the total value of all rolls is as high as possible. Equation (2) ensures that no rolls should (virtually) overlap and Eq. (3) states that each roll must occur exactly once. The cost coefficients in (1) are determined by matching existing quality data with customer/product requirements, also taking into consideration the amount of material. The problem results in an optimal cutting plan with respect to a chosen grid density. For typical jumbo-reel widths of up to 8000 mm an exact grid (e.g. deltaw = 1 mm) would make the problem size intractable. Therefore, a coarser grid (delta_w >10-20 mm) must be selected and all widths need to be rounded to match the chosen discretization. In this case the roll widths have to be rounded down in order to maintain the feasibility of the problem (e.g. 578 mm becomes 570 mm when using a 10 mm grid). Due to the rounding error, a post-processing step is needed in order to ensure that the final solution is realistic and can be implemented on the winder. 2.2. The continuous model (slot-based model) In the continuous/slot-based approach, the jumbo-reel width is divided into quality sectors, s, beginning and ending at pre-defined points, ^S/ and 5'/. Each sector has a respective quality class coefficient for every roll (again: A, B, C), see Fig. 3. The quality value is calculated by combining the quality mapping with some key product roll parameters.

i

1540 mm I

2300 mm

Figure 3. Continuous approach illustration The model is based on the continuous time-slot scheduling approach by Pinto and Grossmann (1995). The slots are ordered from left to right and the borders between the

1398

/. Harjunkoski and M. Fahl slots, Wn^ and fF/, are continuous, i.e. able to adapt to the respective customer roll widths, Wr. Each roll is assigned to a slot and quality sector through binary variables, Xm and Xrs. The sector is selected with respect to the mid-point of the roll, r^'"'^, in order to simplify the quality mapping.

"^^^Ya^rsK'^rs r,s

(4)

Z^™=1 ^'^

(5)

n

r

Wf=K"+I,Xrr.-K

V«

(7)

r

Wf=W„l,

V«|«<|iV|

(8)

r;"<W„'+i-W^+M-{l-xJ

^r,n

r;">Wf+^-W^-M-{l-xJ

\fr,n

X^„=l

Vr

(9) (10) (11)

^m/J

<Sf + M-{\-xJ

\fr,s

(12)

^m/J

>5f-M-(l-x„)

Vr,^

(13)

W„\W„\rr''e^'

^.,^„€{0,1}

The objective function (4) maximizes the quality yield, as in (1) but here with respect to sectors and not with respect to single grid points. Constraints (5) and (6) ensure that each roll is assigned to exactly one designated slot. The distance between the start and end point of a slot depends on the assigned roll, as stated in Eq. (7). Constraint (8) hinders (virtual) overlaps of rolls and defines that the next slot must start exactly at the end point of the previous one (i.e. slots are adjacent). The mid-roll position is defined in Eqs. (9) and (10) and depends on the slot assignment, as well as, the slot starting point. In Eq. (11) each roll is assigned to one quality sector. Finally, constraints (12) and (13) make the decision to which quality sector the mid-point of a roll belongs to. This approach results in a global optimal cutting strategy taking into account the quality distribution along the jumbo-reel. However, the computation times may be very high -

A Novel Solution Approach for Quality-Based Retrimming Optimization

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with some test problems a first solution was found relatively fast (< 1 CPU-min) and finding the optimal solution did also not take more than a few minutes, however, proving the optimality took typically around 1-2 CPU-hours. As the complexity of this approach also grows with an increasing number of quality sectors, a combination of the discretization-based and the slot-based approaches is proposed. 2.3. Combined discrete-continuous approach The proposed two-step strategy makes use of both of the abovementioned approaches. The resulting steps are: 1. Solve the discretization-based problem with a grid of e.g. delta_w=10 or deltaw = 20 [mm] (inexact, but fast). 2. Fix the sequence of the rolls (variables, Xm defining the slot assignments) and solve the exact slot-based problem. Using this strategy, the slot-based problem can be solved within afi-actionof a second and the resulting solution is exact, feasible and close-to-optimal. The mathematical models presented above remain unchanged.

3. Example results In the following some examples are given in order to illustrate the method. The example roll widths are shown in Table 1. Table 1. Example problem data (roll widths) Roll

1

2

3

4

5

6

7

8

9

10

Width (mm)

1790

1100

825

485

770

750

650

580

580

385

The considered jumbo-reel trim width is 8000 mm. The sum of the roll widths to be trimmed is 7915 mm, resulting in a trim loss of 85 mm. Each roll is assumed to have exactly the same quality requirements. Therefore, the example can be simplified by directly dividing the jumbo-reel into various quality zones. If a roll spans over several quality zones, it is valued according to the worst quality. For illustration purposes, three different quality distributions are shown in Fig. 4 (case 1 on the top, followed by case 2 and case 3), where the quality is expressed in terms of A-, B-, and C-quality. For the optimization problems reported below, we calculate the value of each roll based on the following assumptions: Paper weight = 80 g W , price = 500 €/ton, A-quality (full quality) = 100% of price, B-quality (minor defects) = 70% of price and C-quality (rejected) = 0% of price, i.e. no value. 3760

1

1210

1700

A

B

900

1210

C ' B

2020

2220

B

A

A

1985

2345

1900

440

A

B

760

c

B

A

1185 38£

A

c

1160

820

A ' B

Figure 4. Example jumbo-reel quality mapping The example cases are solved using GAMS/CPLEX 9.1 and the results are shown in Table 2. The size of the discretization-based models is constant (1611 constraints, 14417 0-1 variables) and is therefore not shown in the table to save space. The

/. Harjunkoski and M. Fahl

1400

optimization time for all optimization runs was limited to 1000 CPU-s (optimality criteria 0,1 %) and the results obtained until then are reported. Table 2. Example results Discretization-based (deltaw = 5 mm) Case

CPU-s

Combined (deltaw = 20 mm)

Slot-based (continuous)

Obj

con

var(c/d)

CPU-s

Obj

1

130,2

1672,10

310

31/130

1000

1723,05

0,7

1702,88

2

1000

1426,95

348

31/149

1000

1429,43

259,9

1426,95

105,1

1364,45

406

31/178

1000

1364,45 22,8

CPU-s

Obj

1208,44

The specially selected examples show various aspects of the problem: The highly varying CPU-times and the fact that the combined approach is not always giving the best solution can be seen in case 3. In this case the 20 mm discretization causes a rounding error such that a quality sector change is ignored. This behaviour is avoided when using a 10 mm discretization. However, the combined strategy gives a good answer within a reasonable time. An example solution for case 2 is illustrated in Fig. 5.

Figure 5. Resulting trim set (quality: A=white, B=yellow, C=red) 4. Conclusion The two-step quality-based re-trimming solution is able to efficiently generate close-tooptimal solutions. The algorithm can be further tuned for performance/solution quality by e.g. changing the width discretization, CPU-limits and optimality criteria. The same methodology can also be applied to quality-driven sequencing of trim sets in machine direction. Furthermore, the combination of the optimization approaches for width and machine direction can result in an automatic quality-based re-trimming of an entire jumbo-reel.

References Bergman J., Flisberg P. and Ronnqvist M. (2002). Roll cutting at paper mills. Control Systems 2002, pp 159-163. Pinto, J.M. & Grossmann, I.E. (1995). A continuous time mixed integer linear programming model for short-term scheduling of multistage batch plants. Industrial and Engineering Chemistry Research, 34, 3037 - 3051. Ronnqvist M. (1995). A method for the cutting stock problem with different qualities. European Journal of Operational Research, 83, Issue 1, pp. 57-68. Shah, N.E., Pantelides, C.C. and Sargent, R. (1993). A General Algorithm for Short-Term Scheduling of Batch Operations - II. Computational Issues. Computers & Chemical Engineering, 17, pp. 229-244.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Chapter 1

An Integrated Framework Based on Data Driven Techniques for Process Supervision B. Bhushan,^ Jose A. Romagnoli^ ^ Department of Chemical Engineering, University of Sydney, Sydney, 2006, Australia ^Department of Chemical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA. Abstract An integrated framework for process monitoring and supervision is proposed. Firstly, the data is freed from outliers using mean minimum distance clustering technique. A novel technique for unsupervised pattern classification is proposed. It is applied for simultaneous fault detection and diagnosis. A continuous pilot plant is used to check the efficiency of the proposed strategy. The result shows that the proposed framework can be used for process supervision of real time, non-linear systems. Keywords: Fault detection, Fault diagnosis. Unsupervised pattern classification, Mean minimum distance (MMD) clustering 1. Introduction The advent of faster and more reliable computer systems has revolutionized the manner in which industrial processes are monitored and controlled. These advances have resulted in the generation of a large amount of process data. These process data may sometime contain outliers due to faulty sensors, equipment failure or other unmeasured disturbances. For reliable assessment of the process status, it is necessary to remove these outliers from the measurement data. Fault detection and diagnosis are two most important tasks of process supervision. Principal component analysis (PCA) is extensively used for feature extraction and fault detection (Jackson, 1991). On the other hand, all fault diagnosis methods can be broadly classified into two groups; model based methods (observers, SDG, parity space etc.) and process history based methods (PCA/PLS, neural networks, expert systems etc.). Model based methods is been used by many researchers for fault diagnosis (Vaidhyanathan et al., 1995; Vedam et al., 1997). The main disadvantage of these methods is that the complexity of the model increases with increase in size and complexity of the plant. Although research and development in each of these areas is active, little has been done to establish an effective framework for process supervision. Research in interfacing multivariate statistical tools with fault diagnosis has taken off in the last few years (Vedam et al, 1999; Norvilas et al. 2000), yet these integration attempts are qualitative in nature. Though, the main contribution of this work is the integrated framework; self organizing self clustering network (SOSCN) for unsupervised classification is also a novel contribution which is applied for simultaneous fault detection and diagnosis.

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1402

The remaining part of the paper is organized as follows. In section 2, framework for process supervision is presented. Outlier detection based minimum distance clustering is explained in section 3. Section 4 contains SOSCN in detail followed by case study and conclusions in section respectively.

proposed on mean proposed 5 and 6

2. Integrated Framework In this integrated approach for process supervision, following steps are applied in succession: (1) Outlier detection using mean median distance (MMD) clustering, (2) Feature extraction using PCA and (3) Simultaneous fault detection and diagnosis using SOSCN. The overall framework is shown in fig. 1. Process Measurement

Data Preprocessing

^

M

Feature Extraction

) \

Fault Detection and Diagnosis

Fig. 1. Integratedframeworkfor process supervision In data preprocessing step, the outliers are detected and removed from the normal data. These refined data is projected from measurement space to the reduced feature space using PCA. Furthermore, the proposed SOSCN is used for simultaneous fault detection and diagnosis. These steps are discussed in detail in next sections. 3. Outlier Detection Outliers are defined as those measurements in which error does not follow the statistical distribution of the bulk of the data. Normally, outliers are a small fraction of the whole data and have little or no relation with the main structure. In other words, the outlier lies outside the pattern or cluster formed by the normal data. In this work, the non-iterative clustering based on mean minimum distance is used (Yin et al., 1994). Since the variation of individual variables may be different, each variable should be weighted by its own variance (Chen et al., 1998). Therefore, for a given window size M, '\fx^^\ x^^\ ..., x^^ are M measurements in N-dimensional space, the mean minimum distance VM is defined as 1

.i^f-^f?^

^

'M /=1

]*l

(1)

\k=\

where Vk is the k^^ diagonal element of the covariance matrix V. A moving window is used and if minimum distance of any measurement is more that ITM it is considered as an outlier.

4. Self Organizing Self Clustering Network (SOSCN) Let S = {x %..., xJQ) ^ 7 is a sample of Q, n-dimensional feature vectors from a population P such that x "^ = /x;'(I)\ ...,x„'(1)7. The process begins by randomly organizing the order of

An Integrated Framework Based on Data Driven Techniques for Process Supervision 1403 the feature vectors. A node /^^ is generated with the center c^^^ = {x/^\...,xf^} i.e. by assigning the first sample feature vector as the center of the first node and spread a^^^ = { (JP\ ..., aj^^} where oP^ , i = 1, ..-, n are predefined constants. The belongingness or the membership of a vector x^^ to a node /^^ is defined as

F,(x^>) = exp(-7/2)5;

(2)

(f'y

In this approach, the winner is decided based on the highest inclusion of the vector to the node and both center and the standard deviation of the node are learnt. The winner node k* for a sample vector x^^ is defined as ^*=arg max Fj^(x^^)

(3)

i
where K is the number of node at that instant. We propose to update all the nodes in which the feature vector has an inclusion greater than a pre-specified value. However, the amount of the movement of the node is governed by membership value, if the membership value is high, the node will move more towards the exemplar vector than when it is low. Therefore, the function which should control the movement must ((i) be nonnegative, (ii) be zero when the membership between the exemplar vector and the node is 0, (iii) have a maximum value of 1 and (iv) go to zero as number of iteration -^ 00. The function which can satisfy the criterion above can be represented as 0 ) = Fj, (x<^>) *

^

^—-r^

(4)

(7 + exp((F^(^jc^^)*zYer))

where iter is the number of iterations. Therefore based on the evaluation function (Nomura et al., 1995), the update rule for center and standard deviation for each node can be represented as: (k)

c)

^

(k) ^ i,new

(k)

,

j .

(k) _ ^ 7 „ y * i,old

^"^yk

(k)

(k)

.

ioidyy^i

...

((^k_^{k)

^i,oid)

i,oid )

w

Each sample feature vector is presented to the network and the membership of the vector to all the nodes is calculated. The update nodes are selected using arg l
(Fk (X^^)>

/threshold )

(7)

If Fyt(x^^) < /threshold, then a new node is created with centre as the presented vector and spread as cr -{ a/^\..., aj^^} else the centre and spread of all the nodes for which Fiix^^) >/threshold is Updated using (5) and (6). There are two termination criterion decided for the methodology: (1) number of maximum iteration is reached, or (2) the total movement of the nodes is less than a user defined value. This approach applied distributes the nodes in the sample space without specifying the number of nodes as well as gives an idea about the concentration of the feature vector in the space. However, to know the number of clusters present a based on Z- membership function is proposed. A Z-membership function is defined as

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B. Bhushan andJ.A. Romagnoli 1,

x
WH , \"^

'

2\ , [b-aj 0, x>b

(8)

<x
where a and b locate the extremes of the sloped portion of the function, b can be calculated as: K

/

,^km'

km

h=^^

k = lX3,...,K

(9)

/ . km m=l

where K is the number of nodes, D^m is the Euclidean distance of the node m from node k and Skm can be defined as: (10)

km=

(7 + exp(D^)) Since the Euclidean distance is a non negative value a is taken as 0. The membership value of each node with respect to other nodes is calculated. A graph similarity matrix P (Looney, 1997) is created as follows: if juilzT) > r then/7,7 = 1 else p,y = 0, where jUk(/") is the membership value of m^^ node with respect to k node and r is the threshold defined by the user and determines the spread of the clusters. Once the matrix is created, column searching method is applied for finding out the classes. It is a recursive process that comes back to the remaining column after it runs out of rows for the current column. When no more columns remain for searching, all vectors are assigned to the classes. If a new data point is presented to the network, its membership value with each node is calculated. If the membership value is greater than a pre-specified value the data belong to the class of the node, else the data belongs to a new class. 5. Case Study The proposed strategy is applied to a section of an existing twin-CSTR pilot plant. This contains two CSTRs, a mixer, a feed tank and a number of heat exchangers. Each CSTR consists of a reaction vessel, a steam jacket, a cooling coil and a stirrer. Material from the feed tank is heated before being fed to the first reactor and the mixer. The effluent from the first reactor is then mixed with the material in the mixer before being fed to the second reactor. The effluent from the second reactor is, fed back to the feed tank and the cycle continues. Nine variables Fin (feed flow rate in). Tin (temperature of feed in), Tc,in (temperature of cooling water in), Ts,in(temperature of steam in), Lvl (level of the reactor), Fout (feed flow rate out). Tout (temperature of feed out), Tc,out (temperature of cooling

An Integrated Framework Based on Data Driven Techniques for Process Supervision 1405 water out), Ts,out (temperature of condensate) related to first CSTR is considered for this study.

il^^^w^ ^*'SvJU^vV;^A^W

50

100

t50

200

rTn Fig. 2: Outlier detection of all the nine variables using MMD algorithm (* - outliers). Plant was first run under normal conditions followed by six different types of abnormalities. 200 data points at 5 seconds interval were collected for all these condition. Outliers of magnitude 8-10 times the standard deviation of the data was randomly added to the actual measurements.

25 J 20 J 15

+

*

+ o

10

Trx-biaslO T-bias60

• * •

Cool-foul Cat-deact Normal Fin-dist10 T-bias-60

5J

oJ -5 J

Fig. 3. Classification of all the data points in 7 different classes using SOSCN.

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B. Bhushan and J. A. Romagnoli

Mean minimum distance clustering was applied to detect the outliers and the result is shown in figure 2. It can be noted that the algorithm could correctly detect all the outliers. Furthermore, PC A was applied to the preprocessed data for feature extraction. Three principal components could explain 62% of the variance in the data. Proposed SOSCN was applied for classification of the data without specifying the classes. The result is shown in figure 3. It can be seen that all the data were correctly classified in 8 classes. 25 data points of all these conditions were used for validating the capability of the strategy for fault detection and diagnosis. The membership value of these points with the nodes was calculated. If the membership value was greater than 0.75 the point was assigned to that class. All the points were assigned correctly to the appropriate class. 6. Conclusion An integrated framework for process monitoring and supervision is proposed in this work. Firstly, the outliers are removed using mean minimum distance clustering algorithm followed by PCA for dimensionality reduction. A novel technique is proposed in this work for unsupervised classification. The advantage of the proposed method over existing techniques is its ability to automatically determine the number of clusters. Since the methodology is based on learning, it is computationally less expensive and the result is not affected by the initial guesses. This technique is applied for simultaneously fault detection and diagnosis. The proposed fi-amework is applied for process supervision of a continuous pilot plant and the results show promising direction for real time process monitoring and supervision. References A. Norvilas, A. Negiz, J. DeCicco and A. Cinar (2000). Intelligent process monitoring by interfacing knowledge-based systems and multivariate statistical monitoring. Journal of Process Control, 10, 341-350. C. G. Looney (1997). Pattern recognition using neural networks: theory and algorithm for engineers and scientists. Oxford University Press, New York. H. Vedam, and V. Venkatasubramanian (1997). Signed diagraph based multiple fault diagnosis. Computer and Chemical Engineering, 21, S655-S660. H. Vedam and V. venkatasubramanian (1999). PCA-SDG based process monitoring and fault diagnosis. Control Engineering Practice, 7, 903-917. J. Chen, and J. A. Romagnoli (1998). A strategy for simultaneous dynamic data reconciliation and outlier detection. Computer and Chemical Engineering, 22, 559-562. J. E. Jackson (1991). A user's guide to principal components. Wiley, New York. P. Y. Yin and L. H. Chen (1994). A new non-iterative approach for clustering. Pattern Recognition Letters, 15, 125-133. R. Vaidhyanathan. and V. Venkatasubramanian (1995). Diagraph-based models for automated HAZOP analysis. Relaibility Engineering and Systems safety 50(1), 33-49. T. Nomura and T. Miyoshi (1995). An adaptive rule extraction with the fuzzy self organising map and a comparison with other methods. Proceedings of ISUMA-NAFIPS '95, 311-316.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Reliable multi-objective optimal control of batch processes based on stacked neural network models Ankur Mukherjee and Jie Zhang School of Chemical Engineering and Advanced Materials, University ofNewcastle, Newcastle upon TyneNEl 7RU, UK, E-mail:[email protected] Abstract This paper presents a stacked neural network based multi-objective optimal control method for batch processes. Stacked neural networks not only give better generalisation performance than single neural networks but also provide model prediction confidence bounds. In addition to the process operation objectives, the reliability of model prediction is incorporated in multi-objective optimisation in order to improve the reliability of the obtained optimal control policy. The standard error of the individual neural network predictions is taken as the indication of model prediction reliability. The proposed method is demonstrated on a simulated fed-batch process. Keywords: Batch processes, multi-objective optimisation, neural networks 1. Introduction Batch or semi-batch processes are suitable for the responsive manufacturing of high value added products [1]. In the operation of batch processes, it is desirable to meet a number of objectives, which are usually conflicting to each other. The relative importance of the individual objectives usually changes with market conditions. To maximise the profit from batch process manufacturing, multi-objective optimal control should be applied to batch processes. The performance of multi-objective optimal control depends on the accuracy of the process model. Developing detailed mechanistic models is usually very time consuming and may not be feasible for agile responsive manufacturing. Data based empirical models have to be utilised. Stacked neural networks have been shown to possess better generalisation capability than single neural networks [2, 3] and are used in this paper to model batch processes. An additional feature of stacked neural networks is that they can also provide prediction confidence bounds indicating the reliability of the corresponding model predictions. Due to model-plant mismatches, the "optimal" control policy calculated from a neural network model may not be optimal when applied to the actual process [4]. Thus it is important that the calculated optimal control policy should be reliable. This paper presents a reliable multi-objective optimal control method for batch processes. In addition to the process operation objectives, the reliability of model prediction is incorporated in multi-objective optimisation in order to improve the reliability of the obtained optimal control policy. The standard error of the individual neural network predictions is taken as the indication of model prediction reliability. The proposed method is demonstrated on a simulated fed-batch process. It is shown that by incorporating model prediction reliability in the optimisation criteria, reliable control policy is obtained.

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A. Mukherjee and J. Zhang

1408

2. A fed-batch process The fed-batch reactor is taken from [5]. The following reaction system

5 + 5-

-^D

is conducted in an isothermal semi-batch reactor. The objective in operating this reactor is, through addition of reactant B, to convert as much as possible of reactant A to the desired product, C, in a specified time tf= 120 min. It would not be optimal to add all B initially as the second order side-reaction yielding the undesired species D will be favoured at high concentration of B. To keep this undesired species low, the reactor is operated in semi-batch mode where B is added in a feed stream with concentration bfeed = 0.2. At the start of reaction, the reactor contains [^](0) = 0.2 moles/litter of ^, no B ([B](0) = 0) and is fed to 50% (F(0)=0.5). 3. Modelling of the fed-batch process using stacked neural networks Fig. 1 shows a stacked neural network where several networks are developed to model the same relationship and are combined together. Earlier studies show that an advantage of stacked neural networks is that they can not only give better generalisation performance than single neural networks, but also provide model prediction confidence measures [2]. Disturbances Goal _ J Multi-objective optimiser |

_L Batch Process

^^^--l y^-'

Fig. 1. A stacked neural network

Fig. 2. The control system structure

In this study, a fixed batch time of 120 minutes is considered as in [6]. Since it is usually difficult to measure the product quality variables frequently during a batch, it is a general practice to measure the product quality variables only at the end of a batch. The batch duration is divided into 10 equal intervals and within each interval the reactant feed rate is kept constant. The objective in operating this process is to maximise the amount of the final product Ccitf)V(tf) and simultaneously minimise the amount of undesired species CD{tf)V{tf). Neural network model for the prediction of concentration variables Cc(tf}V(tf) and CD{tf)V(tJ) at the final batch time are of the form: (1) y2=f2(U)

(2)

where yi = Cdtf} F(//), y2 = Coitj) V(tf}, (7 = [wi W2 • • • wio]^ is a vector of the reactant feed rates,/i and^ are nonlinear functions represented by neural networks. In this study, simulated process operational data from 50 batch runs were generated with the reactant feed rate randomly distributed in the range [0, 0.01]. Of the 50 batches of

Reliable Multi-Objective Optimal Control of Batch Processes

1409

data, 40 batches were used to develop neural network models and the remaining 10 batches were used as unseen testing data. Gaussian noise with zero mean and a variance of 2.5x10''^ was added to the reactant feed rate to simulate the effect of measurement noise. Two stacked neural networks each containing 20 neural networks were developed for predicting Cc{tj)V{tf) and CoitJ)V{tf). Each individual neural network has a single hidden layer with 10 hidden neurons. Hidden neurons use the sigmoid activation function whereas the output layer neuron uses the linear activation function. The LevenbergMarquardt training algorithm with "early stopping" was used in the study to train the networks. For training each network, bootstrap re-sampling with replacement [7] was used to generate a replication of the 40 batches of process data. Half of the replication was used as training data while the other half was used as the validation data. 4. Reliable multi-objective control of the fed-batch process The objective in operating the process is to maximise the amount of the final desired product Cc(tf)V(tf) and simultaneously minimise the amount of the final undesired species Co(tf)V(tJ). In order to obtain reliable control policyfi*omthe stacked neural network model, minimisations of the standard errors of individual network predictions are introduced as additional objectives in the optimisation problem. This may be formulated in terms of a multi-objective optimisation problem which is solved using the goal attainment method [8].

-c.it^Witf) F(U) =

(3)

e.cAtf) min u. subject to

(4)

F,{U)-W,


0<w^. <0.01

z = l,2,---,10

where is a scalar variable, Wj is the weighting parameter for the /th objective, F*i is the desired goal value for the rth objective, U=[uu W2,..., wio] is the sequence of the reactant feed rates into the reactor, V is the reaction volume, ^ ^ {t^) and ^ ^ {t.) denote the standard prediction errors from the two bootstrap aggregated neural network models for predicting Cc(tf) and Coit/) respectively. The presented objective function F(U) maximises the amount of product, Cc{tf)V(tj), minimises the amount of by-product, CD(tf)V(tf), and also minimise the standard errors of the neural network model predictions (i.e. maximise the reliability of model predictions). Fig. 2 shows the structure of the proposed control system. Two cases having the following goal values are considered: Case I: Goal = [-0.065 0.015 0.001 O.OOlf

1410

A. Mukherjee and J. Zhang Case II: Goal = [-0.075 0.020 0.001 0.001]^

Case I emphasises on less by-product generation whereas Case II stresses on more of the desirable product. The weights for the goal attainment algorithm for the maximisation of Cc(tf)V(tf) and minimisation of CD(tf)V(tf} are taken as being complementary to each other with the summation of them being equal to 1.0 (wtcdv= 1wtccv)' The weights for the standard prediction errors are taken as 0.04 and 0.03 respectively for all the simulations. Experimental studies indicate that these weights usually give good performance. 50 solutions of the multi-objective optimal control problem were computed by varying the weights on Cc {tf)V{tf) and CD{tf)V{t^ randomly and uniformly within [0, 1]. Fig. 3 shows the bootstrap aggregated neural network model prediction errors under the optimal control profiles obtained by considering confidence bounds (o) and not considering confidence boimds (*). It can be seen from Fig. 3 that the neural network predictions under the control profiles calculated by considering the model prediction confidence bounds are generally more accurate than those under the control profiles calculated without considering the model prediction confidence bounds. Accurate model predictions in optimisation will lead to reliable optimal control policy. Case I - Cc(tf)*V(tf)

10

20 30 Solution Number

Case I - Cd(tf)*V(tf)

40

20 30 Solution Number

Case II - Cc(tf)*V(tf)

10

20 30 Solution Number

Case II - Cd(tf)*V(tf)

40

50

20 30 Solution Number

Fig. 3. Stacked neural network model prediction accuracy under different optimal control profiles Fig. 4 illustrates the percentage improvement (increase of product and decrease of byproduct) in the quality variables due to the incorporation of minimising the standard prediction errors as extra objectives. It may be concluded that there is a consequent improvement in the end point quality variables since most of the obtained solutions show a positive improvement in either of the concentration values. Table 1 shows the number of cases (out of 50) where improvement in solutions were achieved through incorporating the minimisation of confidence bounds as extra objectives. Table 2 shows sample optimisation and simulation results for incorporating and not incorporating model prediction confidence bounds. The results shown in Table 2 clearly

Reliable Multi-Objective Optimal Control of Batch Processes

1411

signify the effect of incorporating model prediction confidence bounds. For Case I, though the neural network predicted values of C(itf)V(tf) and CD(tf)V(tf) are better if no confidence bound criterion is included in the objective fiinction, but the resultant control profile is not reliable and may not provide "optimal" performance if it is applied to the actual process. This is verified using the simulation results on the mechanistic model (representing the actual process). The acutal values (mechanistic model simulated) of Ccitf)V(tf) and CoitfjVitf} are better when the standard prediction errors are minimised as part of the optimisation objectives. For the study in Case II, though the neural network predictions for Ccitf)V(tf} is better and for CD(tf)V(tf} is only marginally worse if standard prediction errors are not included in the optimisation objectives, but the resulting optimal control profile is not expected to be reliable. The simulation results on the mechanistic model indicate that optimisation incorporating the standard prediction errors leads to a large improvement in Coit/jVitf) and only a marginal degradation in the value of Cc(/^)F(/^). Table 1. Improvement in solutions by incorporating confidence bounds as extra objectives Criteria considered

Number of cases Case I Case II

Improvement in Cc{tf)V{tf)

38

8

Improvement in CD{tf)V{t^

12

40

Improvement in Cc{tj)V(tj) & CDit/)V{tf)

0

0

No improvement in Cc(tJ)V(tj) & no improvement in Cj)(tj)V(tf) 0

2

Case - 1 50 o o en O 0

^--50

%o ^

%)

o

O

O •^100 E

1 Q.

0

e-150 o ®-200

-250

S-40

o o

0° o

Percentage Improvement in Cc(tf)*V(tf)

Percentage Improvement In Cc(tf)*V(tf)

Fig. 4. Improvement in solutions by incorporating confidence bounds as extra objectives In order to investigate the robustness of the proposed method to process variations, process variations were introduced by randomly varying the values of ki and ^2 assuming a normal random distribution with a standard deviation of 0.05 a mean value (nominal value) of 0.50. Table 3 shows the number of cases (out of 50) where improvement in solutions were achieved through incorporating the minimisation of confidence bounds as extra objectives. It can be seen from Table 3, the proposed

1412

A. Mukherjee and J. Zhang

technique still performs better than optimisation without considering model prediction confidence bounds even under process variations. Table 2. Sample optimisation and simulation results for end point quality Cases I

II

Neural network model

Weights

Mechanistic model

Cdtf)V{tf)

CD{tf)V{tf) CcWitf)

Co{tf)V{tf)

[0.416, 0.384, 0.04, 0.03]^

0.05974

0.02486

0.06023

0.02705

[0.416, 0.384]^

0.05898

0.02067

0.06140

0.02832

[0.416, 0.384, 0.01, O.Olf

0.06082

0.02532

0.06193

0.02707

[0.416, 0.384f

0.06202

0.02639

0.06239

0.03034

Table 3. Performance under process variations Criteria considered

Number of cases Case I

Case II

Improvement in Ccitf)V{tj)

30

0

Improvement in CD(tf}V(tJ)

22

49

Improvement in Cc(tf)V(tf) & CD(tf)V{tf)

2

0

No improvement in Cc{tj)V{tj) & no improvement in CD{tf)V{tj)

6

1

5. Conclusions A reliable multi-objective optimal control strategy using stacked neural network is proposed in this paper. Stacked neural networks can not only provide enhanced model prediction performance, but also provide model prediction confidence bounds. Minimising model prediction confidence bounds is taken as additional objective in calculating the control policy. Application to a fed-batch process demonstrates that the proposed technique can significantly enhance the reliability of the control profiles. Acknowledgement: Ankur Mukherjee thanks the Commonwealth Scholarship Commission for the financial support for his MSc studies in University of Newcastle.

References [1] D. Bonvin, Journal of Process Control, 8 (1998) 355-368. [2] J. Zhang, E.B. Martin, A.J. Morris, & C. Kiparissides, Computers & Chemical Engineering, 21 (1997) sl025-sl030. [3] Y. Tian, J. Zhang, A. J. Morris, Ind. Eng. Chem. Res., 40 (2001) 4525-4535. [4] J. Zhang, Ind. Eng. Chem. Res., 43 (2004) 1030-1038. [5] P. Terwiesch, D. Ravemark, B. Schenker, and D. W. T. Rippin, Computers & Chemical Engineering, 22 (1998) 201-213. [6] J. Zhang, IEEE Transactions on Fuzzy Systems, 13 (2005) 417-427. [7] B. Efron, The Jackknife, the Bootstrap and Other Resampling Plans; Society for Industrial and Applied Mathematics: Philadelphia, 1982. [8] F. W. Gembicki, Vector Optimisation for Control with Performance and Parameter Sensitivity Indices, PhD Thesis, Case Westem Reserve University, Cleveland, USA, 1974.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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A Thermodynamic Based Plant-Wide Control Design Procedure of the Tennessee Eastman Process Luis T. Antelo, Irene Otero-Muras, Julio R, Banga and Antonio A. Alonso Process Engineering Group. IIM-CSIC. C/Eduardo Cahello, 6. 36208-VIGO (Spain) Abstract In this work, we apply the systematic approach to plant-wide control design presented in [1], based on the fundamentals of process networks, thermodynamics and systems theory, to the Tennessee Eastman (TE) Challenge Process, deriving robust decentralized controllers that will ensure the stability of the complete plant. We take one step fiirther in the control design procedure by completing it with the realization of the controllers. The inventory control laws proposed are derived from a set of control loops over the available degrees of freedom in the process. Keywords: Thermodynamics, Plant-Wide Control Design, Process Networks, Inventory Control, Control Realization.

1, Introduction The TE process defined in [2] produces two products (G and H) from four reactants (A, C, D and E). A fiirther inert trace component (B) and one byproduct (F) are present. The process consists of a continuous stirred tank reactor, a condenser, a flash drum and a stripper. In Figure 1, the flowsheet of the TEP is reproduced. The gaseous reactants are fed to the reactor where they are transformed into liquid products. The following reactions take place in gas phase: A(g) + C(g) + D(g)^G(i) (1) A(g) + ^(g) "^ E(g) ^ ^(1) (2) A(g) + E(g)^H(i) (3) 3D(g) ^ 2F(,) (4) These reactions are irreversible and exothermic with rates that depend on temperature through Arrhenius expressions and on the reactor gas phase concentration of the reactants. The reaction heat is removed from the reactor by a cooling bundle. The products and the unreacted feeds pass through a cooler and, once condensed, they enter a vapor-liquid separator. The noncondensed components recycle back to the reactor feed and the condensed ones go to a product stripper in order to remove remaining reactants by stripping with feed stream. Products G and H are obtained in bottoms. The inert (B) and the byproduct (F) are mainly purged from the system as a vapor from the vaporliquid separator. Since the publication of the TE process example, it has been widely used in the literature as a case study due to its challenging properties from a control engineering

1414

L. T. Antelo et al

point of view: it is highly nonlinear, open-loop unstable and it presents a large number of measured and manipulated variables what offer a widely set of candidates of possible control strategies. A review of the most relevant plant-wide control strategies applied to the TE process is given in [3] and references there in. Many of the previous proposed approaches have weaknesses as described in [3]. Our contribution try to overcome these drawbacks by systematizing the development of Figure 1: The Tennessee Eastman Process robust decentralized controllers that will ensure the TEP stability. In this framework, the TEP is represented as a process network and mass and energy inventory control loops for each node are designed, first to guarantee that the states of the plant will remain on a convex invariant region. In these sets, the second law of thermodynamics provides us with a function -the entropy of the system- which has a definite curvature (is concave). In addition, it can be proved that in these compact regions, the system will be passive and therefore stable. At this point, the next step will be the controller realization using the physical inputs-outputs of the process. In the real TE process, inventory fluxes can be the result of combining multiple convective mass or energy flows and the inventory control law has to be obtained as a composition of control loops implemented over the real manipulated variables available in the process. The paper is structured as follows: In Section 2, network fiindamentals and the TEP network representation are described. The thermodynamic formalism used in the design of decentralized controllers is presented in Section 3. Finally, in Section 4 we show the realization of the inventory controllers and we validate the proposed methodology with simulation results.

%^=i^

2. The Tennessee Eastman Process Network As presented in a previous work [1], a process network is defined by a numbery^y,..., n of well mixed homogeneous material regions connected by material and energy fluxes we will refer to as nodes, plus an extra region j = 0 which represents the environment. To each node j in the network we associate a state vector Zj& W^^ of the form Zj=(n^,...,n^,i/f, where n{ denotes the mole numbers of component /, the internal energy is represented by t/ and c stands for the total number of chemical species. With this, the inventory network dynamics can be represented by: = N ^,

= N PI

with

rhi ,Ui

^Pi

(5)

A Thermodynamic Based Plant-Wide Control Design Procedure

1415

where^, = VVcj^y:* and p^ =Y;7. are the mass and energy inventory flows, /€D k=\

ieD

respectively and N e R^""^ is, by definition, a column conservation matrix. In addition, i represents the number of dissipative r i 7 subnetworks (defined as those only interconnected by dissipative fluxes) in which .8 I 9 the original system can be simplified andD is the number of nodes forming each dissipative subnetwork. For further ...^w ^ information about this network formalism, *i ^ see [1]. The network description of the model is depicted in Figure 2 and derives Figure 2: The Tennesse Eastman Network from the TEP flowsheet presented in Figure 1 by assigning one node to each phase. There, the mass and energy flows are represented by solid and dashed arrows, respectively, and the solid circles symbolize the environment. By applying the inventory network concepts presented, the inventory dynamics can be written as: 0

>«

K3

1—'Q-

-o-

o*i^'

"(Xp"

^6,7-0

0

0

^6,7-1

^8,9-1

0

0

0

0

0

-1

0

0

•*0-8,9

0

^6,7-8,9

-1

0

0

0

•^8,9-10,11

a,10,11-8,9 -1

-1 a,

'0-1,2

0

0

-1 1

0

0

0 0 0 0

wA, wy^, - 1 J w.

(6)

•12-10,11

where aij represents the fraction of the inventory flow goingfi-omnode / to node 7. The resulting inventory network is as depicted in Figure 3. In a similar way, we can represent the energy inventory network (Figure 4), where the main difference with the mass network is that there exist more connections with node 0 (inputs from the environment corresponding to coolant and steam flows in the condenser before the LV separator and the stripper). The underlying structures presented in Figures 3 and 4 for mass and energy inventory layers are the basis of the hierarchical control process design presented next. 3. Thermodynamic basis for decentralized controller design In every node of a process network of volume Vj, thermodynamics provides us with a continuous, twice differentiable scalar function Sj [zj,Vj): R^^""^^^ h^ R named entropy. This function is first order homogeneous in all their arguments and strictly concave with respect to the vector Zj [1]. Such property indicates that S has a definite curvature, ensuring that over the set

1416

L. T. Antelo et al

"kii-cf

Figure 3: Mass TEP Fundamental Network

Figure 4: Energy TEP Inventory Network

K[D) = Uz.,v^)].^

(6) yeD

jeD

jeD k=l

has a maximum for a given (m, u, v) constant and for every je 7>. In the last expression, c/ represents the vector of molecular weights for the c components. Once the network states converge to A, we can ensure that the process system is passive and, therefore, stable [5]. This fact motivates a hierarchical control design decomposition in which mass and energy inventory controllers in every node of the network are first designed to drive the system to this compact set A. In particular, the mass and energy inventory control layers consist of linear proportional controllers of the form: ^/ =^/* + ^ ^ ( ^ / -^\);

P\ =p] +^u[ui -u])

(7)

We select a ^j-*e Null(A/'), i.e. yl^^*= 0 and cOn , a\i SLVQ appropriate gain matrices constructed in a way such that the real part of the eigenvalues associated to A/co^ and A/cOu are negative. Finally, taking deviation variables with (ki, POL (hv> Pov respect a given reference and substituting the expressions in (7) into (5), it follows that m^- -^ m^r* and Uj^^ u^r* ^f 9 exponentially fast so the convergence of the process states ^ to the set A is ensured. For the TE inventory networks represented in Figures 3 and 4, there exist enough degrees of freedom (DOF) to implement the mass inventory control, using the total inventory outflow of each node as (kj, Pv the manipulated variables. For the energy layer, the ^r^b PL r additional connections with node 0 provide extra DOF to Figure 5: Dissipative network control this energy inventory. It must pointed out that, as demonstrated in [1], that control laws represented in Eqn. (7) do not prevent the system from exhibiting complex behavior, such as multiplicities, due to the fact that the convergence of the intensive variables to a unique stable point can not be guaranteed. As presented before, the network entropy is strictly concave in A and intensive variable control can be designed by, for instance, methods discussed in [5]. In order to avoid these problems and ensure global stability for the TE plant, we propose additional T

M:.I

V y

A Thermodynamic Based Plant-Wide Control Design Procedure

1417

control loops for mass and energy inventories in every node belonging to the dissipative subnetworks as depicted in Figure 5. For instance, the mass inventory of a vapor phase is related with the pressure and together with the energy inventory - related with the temperature, the composition of equilibrium systems can be modified through these inventories. In order to develop these extra loops, we consider that the control laws in Eqns. (7) can be obtained as a composition of the control laws applied to each node in 7). In Figure 5, two possible manipulated variables (the outflows of each node) are proposed. Using control laws for the mass inventories in each node analogous to (7), we have: ^ i = ^ z + ^ z ( ' W z - ^ I ) ; (l)y=(l)y+COy[my-ml) It follows that the sum of equations in (8) results in ^,

(8) -(I)] -\-co(mi n] ) with

C0L=(0y, ensures convergence of both total and node mass inventories. The formalism concerning the inventory control laws must then be translated into suitable control loops using the available inputs-outputs of the system. This question will be discussed in the next section. 4. Realization of the Hierarchical Control Design for the TE Process Starting from the fundamental process network showed in Figure 2, inventory control loops are developed for every node, except for the condenser and for the nodes representing heat sources or sinks in the energy layer, where there is no mass holdup. The manipulated variables will be the outflows of each node and the flows from node 0 for the mass and energy cases, respectively. The resulting mass and energy inventory control structures (MIC and EIC, respectively) for the TE are presented in Figure 6. Note that for mass inventory in node 1, we do not use the outflow since in the TE there are not valves neither in the stream leaving the reactor nor in the stream from the condenser to the LV separator. Therefore, the next vapor stream (purge) needs to be used. In addition, for every dissipative subnetwork, equilibrium is assumed, which implies ///=//y , Tj^Tj and Pi=Pj for all /, j eX>, This allows us to defiFigure 6: Inventory Control Loops ne a unique energy inventory control loop for each dissipative subnetwork. For the case of node 6 representing the vapor phase in the LV separator, a composition control as depicted in Figure 6 is considered. Since Ts, V and X6 are being controlled and the nodes are in equilibrium, the pressure as well as the mass inventory are fixed. Finally, the mass inventories for the dissipative subnetworks representing the equilibrium trays of the stripper will be constant, under the assumption of a constant molar overflow (CMO). The dynamic performance of the proposed control

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L. T. Antelo et al

structure against changes in the A/C feed ratio (IDVl) is presented in Figure 8, showing stability and fast convergence of liquid and vapor mass inventories (related to level and pressure subplots, respectively) as well as energy inventories (temperature subplots) to their reference values. Similar results were obtained against other disturbances. The controller tuning was made using standard techniques. Note that this tuning can be also carried out by solving an optimization problem consisting in minimizing the cost function presented in [2], using the controller parameters as decision variables.

S30J 201

0

Figure 7: Proposed Control Structure for TEP 5.

10

20

0

10

S25y ''^•"••""•''*"'*"H 1

201

'

1

20

Figure 8: Dynamic Response against IDV(l)

Conclusions

In this contribution, the systematic plant-wide control design methodology presented in [1] has been applied to the challenging benchmark of the Tennessee Eastman Process. We have designed decentralized control structures which simultaneously ensure stabilization of both plant extensive and intensive variables, paying special attention in the realization of the controllers by making use of the available manipulated variables of the process. The proposed control structure has been tested dynamically with good stability results. References [1]. Antelo, L. T., Otero-Muras, L, Banga, J.R., Alonso, A. A. (2005). A systematic approach to plant-wide control based on thermodynamics. 15* European Symposium on Computer Aided Chemical Engineering (ESCAPE-15), 20(B), 1105-1110. [2]. Downs, J.J., Vogel, E.F. (1993). A plant-wide industrial process control problem. Computers & Chemical Engineering, 17, 245-255. [3]. Larsson, T., Skogestad, S. (2000). Plantwide control: A review and a new design procedure. Modeling, Identification and Control, 21, 209-240. [4]. Alonso, A.A., B.E. Ydstie, 2001, Stabilization of distributed systems using irreversible thermodynamics. Automatica, 37, 1739.

Acknowledgements The authors acknowledge the financial support received fi"om the Spanish Government (MCyT Projects PPQ2001-3643) and Xunta de Galicia (PGIDIT02-PXIC40209PN).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Stochastic Optimal Control in Batch Reactive Systems: Developments on Engineering Applications of Real Option Theory Vicente Rico-Ramirez^, Juan F. Cambero-Benitez^, Hector Canada-Jaime^ and Salvador Hernandez-Castro^ ^Instituto Tecnologico de Celaya, Av. Tecnologico y Garcia Cubas S/N, Celaya, Gto., 38010, Mexico Universidad de Guanajuato, Facultad de Quimica, Noria Alta s/n, Guanajuato, Gto., 36050, Mexico Abstract This work focuses on optimal control problems and on the development of an integrated approach based on real option theory and sampling for addressing uncertainties in a systematic way. Our approach considers the general case of having model based uncertainties as well as external uncertainties. Time-dependent uncertainties in model parameters are modeled as Ito processes; external uncertainties are represented through continuous probability distributions, so that a sampling technique is used to generate the realizations of such variables before operation. Also, we have extended the existing theory for stochastic optimal control to include cases with path constraints on state variables in the problem formulation. Furthermore, our integrated approach for solving optimal control problems in the presence of uncertainties considers both theoretical and numerical methods, such as the stochastic maximum principle and the stochastic dynamic programming technique (for singular problems). We use the optimization of batch reactive systems under uncertainty as our case study to validate the theory and methods. Results show the effects of uncertainties on the operation profiles. Keywords: stochastic optimal control, real option theory, sampling, Ito process. 1. Introduction In the light of a growing competition, the optimization of dynamic processes is needed to improve product quality, to reduce costs and to meet environmental regulations. Due to their time-varying nature, the optimization of dynamic processes results in optimal control problems. An optimal control problem involves the task of determining timevarying profiles of decisions variables generally related to operation. This section provides a summary of the existing techniques for deterministic problems. Further, the incorporation of uncertainty leads to stochastic optimal control problems; the sources of uncertainties in dynamic processes are also discussed. 1.1. Techniques for deterministic optimal control problems Optimal control problems have been extensively studied in the literature. The formulation involves optimizing an integral equation subject to a set of differential equations: Maximize

f r L=\k[x,

\ )dt

subject to

dx r — = f[x,

\ )

//) ^^^

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V. Rico-Ramirez et al

where x represents the state variables, are the control variables, t is the time, L is the objective function and T is the final time. The most widely used methods to address these problems are the dynamic programming and the maximum principle approaches. The maximum principle technique (Sethi and Thompson, 2000) includes the definition of a Hamiltonian function, H, and the incorporation of a new set of variables and their corresponding dynamic equations (adjoint variables and adjoint equations). Also, by applying the conditions for optimality, ^Hjd = 0, an expression for the optimal control profile is derived. This approach results in a two point boundary value problem that can be solved through numerical techniques such as the gradient method or the shooting method. On the other hand, the dynamic programming technique is based on Bellman's principle for optimality which results in the Hamilton-Jacobi-Bellman (HJB) equation. It can be shown that, although the mathematics of these two techniques might seem quite different, they are equivalent and lead to the same solution. 1.2. Sources of uncertainty Uncertainty manifests in many forms in dynamic processes. For instance, there could be uncertainty because of poorly characterized model parameters (model based uncertainty); moreover, operational level or external uncertainty (uncertain initial conditions, variations in feed quality, ambient conditions, etc.) can also play a significant role. Hence, handling uncertainties in optimal control problems becomes an important issue and results in the topic of stochastic optimal control. As we will describe, such classification is used in our solution approach.

2. Real Options Theory and Stochastic Optimal Control An approach based on real option theory for the solution of stochastic optimal control problems in engineering was provided by Rico-Ramirez et al (2003). In that work, the time dependent uncertainties of a dynamic system were represented as Ito processes; then, the optimal control profile was obtained by using Ito's Lemma and the optimality conditions developed for stochastic dynamic programming (Dixit and Pyndick, 1994). Later, Rico-Ramirez and Diwekar (2004) generalized such a strategy and developed the theory for the stochastic version of the maximum principle method. Those results are summarized in this section. 2.1. Time Dependent Uncertainties: Ito Process and Ito's Lemma An Ito process is a stochastic process on which the increment of a stochastic variable is represented by the equation:

(2)

dx = a(xj) dt + b(x,t) dw

where dw is the increment of a Wiener process, and a(x,t) and b(x,t) are known ftinctions. Ito's Lemma, the fundamental theorem of stochastic calculus, is needed to differentiate and integrate Ito processes,. Considering a function F that is twice differentiable in x and once in t, the differential dF given by Ito's Lemma is: dF--

dF

(

.dF

1 2.

.d^F

dF clt + b(x,t)—dw dx

(3)

2.2. Stochastic Optimality Conditions It has been reported that (Dixit and Pyndick, 1994), if the state variables are represented as Ito processes: dx = f dt-\- dw

(4)

Stochastic Optimal Control in Batch Reactive Systems

1421

then, because of Ito's Lemma, the optimality conditions for stochastic dynamic programming are given by: 0=

Maximize

L+LJ + — 4,

Maximize r

-,

(5)

LA + ^J

where H is the Hamiltonian function for stochastic optimal control problems. 2.3. Stochastic Maximum Principle Assuming that the optimal control problem has been rewritten in Mayer linear form {k=0), Eq. 5 and Ito's Lemma were used to derive the stochastic maximum principle. Let us name = L^^rvd = L^ as the adjoint variables ( represents new "stochastic'' adjoint variables). The resulting adjoint equations are then (Rico-Ramirez and Diwekar, 2004): 2

H= / + — / ^ dt dt

^

2^

^'

^'"

-^'^

2^

(^^ ^^^

where c is the coefficient of the state variable in the objective function. The optimal control profile will result from extremizing the Hamiltonian with respect to the control variable. Notice that a two-point boundary value problem has to be solved.

3. Recent Developments and our Integrated Solution Approach This section describes our recent theoretical developments and our generalized approach for solving stochastic optimal control problems. From the theoretical point of view, we have worked on cases involving path constraints and on the characterization of the solution. Our approach involves different numerical techniques depending upon the formulation and characterization of the problem. 3.1. Handling uncertainties Both external (static) and internal (time dependent) uncertainties are considered. External uncertainties are represented through continuous probability distributions, so that a sampling technique is used to generate the realizations of such variables before plant operation; in this work we use the Hammersley sequence sampling technique. On the other hand, internal uncertainties are represented as Ito processes. That implies the use of real option theory, as described above. 3.2. Characterizing the solution There are problems in which the state equations (and the Hamiltonian) are linear functions of the control variables. That is: ^ = f{xj)^g{x) (V at Therefore, the optimaHty conditions 3///a = o do not provide an explicit expression for the optimal control profile. This kind of problems is sometimes called singular optimal control problems. In our solution approach, if an expression for the optimal control profile cannot be derived, there are two options. The first one is using a numerical implementation of a dynamic programming based method. The second one is using the

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V. Rico-Ramirez et al.

gradient method (and the maximum principle formulation) to obtain the optimal profiles. 3.3. Path Constraints on Stochastic Optimal Control We have developed some mathematical tools needed to solve stochastic optimal control problems with path constraints in the state and control variables. In particular, we have addressed the cases where the constraints are linear functions of the state variables and the state equations are linear functions of the control variables. In summary, we could show that, if a state variable is represented as an Ito process and is a linear function of the control variable, dx = [f(x, t) + g{x)

\it + b{x, t)dw

(8)

and a constraint C exists such that: C(x, , 0 < 0

=^C(x, j) = x-x^^

(9)

then the expression (E is the expectation operator) Ed[x\ = J?{[/(x,0 + g{x)

]dt + b(xj)dw] = 0 ^ ^{[/(x,0 + gix)

]dt] =0

(10)

can be used to derive the optimal profile when the path constraints provide the solution. 3.4. Optimization criterion for stochastic dynamic programming We have shown that there is a complete analogy between the numerical implementation of the deterministic and the stochastic versions of dynamic programming by using the optimality conditions developed for stochastic dynamic programming (Eq. 5). That is, the maximization criterion provided by the literature for the deterministic case (Luus, 1990) can still be used. The only difference is that Ito processes have to be integrated by an appropriate method for stochastic differential equations, such as the Euler-Murayama method. In mathematical terms, the following equations apply for the stochastic (Eq. 12) and the deterministic (Eq. 11) cases of dynamic programming: J ( x , 0 = max (,,+Ao[^(^/+i'^ + ^0]

0^)

— m a x (^tj+^fy

3.5. The Solution Framework: Integrating Real Option Theory and Sampling Figure 1 describes our approach to the solution of stochastic optimal control problems. This approach intends to integrate recent theoretical and numerical strategies. In particular, the theory and methods described in section 2 and in subsections 3.1 through 3.4 have been included. 4. Illustrative Example and Results We have chosen a problem described by Srinivasan et al. (2002) as our case study. It involves a batch reactive system in which the reactions 2A^B->C are taking place. However, we have added a stochastic behavior to the kinetics parameters ki (internal parameter), so that there are time dependent parameters in the formulation and the resulting problem is a stochastic optimal control problem. The temperature of the reactor, T, is the state variable, the cooling water temperature, w, is the control variable and the objective is to maximize the concentration of B, Q , after an operation time tf. Eq. 13 provides the Ito process expression used for ki. The state variables are defined by

Stochastic Optimal Control in Batch Reactive

1423

Systems

Eq. 15. The problem includes state constraints (E[T(t)JM^umw)' The rest of the model equations are:

i^ei •

i^^di +

^/ -

max^ J ^ E "

dC.=

,Uo/exp

(13) RT (14)

C^{ty)

- 2 ,^„, e x p l - ^ IC! dt-2

Aiexp|-^|ci-

dCr,

dT =

and control variable

(15)

,dw,

A2exp|

RT

(-^^) , , ^ ^ e x p f ^ V >C„^ C„

^tfJ ,ci*;.

\C C

\dt +

^Cl^dw] - ^^BtCaC^wf

A.^-p(^\C.C.. RT

UA _ ^ VC„

^^ VC„

dt

i^ ,C,,Q,^w;

4.1. Results Table 1 provides the data for all of the parameters included in the formulation. The initial conditions are T(0)=2{) C, CA(0)= Imol/lt, CB(0)=Q mol/lt and Cc^t/; =0.005mol/l; tf is 1 hr. Concentration of C is calculated through a mass balance equation. This is a problem with internal uncertainties, in which the optimal control profile cannot be explicitly obtained from the optimality conditions. So, the numerical implementation of stochastic dynamic programming was used. Figure 2 shows the results. In case (b) the state variable is constrained to be less or equal to 35 degrees, and in case (a) does not exist such a constraint. By applying Eq. 11, the control profile due to the state path constraints can be obtained.

INTEGRATED SOLUTION

APPROACH

MODEL FORMULATION STAGE

SOLUTION STAGE

Uncertainty

Optimal Control Problem

Non Singular

Static (time independent) Uncertainties

Time Dependent Uncertainties

Stochastic Optimal Control Problem

Ito Process

^ ^

Stochastic Maximum Principle Dynamic Programming Singular

Figure 1. An integrated solution approach to stochastic optimal control problems

V. Rico-Ramirez et al.

1424 Table l.Data Parameter

ai

0C2

Value

0.95

0.9

Parameter Value

0.05

Hi

^mm

^max

10 C

100 C

35 C

Cp

0.10

H2

500

19

1/molh

1/molh

V

U

A

EAI

EA2

12500

5000

-30

-40

900

4.2

1000

10^

5

J/mol

J/mol

KJ/mol

KJ/mol

g/lt

J/gK

It

J/hm^K

m'

u("C)

u ("C) to

Tij^^.vi'v'HWVW^^ Tlempo (hrs)

U

\j^AVMHWVW^/^

Tlempo (hrs)

Figure 2. Optimal control profiles, (a) No path constraints (b) With path constraints (T<35)

5. Conclusions and Future Work An integrated solution approach for solving stochastic optimal control problems has been described. It incorporates several mathematical tools based on real option theory that have been recently developed, including the stochastic maximum principle, stochastic dynamic programming (numerical implementation), the ability to handle singular problems, path constraints and internal and external uncertainties. So far, we have solved just batch reactive problems, but its potential applications go further, to general dynamic optimization problems.

6. Acknowledgements V. Rico-Ramirez thanks the financial support provided by CONACYT, Mexico.

References Dixit, A.K. and R. S. Pindyck, 1994, Investment under uncertainty, Princeton University Press, Princeton, NJ, USA. Luus, R., 1990, Optimal control by dynamic programming using systematic reduction in grid size., Int. J. Control, 51, 995-1013. Rico-Ramirez, V. and U. M. Diwekar, 2004. Stochastic maximum principle for optimal control under uncertainty, Comp. Chem. Eng., 28, 2845-2849. Rico-Ramirez, V., U. M. Diwekar and B. Morel, 2003, Real option theory from finance to batch distillation. Comp. Chem. Eng., 27, 1867-1882. Sethi, S.P. and G. L. Thompson, 2000, Optimal control theory, Kluwer Academic Publishers, Boston, MA, USA. Srinivasan, B., S. Palanki and D. Bombin, 2002, Dynamic optimization of batch processes: I Characterization of the nominal solution, Comp. Chem. Eng., 27,1.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1425

A Web Service Based Online Optimization and Monitoring System for Chemical Processing Systems Xiangyu Li, Xiuxi Li,Yu Qian* School of Chemical Engineering, South China University of Technology, Guangzhou, 510640, P. R. China

Abstract Many information integration systems implemented in the process industry are based on Agent or CORBA technology. There are limitations in these schemes in terms of real time, security and stabilization in information communication. For a rational and better scheme, a novel web service based optimization and monitoring system is implemented, in which some professional software such as Matlab, G2, GAMS are described as web services by web service wrapper and registered in the UDDI register centre. Finally the practicability and validity of the online optimization and monitoring system are verified through the application in the product development of Nipagin ester. Keywords: web service, real-time optimization, ole process control, data reconciliation

1. Introduction Nowadays automation and process control systems are widely implemented in different operation systems in the process industry. But on the other hand, it is very difficult to implement cooperation among these softwares. For the software for different operation systems are developed at different time by different vendors, the functions of the software are isolated and inconsistent with current technology and/or protocol. The different software may be instituted on different operation systems and different areas; it brings up the problems so called "Automation Island" (Qian, 1999). In the last decade, there have been may works done by the academics and industry in order to solve the information ' Corresponding author: [email protected]. Phone and Fax: +86(20)87113046

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X. Li et al

integration problems among different subsystems, some technology such as Agent/CORBA/XML (Cheng, 2002; Cheng, 2003; Shen, 2002) are used to research the problem, but using them to implement information integration, there are some limitations in the integration system, for example, different subsystems are characteristic of point-to-point connection and difficult to maintain. For lacking agility, they can't cope with the change of integration strategy dynamically. In this paper, a novel web service based optimization and monitoring system is implemented. In this system, some professional software such as Matlab, G2, GAMS are described as web services by web service wrapper and registered in the UDDI register centre (Yu, 2004), then the cooperation among these web services is implemented. 2. Web service based online optimization and monitoring system 2.1 The framework of the online optimization and monitoring system Researching how to shorten the development cycle of chemical with high valueadded, to a corporation, it means the increase of economic benefit. For researching product and process development, a Mini-plant has been built in author's laboratory. In order to control the product development process in the Mini-Plant, an online optimization and monitoring system has developed, and use it to implement the real-time control and optimization of product development process. Thefi*ameworkof the system is shown in Figure 1.

Windows Client

1 Web Client

Other Client

Sparely UDDI Register Register Center •

^

>

Soap Request

Binding W o' o'

^

Wrapper Matlab

1—•

Wrapper IGAMS

•

~

Binding Soap Response Binding!

Wrapper

G2 1

•

Ope xml Server

•

Simense WinCC

Mini Plant

W

Figure 1. The framework of the online optimization and monitoring system

In the system, we use Simense WinCC (Windows Control Center) to control the Mini-plant directly. There are three key web services in the system too; they are implemented by encapsulating Matlab, GAMS, and G2 separately. According to real-time data, the MatlabService can adjust the control model of product development timely; the GAMSService is used to implement the real-time

A Web Service Based Online Optimization and Monitoring Systems

1427

optimization of reactive condition; the G2_Service is used to implement fault analysis, fault diagnosis, alarm and advice. The description information of the three web services is saved in uddi register center, so the application server can find web services and know how to call them to meet client request from different terminals. 2.2 Web services implementation in the system In the paper, a web service wrapper is used to convert traditional softwares to web services. The function of wrapper is to accept client request from different terminals and submit it to traditional software and return result to the client. Figure 2 shows how to covert Matlab to MatlabService. <System.Web.Services.WebService(Namespace:=_ http://tempuri.org/myws/Matlab_Service)>_ Public Class MatlabService Inherits System.Web.Services.WebService <WebMethod()>_ Public Function runMatlab(x() as double, y() as double_ trow as integer, tcol as integer) as double() dim yi(trow,0) as double, xi(trow,tcol) as double dim rr(tcol,0) as double, ri(tcol,0) as double dim matlab as object matlab=createobject("matlab. application") matlab. visible=false matlab.execute("clear") matlab.putfiillmatrix("x","base",x,xpi) matlab.putfullmatrix("y","base",y,ypi) matlab.execute("b=regress(y,x)") matlab.getfullmatrix("b","base",rr,ri) matlab.quitO return rr End Function Figure 2. The description of MatlabService

In MatlabService, runMatlab function can analyze the linear relation between X and y. x is an array with trow row and tcol column, y is an array with trow row and 1 column, and the return result is an array with tcol row and 1 column, it describes the linear relation between x and y. According to the linear relation between x and y, we can adjust our control model and use GAMSService to calculate the optiminal reactive condition at different moment. We can call MatlabService from different terminals, but the main function is provided by

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X.Lietal.

Matlab software. To GAMSService and G2_Service, we get them by encapsulating GAMS, G2 with the same method too. So the web service wrapper is the key to convert traditional softwares to web services. 3. An application of the online optimization and monitoring system Nipagin ester is one of the case studies carried out in the Mini-plant. Nipagin ester is a new anti-mildew and antiseptic agent with the best inhibiting effect on aflatoxin in food, drink, cosmetics and medicine. The reaction equation is as following: CvHfiOj + CH3CH2OH

•

C9H10O3 + H2O

According to the analytic result in the chemic laboratory, for getting the maximal output of Nipagin ester, the control model is built, the description is as following (the initial weight of CH3CH2OH is 350g, CyHgOj is 222g): y=810981 xxf'-'"' XX2''"xx^'

xx/"'xxf"""

xx/"^^

(1)

y< 7.22x1

(2)

0.6x2
(3)

Xrl.21x2-97.5< 0

(4)

Xr2.12x2+130<0

(5)

xi< 400

(6)

X2<250

(7)

0.8<X3<1.1

(8)

60<X4<90

(9)

6<X5<8

(10)

65<X6<80

(11)

In the control model, there are seven variables; the means of the seven variables is as follows: y is the output of Nipagin ester, X] X2 Xs X4 X5 X^ dXQ the quantity of C2H5OH which has participated in reaction, the quantity of C7H6O3 which has participated in reaction, the pressure in reactor, the temperature in reactor, the pH value in reactor, and the stirrer speed in reactor, respectively. MatlabService can analyze the relation among x and y, and according the analysis, we can adjust the control model timely. According to the control model, for getting the maximal output of Nipagin ester, we can work out the optimal value of jci (i=l, 2, 3, 4, 5, 6) by Web service of GAMS. When the

A Web Service Based Online Optimization and Monitoring Systems

1429

reaction is processing in Mini-plant, we can get the real values of x from OPC XML DA Server, and we can set the domain of x near the real-time value, then we use the web service of GAMS to work out the optimal value of x, finally, the operator can set current values of x in Mini-plant to the optimal value, through this way, the whole reaction will process along the goal for getting the maximal output of Nipagin ester. Table 1 show the optimal values of x that are worked out by the web service of GAMS at different time. Table 1 The optimal values of Xi (1=^1, 2, 3, 4, 5, 6) Time

Xx

08:10 09:10

y 235.36 219.27

350.00 350.04

222 222

10:05 11:13

207.00 224.12

368.70 372.75

13:25

235.21

364.24

X4

X5

^6

0.95 0.85

67 67

7.0 7.2

80 80

232 252

1.01 1.05

90 90

7.6 7.2

80 80

224

1.10

82

6.9

80

X2

X3

For finding abnormal reactive condition and equipment faults in Mini-plant, a web service is developed based on G2 develop platform. An interface of the web service is shown in Figure 3. In the interface, the temperature of Reactor 10 and Reactor20 is monitoring, and the trend curve of temperature is shown, to the two reactors, when the temperature reaches the maximal value that is set by operator, the web service will display the warning message, and at the same time, according to rule inferring based rules in rule-database, the operator will get some advices from the web service on how to solve the current problem. I t h e t e m p e r a t u r e o f r e a c t o r R i o r e a c h e s t h i s , s e n d et w a r n i r jl w h e n t h e t e m p e r a t u r e o f r e a c t o r R g O r e a c h e I The

I The

Current Temperature

o f R e a t o r R 1 Ci ! of R e a t o r R g o |

EiD

lEl

Ting: t h e t e m p e r a t u r e of r e a c t o r R I O h a s r e a c h e d t h e m a x ^ a l u e ! , n p e r a t u r e of r e a c t o r R 2 0 h a s r e a c h e d t h e m a x v a l u e , t o o !

the

i^dvice: 1. S e t u p t h e t e m p e r a t u r e of R I O a n d R 2 0 1 3 t h e ma>cimal v a l u e in t h e a l l o w a b l e d o m a i n a g a l n l . 2. A c c e l e r a t i n g t h e s p e e d of c o n d e n s a t e i n t h e c o o l i n g s y s t e m . 3. A c c e l e r a t i o n t h e s t i r r e r s p e e d i n r e a c t o r

Figure 3 An interface of web service of G2

In the platform, G2, GAMS, Matlab, WinCC are encapsulated as web services, and they are integrated effectively. In chemical field, to an enterprise, there are many applications developed by different people and running normally, but they

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can't share information each other, based on web service technology, we can encapsulate them as web services and integrate them as a whole system easily(Liu, 2004; Zhou, 2004). At the same time, the stability and security of these applications are not influenced. 4. Conclusion It is important to research on how to implent the information integration among different process operations in process industry. In this paper, a novel web service based optimization and monitoring system is implemented. In this system, the intrinsic problems that traditional technologies often meet have been solved. Finally the practicability and validity of the online optimization and monitoring system are verified through the application in the product development of Nipagin ester. Acknowledgements Financial support from the China Excellent Young Scientist Fund (No. 20225620), the China National Natural Science Foundation (No. 20376025 and 20476033) are gratefiilly acknowledged. References Cheng, H.N., Y. Qian, X.X. Li, 2003. Agent-oriented modeling and integration of process operation systems, European Symposium on Computer Aided Process Engineering-13, 599604. Cheng, H.N., Y. Qian, X.X. Li, 2002. Agent-oriented approach for integrated modelUng of process systems. Journal of Chemical Industry and Engineering, 54(1), 128-136. Shen, J.Y., J. Hang, 2002."CORBA based tele-collaboration workflow mode. Computer Applications, 19(9), 19.-25 Qian, Y., P.R. Zhang, 1999. Fuzzy rule-based modelling and simulation of imprecise units and processes, Canadian Journal Chemical Engineering, 77(1), 186-192. Yu, K., X.L. Wang, Y. Zhou, 2004. Underlying techniques for web services: a survey. Journal ofSoftware, 15(3), 428-452. Liu, M.J., X. Liu, X.J. Liu, 2004. Role certification-a solution to dynamic integrated web services security, Journal of Sichuan University(Nature Science Edition), 41(2), 319-322. Zhou, Q., T. Wu, 2004. Workflow markup language based on web services. Computer Engineering and Desgin, 25(5), 129-138.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Using the Process Schematic in Plant-wide Disturbance Analysis S.Y. Yim^ H. G. Ananthakumar^ L.Benabbas^ A. Horch^ R. Drath^ N.F. Thomhiir '^Department ofE&E Engieering, University College London, London, UK, WCIE 7JE ^ABB Corporate Research Centre, Ladenburg, Germany, D-68526. Abstract This article describes how isolation and diagnosis of the root cause of a plant-wide disturbance is enhanced when process connectivity is considered alongside the results of data-driven analysis. A prototype software has been designed and implemented which, when given an electronic process schematic of a plant and results from a data-driven analysis of process measurements, allows the user to pose queries about the plant and to find root causes of plant-wide disturbances. The plant topology information is written in XML according to the Computer Aided Engineering Exchange (CAEX) schema. Keywords: Fault diagnosis, plantwide oscillation; plant topology; root cause; XML. 1. Introduction Methods for data-driven, signal-based analysis have been developed recently for finding root causes of plant-wide disturbances using measurements from routine process operations [Ruel and Gerry, 1998; Thomhill et.al, 2003a; Xia and Howell, 2003]. Several authors have observed, however, that data-driven analysis is enhanced if a qualitative model is used as well to capture the fundamental causal relationships of a process [Chiang and Braatz , 2003; Lee et.al., 2003]. The challenge is to represent causal information in electronic form and to manipulate it to draw conclusions. Object-oriented representations of processes are becoming available using computer aided engineering tools. The plant topology can now be exported into an vendor independent and XML-based data format, giving a portable text file that describes all relevant equipments, their properties and directional connections between them [Fedai and Drath, 2005]. The Standard is lEC/PAS 62424, 2005, Computer Aided Engineering Exchange (CAEX). It specifies an XML schema. ISO-15926-7 is a similar standard. A prototype tool called CAEX Plant Analyser that links a CAEX description with a report from a data-driven analysis is reported in this article. The features are (i) capture of process topology using CAEX, (ii) parsing and manipulation of the description, (iii) linkage of plant description and results from data-driven analysis, and (iv) logical tools to give root cause diagnosis and process insights. Section 2 of the paper describes the background and places the work in context. Section 3 introduces the CAEX Plant Analyser and section 4 presents a case study.

2. Background and context Reviews by Venkatasubramanian et. al., [2003^, 2003Z?] discussed detection, isolation and diagnosis of faults in chemical processes, classifying the available methods into quantitative and qualitative model-based methods and quantitative and qualitative process history based methods. This paper concerns a hybrid system using a qualitative models and quantitative process history.

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The signed digraph (SDG) has been widely used as a causal qualitative model. Maurya et. al, [2003, 2004] gave a comprehensive review of graph-based approaches for fault diagnosis of chemical process systems and showed how to develop SGDs from a system of differential-algebraic equations. Quantitative process history based methods involve signal analysis of process data to find the root cause of a disturbance. They include oscillation detection, linear and non-linear time series analysis [Ruel and Gerry, 1998; Xia and Howell, 2003; Thomhill et. al, 2003^?; Choudhury et. .al, 2004; Zang and Howell 2005]. These papers do not, however, exploit the physical connections between the measurements. A knowledge of process topology (connectivity) enhances the diagnosis, for example it shows which loops might disturb one another. Work which has combined qualitative modeling and signal analysis includes Chiang and Braatz [2003] and Leung and Romagnoli [2002] who integrated multivariate statistical analysis with a causal map of a process to help in the diagnosis 5. Graphical User Interface of faults, while Lee et.al, [2003] also C# 4. Interface to combined SDGs with multivariate Reasoning Engine statistical analysis. It is clear that 1.CAEX 1 2. Input parser data signal analysis is enhanced by the parser 3. Reasoning Engine capture and integration of cause and effect information fi*om a process schematic, a step which now can be Prolog translated plant plant automated by CAEX. to C# using P#

t

3. The CAEX Plant Analyser

connectivity (CAEX & XML)

disturbance analysis (text file)

Figure 1. CAEX Plant Analyser

3.1. Overview An overview of CAEX Plant Analyser is presented in Figure 1. One input is the CAEX file which describes the items of equipment in the plant such as tanks, pipes, valves and instruments and how they are linked together physically and/or through electronic control signals. A physical link (or path) is a pipe carrying a flow of mass or energy, while a control link (or path) is a cable connecting a valve to a controller carrying an electronic signal. The data input file contains information about plant-wide disturbances, for instance the period of oscillation, intensity and regularity, the measurement points where it was detected and any non-linearity detected in the time trends. These are given as a text-based report from a tool such as ABB's Plant Disturbance Analyser (PDA) [Horch et.al 2005]. 3.2. The system components, programming and integration The system architecture consists of five components. The purpose of the parsers (blocks 1 and 2 in Figure 1) is to read and deconstruct the XML file containing the CAEX description and the PDA results text file. The XML parsing leads to lists of items of equipment and their connections from which the algorithms of the reasoning engine (3) find physical and control paths in the plant. With additional results from signal-based analysis it also determines root causes for plant-wide disturbances. It can also check that there is a feasible propagation path between a candidate root cause and all the other locations in the plant where secondary disturbances have been detected. The user interface (5) makes it easy to present such queries and to read the answers. Prolog was used to implement the fiinctions of the reasoning engine and C# was used for the remaining parts of the system. Prolog exploits the rule based nature of the connectivity information and the procedural and object-oriented features of C# allow an

Using the Process Schematic in Plant- Wide Disturbance Analysis

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efficient parser and graphical application. The use of P# [Cook, 2004] to translate Prolog code to C# leads to an integrated system as a standard Windows application. Information is represented in different ways due to differing data types which exist in the two languages. The interface to the reasoning engine (block 4 in Figure 1) converts data that must flow between the components into the correct format. 33. Core technologies Extensible Markup Language and CAEX: XML is an open standard developed by the XML Core Working Group, part of the World Wide Web consortium (Quin, 2005). XML uses plain text to represent structured data and uses tags to mark up the information. The structure of an XML document is defined in an XML schema which specifies which tags are allowed and what attributes they can have. A valid XML document must conform to the schema and the schema for representing an industrial plant is defined in the Computer Aided Engineering Exchange (CAEX) standard. Microsoft .Net: Microsoft .Net allows applications to integrate and communicate easily. It consists of the Microsoft .Net Framework, development tools to create software for .Net, and client and server software. The .Net Framework is a run-time environment which allows programs built using different programming languages and running on different supported platforms to exchange data and work together. The C# Programming Language and Prolog: The whole of the CAEX Plant Analyser Application except for the reasoning engine is written in the C# programming language. C# was developed by Microsoft and uses object-oriented features. It is supported by .Net and includes a number of classes that aid working with XML and the creation of Graphical User Interfaces. Prolog (PROgramming LOGic) is a declarative programming language which uses rules called predicates and facts to determine whether a query is true or false. For example, given that pipel is connected to valve 1, and valve 1 is connected to pipe2, and that a path is a list of connections, it is clear that there exists a path between pipel and pipe2. Prolog is able to determine that this is logically the case by applying its rules. The Prolog environment was P# which includes a subset of the Prolog programming language as a native implementation language for the .NET platform and interoperation is achieved by means of C# objects created from Prolog. 3.4. Distinctiveness of the approach The distinctiveness of the CAEX approach should be considered relative to successful developments in SDGs e.g. Maurya et. al, [2004]. A CAEX process representation has less predictive capability than an SDG because it is derived from a process schematic and not from a mathematical model. It gives binary (yes/no) answers to queries about presence or not of physical links and control paths, but not the signs of deviations from a nominal operating point. On the other hand, it is easy to generate an XML file describing the plant topology and the binary nature works well to verify or falsify hypotheses about disturbances and to infer root causes and propagation paths.

4. Case study 4.1. The application and plant object model The application is from Thomhill et.al, [2003a]. Figure 2 shows the process schematic in which the spots (placed by hand) indicate the locations where a plant-wide disturbance was detected. The larger spots show the locations of measurements whose time series were non-linear and therefore are candidates for the root cause. As described in Thomhill et. ah, [2003a], the key logical reasoning step is that the non-linearity reduces as the disturbance propagates away from its source. The task for CAEX Plant

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Analyser is to locate the area of the process where the signals are most non-linear, and to use the connectivity information to identify the location furthest upstream as the most likely candidate for the root cause. Loading the CAEX file from the File menu creates a plant object model and populates the Elements and Internal Links tabs of the GUI. The Elements tab shows the elements found in the topology and whether they are controllers, indicators or neither. The Internal Links tab shows the internal links and the elements connected by the link. Opening of a data file containing the PDA report causes the Input Data tab of the application to be populated with the table of data read from the file. 4.2. Queries Analyses done in the Perform Queries tab (Figure 3) relate to the CAEX plant description. The following functions are available: • Find a physical path between two elements in the topology; • Find a control path between two elements in the topology; • Find out what other elements are directly connected to another element; • Find out whether an element is in a control loop. For the purposes of visualization, the two physical paths found in Figure 3 are shown as heavy lines in Figure 2. The lower half of the window allows queries on individual elements in the topology. The Root Cause tab (Figure 4) is the part of CAEX Analyser where the process connectivity information is linked with the data-driven analysis. The panels in the display show the results of queries about: • Working controllers (i.e. those achieving disturbance rejection); • Non-linear controllers (those with non-linearity in PV or OP or SP); • Non-linear indicators (those with non-linearity in the PV); • Possible root cause controllers (those with nonlinearity in PV and OP, and which are upstream of other non-linear controllers and indicators). The results show that Valve-003 (the level control of the decanter marked A) is the root cause of disturbance 1, which is the correct result. It has also identified the causes of two more oscillating disturbances which were discussed in Thomhill et.al. (2003a).

Figure 2 Process schematic (courtesy of J.W. Cox, Eastman Chemical Company). See sections 4.1 and 4.2 for explanations of the spots and heavy lines.

Using the Process Schematic in Plant-Wide Disturbance Analysis

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5. Critical evaluation and conclusion Requirements for a plant-wide control loop performance analysis system have been given by Paulonis and Cox (2003) and Desborough and Miller (2002), including: 1. Facility-wide approaches including behaviour clustering; 2. Detection of the presence of one or more periodic oscillations; 3. Detection of non-periodic disturbances and plant upsets; 4. Determination of the locations of the various oscillations/disturbances; 5. Incorporation of process knowledge such as the role of each controller; 6. Automated model-free causal analysis to find the most likely root causes. Requirements 1 to 4 have been substantially addressed by the quantitative process history based methods reviewed earlier. While it is possible to interpret the results from topics 1-4 by hand, the benefit of automation is that it speeds up the analysis by helping process control engineers to make queries and document the results. The CAEX Plant Analyser now demonstrates the feasibility of requirements 5 and 6. In summary, the CAEX Plant Analyser has given a new way forward to allow a user to pose queries about the plant and to gain insights into the root causes of plant-wide disturbances.

References Chiang, L.H., and Braatz, R.D., 2003, Process monitoring using causal map and multivariate statistics: fault detection and identification. Chemometrics and Intelligent Laboratory Systems, 65, 159-178. Choudhury, M.A.A.S., Shah, S.L., and Thomhill, N.F., 2004, Diagnosis of poor control loop performance using higher order statistics. Automatica. 40, 1719-1728. Cook, J.J., 2004, P#: A concurrent Prolog for the .NET Framework, Software: Practice and Experience, 34(9):815-845. Desborough, L., and Miller, R, 2002, Increasing customer value of industrial control performance monitoring - Honeywell's experience, AIChE Symposium Series No 326, 98, 153-186. Fedai, M., and Drath, R., 2005, CAEX - A neutral data exchange format for engineering data, ATP International Automation Technology 01/2005, 3, 43-51. Horch, A., Hegre, V., Hilmen K., Melbo, H., Benabbas, L., Pistikopoulos, E.N., Thomhill, N.F, and Bonavita, N., 2005, Root Cause - Computer-aided plant auditing made possible by successfiil university cooperation, ABB Review 2/2005, 44-48. Lee, G.B, Song, S.O.,. and Yoon, E.S., 2003, Multiple-fault diagnosis based on system decomposition and dynamic PLS, Industrial & Engineering Chemistry Research, 42, 6145-6154. Leung, D., and Romagnoli, J., 2002, An integrated mechanism for multivariate knowledge-based fault disLgnosis, Journal of Process Control, 12, 15-26. Maurya, M.R., Rengaswamy, R., and Venkatasubramanian, V., 2003, A systematic framework for the development and analysis of signed digraphs for chemical processes. 1. Algorithms and analysis. Industrial and Engineering Chemistry Research, 42, 4789-4810. Maurya, M.R., Rengaswamy, R., and Venkatasubramanian, V., 2004, Application of signed digraphs-based analysis for fault diagnosis of chemical process flowsheets. Engineering Applications of Artificial Intelligence, 17,501-518. Paulonis, M.A., and Cox, J.W., 2003, A practical approach for large-scale controller performance assessment, diagnosis, Siud impvoYQmQnt, Journal of Process Control, 13, 155-168. Quin, L., 2005, Extensible Markup Language (XML), On-line: http://www.w3.org/XML/ Accessed: 7th Sept 2005. Ruel, M., and Gerry, J., 1998, Quebec quandary solved by Fourier transform, Intech (August), 53-55. Thomhill, N.F., Cox, J.W., and Paulonis, M., 2003a, Diagnosis of plant-wide oscillation through data-driven analysis and process understanding. Control Engineering Practice, 11, 1481-1490. Thomhill, N.F., Huang, B., and Zhang, H., 2003Z7, Detection of multiple oscillations in control loops. Journal of Process Control. 13, 91-100. Venkatasubramanian, V., Rengaswamy, R., and Kavuri, S.N., 2003a, A review of process fault detection and diagnosis Part IL Qualitative model and search strategies. Computers and Chemical Engineering, 27, 313326. Venkatasubramanian, V., Rengaswamy, R., Kavuri, S.N., and Yin, K., 2003^, A review of process fault detection and diagnosis Part IH: Process history based methods. Computers and Chemical Engineering, 27, 327-34.

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Xia, C. and Howell, J., 2003, Loop status monitoring and fault localisation. Journal of Process Control. 13, 679-691. Zang, X.Y., and Howell, J., 2005, Isolating the root cause of propagated oscillations in process plants. InternationalJournal ofAdaptive Control Signal Processing, 19, 247-265.

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5

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Constrained Control for Chemical Processes Using Reference Governor Kiminao Kogiso, Masaru Noda, and Hirokazu Nishitani Department of Information Systems, Nara Institute of Science and Technology, 8916-5 Takayamacho, Ikoma 630 0192, Nara, Japan Abstract A problem exists for nonlinear control systems with state and control constraints in that the violation of such constraints causes degradation of control performance and system destabilization. This paper considers a reference governor control approach for such constrained nonlinear control systems. The main feature of the reference governor approach is the management of a set-point reference to escape constraint violations. Introducing a connection among invariant state sets can achieve the management based on a state estimated by an observer. Moreover, we demonstrate its effectiveness using the example of a continuous stirred tank reactor (CSTR). Keywords reference governor, state and/or control constraint, constraint fulfillment, CSTR 1. INTRODUCTION Constraints are inherent characteristics in almost all practical control systems. They appear commonly in control variables, but also in state variables. Violations of such constraints drastically degrade system performance, and in the worst cases, may lead to the instability of the system or even to its failure [3]. To avoid such situations, some researchers have proposed control techniques that use a reference governor [1,4,5]. A reference governor is an auxiliary nonlinear device whose main operations is managing set-point references to avoid violating constraints. A reference governor is not a controller, but is installed in an already designed control system. This means that the design problems of a reference governor can be separated from a closed-loop system design. In terms of implementation, since a reference governor control approach does not require any changes in an existing closed-loop system, the approach is useful for practical control systems. In the process control area, there are few studies about reference governor control and its numerical application for chemical processes. This paper, therefore, considers the reference governor construction method for nonlinear control systems with state and control constraints, and applies this method to a CSTR to validate its effectiveness. We assume that a feedback controller has already been designed using a conventional design tool. This paper's goal is to achieve the constraint fulfillment of both transient and steady-state responses using a reference governor. 2. CONSTRAINED NONLINEAR CONTROL SYSTEMS Control systems are designed using a proper conventional design method for a chemical process as indicated in Fig. 1. In this paper, we consider the already designed control system and formulate it as the following nonlinear system with polynomial vector fields: x{t) = f{x{t)Mt)).

zi{t) = CMt)),

zo{t)=Co{x{t)Mt)).

(1)

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K. Kogiso et al set-point reference

w(t)

Reference Governor

Designed Controller

Chemical Plant

output

zi(t)

Figure 1. Control system equipped with a reference governor. where time f G Z+ is non-negative, / : St" x 9t^i -> 9t", Ci : 9t" ^ Sl^i, Co : 5t" x 9t^i ^ 9t^o, and / , Co and C\ are smooth function of their arguments, x G 9t" is a state vector of the control system that consists of plant and controller states, jco G 9t" is initial, and w G 9t^i is a set-point reference. Supposing thatXe G 9t" is an equilibrium state, i.e., f{xe,We) = 0 with an appropriate set-point We G 9t^i and that it is locally stabilized using conventional design methods. z\ G 9t^^ is an output vector and partially measurable. In addition, zo G 91^^ is a vector constrained within the prescribed subset 2f C 9t^o as zo{t)e^

(2)

V^GZ+,

where ^ = {zo e 5t^^ iM^Zo ^ ^z} includes an origin, i.e., m^ > 0, where M^ G St^^^^o and m^ G 9t^^ are an appropriate matrix and vector. Note that the above inequality implies that it is component-wise. Our interest is focused on an additional reference management technique applied to the primary designed control system (1). As Fig. 1 illustrates, the operation of a reference governor manages a given set-point reference We into an appropriate signal r to escape the violations of constraint (2), which is mainly performed in transient responses. Therefore, at least, the constraint fulfillment in the steady state is indispensable in enabling us to apply a reference governor. Under We, an output in steady state Xe denotes zoe ^ ^^^- Then, we make the following assumption. Assumption 1 zoe — Co{xe,We) G ^. Remark 1 For nonlinear systems the simplest design method is a linearization technique around the equilibrium state Xg of (I). As a result, we can easily stabilize the equilibrium locally, and in the absence of specified constraints, the controller achieves the control specification. 3. NONLINEAR OBSERVER Before explaning the reference governors, this approach employs a nonlinear observer for state estimation, since the measurement of full state information is generally impossible. Following the procedure in [2], we construct a G-invariant observer for a nonlinear system. The key idea of this method is based on the fact that the observer dynamics remain unchanged up to any transformation of the state coordinates belonging to a Lie-group of synmietries. We now briefly introduce definitions about the observer. For clarity's sake, under w{t) = We, the term We is dropped in function / . Definition 1 Dynamics x{t) = f{x{t)) is called the G-invariant if for all g ^G, andx G <^,

/((p,WO)) = ^WO)/(x(0), where G is a Lie-group acting locally on an open subset ^ C 9t" OM which (pg is a dijfeomorphism.

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Definition 2 Take a G-invariant dynamic x{t) = f{x{t)) with a smooth function C\. The dynamical system i ( 0 = /(x(0,Ci(x(f))) is called a G-invariant observer if, and only if for all g ^G and for all x andx in ^ we have /(x(f),Ci(x(0)) = / ( x ( 0 ) , ^ ( x ( 0 ) M f ) , C i ( x ( f ) ) ) =/(

t\.

It is essential to use the invariance of the sets to assure constraint fulfillment for an infinite time. It is known that a level set from the Lyapunov functions is one such invariant set. A level set is defined using the Lyapunov functions as follows: denote y(x) : 9t" ^ 9t as a candidate of Lyapunov functions for (1), where V{xe) = 0, and the level set can be denoted by Vy, i.e., Vy = {x G 9t" |y(x) < 7}. Since the only level sets cannot achieve constraint fulfillment, we consider the following situation: Vy C CX := {jc G 9t" |Co(jc,w^) G ^ } ,

(3)

where level set Vy has properties of its invariance as well as constraint fulfillment for an infinite time. Under inclusion (3), it is clear from Definition 3 that if level set Vy includes both the initial state XQ and equilibrium state Xe corresponding to a set-point reference w^, then the state x of system (1) moves from the XQ to the Xe fulfilling constraint (2). Figure 2(a) illustrates such a situation using an example planar system, where the level set is labelled by V^. The reference governors need not alter the reference w^, i.e., r{t) = We Vr, and as a result any violation of constraint (2) does not occur for an infinite time. A problem exists, however, when an initial state cannot be included in the level set VS^ due to the narrow and small region of CX. Then, a constraint violation necessarily occurs in the transient response. To escape such constraint violations, inside the state region CX we construct some level sets Vy. with the corresponding Lyapunov function Vi{x) that satisfies Vi{xei) = 0, where Xet is an equilibrium under set-point r^/. Next, these sets are arrayed so that the state remains in the connected sets and attains the final goal state Xe.

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K. Kogiso et al

H

(a) Level set V^ on the state space under satisfaction of (3).

(b) Arraying level sets, Vy^, V^ and V^, instead of V^.

Figure 2. The basic idea for a management technique for reference governors with illustrations of level sets on state space. Constraint (2) holds if a state exists inside a state region between two solid lines in (b). For example, in Figure 2(b), three level sets, Vy^, y ^ , and V^ are computed to connect them. If the state is inside V^^, then a reference governor manages a set-point reference We into We\, and if the state steps are inside the common region of Vy^ and V^, then the reference governor selects the input signal Wei to system (1). Finally, the reference governor manages the set-point reference w{t) — We Vf into signal r{t) that consists of r^i, r^2. and r^3(= We) with two appropriate changes. Additionally, we know that such a Lyapunov function for nonlinear systems is automatically generated using a method [6], which is powerful analyzing of nonlinear systems. Since each level set satisfies inclusion (3), the connected sets hold the following inclusion:

U^^^^^-

(4)

Therefore, under inclusion (4), the state can move from the initial state XQ to a final goal state Xe without going outside the connected level sets. This means that the constraints are fulfilled for an infinite time due to the operation of a reference governor. Theorem 1 Under Assumption 7, if there exists level sets subject to inclusion (4), then the constraints (2) are fulfilled for an infinite time. Remark 3 The construction of a series of level sets considering control performance should be ideal. However, due to the nonlinearity of the constrained systems (1) and (2), this is generally a difficult problem and remains a future task. 5. NUMERICAL EXAMPLE: CSTR A numerical example is the same as one in [5] but is an approximated value of an Arrhenius term. In a CSTR, the raw materials A and B react to form yields C and D. The chemical reaction is as follows: A-\-B -^C-{-D, and the corresponding dynamic equations are as follows: Ni{t) = HR{t)

ViV{t)Rate{t)+Fin{t)CinXt)-Fout{t)Couti{t), 4 = aoV{t)Rate{t)QcR + £ « , • {(7;„(r) - Tref)Fin{t)CinXt) i=\

" (7}e(0 - Tref)Fout{t)CoutXt)}

where the indices / = 1,2,3,4 respectively correspond to components A, B, C, D, and Ra,e{t) = a i C i ( 0 C 2 ( f ) e « 2 / ^ « « ,

TR{t) = Tref +

4

^ '

QcR{t) =

a^U^Fc^it),

.

Constrained Control for Chemical Processes Using Reference Governor

1441

Note that the system has an exponential term, and that by setting AR{t) = e"^^^"^'^ — 1 with new variable AR the term can disappear. Differentiating AR, we obtained the following extra dynamical equation in the analysis:,

A>^(o-«.-"—-c:':-,-^='r''^'^(^^(o+i). -'*^^«>-iHRjt) I t 1 «/^K0 - HRJt) I t i ajNiit), {TrefLUaiNi{t)+HR{t)y

Therefore, the CSTR dynamics are written by equations related to TV, HR, and AR without any approximations. Here, the meaning of each variable is as follows: Ni [mol] denotes the mol holdup of the /th component, HR [J] is the total enthalpy of CSTR, Fin [mol/s] is the feed flow rate, Tin [K] is the feed temperature, Few [g/s] represents the flow rate of the cooling water, Fout [mol/s] is the outlet flow rate, Couti is the mold fraction of the yield, TR [K] represents the reaction temperature, V [m^] is the volume in reactor. Rate [mol/s] is the reaction rate, QCR [J/S] is the heat removal rate, Q«. represents the feed mol fraction, V/ is the stoichiometric coefficient, Pi [mol/w?] is the mol density, and ai is the specific heat. The control problem considered here is the tracking control of set-points Vset = 1.1 [m^] and TRset = 399 [K] under the following constraints: V^ 0.2 < V{t) < 1.2, 350 < TR{t) < 400, and Fcw{t) > 0, Fout{t) > 0. To construct a G-invariant observer we consider the scaling of Ni -^ PNi with j8 > 0. Take the observer's state XQ = [NI,N2,N3,N4,HR,TR,V] and the G action is defined for each JS > 0 via the transformation: Xo -^ (PpM = [^Ni,pNz,PN3,liN4,PHR,TR,V/P]. As a result, we can estimate the plant state by the G-invariant observer, and see more details in [2]. After making a conventional design of two PI controllers - one from the error TRset ~TR{t) to theflowrate of the cooling water /Vw(0 ^^^ the other from the error Vset — V(^) to the outlet flow rate Fout{t) - the controller states are respectively denoted as Xpn G 9t and Xpi2 G 5t. Next, we obtain the formulation of a constrained nonlinear control system (1). The state vector is defined as X = [Ni,N2,N^,N4,HR,AR,Xpi\,Xpi2] G St^, the set-point reference as w = [Vset^ TRset] ^ ^^» the output as zi = [V,TR] G 9t^, and the constrained variable as zo — \y'>TR^Fcw',T'out\ ^ ^'^• In this simulation, we set an initial state JCQ = [2.5 x 10^,2.5 x 10^, 1.2 x 10^, 1.2 x 10^, 1.0 x 10^,2.284 X 10~^5.0 x 10,2.0 x 10"^]. These controllers can stabilize the equilibrium state jc^ - [6.342X 102,6.342x102,5.266xl0^5.266xl0^,5.234xlO^3.255 xlO-^5.821,9.993 X 10] corresponding to the set-point reference We = \yset->TRset\' However, when a set-point with rapid changes is input to a constrained system (1) without any reference management by a reference governor, the constraints will be violated in the time-responses from the initial state JCQ to the goal state Xe. A simulation result of such a time-response is shown in Figure 3. Figure 3 respectively shows for the given set-points We = [Vset, TRset] the time-responses of liquid volume V and the reaction temperature TR under the prescribed constraints. From these figures, we can see that the constraints are violated, where in this simulation the constraints are considered soft constraints. In the case of the saturation of time-responses of V and TR by the upper bound, we numerically confirm that the CSTR control system is destabilized due to saturation. On the other hand, the set-point reference Vset and TRset are managed by the reference governor into the actual input signals Vmset and TRmset, which are both thin solid lines in Figure 4 (a)(b). As a result, from Figure 4 we can see that both constraints about V and TR are fulfilled, where the no time-responses of Few and Fout are introduced, but where we confirmed the constraint fulfillment. From the simulation results, we can see that the governor only manages the given set-point and that constraint violations do not occur. Therefore, it is obvious that our proposed control technique with a reference governor is more effective for control systems in which constraint

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h^ aj 398 Q.

Vset

I

'

E *1 3961

V upper bound on V 1000 time

upper bound on TJj

1500 [s]

1000 time

15.00 [s]

2000

(a) Set-point V^et of liquid volume V(t) and (b) Set-point TR^et of reaction temperature tank capacity. TR (t) and temperature limitation. Figure 3. Simulation results of CSTR time-responses without a reference governor. 4021

^

i

2^ 400[

I r S.398 T/jmsef(managed Taset)


TR

c 396

upper bound on TR iOO

1000 " 1300 time [s]

2000"

:

(a) Managed set-point Vmset{t), liquid volume (b) Managed set-point TR^set(t), reaction temV{t), and tank capacity. perature TR{t), and temperature limitation. Figure 4. Simulation results of CSTR time-responses with a reference governor. violations or saturation affects control performance and stability. 6. CONCLUSION This paper considered reference governors for constrained nonlinear control systems using a G-invariant observer. For constraint fulfillment we introduced the connection of level sets obtained from Lyapunov functions. If the connected level set includes the initial and an equilibrium states of the system, then the state can move through the connected set to the equilibrium fulfilling constraints. In a numerical CSTR simulation, we confirmed the effectiveness of the reference governor, comparing the time-response results of a PI closed-loop control system with and without a reference governor.

REFERENCES 1. A. Bemporad. Reference governor for constrained nonlinear systems. IEEE Transactions on Automatic Control, 43(3):415-419, 1998. 2. S. Bonnable and R Roucho. On invariant observers. In T. Meurer, K. Graichen, and E. D. Gilles, editors, Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems, Lecture Notes in Control and Information Sciences 322, pages 53-65. Springer, 2005. 3. E. G. Gilbert. Linear control systems with pointwise-in-time constraints: What do we do about them? In Proceedings of the 1992 American Control Conference, page 2565, Chicago, 1992. 4. E. G Gilbert and I. Kolmanovsky. Nonlinear tracking control in the presence of state and control constraints: A generalized reference governor. Automatica, 38(12):2063-2073, 2002. 5. K. Kogiso, M. Noda, and H. Nishitani. A reference governor for constrained nonlinear systems based on a Lyapunov function. In Proceedings of The 3rd International Symposium on Design, Operation and Control of Chemical Processes, pages FrA02-5, Seoul, 2005. 6. A. Papachristodoulou and S. Prajna. On the construction of Lyapunov funtions using the sum of squares decomposition. In Proceedings of the 41th IEEE Conference on Decision and Control, pages 3482-3487, Las Vegas, 2002.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Agent-based diagnosis for granulation processes Rozalia Lakner^, Erzsebet Nemeth^'^, Katalin M. Hangos*^ and Ian T. Cameron^ ""University ofVeszprem, Egyetem u. 10, Veszprem H-8200, Hungary ^Computer and Automation Research Institute HAS, Kende u. 13-17 ,Budapest H-lUl, Hungary ^The University of Queensland, St Lucia, Brisbane, QLD, Australia, 4072. Abstract A multiagent diagnostic system implemented in a Protege-JADE-JESS environment is described in this paper. It enables the use of a combination of diagnostic methods from heterogeneous knowledge sources. The system is demonstrated on a case study for diagnosis of faults in a granulation process. Keywords: diagnosis, multiagent system, granulator.

1. Introduction For complex multiscale process systems that are difficult to model, a combination of model-based analytical and heuristic techniques is usually needed to develop a diagnostic system. The approach of multiagent systems (MAS) [1] which emerged in AI represents a promising solution for such a diagnosis task, being based on information from heterogeneous knowledge sources [2]. A multiagent system can then be used for describing the behavior and the structure of the elements in a diagnosis system. These elements include the system model, the observations, the diagnosis and loss prevention methods with each element being established through formal descriptions. This work investigates the use of the architecture and algorithms of multiagent systems for diagnosing faults in process plants. In particular we consider a granulation process and the advice to operators in order to reduce potential losses. The significance of this work lies in a coherent fault detection and loss prevention framework based on a well-defined formalization of complex processes and the diagnostic procedures. Its novelty lies in the interesting combination of tools and methodologies that can be generalized to other process-related applications. 2. Main processes in fault detection and diagnosis Early detection and diagnosis of process faults while the plant is still operating in a controllable region can help to avoid abnormal events and reduce productivity losses. Therefore diagnosis methods and diagnostic systems have practical significance and strong traditions in the engineering literature. Fault detection and diagnosis methods can be classified in two main categories: the analytical model-based methods [3] and the logical and/or heuristic methods. 2.1. HAZOP andFMEA analysis HAZOP [4] is a systematic procedure for determining the causes of process deviations from normal behavior and consequences of those deviations. The main idea behind HAZOP is that hazards in process plants can arise as a result of deviations from normal operating conditions. The result of a HAZOP analysis is collected in a HAZOP table, where a possible cause can be regarded as root cause if it refers to a failure mode of a

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physical component in the system or a result of operational error. For example, cause (2): failure in valve actuator, in the second major row of the HAZOP table in Figure 3 is such a component failure. When such a root case is found we can complement or refine the diagnosis result by using the complementary information from an FMEA table. Failure mode and effects analysis (FMEA) [5] is a qualitative hazard identification (HAZID) tool, universally applicable in a wide variety of industries. FMEA is a tabulation of each system component, noting the various modes by which the equipment can fail, and the corresponding consequences (effects) of the failures. System components not only include the physical equipment but can encompass software and human factors. It is regarded as one of the most comprehensive hazard identification techniques. HAZOP and FMEA provide a comprehensive analysis of the key elements that help constitute an effective diagnostic system. 3. The structure of the multiagent diagnostic system The proposed framework for a multiagent diagnostic system consists of an ontology design tool and a multiagent software system. The domain specific knowledge is represented as modular ontologies using the ontology design tool Protege [6]. This knowledge is integrated into a multiagent software system where different types of agents cooperate with each other in order to diagnose a fault. 3.1. The ontologies of the diagnostic system In order to facilitate the modularity and general applicability of the system, two set of ontologies are developed: 1. A process-specific ontology that describes the concepts of the processes in question similar to the general ontology for process systems given by OntoCAPE [7]. 2. A diagnostic ontology that contains the semantic knowledge on diagnostic notions, tools and procedures. From the common part of the two different types of ontologies a real-time database is formed storing the values of process variables, actuator variables and related variables. 3.2. The structure of the multiagent diagnostic system in JADE (integrated with JESS) Several agent construction and simulation tools have been proposed by a number of researchers and commercial organizations. A non-exhaustive list of them is ABLE [8], AgentBuilder [9], FIPA-OS [10], JADE [11] and ZEUS [12]. We have chosen JADE (Java Agent DEvelopment Framework) as the multiagent implementation tool, because it has integration facilities with the Protege ontology editor and the Java Expert System Shell (JESS) [13]. Similar to the ontology classification, the agents of the diagnostic system belong to the following main categories: 1. Process agents that assist the user and the other agents in modelling and simulation of the process in question. This can be under different, faulty and non-faulty circumstances. The most important process agent diXQ process output predictors^ (PPs) that perform prediction with or without preventive action(s). 2. Diagnostic agents that perform measurements, symptom detection, fault detection [14], fault isolation and advice generation for avoiding unwanted consequences. These agents may perform logical reasoning and/or numerical computations. Some types of diagnostic agents and their main tasks are as follows: • Symptom generator and status evaluator: checks whether a symptom is present or not.

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Fault isolators (FIs): isolate the fault based on different techniques (fault-tree, HAZOP, FMEA, etc.). Loss preventors (LPs): suggest preventive action(s) based on different techniques that have been used for the HAZID and remedial actions (HAZOP, prediction, etc.). Completeness coordinator: checks completeness of the result (detection, isolation or loss prevention) and calls additional agents if necessary. Contradiction or conflict resolver (CRES): calls additional agents in case of contradiction.

Monitoring Agent

Real-time database (Blackboard)

V

Corroborating Agent

Preprocessor Agent

Real-time agents Based on Real-time database ontology

Control Agent

i

/

ACLj messages Remote Monitoring Agent (GUI)

Agent Management System

Directory Facilitator

Blackboard Agent

RMI server (for communication) + BB agent \

=^

\ACL

ACL messages

Diagnostic Agent

Diagnostic Agent

Diagnostic Agent

Diagnostic agents Based on Diagnostic ontology (HAZOP, FMEA)

Process Agent

Process Agent

Process Agent

Process agents Based on Process-specific ontology

Figure. 1. The structure of the multiagent diagnostic system Beside the two main categories, the diagnostic system contains the following Real-time agents for controlling and monitoring the process environment: • Monitoring agents: access and/or provide data from real world or from simulation. • Pre-processor agents: detect the deviances which are the possible symptoms. • Corroborating agent: acts on request from diagnostic agents and provides additional measured values or information on request. 4. Case study The proposed methods and the prototype diagnostic system are demonstrated on a commercial fertilizer granulation system [15]. 4. L The granulation process The granulator circuit contains the granulator drum where fine feed or recycle granules are contacted with a binder or reaction slurry. The slurry can be added at various points along the axial direction of the drum, thus controlling moisture content in the drum and thereby growth. Growth occurs depending on a complex set of operational and property factors. Drying, product separation and treatment of recycle material then occurs. A simplified schematic of a typical granulation plant is shown in Figure 2. 4.2. The investigatedfault scenario The diagnostic process performed by the proposed agent-based diagnostic system is illustrated on the example of a symptom, where the mean particle size, measured as D50

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is less than its nominal value. This situation corresponds to the first row of the HAZOP table seen in Figure 3. Many other more complex, interacting faults are possible in such an industrial system. The aim of the case study was to investigate the cooperation of the diagnostic agent-set, based on logical reasoning complemented with a process simulator. This provides a holistic approach to the diagnosis problem.

Initial stream cJeviatton "d50 LOW"

sirH^^iiillj)

Figure. 2. Granulation pilot plant schematic 4.3. Simulation results A detailed simulation model of the plant has been implemented in MATLAB/SIMULINK that provided the "measured data" with and without faults. Simplified dynamic models implemented also in MATLAB/SIMULINK served as tools for the Process output predictor agent to check the validity of the preventive actions suggested by the Loss preventor agent. Based on the variable-values supplied by the Monitoring agent the Pre-processor agent determines the deviances in the system. In the case of a detected deviance, the Symptom generator agent checks the presence of symptoms and sends it to the Completeness coordinator agent which calls the Fault isolator- and Loss preventor agents if necessary. These agents determine the possible faults and suggest preventive actions. Based on the suggestions of these agents the Completeness coordinator agent orders the operation of the Process output predictor agent for predicting the behaviour of the system with the preventive action. In the case when the symptom "Mean particle diameter is LESS" is detected, first the Fault isolator agent based on HAZOP knowledge is called. It uses the formally described version of the HAZOP table and performs reasoning to deduce the root cause(s) of the symptom. The reasoning order of this Fault isolator agent is indicated in both Figure 2 (the flowsheet) and in Figure 3 (the HAZOP table). It is seen from the HAZOP table that the reasoning finds three possible root causes (i.e. causes that corresponds to failure modes in elementary system components) as indicated in the second row of the table. This set of diagnosis and loss prevention results has been refined based on the FMEA analysis (by the Fault isolator agent based on FMEA knowledge; its reasoning is also indicated on the FMEA table in Figure 3) initiated by the Completeness coordinator agent and the root cause "Slurry flow control valve fails Closed" has been deduced. Thereafter the Loss preventor agent determines the possible preventive actions, where again three possibilities are found in the last column of the first row in the HAZOP

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table. Again, the Completeness coordinator agent requests the operation of the Process output predictor agent to determine the effect of the suggested preventive actions. An example of the predicted output behaviours of the particle mass flows of different sizes with the action "increase in fresh feed size" (option a)) is shown in Figure 4. Action requiied

Consequences

decrease in system holdup change in granulation c ondition change in recycle P S D * reduced liquid phase in granulator

a) increase fresh feed size b) change to orignal feed type c) increase slurry flow a) re'view training regimes for operators and ensure crosschecking on setpoint changes

* lack of gran^4lation

b) actuator preventati-re maintenance plan check * lower product size range c) activate slurry control'valve byp a s s and s e t flow manually flow from granulator d) check slurry feed density and adjust preneutralizer The relevant part of the HAZOP table

The relevant part of the FMEA table

Figure. 3. A part of the corresponding HAZOP and FMEA tables

- ^

0.9

IVI^12) 1

0.8 0.7

^.0.6

1'

|0.4

^0.3

^0.2

0.1

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Figure. 4. The effect of suggested preventive action "increase in fresh feed size"

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5. Conclusion and discussion The prototype multiagent diagnostic system implemented in a Protege-JADE-JESS environment interfaced with a dynamic simulator is proposed in this paper. The proposed system architecture enables the use of a combination of diagnostic methods from heterogeneous knowledge sources. The process ontology and the process agents are designed based on the structure of the process system, while the diagnostic agents implement the applied diagnostic methods. A specific completeness coordinator agent is implemented to coordinate the diagnostic agents based on different methods. The proposed multi-agent diagnostic system has clearly shown the advantages of such a technology in building complex diagnostic systems based on heterogeneous knowledge sources. However, the interfacing of the system components with each other and with other system components, such as dynamic simulators is far from trivial. In addition, the reliability of such a complex software system has not reached a sufficient level to be ftilly deployed into an industrial application. Further work is needed to enhance interoperability and to provide a more comprehensive set of analysis tools to attain high reliability systems. Acknowledgements This research has been supported by the Hungarian Research Fund through grants T042710 and T047198, which is grateftilly acknowledged, as well as the Australian Research Council International Linkage Award LX0348222. References [1] N. R. Jennings and M. J. Wooldridge, 1998, Agent Technology, Springer-Verlag, Berlin. [2] H. Worn et al., 2004, DIAMOND: Distributed Multi-agent Architecture for Monitoring and Diagnosis, Production Planning & Control, 15, pp. 189-200. [3] Blanke, M., Kinnaert, M., Junze, J., Staroswiecki, M., Schroder, J., Lunze, J., Eds, 2003, Diagnosis and Fault-Tolerant Control. Springer-Verlag. [4] R.E. Knowlton, 1989, Hazard and operability studies : the guide word approach, Vancouver: Chematics International Company [5] W. Jordan, 1972, Failure modes, effects and criticality analyses. In: Proceedings of the Annual Reliability and Maintainability Symposium, (IEEE Press 1972) pp. 30-37 [6] The Protege Ontology Editor and Knowledge Acquisition System, 2004, http ://protege. stanford.edu [7] Yang, A., Marquardt, W., Stalker, I., Fraga, E., Serra, M., and D. Pinol, 2003, Principles and informal specification of OntoCAPE, Technical report, COGents project, WP2. [8] Agent Building and Learning Environment (ABLE), http://www.research.ibm.coni/able [9] Reticular Systems. AgentBuilder - An integrated Toolkit for Constructing Intelligence Software Agents. 1999. http://www.agentbuilder.com. [ 10] FIPA-OS. http://www.nortelnetworks.com/products/announcements/fipa/index.html. [11] JADE - Java Agent DEvelopment Framework, http://jade.tilab.com. [12] H.S. Nwana, D.T. Ndumu and L.C. Lee, 1998, ZEUS: An advanced Tool-Kit for Engineering Distributed Multi-Agent Systems. In: Proc of PAAM98, pp. 377-391, U.K. [13] JESS, the Rule Engine for the Java platform, http://herzberg.ca.sandia.gov/jess/ [14] Venkatasubramanian, V., R. Rengaswamy and S. N. Kavuri, 2003, A review of process fault detection and diagnosis Part II: Qualitative models and search strategies. Computers and Chemical Engineering 27, pp. 313-326. [15] Balliu, N., 2004, An object-oriented approach to the modelling and dynamics of granulation circuits, PhD Thesis, School of Engineering, The University of Queensland, Australia 4072.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

An application of metamodels for process optimization Marcos V.C. Gomes^ I.David. L.Bogle^, Darci Odloak'', Evaristo C. Biscaia Jr.^ "Petroleo Brasileiro S.A. - PETROBRAS/CENPES, Ilha do Fundao Q. 7, Rio de Janeiro, 21910-940, Brazil ^Chemical Engineering Department, UniversityCollege London, London WCIE 7JE, UK ^Departamento de Engenharia Quimica, Universidade de Sao Paulo, Sao Paulo 05508900, Brazil ^Programa de Engenharia Quimica-COPPE, Universidade Federal do Rio de Janeiro,P.O. Box 68502, Rio de Janeiro, 21945-970 Brazil Abstract In this work, a methodology for the use of metamodels in the steady-state optimization of chemical processes is proposed. The methodology is focused on automatic procedures for use in real-time optimization (RTO). The methodology is tested with a NLP problem from the literature. Keywords: Process optimization; Metamodels; Kriging; Real-time optimization 1. Introduction Although widely used in the industry, real-time optimization (RTO) tools based on rigorous models still present challenges related to attaining solutions with desirable levels of robustness, accuracy and computational efficiency [1,2], when complex process models are being considered. Moreover, more detailed and mathematically complex new models are always being developed. In order to solve simulation, design and optimisation problems with complex models in a tractable way, many solutions based on reduced models and approximation techniques have been proposed in the literature [3,4]. Kriging models have been used in a number of engineering applications [5], to approximate rigorous models when those computer codes become too time-consuming to be used directly. In this context, they are called surrogate models or metamodels [7,8]. This family of models has shown high flexibility to fit non-linear, complex functions, typically requiring less information than other approaches like neural nets or regression splines [3]. One of the most common uses of kriging models in engineering has been in optimal design, in sequential and interactive procedures where many optimization problems related to many scenarios can be solved, due to the fast computation of the metamodel. Despite the use of kriging models in many engineering fields, there are few known applications in Chemical Process Engineering [3,4,6]. This work addresses the development of a methodology where kriging models are combined with rigorous process models in a procedure that aims for use in RTO. Taking advantage of the good combination of flexibility and fast computation of kriging models, process optimization can be accomplished with minimum computations of the rigorous model, therefore attaining robustness and computational efficiency.

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2. Kriging models The kriging model structure presented here is the most frequently used in the literature (for more details on the kriging models structure and theoretical background refer to the book of Santner et al. [10]). Let the set of functions y(x,u) be a rigorous mathematical description of a process, where x are independent variables and u model parameters. The kriging models that approximate the rigorous one are built from a set of design points (X,Y) obtained from runs of the rigorous model for a set of parameters u©. They are composed by a linear regression model and a random function: yi(x,uo) = P ^ f ( x ) + Zi(x),i = l...nY

(1)

The first term is usually a low-order polynomial. The random functions Zi have the following form: Zi = r J ( x ) R " ^ ( Y - F P i )

nX ri(x) =

where

nX andR.,(Xi) = f|9tij(eij,Xkj-Xij)

(2)

k,l = l...nP

(2a)

The matrix F is obtained by computing the f(x) values for the design inputs X. 91 are correlation models, usually built as functions of the difference or distance between two points. Hence, Rii (Xi) is a matrix that contains the covariances of all design data, and ri(x) is a vector that contains the covariances of a point x and the design points. For expressions of correlation models used in the literature, refer to [9,10]. 3. A methodology for Real Time Optimization with metamodels Most of the approaches for optimization based on metamodels follow the general steps below [9-11]: L Generate a first (base) metamodel; II. Run the optimization problem; III. Check if the solution is acceptable. If it is, FINISH; IV. If the solution is not acceptable, improve the metamodel and return to 2. During the design, all steps are subject to human intervention. For real-time applications, however, graphics and other non-automatic tools cannot be used. Another important issue in a real-time environment are the frequent changes in process behavior due to changes in feed composition, catalyst properties, etc.. Even when these properties are not measurable in real-time, a rigorous process model will have parameters that, when adjusted, will allow reliable answers. However, a metamodel built from responses of a rigorous model for a specific set of parameters cannot provide acceptable responses for a new set. Therefore, real-time applications based on metamodels require adjustments to their answers whenever changes in the process or the rigorous model occur. The main features of the proposed methodology are presented below. 3.1. Generation of the base metamodel The base metamodel generation step is not accomplished in real time. It is composed of the following procedures: Generation of data with the rigorous model: Data is obtained through successive runs of the rigorous model for different sets of values of the independent variables. As for

et al

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physical experiments, experimental design techniques can be used to define these sets of values (frequently referred as computer experiments), providing reliable models while minimizing the computational effort (See [10] for experimental design techniques suitable for computer experiments). The Latin Hypercube Design (LHD) is widely used with kriging models [13], and has been used in this work. Selection of variables: This step is important to assure model consistency, avoiding the inclusion of unreal influences. A smaller number of independent variables will require a smaller group of parameters and therefore a smaller set of data to be obtained from the rigorous model. Forward Stepwise Regression is used to define the regression model structure based on statistical criteria. First and second order terms are included. Parameter estimation: The MatLab toolbox DACE [9] provides tools for estimation of the kriging model parameters, given the design (LHD) data and the regression model structure. As the objective fimction (based on the maximum likelihood method) can be multimodal, the heuristic optimization algorithm Particle Swarm available in the MATLAB toolbox PSOt [14], has been used along with the DACE toolbox. Starting from an initial set of estimates, the PSO algorithm generates a set of parameters that are then enhanced by the original parameter estimation algorithm of DACE toolbox. Correlation model selection: The DACE toolbox provides seven correlation models. In this work, a subset of the available correlation models is selected, and the parameter estimation procedure described above is accomplished for each one. The correlation model that provides the smallest value for the objective function is selected. Assessment of the base metamodel: A new, independent data set is used to perform the base metamodel assessment. Though some techniques that avoid the need for generating new data have been proposed, recent works [5,7] suggest that the use of independent data would be more effective. 3.2. The adaptive optimization procedure As justified before, an optimization procedure based on metamodels should have adaptive features. The main aspects of the procedure under development are presented below. Metamodel update: If the set of rigorous model fiinctions y(x,u) are approximated for a set of metamodels y(x,u), there will be a prediction error: yi(x,u)=yi(x,uo)+£i(x,u),

i = l...nY

(3)

It is assumed that the prediction error 8i is a consequence of two contributions: changes in the intrinsic parameters of the rigorous model (u ^ uo), and errors related to the lack of capacity of the metamodel to fit the original function perfectly, even when u = UQ: 8i (x, u) = z\^^ (u) + ^™ (x)

i = 1,..., nY

(4)

There are several ways to describe the prediction error. In this work, mathematical expressions are proposed for each contribution and the parameters can be estimated during the optimization procedure, correcting the base metamodel response for different areas of the search space or whenever changes in the rigorous model responses occur. Some of the possible corrections are: (I) Z\^^ is negligible and 8i^^^ linear: e f ^ ( u ) = 0; ef"T(x) = b T . x (II) Si'''^ and 8i''" linear:

(5)

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z\^{vL) = J^n-

z™{±) = hj-x

(6)

(III) Include the rigorous model parameters into the kriging model and assume 8i^^^ is linear: yi(x,u)=yi(x,u)+8pT(x,u),

i = l...nY;

£p'r(x) = b ^ x

(7)

(IV) Describe the prediction error by a secondary kriging model:

£,(x,u) = wf^(x,u)

(8)

Correction I (5) was used by [11] in the optimal design of electric circuits, and [15] used a secondary kriging model as in (8) to prevent ill-conditioning during the optimization. Model assessment: The assessment of the adjusted metamodel during the adaptive optimization procedure is accomplished by the comparison of the prediction errors with previously defined maximum error values for each dependent variable. 3.3. Implementation of the optimization problem with metamodels Optimization with rigorous and with kriging models was implemented in a similar manner, except that in the latter case y was replaced by the approximations y in the general problem : min

f(y,x)

X

Jh(y,x) = 0 [g(y,x)<0

subject to {

(9)

Only convex problems have been addressed in this work, although kriging models can be used to approximate non-convex functions. The package NPSOL, based on the SQP algorithm, has been used. Termination criteria'. The termination of the optimization procedure depends on the desired accuracy. Two extreme alternatives are being considered: I. Assume the optimal solution for the optimization with metamodels acceptable if the prediction error of the solution is acceptable; II. Solve the optimization problem based exclusively on the metamodel with corrections. Then, starting from the attained solution, solve the problem using the rigorous model. 4. Example A modified version of the alkylation process problem in [15] has been used to illustrate the proposed methodology. The original problem was modified (eqs. 10a - lOi) so that only two decision variables are used: xl and x8. The variable 8, arbitrarily introduced into equation lOd plays the role of a model parameter, along with variables x2 and x6. Eq. (10b) is an equality constraint, while the remaining inequality constraints define the operating limits to the dependent variables x3, x4, x5, x7, x9 and xlO (See [15] for a complete description of the problem). A set of kriging base metamodels was generated from a LHD design with 15 data points to approximate all dependent variables (xs, X4, X5, Xy, X9, xio). To simulate changes in the rigorous model response, each model parameter (8, x2 and x6) received a base and a second value. The eight possible combinations (see Table 1) of these values were used as cases for which the optimization was performed with the rigorous model and with the

etal

An Application of Metamodels for Process Optimization

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adaptive procedure based on the kriging models, with model correction of type I (eq. 5) and termination criteria type I. min f(x)= C1X4X7 -C2X1 -C3X2 -C4X3 -C5X5 XpXg

subject to: h = xixg-(x2+ X5) xf"" < Xi < x P ^ , i = 3,4,5,7,9,10 where:

X4 =xi(1.12 + 0.13167xg 0.0067xg) + 5 X5 =1.22x4-XI ^

^

^

(lOa-i)

X4X6X9

1000(98-X6)

X7=86.35 + 1.098x8-0.038x^+0.325(x6-89) xio=-133 + X7 X9 =35.82-0.222x10

Table 1. Cases of study for the alkylation process CASE

X2

BASE ClOO COlO COOl

15819 15000 15819 15819 15000 15000 15819 15000

Clio ClOl

coil

cm

5

X6

90.115 90.115 86.200 90.115 86.200 90.115 86.200 86.200

0.0 0.0 0.0 100 0.0 100 100 100

The results are presented in figure 1. A comparison of the results obtained for the optimization with rigorous model and the adaptive procedure was made by (1) relative difference between the objective function values; (2) scaled quadratic norm between the -Objectivie function error •

-Reduction onsimulations, %

-IMI2

1.0 BOO

1.0 EOI

1.0 E03

1.0 ED4

1.0E05 BASE

C100

C010

COOl

C110

C101

C011

C111

Figure 1 - Comparison of optimal solutions with the rigorous model and with the adaptive optimization procedure based on kriging models.

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values of the decision variables at each solution; (3) the amount of reduction in the number of simulations with the rigorous model with the adaptive procedure. The resuhs show that the solutions based on the metamodels are quite accurate and were obtained with less than 30% of the number of simulations required when only the rigorous model was used. 5. Conclusions This work proposes a methodology for RTO based in kriging metamodels. The methodology is comprised of automatic procedures suitable for real-time applications involving data generation, independent variables and model structure selection, model assessment and updating. A modified version of the alkylation process NLP problem has been used to illustrate the methodology. The results show that accurate solutions were obtained with less then 30% of the simulation runs required by the optimization with the rigorous model for all cases studied. The next steps shall involve the implementation of the other proposed strategies for metamodel updating and termination criteria, the enhancement of the adaptive algorithm, and tests with more complex examples. References [ 1] D. Alkaya, S. Vasantharajan, L.T. Biegler, Eng. Chem. Res, 39 (2000) 1731. [ 2] J.E. Tolsma, J.A. Clabaugh, P.L Barton, Ind. Eng. Chem. Res, 41 (2002) 3867. [ 3] K. Palmer, M. Realff, Trans IchemE, 80 (2002) Part A 760. [ 4] K. Palmer, M. Realff, Trans IchemE, 80 (2002) Part A 773. [ 5] J.D. Martin, T.W. Simpson, AIAA Joumal, 43 No. 4 (2005) 853. [ 6] X. Van, J.F. Pekny, Computers and Chemical Emgineering, 29 (2005) 1317. [ 7] M. Meckesheimer, A.J. Booker, R.R. Barton, T.W. Simpson, AIAA Joumal, 40 No. 10 (2002) 2053. [ 8] S.M. Clarke,J.H. Griebsch, T.W. Simpson, Analysis of support vector regression for approximation of complex engineering analyses. Proceedings of ASME-DETC 2003, 2003. [ 9] S.N. Lophaven, H.B. Nielsen, J. Sondergaard, DACE - A MATLAB Kriging Toolbox. Technical University of Denmark, Technical Report IMM-TR2002-12, 2002. [10] T.J. Santner, B.J. Williams, & W.I. Notz, Springer-Verlag New York, Inc. The Design and Analysis of Computer Experiments. New York, 2002. [11] M.C. Bernardo, R Buck, L. Liu, W.A. Nazaret, J. Sacks, W.J. Welch, IEEE Transactions on Computer-Aided Design, 11, no.3 (1992) 361. [12] G.G. Wang, Transactions of the ASME, Joumal of Mechanical Design, 125(2003) 210. [13] J.P.C. Kleijnen, European Joumal of Operational Research, 164 (2005) 287. [14] B. Birge, PSOt, A Particle Swarm Optimization Toolbox for MatLab. IEEE Swarm Intelligence Symposium Proceedings, April 24-26, 2003 [15] A.J. Booker, Well Conditioned Kriging Models for Optimization of Computer Simulations. The Boeing Co., Tech Report M«&CT-TECH-00-002, Bellevue, WA, Feb 2000. [16] T.F.Edgar, D.M.Himmelblau, L.S. Lasdon, McGraw-Hill, Optimization of Chemical Processes. 2^^ edition. Singapore, 2001.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

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Time scale separation and the link between open-loop and closed-loop dynamics Antonio Araiijo^, Michael Baldea*^, Sigurd Skogestad^ and Prodromes Daoutidis^ ^Department of Chemical Engineering, Norwegian University of Science and Technology, Trondheim, Norway ^Department of Chemical Engineering, University of Minnesota, Minneapolis, MN, USA ^Department of Chemical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece This paper aims at combining two different approaches ([1] and [2]) into a method for control structure design for plants with large recycle. The self-optimizing approach ([1]) identifies the variables that must be controlled to achieve acceptable economic operation of the plant, but it gives no information on how fast these variables need to be controlled and how to design the control system. A detailed controllability and dynamic analysis is generally needed for this. One alternative is the singular perturbation framework proposed in [2] where one identifies potential controlled and manipulated variables on different time scales. The combined approaches has successfully been applied to a reactor-separator process with recycle and purge. Keywords: singular perturbation, self-optimizing control, regulatory control, selection of controlled variable. 1. I N T R O D U C T I O N Time scale separation is an inherent property of many integrated process units and networks. The time scale multiplicity of the open loop dynamics (e.g., [2]) may warrant the use of multi-tiered control structures, and as such, a hierarchical decomposition based on time scales. A hierarchical decomposition of the control system arises from the generally separable layers of: (1) Optimal operation at a slower time scale ("supervisory control") and (2) Stabilization and disturbance rejection at a fast time scale ("regulatory control"). Within such a hierarchical framework: a. The upper (slow) layer controls variables (CV's) that are more important from an overall (long time scale) point of view and are related to the operation of the entire plant. Also, it has been shown that the degrees of freedom (MV's) available in the slow layer include, along with physical plant inputs, the setpoints (reference values, commands) for the lower layer, which leads naturally to cascaded control configurations. b . The lower (fast) variables implements the setpoints given by the upper layer, using as degrees of freedom (MV's) the physical plant inputs (or the setpoints of an even faster layer below). c. With a "reasonable" time scale separation, typically a factor of five or more in closed-loop response time, the stability (and performance) of the fast layer is not influenced by the slower upper layer (because it is well inside the bandwidth of the system). d. The stability (and performance) of the slow layer depends on a suitable control system being implemented in the fast layer, but otherwise, assuming a "reasonable" time scale separation, it should not depend much on the specific controller settings used in the lower layer. e. The lower layer should take care of fast (high-frequency) disturbances and keep the system reasonable close to its optimum in the fast time scale (between each setpoint update from the layer above). The present work aims to elucidate the open-loop and closed-loop dynamic behavior of integrated plants and processes, with particular focus on reactor-separator networks, by employing the approaches

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of singular perturbation analysis and self-optimizing control. It has been found that the open-loop strategy by singular perturbation analysis in general imposes a time scale separation in the "regulatory" control layer as defined above. 2. SELF-OPTIMIZING CONTROL Self-optimizing control is defined as: Self-optimizing control is when one can achieve an acceptable loss with constant setpoint values for the controlled variables without the need to re-optimize when disturbances occur (real time optimization). To quantify this more precisely, we define the (economic) loss L as the difference between the actual value of a given cost function and the truly optimal value, that is to say, L{u,d) = J{u,d)-Jopt{d)

(1)

During optimization some constraints are found to be active in which case the variables they are related to must be selected as controlled outputs, since it is optimal to keep them constant at their setpoints (active constraint control). The remaining unconstrained degrees of freedom must be fulfilled by selecting the variables (or combination thereof) which have the best self-optimizing properties with the active constraints implemented. 3. T I M E SCALE SEPARATION B Y S I N G U L A R P E R T U R B A T I O N ANALYSIS In [2] and [3] it has shown that the presence of material streams of vastly different magnitudes (such as purge streams or large recycle streams) leads to a time scale separation in the dynamics of integrated process networks, featuring a fast time scale, which is in the order of magnitude of the time constants of the individual process units, and one or several slow time scales, capturing the evolution of the network. Using singular perturbation arguments, it is proposed a method for the derivation of non-linear, non-stiff, reduced order models of the dynamics in each time scale. This analysis also yields a rational classification of the available flow rates into groups of manipulated inputs that act upon and can be used to control the dynamics in each time scale. Specifically, the large flow rates should be used for distributed control at the unit level, in the fast time scale, while the small flow rates are to be used for addressing control objectives at the network level in the slower time scales. 4. C A S E S T U D Y O N R E A C T O R - S E P A R A T O R PROCESS In this section, a case study on reactor-separator network is considered where the objective is to hierarchically decide on a control structure which inherits the time scale separation of the system in terms of its closed-loop characteristics. This process was studied in [3], but for the present paper the expressions for the flows F , L, P , and R and economic data were added. 4 . 1 . The r e a c t o r - s e p a r a t o r process The process consists of a gas-phase reactor and a condenser-separator that are part of a recycle loop (see Figure 1). It is assumed that the recycle flow rate R is much larger than the feed flow rate Fo and that the feed stream contains a small amount of an inert, volatile impurity yj^o which is removed via a purge stream of small flow rate P. The objective is to ensure a stable operation while controlling the purity of the product XB • fci A first-order reaction takes place in the reactor, i.e. A-^ B. In the condenser-separator, the interphase mole transfer rates for the components A, B, and / are governed by rate expressions of the form Nj = pS

Kja{yj — p ~ ^ j ) ^ ^ where Kja represents the mass transfer coefficient, yj the mole fraction in the gas phase, Xj the mole fraction in the hquid phase, P^ the saturation vapor pressure of the component j , P the pressure in the condenser, and PL the liquid density in the separator. A compressor drives the flow fi:om the separator (lower pressure) to the reactor. Moreover, valves with openings Zf, zi^ and Zp allow the flow through F , L, and P , respectively. Assuming isothermal operation (meaning that the reactor and separator temperatures are perfectly controlled), the dynamic model of the system has the form given in Table 1.

Time Scale Separation and the Link Between Open-Loop and Closed-Loop Dynamics 1457

Figure 1. Reactor-separator process.

4.2. Economic approach to the selection of controlled variables: Self-optimizing control computations The open loop system has three degrees of freedom at steady state, namely the valve at the outlet of the reactor (2/), the purge valve (^p), and the compressor power iWs)- The valve at the separator outlet {zi) has no steady state effect and is used solely to stabilize the process. The profit (—J) = {PL —pp)L —pw^s is to be maximized where PL, PP, and pw are the prices of the liquid product, purge (here assumed to be sold as fuel), and compressor power, respectively. The profit should be maximized subject to the following constraints: The reactor pressure Preactor should not exceed its nominal value and the product purity Xb should be at least at its nominal value. In addition, there are bounds on the valve openings which must be within the interval [0 1] and the compressor power should not exceed its upper bound. For optimization purposes the most important disturbances are the feed flow rate Fo, the feed compositions 2/A,o5 2/B,o and ^/,o, the reaction rate fci, and the reactor temperature TreactorTwo constraints are active at the optimal through all of the optimizations (each of which corresponding to a diff'erent disturbance), namely the reactor pressure Preactor at its upper bound and the product purity Xb at its lower bound. These consume two degree of freedom since it is optimal to control them at their setpoint (active constraint control) leaving one unconstrained degree of freedom. To find the remaining controlled variable, it is evaluated the loss imposed by keeping selected variables constant when disturbances occur and then picking the variable with the smallest average loss. Accordingly, by the self-optimizing approach, the primary variables to be controlled are then y ~ [Preactor Xb Wg] with the manipulations u = [zf Zp Wg]4.3. Singular perturbation approach for the selection of controlled vsiriables According to the hierarchical control structure design proposed by [2] based on the time scale separation of the system, the variables to be controlled and their respective manipulations are: MR (Preactor) "^ F {Zf)\

Mv

(Pseparator)

^

R (Zp)] ML ^

L (zl); Xb ^

MR^setpoint

[Preactor,setpoint)]

yi,R

^

P-

It

is important to note that no constraints are imposed in the variables in contrast to the self-optimizing control approach. 4.4. Control configuration arrangements The objective of this study is to explore how the configurations suggested by the two different approaches can be merged to produce an effective control structure for the system. Thus, as a starting point, the following two "original" configurations are presented: 1. Figure 2: This is the original configuration ([2]) from the singular perturbation approach. 2. Figure 3: This is the simplest self-optimizing control configuration with control of the active constraints {Preactor and Xb) and self-optimizing variable Ws-

A. Araujo et ah

1458 Table 1 Dynamic model of the reactor-separator network. Differential equations Algebraic equations ^reactor

'-^^

= M^o{yA,o-yA,R)

+ R{y

^

-^ w^[Fo{yi,o - yi,R) + Riyi - yi,R)]

^

=

- yA,R)

MyRga

^sevarat

-kiMRyA,R]

^ ^

NA = KAa(yA - ^-^^XA) y"^

F-R-N-P

Ni = Kia(yi NB

J i

- p "^^ ^ xi) ^

\

^ ^lFiyA,R - yA) -NA+ yAN] = lk[F{yi,R-yi)-Ni + yiN]

-

i^separator ^separator

= KBOL\yB \

-

p

) i

^^

XB\-

^separator

J i

N = NA-\-NB+NI ^ ^ ^V fZf y/r reactor

i'^separator

^ ^^ ^VlZly^

-t^^downstream

J^^separator

-t ^ ^VpZpyJ rseparator

R-

1

iRgasTs,

-i downstream

^)

MR, MVI and ML denote the molar holdups in the reactor and separator vapor and liquid phase, respectively. Rgas is the universal gas constant; 7 = ^ is assumed constant; Cv/, Cvi, and Cvp are the valve constants; Pdownstream IS the prcssure dowustrcam the system; e the compressor efficiency; and Preactor.max is the maximum allowed pressure in the reactor.

yjE£u(cc)-

i ^

'—SK

Figure 2. Original configuration based on singu- Figure 3. Simplest self-optimizing configuration lar perturbation with control of Xh^ Pseparator-, and with control of Xh^ Preactor, and Wg. yi,R-

None of these are acceptable. The configuration in Figure 2 is far from economically optimal and gives infeasible operation with the economic constraints Preactor exceeded. On the other hand, Figure 3 gives unacceptable dynamic performance. The idea is to combine the two approaches. Since one normally starts by designing the regulatory control system, the most natural is to start from Figure 2. The first evolution of this configuration is to change the pressure control from the separator to the reactor (Figure 4). In this case, both active constraints {Preactor and Xb) are controlled in addition to impurity level in the reactor {yi^n). The final evolution is to change the primary controlled variable from yi^R to the compressor power Wg (Figure 5). The dynamic response for this configuration is very good and the economics are close to optimal.

Time Scale Separation and the Link Between Open-Loop and Closed-Loop Dynamics 1459 ^^aw^

Figure 4. Modification of Figure 2: Constant pressure in the reactor instead of in the separator.

Figure 5. Final structure from modification of Figure 4: Set recycle {Ws) constant instead of the inert composition (yi,R).

4.4.1. Simulations Simulations are carried out so the above configurations are assessed for controllability. Two major disturbances are considered: a sustained reduction of 10% in the feed flow rate Fo at t = 0 followed by a 5% increase in the setpoint for the product purity Xb a^t t = 50h. The results are found in Figures 6 through 9.

1

I'

. \n

i u.

50

100

i4

1] 1

1

|-4-f i — [7

t / [7 / "

Time (h)

6

'

'

Figure 6. Closed-loop responses for configuration in Figure 2: Profit = 43.13A:$//i and 43.32fc$//i (good but infeasible).

Figure 7. Closed-loop responses for configuration in Figure 3: Profit = 43.21/c$//i and = 43.02fc$//i.

The original system in Figure 2 shows an infeasible response when it comes to increasing the setpoint of Xb since the reactor pressure increases out of bound (see Figure 6). With Preactor Controlled (here integral action is brought about) by Zf (fast inner loop), the modified configuration shown in Figure 4 gives infeasible operation for setpoint change as depicted in Figure 8.

A. Araujo et al

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§T 5 ,L

IT

v.. 100

150

§ y^

i\

|5

[-V

^ \ •

50

100

1.

Time (h)

o: ^

Figure 8. Closed-loop responses for configuration in Figure 4: Profit = 43.20/c$//i and = AZmk%/h.

Figure 9. Closed-loop responses for configuration in Figure 5: Profit = 43.21A;$//i and = 43.02fc$//i.

The proposed configuration in Figure 3, where the controlled variables are selected based on economics presents a very poor dynamic performance for setpoint changes in Xh as seen in Figure 7 due to the fact that the fast mode xi) is controlled by the small flow rate Zp and fast responses are obviously not expected, indeed the purge valve {zp) stays closed during almost all the transient time. Finally, the configuration in Figure 5 gives feasible operation with a very good transient behavior (see Figure 9). The steady state profit for the two disturbances is shown in the caption of Figures 6 through 9. 5. C O N C L U S I O N This paper contrasted two different approaches for the selection of control configurations. The selfoptimizing control approach is used to select the controlled outputs that gives the economically (near) optimal for the plant. These variables must be controlled in the upper or intermediate layers in the hierarchy. The fast layer (regulatory control layer) used to ensure stability and local disturbance rejection is then appropriately designed (pair inputs with outputs) based on a singular perturbation framework proposed in [2]. The case study on the reactor-separator network illustrates that the two approaches may be successfully combined. REFERENCES [1] S. Skogestad. Plantwide control: The search for the self-optimizing control structure. Journal of Process Control, 10:487-507, 2000. [2] M. Baldea and P. Daoutidis. Control of integrated process networks - a multi-time scale perspective. Computers and Chemical Engineering. Submitted, 2005. [3] A. Kumar and P. Daoutidis. Nonlinear dynamics and control of process systems with recycle. Journal of Process Control, 12(4):475-484, 2002.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubHshed by Elsevier B.V.

Fault Detection and Diagnosis of Pulp Mill Process Gibaek L e e \ Thidarat Tosukhowong^, Jay H. Lee^ Chungju National University, Chungju 380-702, Korea^ Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.^ Abstract The hybrid fault diagnosis method based on a combination of the signed digraph and the partial least-squares (PLS) has the advantage of improving the diagnosis resolution, accuracy and reliability, compared to those of previous qualitative methods, and of enhancing the ability to diagnose multiple fault. In this study, the method is applied for the fault diagnosis of the pulp mill process, which is used to produce pulp from wood chips. It is one of the biggest processes tested in the fault diagnosis area, and it will be a new benchmark process to test the fault diagnosis method. In order to consider large time delay in the process, the diagnosis model was modified to include the information of the time delay between process vaiables. Through case studies, the proposed method demonstrated good diagnosis capability compared to the previous hybrid method not considering time delay. Keywords: fault detection, fault diagnosis, partial least squares, pulp mill process, transporation lag 1. Introduction In order to improve the safety of the chemical plant and their plant personnel, automatic fault diagnosis system analyzes process data on-line, monitors process trends, and diagnoses faults when an abnormal situation arises. Among a variety of fault diagnosis approaches for chemical processes, expert system, state estimation such as observer and EKF (extended Kalman filter), signed digraph (SDG), fault tree, qualitative simulation, statistical method, and neural network have been developed\ Tennessee Eastman challenge process^ created by the Eastman Chemical Company has been used for evaluating fault diagnosis methods during the past several years. Recently, Castro and Doyle introduced a pulp mill simulator as a benchmark process for process system engineering studies^. The simulated process can be an alternative realistic testbed for plantwide fault diagnosis methods because it includes approximately 8200 states, 140 inputs, 114 outputs, and 6 operating modes. Also, the process shows common characteristics of the industrial procesess such as long measurement delay, many components, several recycle streams, and high nonlinearity. Safe and reliable operation of pulp mill process has been important for survival in a very competitive international market. Although automatic fault diagnosis systems are in demand, to our knowledge there currently exists no literature on fault detection and diagnosis of a pulp mill. For the fault diagnosis of the process, this study uses the PCA (principal component analysis), which is compared with the hybrid method combining SDG and the PLS"^. The hybrid method has the advantages of improving the diagnosis resolution and accuracy compared to previous qualitative methods. Moreover, it enhances the reliability of the diagnosis for all predictable faults, including multiple fault. Although it is based on statistical process data, it allows the diagnosis model to be built based on easily obtainable data sets, and does not require faulty case data sets.

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2. Pulp Mill Process 2.1. Process Description Pulp mills produce pulp of a given Kappa no. or brightness from wood chips while satisfying the criteria of productivity and environmental impact. The Kappa no., which is a major variable of interest, is a measure of the amount lignin remaining in the wood. Pulp mills can be divided into two major sections of the fiberline and the chemical recovery. It includes key unit operations such as digester, brown stock washers, oxygen reactor, bleach plant, evaporators, recovery boilers, smelt dissolving tank, green liquor clarifiers, mud washers, causticizers, white liquor clarifier, and lime kiln. The fiberline removes lignin from the wood to achieve fibers of a certain brightness and strength target. It uses chemicals such as NaOH, NaSH, oxygen and CIO2. The process is sequential from the digester to brown stock washers, and the bleaching plant. Wood chips enter the digester with the white liquor (WL, mixture of NaOH and NaSH). Pulps delignified in the digester are sent to the brown stock washer where dissolved lignin and chemicals are removed and sent to chemical recovery area to regenerate NaOH and NaSH from the spent liquor. The final operations in the fiberline are to further delignify and brighten the pulp in the bleaching section of an oxygen reactor (O) followed by a chlorine dioxide tower (Di), a sodium hydroxide tower (E), and the second chlorine dioxide tower (D2). 2.2. Process and Fault Simulation The pulp mill simulator developed by Castro and Doyle^ is used for this study. It is written in the C language using Matlab® s-function format with Simulink®, and can be downloaded from the homepage of Doyle's group"^. This study modified the simulator to get more realistic fault diagnosis problem as follows. At first, white noises were added to the simulator input and output variables. The maximum and minimum values of the noise are ±0.1% of the maximum expected errors or changes of the variables, which were specified in the original simulator. 114 measured output variables and 82 controller outputs are used for fault diagnosis. It is assumed that the measurements of manipulated variables and disturbances are not available. The name of the process variables are formed as adding the type of the variables to the consecutive number of the variables. For instance, KN4 means the Kappa number sensor of y(4). Also, the simulator has been modified to include sensor or control valve faults. The faulty sensors and control valves were chosen with regard to the most important variables of the process. 4 sensors of digester Kappa no., final brightness of the pulp in the D2, [0H-] concentration before the E washer, D2 production rate were selected, and 3 sensor faults of bias (-5% and +20%), precision degradation {±\% and ±10%), and drift (0.025%o x time) were simulated for the sensors. Also, 4 control valves of wood chips flow, Dl caustic flow, caustic flow 3, and oxygen WL flow. For each sensors, 2 valve faults of bias (P/o and -5%) and sticking were defined. In addition, 5 disturbances already included in fiberline of the original simulator became the target faults. They are the variations in wood temperature and densities, changes in operating temperature and compositions of the caustic in the E tower, and change in the wash water temperature of the E tower, and the number of target faults is 37. Adding the type of fault to the name of the equipment composes the name of the fault. For example, KNmSBias means -5% bias for the digester Kappa number sensor. The sampling interval is 5 minutes and the total simulation time is 1500 minutes. Fault or setpoint change occurs at 600 minutes.

Fault Detection and Diagnosis of Pulp Mill Process

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3. Fault Diagnosis Method 3.1. System Decomposition based on SDG The first step of building fault diagnosis model is the system decomposition centering on measured variables in SDG. In SDG, each arc represents the instantaneous effect produced from the source node to the target node. All source nodes connected to a particular target node by means of the arcs have a direct influence on that target node. That is, only the source nodes connected to a target node can affect the particular target node. Because unmeasured nodes among the source nodes cannot demonstrate the direct effects from faults, unmeasured nodes are removed, resulting in the reduced digraph that contains only the measured nodes of the original SDG. Each decomposed subprocess includes a central measured variable (target variable) as well as measured variables (source variables) and faults connected to the target variable. In order to diagnose 37 faults defined in the pulp mill process, the process is decomposed centering on 16 measured variables and the reduced digraph for the decomposed subprocess is obtained. The variables are Fl, F3, KN4, CD 10, T12, T15, and T16 around the digester, KN19, T21, KN22, T23, OH24, T25, BR26, and P37 around the bleaching plant, and CI22 in the recausticizing section. Local Fault diagnosis can be performed for each decomposed subprocess. Because fault diagnosis is locally executed for each target variable, the fault diagnosis method using the system decomposition can diagnose all types of multiple faults except for those multiple faults that affect the same measured nodes. 3.2. Fault Diagnosis based on Dynamic PLS Models This simple diagnosis on the based on the decomposition technique is to estimate the value of each target variable using the measured values of the source variables connected to the target variable. A substantial difference between the estimated and measured values implies the occurrence of one or more faults. The sensor faults occurred in the sensor corresponding to the source variables used for the estimation, produce errors in the estimated values. The faults added to the target node give rise to errors in the measured values. The estimation of our previous study used the PLS model built for each decomposed subprocess. The input X of the model contains the source variables connected to the target variable, and the output Y is the estimated value of the target variable. To handle the process dynamics accurately, we used DPLS that is integrated with ARMAX. In addition to the past values of the source variables, the resulting input of DPLS for a target variable includes the past values of the target variable, as well as the source variables. The necessary number of past values (time lags) / and the principal components (PC) are determined from the learning data. The number / is usually 1 or 2 which indicates the order of the dynamic system. Each DPLS model can be built from the operation data set representing local relations between the input variables and the output variable of the DPLS models. Therefore, the required data set for each DPLS model can be easily obtained. The available data sets can be obtained in the presence of set-point changes or external disturbances, which occur frequently. Therefore, the proposed method does not need a faulty case data set, which would otherwise be difficult to obtain. Fault detection is performed by the observation of the residual, which is the difference between the estimated value determined by the DPLS model and the measured one.

^=J/-i>/

(1)

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Where r/ is the residual of output variable /, and yt and y. are the measured and estimated values of variable /, respectively. A qualitative state, which corresponds to ranges of possible values for the residual, becomes an attribute of the residual. We will consider methods that use three ranges: low, to which the qualitative state (-) is assigned; normal, assigned (0); and high, assigned (+). If a fault occurs, the qualitative state for the residual may be (+) or (-). The abnormal qualitative state for the residual becomes a symptom, which is expressed as the pair of the target variable and the qualitative state of the residual. Those faults inducing the abnormality of each residual are classified along with their symptoms, and the classified faults are stored in a set (called a fault set). Also, faults can be classified into two types: one is the faults added to the target variable and the other is the sensor faults that occur in the sensor corresponding to the source variables in the DPLS model. The first step of on-line fault diagnosis is the monitoring of the residuals, in order to detect their qualitative change of state. The detected residual of a variable becomes an element in the set. The fault of sensor degradation can make the signs of the symptoms fluctuate between (+) and (-), which greatly decreases the diagnosis accuracy. In order to make a stable diagnosis, CUSUM monitors the squared residuals as well as the residuals of each variable. The next step of fault diagnosis is to obtain the minimum set of faults that can explain all of the detected symptoms. 3.3. Incorporation of Time Delay into the DPLS Model Time delay can be defined as the time interval between the start of an event in one point and its resulting action at another point. It is also referred transportation lag, time delay, dead time or distance-velocity lag. In the target process, large time delay can be found in the equipments such as the digester, the storage tank, and the D2 tower. As the model assumes that the change of the source variable effects instantly the target variable, these large time delay may make the estimation model inaccurate. In order to increase the accuracy of the estimation, the information on the time delayfi*omthe input variables to the output variables should be incorporated into the dynamic PLS model. The input matrix, X of the DPLS model for the variable / is modified as follows. x..(/t

+ T

- T^^^ •

0,y }

(2)

x=

^iN:

Where, T

V "*" ^MD,Ni

MD,i

^^MAX(t

^MD,i

-lAt^\

^Ti

is the measurement delay of /, r ^ is the time delayfi-om/ to /, and t ^ . •'

TD,ij

•'

J

7

" MD,i

, r Twr. )• The measured value of variable / to calculate the residual / is

\ MD,j=\,Ni'

MD,i I

yk'^'^MDi~'^1!^i)' ^^^ study used the data obtained with the set-point change in order to determine the dead time.

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4. Result and Discussion Consider the example of Dl Stuck (sticking of Dl CIO2 flow control valve). While the +10% setpoint change of wood chips flow at 600 minutes, the Dl CIO2 flow control valve stuck also at the same time. Using the DPLS model, the detection sequence of symptoms is KN22(+) and KN22^ from 915 minutes, T21(+) from 920 minutes, T21^ from 935 minutes, BR26^ from 1185 minutes, and BR26(-) from 1190 minutes (Figure 1 (a)). The bounds of Figure 1 are the minimal jump size of CUSUM. There is no false detection, and DlCm5Bias, DlCStuck, WLlBias, WLStuck, and WoodDens obtained as the solution from 920 minutes. Although the resolution is 5, operator may easily judge that WLlBias, WLStuck, and WoodDens are not fault candidates because there are no detected variables in the digester and oxygen reactor section. Also, the resolution can be reduced by using the dynamics of the residuals. Figure 1 (b) shows the residual obtained by the previous hybrid model without time delay. KN22(+) is detected from 995 minutes, KN22^ and T21(+) from 1005 minutes, T21^ from 1035 minutes, BR26^ from 1055 minutes, and BR26(-) from 1105 minutes. The detection is 80 minutes later than the proposed method with time delay. Figure 1 shows evidently that the models with time delay generate clearer and more accurate residuals denoting fault occurrence than the ones without time delay. 0.05 0.04 H o

g 0.02

0.03

•2 0.01

1 0.02

'So

f2

'% 0.01 0 -0.01

, 500

700

0 -0.01

900 1100 Time (min)

500

1300 1500

700

900

1100

1300

1500

1300

1500

Time (min)

(a)

700

900

1100

1300

900

1500

1100

Time (min)

Time (min)

(b)

Fig. 1 (a) residuals obtained by the DPLS models with time delay, (b) residuals obtained by the DPLS models without time delay for DlCStuck.

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fault BR0025Drift BRlODeg BR20Bias BRm5Bias CFmSBias DlCStuck DlCmSBias EbackTemp ECausComp ECausTemp KN0025Drift KNlODeg KN20Bias KNmSBias

detection delays 215/40/560 20/20 20/65/65 315/395 20/20 15/15 65/65 15/15 115/105 55/45 35/25 35/35

accuracy

fault

lA 1/1 1/1 1/0 1/1 1/1 0.99/0.94 0.22/0.1 1/1 1/1 1/1 1/1 1/0.97 1/1

OH0025Drift OH20Bias 0Hm5Bias PR0025Drift PR20Bias PRm5Bias WClBias WCStuck WCm5Bias WLlBias WLStuck WLm5Bias WoodDens WoodTemp

detection delays 65/85 25/25 25/25 145/175 35/35 45/45 35/35 35/35 15/35 25/25 10/10 10/20 110/60 115/70

accuracy

iTl 0.99/1 1/1 1/1 1/1 1/1 1/1 1/1 0.02/0.0 1/0.94 1/1 0.85/0.28 0.26/0.17 1/1

To compare the diagnostic performance, accuracy and detection delay are used. The accuracy is 1 if the diagnosis is accuate; that is, the true fault is included in the final fault candidates set. Otherwise, the accuracy is 0. The detection delay refers to the time from fault occurrence to fault diagnosis. In Table 1, the former performance parameter value is the diagnosis result obtained by the proposed method, and the latter is the one by the previous hybrid method not considering time delay. The 1% faults of BRlDeg, CFlBias, DlClBias, KNlDeg, and PRlDeg are too small to be diagnosed by two methods, and are not shown in Table 1. In addition, the method failed to diagnose CFStuck and PRlODeg. Though T15 is independent with WCm5Bias and WoodDens, the proposed method detected wrongly T15 for these faults. Because WLStuck can explain all detected symptoms, the accuracy was very low. The diagnostic performance by the new method for all cases except WoodDens and WoodTemp are much better than the previous one. The faster detections of two cases are due to fast and wrong detection of T12. In WoodTemp case, wood temperature increases and the symptom of T12(+) should be detected. However, the previous method detected T12(-), and the detection was earlier than the new method.

Acknowledgement This work was supported by grant No. (R05-2002-000-00057-0) from the Basic Research Program of the Korea Science & Engineering Foundation.

References [1] V. Venkatasubramanian, R. Rengaswamy, K. Yin and S.N. Kavuri, Comp. Chem. Engng., 27, (2003) 293 [2] JJ. Downs and E.F. Vogel, 1993, Comp. and Chem. Engng., 17, (1993) 245 [3] J.J. Castro and F.J. Doyle III, Journal of Process Control, 14, (2004) 17 [4] G. Lee, S.-O. Song, and E.S. Yoon, 2003, Ind. Engng. Chem. Res., 42, (2003) 6145 [5] http://www.chemengr.ucsb.edu/~ceweb/faculty/doyle/docs/benchmarks/mill

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment Richard Faber^, Harvey Arellano-Garcia^, Pu Li^, Gunter Wozny ^ ^Berlin University of Technology, Department of Process Dynamics and Operation Sekr. KWT-9, Strasse des 17. Juni 135, 10623 Berlin, Germany ^Technische Universitdt Ilmenau, Simulation and Optimal Processes Department, P. O. Box 10 05 65, 98684 Ilmenau, Germany Abstract An optimization-based approach is proposed to improve the observabihty of large-scale systems with iterative adjustment of the weighting matrix. The approach is based on a rigorous process model making it applicable to nonlinear systems. The result of the state estimation is improved by introducing sensitivity information into the weighting of the objective function. The needed sensitivity information is iteratively computed and adjusted during the optimization run. The approach has been applied to a large-scale nonlinear industrial process to estimate the unknown feed composition from scarce measurement data. Keywords: model-based optimization, state estimation, weighting adjustment 1. Introduction Many advanced approaches have been developed for on-line optimization and control of industrial processes. The realization of these approaches requires the information about the current state of the processes. It is usually assumed that the state can be gained through measuring essential process variables. In many cases, however, it is not possible to measure all required variables, especially in on-line applications where a frequent update of measurements is necessary. Therefore, it is necessary to use techniques which are able to estimate as much unmeasured variables as possible from the set of available measurements. One possibility to determine the unmeasured variables is to use the concept of observability classification to identify the observable variables from a given set of measurements using balance equations. A broad variety of methods have been proposed to address this problem. Basically two categories can be distinguished in equation oriented observability analysis: structural and non-structural techniques. The nonstructural techniques are based on calculations made using model coefficients. In general, the nonstructural equation-oriented techniques can be applied to linear [1] and bilinear relationships [2]. On the other hand, structural techniques use the process occurrence matrix to classify the variables into observable and unobservable [3,4]. The occurrence matrix is rearranged and observable subsets are generated from the whole variable space. Therefore, these methods are very useful if complex industrial plants with a large number of units are analyzed. A broad variety of state observers such as the extended Kalman filter (EKF) have been developed to estimate unmeasured process states continuously from frequent measurements [e.g. 5]. A weakness of the standard EKF is that it performs poorly when applied to highly nonlinear processes. Another approach uses the techniques of parameter estimation where nonlinear optimization techniques are used to minimize the difference between the experimentally

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measured output and the predictive output of the rigorous mathematical model. In this case the result of the optimization strongly depends on the definition of the objective fimction and the nonlinearity of the process model. In this work, we propose an optimization-based approach where the weighting matrix in the objective function is iteratively adjusted to improve the result for state estimation of nonlinear large-scale systems. The basic idea is to utilize the available measured variables to infer unmeasured variables based on a detailed nonlinear process model. The residual of the measured data to the computed values according to the model is to be minimized. Due to the nonlinearity of the model the sensitivity of the measured variables to the unmeasured variables strongly influences the estimation quality and a poor scaling may even lead to a wrong result. Therefore, the weighting matrix in the objective ftmction plays a key role and has to be carefully chosen. We include the information of the sensitivity of the measured variables to the unmeasured variables in the estimation procedure. The approach has been applied to small-scale example problems as well as to an industrial gas purification process described by a rigorous non-equilibrium model. 2. Optimization-based state estimation The problem of estimating the unmeasured variables firom a set of available measurements can be stated as an optimization problem where the unknown variables are the optimization variables. An objective function consisting of the difference between the experimentally measured output and the predictive output of the rigorous mathematical model is to be minimized. For the case of unknown input variables the optimization problem takes following form: min / = Ay'W;^ Ay + Aw'W;Aw u

^

s.t. g(x,u,0) = O h(x,u,0)>O

where u e U c S t ' " are the (independent) input variables of the process model, X c X G 9t' are the (dependent) output variables, y c xe 9t'~'" is the measured subset of the output variables and w c u e 9t'"^ is the measured subset of the input variables. Ay is the residual of the measured data to the computed values according to the model for the output variables and Aw is the residual of the measured data to the estimated values of the input variables. W^ and W^ are weighting matrices, g and h are the equality and inequality constraints, representing the model equations and restrictions and 0 are model parameters which in this case are assumed to be known. If the optimization problem is analyzed it is obvious that if the number of measured variables is smaller than the degree offi*eedomof the optimization problem (number of inputs of the process model) the solution is not unique and it strongly depends on the initial values given. On the other hand, for highly nonlinear processes even if there are theoretically enough measurements available the result of the optimization might be still very poor because the sensitivity of the measured variables to the estimated variables is

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very small. In this case the result strongly depends on the nonlinearity of the process model and the scaling of the objective function introduced by the weighting matrices. The standard procedure for weighting the individual terms of the objective function is to use the variance-covariance matrix of the measurement errors to compensate for different scaling of the variables and different accuracies of the measurements. If the measurements are assumed to be independent from each other, the diagonal elements of the weighting matrix consist of the standard deviation of the individual measurements: ^/ag(W) = ^/ag(WJ = G

(2)

with a being the vector of standard deviations of the measurements. In this formulation the sensitivity information that links the measured and the unmeasured variables is only available in the process model g. The objective function only depends on the residuals of the measured variables and the (mostly constant) weighting of the individual variables. It is, however, also possible to include information about the measurement variances, the variances of the measured variables and the sensitivity of these variables with respect to the unknown variables. 2.1. Using the variance-covariance matrix for model predictions In several research works for model discrimination and experimental design an alternative weighting matrix formulation is used [e.g. 6]: W = (W,+W,J

(3)

where W^^ is the variance-covariance matrix for model predictions: W,, =S-V„-'-S^

(4)

V^ is the variance-covariance matrix for the optimization variables and S being the sensitivity matrix of the measured variables with respect to the unknown variables:

s = | ^ z-[y.wr

(5)

an By using the information of W^^, the sensitivity information of the measured variables to the unknown variables is additionally introduced into the objective function. In previous applications V^ has to be approximated using historical experimental data or has to be calculated at the solution of a previous optimization [7]. In the presented approach the variance-covariance matrix V„ has been iteratively calculated using first order approximations during runtime following eq. 6:

dn da' W. dz dz

(6) f)ll

The needed sensitivities — are calculated at each iteration of the optimization using first order approximations of the Hessian matrix of the objective fimction [8]. This approximation is strictly speaking only valid at convergence u = u*, but with the

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iterative recalculation during runtime the approximation of V„ converges to the actual solution. The model-based optimization with iterative weighting adjustment has been used to identify unmeasured quantities from only few measurements for the separation process presented in the next section. 3. The process model The estimation procedure described in the previous section has been applied to the ammonia hydrogen sulfide circulation scrubbing process which is a common absorption/desorption process for coke-oven-gas purification. This process selectively removes NH3, HCN and H2S in the effluent gases from the coking oven while simultaneously suppressing the absorption of C02. A rigorous rate-based tray-by-tray model for the mass and heat transfer is used to describe the process. The model is composed of a large-scale highly nonlinear equation system with up to 636 variables per ab-/desorption unit. The enhanced mass transfer due to the chemical reactions in the liquid phase is accounted for by the use of enhancement factors. The mass transfer is described by a mass transfer coefficient and the interfacial area. The correlations of the mass transfer coefficients and the interfacial area are taken from Billet and Schulte's correlations [9]. A detailed description of the process model can be found in [10]. 3.1. Estimation setup To test the performance of the proposed estimation approach the absorption part of the described coke-oven-gas purification process has been chosen. The setup was chosen according to the specifications of the deacidifier unit in the real scrubbing plant where the enriched water from the H2S washing unit is freed from sour components. The absorption plant with the in- and outgoing streams is shown in Fig. 1.

V^.x^.D^J,

Fig. 1: Absorption column setup The enriched water enters at the top of the column containing 6 components (H2O, CO2, NH3, H2S, HCN and traces of coke oven gas-COG). The same components can be found in the vapor stream entering at the bottom of the column. The aim of this investigation was to determine the unknown variables for the mass streams (Vj, V2) and component concentrations (Xj,X2) of the ingoing streams. It was assumed that the temperatures of all streams can be measured (T1-T4) and that 5 additional temperature measurements inside the column are available (T5-T9). In addition mass stream of 3 streams ( X - V3) and the densities of the liquid streams (DijDs) are assumed to be known. This means

Improving Observability of Large-Scale Systems by Iterative Weighting Adjustment

1471

that 14 measurement can be used to estimate 14 variables (Note that although Vj, V2 can be measured they are also included in the set of variables to be estimated. This is reasonable to account for measurement errors in these variables). Although enough measurements are available to reduce the degree of freedom of the problem to 0 the result is still very poor because the temperature measurements hold very little information about the composition of the feed streams especially in multicomponent absorption case. 3.2. Decision about additional measurements To decide which measurements are to be additionally included, the Fischer-InformationMatrix is used:

F = s'-w;^-s

(7)

The Fischer-Information-Matrix is often used in experimental design to gain more information about the estimated parameters. In this case measurements have been chosen which maximally increase the determinant of the information matrix. This is equivalent to making the elements of the variance-covariance matrix V^ small. In this investigation additional heuristical information about possible measurement in the real plant have been used to decide on which variables should be additionally measured. Combining both information, the NH3 and CO2 concentration of the outgoing vapor stream (4) and the H2O saturation of the ingoing vapor stream (2) have been chosen. The measurements have been used to determine the unknown quantities using the described optimization procedure. 4. Results To investigate the effect of introducing sensitivity information into the objective ftinction by iteratively adjusting the weighting matrix using Wj,^, the process model setup described in section 3 is used and the input variables given in Table 1 and Table 2 are to be estimated. The values in Table 1 and Table 2 are given in mole fractions and kmol/h. Table 1. Vapor feed conditions H20

NH3

CO2

H2S

HCN

COG

V

2.43E-2

7.93E-3

1.72E-2

3.28E-3

1.14E-3

0.946

3.66

vap

Table 2. Liquid feed conditions H2O

NH3

CO2

H2S

HCN

COG

<

0.985

l.llE-2

2.18E-3

8.36E-4

8.46E-4

0.

2.03

A standard SQP algorithm has been used for optimization and the convergence criteria has been set to 1 .e-6 for all runs. As the result of the optimization is influenced by the initial values, for each investigation, 10 estimation runs have been performed. For each run randomly generated values in a 20% neighborhood of the actual values have been used as initial values. The performance of the estimation has been analyzed by comparing the mean weighted residual and the maximum mean residual for the feed variables. These values are given in Table 3.

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w == w, Mean residual 7.67%

Maximum residual 22.01%

w = (w,+w,,) Mean residual 1.9%

Maximum residual 7.12%

From Table 3 it can be seen, that by using the additional information provided by the weighting matrix W^^ the result of the identification of the unknown feed conditions is significantly better than by using only W^. 5. Conclusions An optimization-based approach is proposed to improve the observability of nonlinear large-scale systems. The approach is used, to estimate unmeasured quantities from available measurements based on a rigorous process model. The variance-covariance matrix for model predictions has been included into the objectivefixnction,to introduce the coupling of the variables into the objective fiinction. The needed sensitivity information is iteratively computed and adjusted during convergence of the optimization algorithm. Due to the iterative adjustment of the weighting matrix it is not necessary to determine the variance-covariance matrix of the optimization variables from historical experiments or previous optimization runs. The proposed approach has been used to estimate unknown feed conditions from scarce measurement data for an industrial cokeoven-gas purification process. A rigorous process model has been used to describe the ab-/desorption process leading to a large-scale, highly nonlinear problem. The result of the estimation procedure using the proposed approach has been compared with the standard definition of the objective function using only the standard deviation of the individual measurements for weighting the residuals. By using the adjusted weighting, the result of the estimation of the unmeasured variables is significantly better. References [1] CM. Crowe, YA. Garcia Campos, A. Hrymak, 1983, Reconciliation of Process Flow Rates by Matrix Projection Part I: Linear Case, AIChE J., 29, 881-888 [2] CM. Crowe, 1986, Reconciliation of Process Flow Rates by Matrix Projection Part II: The Nonlinear Case, AIChE J., 32, 616-623 [3] J A. Romagnoli, G. Stephanopoulos, 1980, On the Rectification of Measurement Errors for Complex Chemical Plants, Chem. Eng. Sci., 35, 1067-1081 [4] I. Ponzoni, M.C Sanchez, N.B. Brignole, 1999, A New Structural Algorithm for Observability Classification, Ind. Eng. Chem. Res., 38, 3027-3035 [5] M.F. Ellis, T.W. Taylor, K.F. Jensen, 1994, On-line Molecular Weight Distribution Estimation and Control in Batch Polimerization, AIChE J., 40, 3,445 [6] H. Chen, S.P. Asprey, 2003, On the design of optimally informative dynamic experiments for model discrimination in multiresponse nonlinear situations, Ind. Eng. Chem. Res., 42, 13791390 [7] A. Kremling, S. Fischer, K. Gadkar, F.J. Doyle, T. Sauter, E. Bullinger, F. Allgower, E.D. Gilles, 2004, A Benchmark for Methods in Reverse Engineeing and Model Discrimination: Problem Formulation an Solutions, Genome Res., 14, 1773-1785 [8] Y. Bard, 1974, Nonlinear Parameter Estimation, Academic Press, New York and London [9] R. Billet, M. Schultes, 1993, Predicting mass transfer in packed columns, Chem. Eng. TechnoL, 16, 1, 1 [10] O. Brettschneider, R. Thiele, R. Faber, H. Thielert, G. Wozny, 2004, Experimental Investigation and Simulation of the Chemical Absorption in a Packed Column for the System NH3-C02-H2S-NaOH-H20, Sep. Pur. Techn., 39, 3, 139-159

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 Pubhshed by Elsevier B.V.

Simulation based engineering - from process engineering to automation engineering Horst Fischer,^ J.-Christian Toebermann,'' "Siemens AG, Industrial Solutions and Services, 91502 Erlangen, Germany ^Siemens AG, Region Deutschland, 52066 Aachen, Germany Abstract Automation systems and their engineering are essential for an economic and safe operation of a plant and make up a significant portion of the total cost for a new plant. Nevertheless it is common practice to work on process design, plant design, automation design and operational concept in separated ways. In general, information is passed on to the next work package in paper form only. Simulation based (automation) engineering means an approach to support a more continuous engineering workflow and a more integrated process view. In this paper we will outline corresponding developments and trends to reduce time and cost in automation engineering. Keywords: Simulation, Automation, Integration, Workflow. 1. Introduction Automation systems and their engineering make a portion of up to 25% of the total cost for a new plant in the process industries [1]. Additionally, both the automation and the operational concept play a very significant role besides the process design to achieve an economic and safe operation of a plant. Nevertheless it is common practice to work on process design, plant design, automation design and operational concept in separated ways. In general, information is passed on to the next work package in paper form only. Accordingly, the automation concept is developed and tested on this basis, even if modeling and simulation was already done in process design. How well process design and automation design matches is then not determined until the commissioning phase with the risk of significant delays and extra cost. This situation is unsatisfactory, in particular because use of simulation in process design is common practice - at least steady-state simulation is standard and also dynamic simulation is more and more applied. However, a more common view on process and automation and their integration within one simulation is not a new task. Several typical approaches exist: • Operator Training System (OTS) [2, 3] as simulation system for a specific plant with a dedicated modeling of process and automation behavior. Unfortunately, the dedicated modeling approach leads to high cost and is obviously unsuitable to support the usual engineering workflow • Extended use of dynamic process simulation: it is common usage to model at least control structures in a dynamic process simulation (e.g. Hysys OTS, Aspen Dynamics). However, the automation system part must then be manually reimplemented in the "real" automation system • Extended use of automation test system: it is possible to model basic process behavior in the automation system, e.g. PLC programs. This becomes difficult for

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complex process behavior and leads always to "re-inventing" part of the process simulation • Combined use of process simulation and automation simulation system coupled to each other via an interface, e.g. OPC-variables The last approach is most promising. The interface is simple, defined, and easily adaptable and the approach offers: • use of the common tools for process design and automation engineering • efficient coupling for an integrated simulation, e.g. to evaluate conceptual designs, to discuss HMI faceplate information during a running process, etc. • extension to an OTS is also readily possible - without changes to the original automation engineering and process simulation Following this approach, we use our simulation platform SIMIT [4] as base tool for simulation based engineering. Besides its own simulation engine STMT has interfaces to different engineering environments, supports standard interfaces, is scalable, and integrated within the SIMATIC automation engineering (Fig. 1).

Automation engineering

Automation engineering oriented process simulation

||*^««iiiite1

Process design oriented process simulation

Figure 1: Simulation architecture with SIMIT 2. Developments and trends In practice the advantages of using such a simulation system are clearly proven. However there is still reluctance observed to actually make use of simulation in medium and small projects. This is due to some obstacles: the necessary effort to set up a simulation, process simulation sometimes does not exist, and bridging the gap between process and automation engineering needs interdisciplinary work of different engineering departments or even different suppliers. In the following, we will outline developments and trends to reduce or overcome some of these obstacles.

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2.1. Thermodynamics and physical properties within automation simulation Often an appropriate process simulation is not available. For example Fig. 2 shows parts of a multistage discharge system. In the reactor a gas-/solid reaction occurs. The solid product is then transported via the network to the product chamber. Because no reaction, no recycle and no other change of material properties occurred, a process simulation was not done. However, discharging the hot and pressurized material properly was a significant issue. For this purpose the thermodynamic state equations of the gas/solid-system had to be integrated in the automation simulation system, either via an extensible modeling language or by properly linking a thermodynamic package. In the actual project the powerfiil modeling language of SIMIT was used. However, to reduce effort in ftiture projects, alternatives to integrate thermodynamic packages are investigated. Consequently, thermodynamic aspects become relevant and solvable also for automation simulation systems, i.e. the clear boundary between process and automation simulation is going to dwindle.

Figure 2: Discharge System 2.2. Generation of simulation model In practice the effort to setting up a simulation system should be as small as possible. In particular much necessary information exists already in other systems. The generation of lO-information for the gateway to the automation system is standard. Also the generation, parameterization and linking of simulation typicals based on typicals and additional information of the automation system is nowadays standard. Beyond that, XML is also a generally available information exchange format. Further parsing makes it possible to use the standard XML-Import to generate also the skeleton of the process-oriented part of the automation simulation system and where applicable, the interface to an additional process simulator. The effort to set up a simulation system therefore strongly decreases - making such simulation system profitable also for medium to small projects. 2.3. Integrated system for design, engineering and simulation The workflow between design, engineering and simulation can be further simplified by extended integration. For example as an add-on to our simulation environment we

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developed a tool for a process engineer to design technology documents (similar to P&I flowsheet) [5], Fig. 3. The results are being used to generate an automation system, i.e. hardware planning and a skeleton pic program, and in a second step to generate a simulation project for the plant behavior. Changes within a subsystem are propagated to the other subsystems. The automation system and the simulation system can be used together to simulate and evaluate the interaction of the automation and the plant. This way the technology expert can already easily check the feasibility of plant concepts and the automation engineer gains a dependable test object. A first prototype was successfiilly used in some pilot projects.

Automation system (HW, SW) generation

Simulation system change propagation

Figure 3: Workflow within the design, engineering and simulation systems

3. Conclusions Simulation based (automation) engineering means an approach to support a more continuous engineering workflow and a more integrated process view. A simulation environment allows the (re-)use of models and data from the design phase, through automation engineering and the start-up phase up to the production phase. Tasks like controller tuning, production planning and validation of quality management or manufacturing execution system can also be tackled within the simulation environment. All this leads to shorter commissioning phases and improved quality and plant efficiency. The interest, especially of plant owners, in such integrated systems is increasing. However, in practice an easier and more automated integration of subsystems is still desirable. We discussed actual developments and trends to reach this goal. Major improvements are already achieved.

References 1. 2. 3. 4. 5.

H. Schuler, CIT plus, 12-2003,4 A. Kroll, atp, 45, 02-2003, 50 A. Kroll, atp, 45, 03-2003, 55 http://www.siemens.com/simit European Patent Application No. 05007417.8

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Large-Scale Dynamic Optimization of an Integrated Cryogenic Process Mariela Rodriguez, M. Soledad Diaz Planta Piloto de Ingenieria Quimica, PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km 7, Bahia Blanca (8000), Argentina

Abstract This work addresses the dynamic optimization of an energy integrated large-scale cryogenic separation plant through a simultaneous approach. The differential algebraic (DAE) optimization problem has been solved through a simultaneous dynamic optimization approach that transforms the problem into a large scale nonlinear program (NLP) through discretization of the entire set of variables by collocation on finite elements. Prior to this discretization, the Method of Lines is applied to transform the original distributed parameter problem into an initial value one. The dynamic model provides profiles of controlled and manipulated variables. Numerical results show the advantage of using a simultaneous approach to solve models of highly integrated processes that include path constraints. Keywords: Dynamic Optimization, Simultaneous Approach, Turboexpansion 1. Introduction Dynamic simulation and optimization of chemical processes constitutes a powerfiil tool for engineering studies that include process control and optimal operation. However, the complex and large-scale nature of these problems has prevented their application to actual plant cases until the recent availability of appropriate resolution techniques. Between sequential and simultaneous optimization strategies, the last ones are preferred for addressing problems of highly integrated processes with path constraints. Cervantes et al. (2000) and Biegler et al. (2002) proposed advanced simultaneous strategies for dynamic optimization of large-scale problems. Natural gas processing plants are integrated cryogenic processes which have received increasing attention regarding optimal steady state design and flexibility (Diaz et al., 1995). Dynamic optimization models have been presented for the demethanizing column (Diaz et al., 2003) and cryogenic heat exchangers (Rodriguez et al., 2005). In this work, the dynamic optimization of the entire cryogenic sector, which includes separation tanks, turboexpanders, distillation columns and countercurrent shell and tube heat exchangers with partial phase change, has been addressed. The differentialalgebraic equation (DAE) model comprises differential energy and mass balances; rigorous thermodynamic predictions in process units with a cubic equation of state (Soave, 1972); hydraulic correlations; pressure drop correlations and design equations for the involved equipment. The DAE optimization problem has been solved through a simultaneous dynamic optimization approach (Biegler et al., 2002) in which the resulting Nonlinear Programming (NLP) problem is solved with a Reduced Space Interior Point strategy (Cervantes et al., 2000). Optimal profiles have been obtained for main operating variables to achieve an enhanced product recovery.

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2. Cryogenic Process Natural gas processing plants that extract that ethane from natural gas are currently based on turboexpansion processes. In these plants, cryogenic temperatures required for methane-ethane separation are obtained by natural gas expansion through a turboexpander in several cryogenic trains. A typical cryogenic sector of a turboexpansion plant is shown in Figure 1. In this process, the feed gas is cooled both in countercurrent cryogenic heat exchangers with residue gas from the demethanizing column and in demethanizer side and bottom reboilers. The partially condensed gas feed is sent to a high-pressure separator. The vapor is expanded through a turboexpander to obtain the low temperatures required for high ethane recovery and is fed to a demethanizer column. The liquid from the high-pressure separator enters the demethanizer at its lowest feed point. Methane and nitrogen constitute top product and ethane and heavier hydrocarbons are obtained as bottom product. Carbon dioxide distributes between top and bottom streams.

Figure 1. Basic Turboexpansion Process

3. Methodology Dynamic modeling and optimization of the cryogenic sector has been performed within a simultaneous dynamic optimization approach (Biegler et al., 2002). Dynamic mass and energy balances in main units give rise to a partial differential algebraic equations system. In particular, energy balances around shell and tube heat exchangers constitute a distributed parameter problem, which has been transformed into an ordinary differential equation system by applying the Method of Lines to spatially discretize the PDE into sets of ordinary differential-algebraic equations (DAE). The DAE optimization problem is then transformed into a large nonlinear programming (NLP) problem by representing state and control variables profiles by a family of polynomial functions over finite elements in time. The differential algebraic system of equations (DAE) is dicretized using orthogonal collocation over these elements. The resulting NLP problem is solved with an Interior Point method with reduced Successive Quadratic Programming (SQP) techniques within program IPOPT (Cervantes et al., 2000), in which succesive parametric NLP subproblems are solved for decreasing values of the barrier parameter. The spatial discretisation, together with the temporal discretisation in the simultaneous optimization approach applied in this work, represents a full discretisation method. Recent applications of this methodology to simulated moving bed processes, with spatial discretisation with finite differences (Kawajiri and Biegler, 2006a,b), have proved to be reliable and efficient.

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4. Model Description The cryogenic plant model includes integrated rigorous dynamic models for countercurrent heat exchangers where partial phase change takes place, high pressure separation tanks, turboexpanders and demethanizer columns. A brief description of the models is given below. 4.1. Countercurrent Cryogenic Heat Exchangers In these heat exchangers, inlet natural gas is cooled and partially condensed in countercurrent with residual gas coming from the top of the demethanizer column. There are two consecutive heat exchangers and partial condensation of natural gas takes place within the second one. 4.1.1. Single phase flow In the heat exchangers with single phase flow, energy balances on microscopic elements on both shell and tube sides give rise to a first order hyperbolic partial differential equation (PDE) system. We have applied the Method of Lines to spatially discretise this PDE system into sets of ordinary differential equations (ODE). Six cells have been considered for a total heat exchanger length of 12m. Backward finite differences have been applied to the convective term, resulting in two sets of equation for tube and shell side fluids, respectively. For tube side fluid, the energy balance in grid point z, is: dTti dt

=

vti . . ht'^A.^p -\Tti -Tti^\) + Az' ' ' fyti^Cpt^At^L^

[Tsf -Ttj) ' '^

(J)

There are additional algebraic equations for density, fluid velocity and compressibility factor as functions of temperature at each grid point /. A linear function based on rigorous mixture predictions with SRK equation of state has been derived for the compressibility factor of both side fluids. 4.1.2. Two-phase flow For shell side fluid (natural gas feed), partial condensation takes place and a two-phase flow model has been formulated. Basic assumptions are that vapor and liquid are in thermodynamic equilibrium, but they may have different velocities; the flow is onedimensional and the void fraction is used to describe the ratio of cross-sectional area occupied by the vapor to the total cross-sectional area. Mass balances for both the liquid and vapor phase have been formulated. Since thermodynamic equilibrium between phases is assumed at any instant, only one energy balance for the mixture is required. Pressure profile along the heat exchanger has been calculated at each grid point assuming steady state conditions through the Bell-Delaware method (Bell, 1960), which takes into account exchanger geometry and fluid type and velocity. Algebraic equations also include rigorous thermodynamic predictions for compressibility factor, components fiigacity coefficients and residual liquid and vapor enthalpy with the SRK (Soave, 1972) equation of state for each cell on the shell side fluid. A detailed description of the model can be found in Rodriguez and Diaz (2005). 4.2. High Pressure Separator The partially condensed mixture from the cryogenic heat exchangers enters a highpressure horizontal tank where vapor and liquid streams are separated. The model includes an overall dynamic mass balance and geometric equations relating liquid content in the tank to liquid height and liquid flowrate as frmction of pressure drop over the liquid stream valve. Detailed equations are presented in Rodriguez and Diaz (2005a,b).

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4.3. Turboexpander The vapor from the high pressure separation tank (about 80% of feed gas) is expanded in the turboexpander. This unit has been modeled through a polytropic expansion, static mass and energy balances and thermodynamic predictions with SRK (Soave, 1972) equation of state. 4.4. Demethanizing column The demethanizing column model comprises dynamic component mass and energy balances at each stage. In particular, vapor holdup as well as liquid holdup have been taken into account at each stage. Component frigacity coefficients, residual enthalpy and compressibility factor have been calculated for both vapor and liquid phases at each column stage through the SRK (Soave, 1972) equation of state. Additional algebraic equations are the equilibrium ratio calculation, summation equation and hydraulic equations (sieve plates). The resulting DAE system is index one and detailed model equations are presented in Diaz et al. (2003). Carbon dioxide precipitation conditions may occur in the upper stages of the column due to cryogenic temperatures. The simultaneous approach allows a straightforward handling of precipitation conditions through the inclusion of carbon dioxide fugacity constraints as follows. If there is vapor-liquid-solid equilibrium at uniform pressure and temperature, the fugacity of each distributed component must also be uniform throughout the coexisting phases (isofrigacity criterion). Therefore, to avoid the formation of a solid phase, the following condition must hold: fi,C02 - fi,C02'

(^)

which corresponds to C02 fugacity in the vapor mixture and in the solid phase at stage /, respectively. C02 fugacity in vapor phase (f^^^^) ^^ stage / is calculated through stage vapor-liquid equilibrium variables. Ji,C02~yi,C02^i^i,C01

(^)

Assuming no hydrocarbon in the solid, i.e., only carbon dioxide in the solid, its fugacity in solid phase (^^^2) ^^ ^^^S^ ^ i^Ji.C02~^i,C02^i,C01'

W

where carbon dioxide saturation pressure at stage / (P,^co2) i^ calculated as function of triple point temperature ( T^Q2 ) ^^'^ pressure ( PQOI )• 5. Optimization Problem The DAE open loop optimization problem has an integral objective function (minimize the offset between current ethane recovery and a set point value), which is included as an additional differential equation. The DAE model for the integrated cryogenic sector, bounds on control and state variables and solubility inequality constraints constitute optimization constraints. Analytical derivatives for all functions with respect to the entire set of differential and algebraic variables have been provided. The model has been implemented within a Fortran 90 environment.

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6. Numerical Results A decrease in natural gas feed flowrate is analyzed, which causes an initial increase of condensed fraction in cryogenic heat exchangers (Fig. 2) with the corresponding decrease of vapor stream sent to the turboexpander and then to the demethanizing column (and increase of liquid feed to the column), as it is shown in Fig. 4. The objective is to increase ethane recovery while remaining within feasible operating conditions. Figure 3 shows spatial and temporal profiles for pressure in shell side fluid. The optimization variable is demethanizer top pressure, which is 21.5 bar at initial time. As natural gas feed has an important content of carbon dioxide (ten components, 89% methane, 4.4% ethane, 1.3%) C02, mole basis), solubility constraints remain active and prevent a further decrease in demethanizer top pressure, whose optimal final value is 20.49 bar (Fig. 5). Consequently, ethane recovery can be only increased from 73.1 to 73.52%) (Fig. 5). C02 concentration profile in the vapor phase is shown in Fig. 6. Throughout the transient, C02 mole fraction is below the corresponding solubility values at each stage temperature, pressure and composition, as imposed by Eqns. (2) to (4). The DAE optimization problem for the integrated cryogenic sector includes 202 differential and 680 algebraic equations. The resulting NLP, when discretising with 20

o E

Figure 2. Partial condensation in shell and Tube Heat exchanger

Figure 3. Optimal pressure profile in shell and tube Heat exchanger

295285 Top stream

275

I I

265

— - -Feed from TE

255

^* •

'

30

40

"V

245 235 0

10

20 Time (min)

Figure 4. Two phase feed from turboexpander to column and optimal top vapor flowrate to heat exchangers

10

20

30

40

Time (min)

Figure 5. Optimal top pressure profile and ethane preduction in demethanizing column

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finite elements and 2 collocation points, has 39583 discretised variables. The problem has converged in 54 iterations (4 barrier problems). Initial barrier parameter value: 0.01. 0.25 0.20 o E,

5

0.15 H 0.10 H 0.05 0.00 20

30

Time (min)

Figure 9. Carbon dioxide composition profile in demethanizer 7. C o n c l u s i o n s A rigorous dynamic optimization model has been developed for the cryogenic sector of a natural gas processing plant. The model is of special interest due to the process high energy integration. The optimization DAE problem has been solved with a simultaneous approach. This strategy has allowed the inclusion of path constraints, such as carbon dioxide solubility constraints in the transient. Rigorous thermodynamic models, while providing a realistic representation of the process, add important nonlinearity to the model, which has been efficiently handled within the simultaneous approach. Model resolution provides optimal temporal and spatial profiles and information describing the two-phase flow and fluid separation in a highly integrated large scale plant.

References Bell K.J., Delware method for shell side design. Petro/Chem Engineering. 1960; 1: 26-40. Biegler L.T., Cervantes A, Waechter A,, Advances in Simultaneous Strategies for Dynamic Process Optimization. Chem. Eng. Sci. 2002; 57: 575-593. Cervantes, A., Waechter, A., Tutuncu, R., Biegler, L.T., 2000, A Reduced Space Interior Point Strategy for Optimization of Differential Algebraic Systems, Comp. & Chem. Eng , 24, 39-51. Diaz M.S., Serrani A., de Beistegui R, Brignole E., An MINLP Strategy for the Debottlenecking problem in an Ethane Extraction Plant. Comp. & Chem. Eng. 1995; 19s: 175-178. Diaz M.S., Tonelli S., Bandoni A., Biegler L.T., Dynamic optimization for switching between steady states in cryogenic plants. Founds. Comp. Aided Process Operations, 2003; 4: 601-604. Kawajiri Y., Biegler, L.T., Large-scale Optimization Strategies for Zone Configuration of Simulated Moving Beds, PSE/ESCAPE 2006. Kawajiri Y., Biegler, L.T., Optimization Strategies for Simulated Moving Bed and PowerFeed Processes, AIChE Journal, to appear (2006). Raghunathan A., Diaz M.S., Biegler L.T., An MPEC Formulation for Dynamic Optimization of Distillation Operations. Comp. & Chem. Eng. 2004; 28: 2037-52. Rodriguez, M., M.S. Diaz, "Dynamic Modelling And Optimisation Of Cryogenic Systems", Applied Thermal Engineering, in press, 2005a. Rodriguez, M., J. A. Bandoni, M. S. Diaz, "Dynamic Modelling and Optimisation of Large-Scale Cryogenic Separation Processes", Chemical Engineering Transactions, 6, 179-184, 2005b. Soave G. Equil. Constants for a Modified Redlich-Kwong Eq. of State. Chem. Eng. Sci. 1972; 27: 1197-1203.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Scheduling of Storage and Transfer Tasks in Oil Refineries by Using Fuzzy Optimization L. C. Felizari^ and R. Liiders^* ^UTFPR/CPGEI, Av. Sete de Setembro 3165, 80230-901 Curitiba-PR, BRAZIL [email protected]; luders@dainf. cefetpr.br A decision process is often used to schedule a task that should be executed taking in account its processing sequence and time. However, the processing time of an operation is often uncertain. Therefore, a scheduUng program should be supported by decision systems, especially those using optimization techniques. In this work, time delays are also considered by using fuzzy optimization techniques. This paper proposes a scheduling for transfer and storage operations by considering products to be transferred among tanks through pipehnes in a refinery. The proposed model is based on mixed integer linear programming (MILP) with continuous time representation. However, delays in operation times were represented by soft constraints, not found in traditional MILP models. 1. Introduction In oil refineries, transfer tasks of products are necessary to store products received from a process unit or to transfer final products from a tank to a customer. The operational complexity of such tasks, associated to a not efficient employment of computational resources, can be considered as a critical factor that makes difficult to reach a better performance under certain criteria [1]. The main aspect to be considered is allocation of shared resources, such as tanks, pipes, and pumps. This work aims to study the effect that time uncertainty in allocation of shared resources can cause to an initially established production program. This program assigns shared resources to a product transfer that should be accomplished under time constraints. However, this work assumes that a transfer task is completed within a time interval. This is quite different from many other applications, where the processing time of a task is assumed to be constant. The processing time of an operation can vary depending on the involved activities, the equipment to be used, or the task to be done. Therefore, the execution time is often subjected to uncertainties [2-4]. The final target is to obtain a solution that incorporates some flexibility to accommodate small changes in the execution time of a particular operation, not causing a great perturbation on the total execution time (makespan). This is accomplished by using concepts of fuzzy set theory [5]. * Financial support from Agencia Nacional do Petroleo- ANF - and from Financiadora de Estudos e Projetos - FINEP - by means of the Human Resource Program of ANP for the Sector of Oil and Gas PRH-ANP/MCT (PRHIO-UTFPR).

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The optimization model is developed using mixed integer linear programming (MILP) with a continuous time approach and uncertainty in parameters. 2. Linear fuzzy programming Fuzzy optimization can be defined as a set of techniques that use fuzzy sets to formulate optimization models with some degree of flexibility or uncertainty in constraints and/or cost [6,7]. Considering a cost function with n decision variables and m constraints, the general model of linear fuzzy programming (LFP) with flexibility in constraints can be stated according to expression (1): fuzzy max

f{x) = cFx

(1)

X

subject to: ajx 0

i = 1,..., n

where c, x and a^ G 7^^, bi G IZ. The symbol < represents that some flexibility is admissible in constraints. This optimization model can be solved by using a method for fuzzy decision-making [8]. The optimal decision vector jf is such that: ^(x*) - max{F(c^:r) A Gi{a^x) A • • • A Gm{a^x)}

(2)

X

where F and Gi are membership functions, assuming values within the interval [0,1]. They represent a measure of how much the cost function can be improved with a more strong constraint violation (F for the cost function and Gi for each constraint). The symbol A denotes the minimum operator. For example, values of Gi near to zero represent a strong constraint violation, but with a potencial better cost function (values of F close to one). The optimization process has to satisfy both: the best cost function considering the minimum violation in constraints. By introducing an additional variable A, the solution of the optimization model described by Eq. 2 can be found through conventional linear programming as follows [9]: max

A

(3)

A

subject to: c^x > Zp — {1 — A)po ajx < 6^ -h (1 - X)pi, ^ == 1,2,..., m X > 0, A > 0, A < 1 3. Initial problem The considered process can be found in an oil refinery, where many units produce different refined products in a continuous way. These products should be stored on intermediary tanks until they are ready to be sent to final customers. An overview of a storage and distribution area is illustrated in Fig. 1. In this scenario, a transfer for receiving a continuous production occurs among units and tanks or among tanks, for releasing space or sending products to a customer. Initially,

Scheduling of Storage and Transfer Tasks in Oil Refineries

1485

a mathematical programming model in discrete time is established, considering tanks as resources and a product demand as an input parameter [10]. The output generated yields a scheduling of operations. Such scheduling considers the refinery mass balance and it defines the starting pumping associated with a target tank (see Fig. 2). In this decision level, the necessary pipes for a transfer are not considered. This assumption is made to simplify the model, since the association of a pipe for each pumping operation may increase the number of binary variables. Starting from this scheduling, we propose an approach to deal with pipe allocation through a job-shop formulation [11] with additional constraints. Hence, tasks can be eventually reordered to adequate available pipes. Our approach also considers time delays in tasks, yielding a model with uncertainties in parameters. This results in a very different scheduling from the initial one. Time Horizon Pi storage

1

m-9

• : • : • : • , • : : • : • : • ; . : : : : :

1

J

^

1

J

,,,

'^fZiii^yyyyyy^Mii^ Wiz^A^yyAiyyyyyyyyyyAyAyAyyyyyyyy

1

p Local Market

Production Line ~ Pipeline Receivership

ta

yyyyyy)

TQ5

Pipeline Disoatch TQ8

Pices

[—]

Taskl

Task 3

I Task 2

Task 4

P2 Storage I

Figure 1. A storage and distribution area.

Figure 2. Initial scheduhng.

Fig. 2 shows an initial scheduling obtained by another optimization program [10] that only considers mass balance among tanks and demand requirements. In this case, no pipe allocation is considered. For example. Task 2, which is for sending product P I to a customer, is composed by five operations ( O P l to 0 P 5 ) . Each of these operations is related to each transfer of P I from tanks T Q l , TQ7, TQ6, TQ2, and T Q 7 in order to provide the whole demand. This is the initial scheduling to be executed. Note that it yields five simultaneous operations, which obviously need five pipes. 4. O p t i m i z a t i o n m o d e l As established in sections 2 and 3, t h e general model of hnear fuzzy programming, is reduced to a model of mixed integer linear programming (MILP) in continuous time. The decisions are modeled by binary and continuous variables, linked by linear restrictions. The objective is to minimize the total execution time or makespan with operations starting as soon as possible. The restrictions basically define the order among operations and the allocation of resources, tanks, and pipes in the considered scenario [12]. The mathematical model is henceforth explained:

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L. C. Felizari and R. Luders

The cost function (expression 4) is defined by the minimization of makespan and the sum of starting times (TSoj) for all operations. fuzzy min

{Mak + X l X l ^ ' ^ ^ ' ^ ^ OEO

(^)

jeJ

The optimization model is also subjected to constraints, stated in Eq. 5 to Eq. 16, as follows: The finishing time (TFoj) of any operation must be lower than the makespan: Mak - TFoj >0

yoeO.jeJ

(5)

The order among operations, for the same task, must be assured: TSoj > TFo-i,j

V o > 1 G O, j G J

(6)

The duration time of the operations {TPoj) is subjected to uncertainties. The fuzzy restriction allows us to consider possible delays in the processing time of the operations: TFoj - TSoj = TPoj

yoeO.jeJ

(7)

The order among operations of different tasks, for the same tank, must be respected: TFp,, - TSoj > - ( 1 - Do,j,p,k)' M

\/o,p e O, (j ^k)eJ,

Qoj = Qp,k

(8)

TFoj - TSp^k > -Doj,p,k 'M

^o.peO,

{j ^k)eJ,

Qoj = Qp,k

(9)

TFp^k - TSo,j < -Do,j,p,k 'M

"io.peO,

{j ^k)eJ,

Qoj = Qp,k (10)

TFoj - TSp,k < (1 - Doj,p,k) • M

"^o.peO,

{j ^k)eJ,

Doj,p,k + Dp^k,oj = 1

yo,peO,

{j ^k)e

Qo,j - Qp,k (11) J, Qoj = Qp,k (12)

The allocation order of the pipes of the same type among the operations, for different tasks, must be considered: TFoj - TSoj

< (1 - Doj,p,k) X M + TD,,d,p,fc +

(1 - K,d,p,fc) X M + (1 - K,d,o,j) X M -

Wo.p e O, (j ^k)eJ,ne

TDn,d,oJ

NDd, de

D

TFp^k - TSp,k < Doj,p,k X M + TDr,,d,oj + (1 - VnAoj) X M + (1 - Vn,d,p,k) X M - TDn,d,p,k

\fo,p e O, (j ^k)eJ,ne

(13)

NDa, de

(14)

D

For the occurrence of a transference operation, a tank and a pipe must be allocated at the same instant of time: TDn4,oj = TSoj

\/oeO,jeJ,ne

NDd, deD

(15)

At any given time, it must be assured that only one pipe is allocated for each operation:

Y,

VnAoj = i

yoeO,jeJ,deD

(16)

neNDd

As follows, the main results obtained for the case study with four pipes and flexibihzation in the operating time of operation O P l of the task T2 are presented.

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Scheduling of Storage and Transfer Tasks in Oil Refineries 5. R e s u l t s

Returning to the initial scheduling presented in Fig. 2, we now consider a delay in the dispatch process of product P I to a customer. It will be considered that operation O P l of task T2 will be able to be delayed up to 3 time units {TAi^2 — 3), representing uncertain processing time during a product pumping operation. The time delay can represent the difficulty of the involved maneuvers, established by the operator's skills. Using the proposal of [9], and the Eq. 3, we get the following model of linear programming: maximize subject to:

Mak + ^

^

A

(17)

TSoj < 75 + (1 - A) x (84 - 75)

o£0 jeJ

TFi,2 - T5'i,2 > 5 - (1 - A) X (5 - 2) A > 0,

A< 1

The complete model is composed by the Eq. 17, in addition to Eq. 5 to Eq. 16, with exception of the Eq. 7 that is replaced by the time restriction already in the Eq. 17. Note that the initial solution of Fig. 2 requires five pipes to be implemented. Assuming that only four pipes are available. Fig. 3 shows the obtained scheduling. The variation in the processing time was considered to the fuzzy optimization proposal, resulting in A = 0, 5833. In this case, the start task T2 was postponed and the solution considered an intermediate duration for O P l of T2. Moreover, if the duration of O P l changes from 2 up to 4 time units, the programming will remain feasible and it will have little impact on the total scheduling time. The allocation of pipes during the scheduling horizon is shown in Fig. 4. Time Horizon

Time Horizon —^

1

^J

^—r 1 "'l' '

V r' •

Y^/yy//;/xxyy//)yx/>

1 1 1 1 1 1 1 ^-^

'\

y/y//)''////Aoyyyyy /////////////////////////>

'^^2^^^m22m^^^m^^^^^ E zz^

^^1

iillii Task 1

Tasks

l:;::x::;:::;l Task2

Task 4

E Eia

D4 1 1

I Task 5

T ' T" 'f'''x"'l ' Y''^Y^Y"*\ "i

liilM^^^ Task 1

Tasks

l;^;^:^;^;^:;^;l Task2

Task 4

Figure 3. Gantt graph for the final scheduling.

Figure 4. Pipes allocation.

The solver Lingo/PC release 8.0, installed in a computer Pentium III lGHz/256MB RAM was used for the solution of the optimization. In a general way, the models used about 37 continuous variables, 328 binary variables and 1,700 restrictions, consuming an average processing time of some minutes.

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6. Conclusions This paper considers the use of fuzzy optimization in scheduling of storage and transfer tasks characterized by delays in their processing times. Another considered aspect is the allocation of shared resources. In particular, the allocation of pipes that allow parallelism of tasks. Starting from an initial scheduling, obtained by not considering pipe allocation, the introduction of operational delays yields a significant increase on the makespan. However, as the maximum delay may not occur, an intermediate solution would be more suitable. Therefore, the solution considering processing times as fuzzy quantities make it possible to obtain a scheduling that admits some flexibility, since an intermediate case is found. REFERENCES 1. J. M. Pinto, M. Joly and L. F. L. Moro (2000). Planning and scheduling models for refinery operations. Computer and Chemical Engineering (Oct), Vol. 24, pp. 22592276. 2. J. Balasubramanian and I. E. Grossmann (2002). A Novel Branch and Bound Algorithm for ScheduHng Flowshop Plants with Uncertain Processing Times. Computers & Chemical Engineering (Jan), Vol. 26, No. 1, pp. 41-57. 3. J. Balasubramanian and I. E. Grossmann (2003). Scheduling Optimization Under Uncertainty - An Alternative Approach. Computers & Chemical Engineering (Apr), Vol. 27, No. 4, pp. 469-490. 4. X. Lin, S. L. Janak; C. A. Floudas (2004). A new robust optimization approach for scheduling under uncertainty: I. Bounded uncertainty. Computers & Chemical Engineering (Jun), Vol. 28, No. 6-7, pp. 1069-1085. 5. L. A. Zadeh (1965). Fuzzy sets. Information and Control (Jun), Vol. 8, No. 3, pp. 338-353. 6. W. Pedrycz and F. Gomide (1998). An Introduction to Fuzzy Sets: Analysis and Design. MIT Press, Cambridge. 7. U. Kaymak and J. M. Sousa (2003). Weighted constraint in fuzzy optimization. Constraints (Jan), Vol. 8, No. 1, pp. 61-78. 8. R. E. Bellman and L. A. Zadeh (1970). Decision-making in a fuzzy environment. Management Science (Dec), Vol. 17, No. 4, pp. 141-164. 9. H. J. Zimmermann (1993). Fuzzy Set Theory and its Application. Kluwer Academic Press, Boston. 10. S. L. Stebel (2001). Modelagem da Estocagem e Distribuigao de GLP de uma Refinaria de Petroleo. Masters thesis. CEFET-PR/CPGEI, Curitiba, Parana, Brazil, (in Portuguese). 11. E. Teixeira and A. Mendes (1998). An Extension of the Model for the Problem of Workpieces Scheduhng in a Flexible Manufacturing Cell. Production Planning & Control, Vol. 9, No. 2, pp. 176-180. 12. L. C. Felizari and R. Liiders (2004). Estudo do escalonamento de operagoes de transferencia de produtos em refinarias usando otimizagao fuzzy. CBA 2004 - XV Brazilian Automation Conference (Sep), Gramado, RS, Brazil, (in Portuguese).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

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Degrees of freedom analysis for process control M.RodriguezJ.A.Gayoso Universidad Politecnica de Madrid Jose Gutierrez Abascal, 2 Madrid 28006, Spain Abstract In defining the control structure of a system it is very important to knovs^ how many variables we can regulate. The degrees of freedom (DOF) analysis of the system allows establishing the maximum variables that need to be fixed to have a completely determined system. Of all the DOF some of them will be disturbances (i.e. they are fixed externally) and the rest indicates the maximum number of variables to control. The procedure developed in this paper is based on, and extends and generalizes, the one presented by Ponton (Degrees of freedom analysis in process control.). The procedure presented derives, from the analysis of a general unit (system), a formula to compute the DOF. This formula is then generalised to be applied to any process. Keywords: Process control, degrees of freedom. 1. Introduction The degrees of freedom analysis is used in the development of control (and plantwide control) strategies. Writing all the equations for a process and counting the variables is a tedious and error prone process, so it is important to have a simple but generic method to compute the degrees of freedom for control of a whole process. The first approach in this direction was developed by Kwauk [1] in the context of process design. The procedure developed in this paper follows somehow the one described by Ponton [2] but instead of applying the Kwauk method as in Ponton's it starts from a generic system and applies on it first principles equations.. Degrees of freedom (DOF) can be defined as: DOF = Number of variables of the system - Number of equations of the system The degrees of freedom are the variables that have to be set to have a completely determined system DOF can be set by the environment (disturbances) or by the control system. The following method will compute the overall DOF of the system and will differentiate between disturbances and control variables. An expression is deduced to compute in a simple way all the available DOF for control. 2. Degrees of freedom analysis The degrees of freedom analysis will be performed on a generic system as the one described in figure 1. First the amount of system variables will be calculated and then the equations that can be applied to the system. 2.1. System variables (streams and unit) • Input streams There are Si input streams, each stream has C variables corresponding to the components and one pressure and temperature, so the total amount of variables in input streams are: Si*(C+2)

M Rodriguez andJ.A. Gayoso

1490 C components

Su Hi!

PiT]

vapor

-•So.l —^Ei

liquid 1

->So,2

liquid2

•^So.3

Si.; ni2 P2,T2

Sio ni3 P3,T3 Si,p

-•s,

•o,J

InipPiTf %"*'""

-^Saf —

•

*

•

-»-Ek

\ y Energy Fig 1. Generic system. Where Si are the input streams, So are the output streams (one per phase), E are additional output streams from a phase, nij are the moles of component i in phase j . Pi and Ti are the pressure and temperature of phase i. The system has C different species and F different phases. One or more energy streams can interact with the system. • Unit (system) There are C variables corresponding to component accumulation and a pressure and a temperature in each phase. Total variables in the unit are: F*(C+2) • Output streams We have So plus E output streams from the system, but the only new (not accounted for) variables that they add are the flows, because composition, pressure and temperatures have been taken into account in the unit (and they are the same as the composition, pressure and temperature of the outlet streams) Total variables in output streams: So+E • Energy stream If there is an energy stream to (or out of) the system then an additional variable has to be taken into consideration. This energy flow can be thermal or mechanic... It only adds one variable (even if more than one energy stream exists) from a control point of view and it is the amount of energy that will be added in the propoer balance (this will affect to system temperature or pressure). If the energy is transferred inside the system boundaries as it happens with a process to process heat exchanger then this variable will not add any DOF.. To consider the energy a variable called H is added, which will take the value 1 if there is energy flow to (out of) the system and 0 otherwise. Variables in the system: Si*(C+2) + F*(C+2) + Sout + H 2.2. System equations • Mass balance There is one mass balance per component, so we have C equations.

Degrees of Freedom Analysis for Process Control

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• Energy balance There is one energy balance to be applied to the system • Momentum balance Although the momentum balance is vectorial and three individual balances can be established, in the process industry only one of them is generally significant (in the flow line, Bemouilli's equation) so we have an additional equation (if more balances are applied more variables have to be considered so it doesn't affect to the degrees of freedom computation) • Equilibrium or transport equations We can have the system in equilibrium or not, in any case the same amount of equations arise. If we have equilibrium we can establish composition, pressure and temperature equalities in the interfaces. If we don't have equilibrium we can establish in each interface C mass transport equations (Pick's law) for the compositions, one heat transport equation (Fourier's law) for temperature and one momentum (only one out of the three possible is generally meaningfiil) transport equation for pressure (Newton's law). Total equilibrium or transport equations (there are F-1 interfaces) : (F-l)*(C+2) Equations in the system: C+l+l+(F-l)*(C+2) (In the case of vapor phase we have an additional equation, the gas law, but we have another variable, V. so it doesn't change the degrees of freedom) 2.3. Degrees of freedom The computation of the degrees of freedom using the above decomposition results in: DOF= Si*(C+2) + F*(C+2) + So„t + H-[ C+l+l+(F-l)*(C+2)]= Si*(C+2) + Sout + H From these degrees of freedom in the input streams we can only act upon the flow so the remaining degrees of freedom of these streams are disturbances (from a control point of view). This results in: DOF= Si + Sout + H Up to this point all the inventories have been considered, but many times we are not interested in or we cannot control them (like when a pipe is splitted into two or when the output of a tank is through a weir). We introduce now an additional variable, A. This variable is the amount of inventories (liquid or gas) that are not considered. This variable (A) removes one DOF if there is one process variable available to control the inventory that is not used (or cannot be used). For example if we have a tank with one input and one output and both are flow controlled then A does not remove any degree of freedom as the inventory is taken into account indirectly. The final DOF equation is: DOF= Sj + Sout + H-A We have not considered the case of the existence of a reaction as it does not vary the DOF analysis (it adds one variable the extent of the reaction, and one equation, the kinetic expression which is composed of variables that are already considered). One final consideration must be noticed. In the system only a temperature, pressure,., is considered in each phase. This is true only in completely mixed systems, in the case where a gradient of variables exists it doesn't modify the DOF analysis. We can decompose each phase in N compartments, this will add N-1 new variables for temperature, pressure and each composition. But we can set an equation for each of the new N-1 interfaces for temperature, pressure and each composition through the transport laws, so the DOF is not altered as stated before.

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and J. A.

Gayoso

2.4. Examples The expression is applied to some common process units. Table 1. Process units degrees of freedom Unit

Degrees of freedom Si + Sout + H-A

Heater

D0F-1 + 1 + 1 - 1 = 2

Process heat exchanger

D0F=2 + 2 + 0 - 2 = 2 (in this case the inventories in shell and tubes are not controlled) Energy is cero because the flow is between process streams.

Pump Compressor

DOF=l + 1 + 1-1=2^^^ DOF=l + 1 + 1 - 1 = 2 ^ ^ ^

Vaporizer DOF= 1+1 + 1 - 0

CSTR DOF=2+1+0-0 =3

Distillation Column: 1+3+0-2 = 2 Condenser: 1+2+1-0 = 4 Reboiler: 1+1+1-1 = 2 (pressure is not controlled) The distillation has been decomposed as a process so the general expression (next section) is applied: DOF= l+[3-2]+[2+l]+[l+l-l]= 6 Furnace D0F=3+ 2 + 0 - 1 = 4 The inventory in the tubes is not considered, and the energy term is cero because it both are considered process streams. (1) Although it is possible to control (specify) two different variables in pumps, the flow and the head, it is not desirable from a process point of view. The speed of the pump and the impulsion valve can be manipulated to fix the flow and the head, but it doesn't make sense to work with other than the minimum necessary head so only one degree of freedom is used in pumps. (2) The same happens to compressors. Usually the pressure is the only variable to control, but both pressure and flow can be controlled.

3. Degrees of freedom analysis of a process 3.1. DOF expression The expression deduced can be extended to compute the degrees of freedom of a complete process. The total DOF of the process will be the sum of the DOF of the units, but removing all the DOF related to input streams but the inputs to the process (any input of a unit is the output of an upstream unit so it is already accounted for in the DOF expression).

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The DOF expression for a complete process is: DOF=Sip+X"(Sout + H-A) Where Sip are the inputs to the process, and S is the sum of all the units in the process. This is the maximum SISO control loops that can be established, although other process constraints can make this number lower. This expression is easily applied to any process, and it finally derives in counting all the process streams, adding all the energy flows (one per unit) and removing all the inventories not to be considered. Following the expression is applied to several processes. 3.2. Example 1. Vinyl Acetate process The flowsheet of fig. 2 represents the industrial process for the vapor phase manufacture of vinyl acetate monomer. The process is based on the description in [3] and [4]. The process has three feeds: oxygen, ethylene and acetic acid. The main reaction takes place in a plug flow reactor and the heat is removed generating steam. This process is described by Luyben in [5] when exposing his methodology for plantwide control. He presents 26 degrees of freedom.

Figure 2. Vinyl acetate monomer process Degrees of fi-eedom: Number of process streams: 39 Number of inventories not accounted: 20 (9 mixers or splitters, 4 heaters, 2 heat exchanger, reactor, CO2 removal, 1 reboiler, 2 distillation column) Number of energy streams: 8 (4 heaters, 1 vaporizer, 1 reactor, 1 reboiler, 1 condenser) DOF=39-20+8=27 The difference with Luyben is the compressor. Usually (as in this case) only the pressure is controlled, so one degree offi"eedomis removed (as explained in the previous section when commenting the DOF of pumps and compressors). One possible set of manipulated variables is shown in the figure. 3.3. Example 2. Vinyl Chloride Monomer(VCM) process Vinyl chloride (VCM), which is made from ethylene and chlorine, is used as feedstock in the production of the standard plastic material PVC. Figure 3 shows the balanced process with no net consumption or generation of HCl as described in [6] and [7]. It combines direct chlorination, ethylene dichloride (EDC) pyrolysis and oxychlorination (which consumes all the HCl generated in EDC pyrolysis. Chlorination and oxychlorination reactions takes place in tubular reactors and the pyrolysis takes place in a furnace type reactor.

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M Rodriguez and J.A. Gayoso

Figure 3. Vinyl chloride monomer process Degrees of freedom: Number of process streams: 57 Number of inventories not accounted: 24 (4 in mixers or splitters, 2 in heaters, 3 in reactors, 5 in reboilers, 10 (2*5) in distillation columns) Number of energy streams: 14 (2 in reactors, 2 in heaters, 5 in reboilers, 5 in condensers) DOF=57-24+14=47 One possible set of manipulated variables is shown in figure 3.

4. Conclusions In this paper a simple expression to compute the (maximum) degrees of freedom available for control of any process. There is no need to write any equations as the expression has been obtained through a rigorous application of the available physicochemical equations (conservation laws, transport laws, equilibrium constraints) to a generic system. This method takes into consideration the inventories which are very important in establishing any control strategy for a process. The expression has been tested in numerous processes as the two presented in this paper. The expression is easily implemented in a software application and can be used as an initial step in any Plantwide control method. It can help to sort out the available controlled variables of the process. This expression is currently implemented and is being used in a plantwide control methodology that is being developed by the authors and that will be presented elsewhere.

References [l]Kwauk,M. AIChE Journal 2 (1952), 40 [2]Ponton, J.W., Degrees of Freedom Analysis in Process Control, Chemical Engineering Science, Vol. 49 (1994), No. 13, pp 1089- 1095. [3]Vinyl acetate, Stanford Research Insititute (SRI) Report 15B, 1994 [4]Ullmann' s Encyclopedia of Industrial Chemistry, 2003. [5]Luyben,L., Tyreus, B. and Luyben, M.. Plantwide Process Control. Mc Graw-Hill, 1999, pp. 321 [6] Kirk-Othmer Encyclopedia of Chemical Technology, 4^^ edition, 2001, Vol 24 [7] http://www.uhde.biz/informationen/broschueren.en.html. Technical Report on Vinylchoride and Polyvinylchloride.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction Michael Baldea^, Prodromos Daoutidis^'^ ^Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA. Email: daoutidi&,cems. umn. edu ^currently at Aristotle University of Thessaloniki, Thessaloniki, Greece Abstract In the present work, we propose a generic dynamic analysis and model reduction framework for autothermal reactors. We document the existence of a two time scale dynamic behavior, and identify the variables that are associated with each time scale. We derive reduced-order, nonlinear models for the dynamics in each time scale. We show that the derived slow model corresponds to generally accepted empirically derived simplified models for the class of reactors considered. Subsequently, we present illustrative simulation results on a hydrogen production reactor. Keywords: Nonlinear Model Reduction, Autothermal Reactors, DAE systems. 1. Introduction The interest in economically efficient hydrogen production has been steadily increasing, and even more so recently, given the progress made in the development and implementation of fuel cell technologies. Autothermal reactors, combining exothermic and endothermic reactions are one of the most promising hydrogen production technologies, featuring in-situ heat generation, which allows for increased fuel efficiency and a compact size. From a design and operation point of view, autothermal reactors rely either on a constant, unidirectional flow, whereby the raw material for hydrogen production and the necessary fuel flow in different, parallel channels of the reactor (either in co-current or counter-current), or on flow reversal, in which case the catalyst bed within the reactor acts as a heat trap (Frauhammer et al, 2000). The majority of the research studies concerning the design and operation of autotermal reactors of either category investigate the steady-state behavior of the system. However, in the context of integrating such reactors in larger systems that include fuel cells for power production, the transient operation of the autothermal reactor, enabling variable levels of hydrogen supply in response to varying power requirements to a fuel cell downstream becomes much more interesting (Gorgun et al, 2005). Thus, the availability of accurate, reliable and, at the same time, computationally efficient models is of great importance for dynamic analysis and control (Vanden Bussche et al., 1993). Due to the inherent multiple-time scale behavior of autothermal reactors (Kaisare et al., 2005), their dynamic models are ill conditioned and challenging to simulate, and recent efforts have been aimed at providing an approach for deriving models of reduced dimensions that capture the salient dynamic features of the original system (Gorbach et a/., 2005). In the present work, we propose a generic dynamic analysis and model reduction framework for autothermal unidirectional flow reactors. Using mathematically rigorous

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M. BaldeaandP. Daoutidis

arguments, we document the existence of two distinct time scales in their dynamic behavior, and identify the variables that are associated with each time scale. We derive reduced-order, nonlinear models for the dynamics in each time scale. In particular, the derived slow model corresponds to generally accepted empirically derived simplified models for the class of reactors considered. Subsequently, we present illustrative simulation results on a hydrogen production reactor. 2. Modeling and Model Reduction of Autothermal Reactors We consider 2 channels of an autothermal reactor with catalytic walls (Fig. 1).

a

b

Figure lAutothemal reactor (a) countercurrent flow; (b) co-current flow A gaseous reacting mixture with components j

of molecullar mass M.,

and a

composition given by the weight fractions W-, density p ^, and temperature T^ ^ enters channel A:, /r = 1,2 at an inlet velocity K^. We assume that reactions r., with stoechiometric coefficients l>.. only occur on the catalytic wall, and that the reaction rates account for the diffusion of components to and from the catalyst surface (diffusion is not modeled explicitly). Under these assumptions, the mass and energy balance equations describing the reactor behavior take the form:

dw,.

^w,.

-

d

H^;

U)

Ot

OZ

k=l,2

k=l,2 1=1

which, by defining the dimensionless timeT = F j / L / , the dimensionless spatial coordinate ^ = zlL

(with L

temperature 0 = T IT .

kj^ = M^ IM^, ^wj k ~^jk^^jk^

being the spatial coordinate), the dimensionless

and the quantities A:^. = Kj / K2 , A:^ = J^eff,! ^ ^eff,i»

with M ^ being the average molecular mass in channel k,

and

^^^^ ^j k being the mole fraction of component I in channel k ,

and using the standard definitions of the Peclet number Pe, the Stanton number

St,

Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction

1497

the Fourier number for the gas, Fo^^, the Fourier number for the soUd, Fo^, the Damkohler number Da. and the adiabatic temperature rise B^ for the I th reaction, takes the following dimensionless form:

£

OT

OT

it=l,2

i^a

k=\,2 i=l

with £ = PgiCpg^ ^ Ps^ps- Taking into account that ky.kj^.k^.k^,

kj^ are typically

0 ( 1 ) quantities, the more compact form arises:

^=-MA|^+^'''|^+^^'('?.-^.)

€

oT

oT

k=\,i

^'-^

k=i,2 1=1

where the prime superscript denotes appropriately modified dimensionelss numbers. Generally, due to the large differences in heat capacity and density between the gas phase and the solid catalyst, € <^l and, consequently, the model in Equation 3. is in a singularly perturbed form and will exhibit a two time scale behavior. We investigate this behavior within the framework of singular perturbations below. To this end, let us consider the limit ^ - ^ 0 , corresponding to an infinitely low heat capacity and density of the gas phase, in which case the model (3.) becomes: dw-.

3iv..

1 d^W;^

J^

dt Equation (4.) then represents a description of the fast dynamics of the autothermal reactor. Let us now define the (slow) time scale T^ = €t, and consider the same limit for the model (3.). In this case, the following description of the slow dynamics of the autothermal reactor is obtained:

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M Baldea and P. Daoutidis

dC

PC dC

i=\

o=-Kk^k^-jr+Fo'—^+st\e,-e,) dC dC OTg

(JT

k=i,2

k=i,2 1=1

Note that the above models are, respectively, systems of coupled partial differential equations (4.), and a partial differential equation under ordinary differential equation constraints (5.). Generally, an analytical solution of (4.) and (5.) is not readily available, but a solution of these equations can be obtained numerically. In what follows, we refer to the numerical solution of (4.) and (5.) using the method of lines for time integration and finite difference approach for discretizing the spatial derivatives. In such cases, the fast dynamics (4.) is described by a system of ordinary differential equations (whose order depends on the discretization of the spatial domain), while the slow dynamics (5.) is described by a system of Differential Algebraic Equations (DAEs). Notice that, in general, the algebraic equations associated with the discretization of (5.) can be solved for the quasi-steady state values of the gas phase temperatures and concentrations, thus setting the index of the DAE system to 1. Remark 1: The analysis presented above can be verified to be valid both for co-current and for countercurrent reactors. Remark 2: Within the model reduction framework proposed above, one can also obtain a consistent initialization of the DAE system describing the slow dynamics. Specifically, presuming that a numerical approximation of the full-order model is available, its states can be used to initialize both the algebraic and the differential variables when simulating the slow model. 3. Illustrative simulation study We consider the case of co-current autothermal reactor for generating hydrogen from methane and steam via methane/steam reforming (MSR) and water-gas shift (WGS):

CH, + IH^O T

^ AH^ + CO^

(^-^

Reactions 6. are endothermic, and the necessary heat is provided by catalytically oxidating methane (MCO) in parallel channels: CH4 + 2O2 - > CO2 + 2 H 2 O

(7.)

For WGS/MSR we considered the kinetics proposed in (Xu and Froment, 1987), while MCO kinetics were taken from (Trimm and Lam, 1980) as modified by (van Sint Annaland et al, 2002) to consider mass transfer limitations. We assumed that the reactor is 0.5 m in length, with square channels of 26mm cross section. The MSRTWGS

Autothermal Reactors for Hydrogen Production: Dynamics and Model Reduction

1499

side was assumed to be fed with a 25% methane 75% steam (by weight) mixture at 1.5 bar, and the MCO side with air containing 2% (by weight) methane. Inlet velocities were chosen so that the residence times were 5s for the WGS/MSR side and 2s for the MCO side. The model was solved numerically using the method of lines and finite differences over a spatial grid with 30 nodes, resulting in an equivalent full order ODE system of 300 equations. Dirichlet boundary conditions were used for the gas phase temperatures and mass fractions at the inlet, while the catalytic wall temperature and all states at the reactor output were subject to Neumann boundary conditions. Figure 2 depicts the eigenvalues of the linearized system at the nominal operating point. Notice that the eigenvalues separate in two groups, with exactly 30 eigenvalues concentrated in the right hand side of the plot, and the remaining eigenvalues further left in the complex plane. The plot confirms the results of our theoretical analysis, which predicts the existence of slow modes corresponding to the temperature of the solid wall (in this case, the 30 states correspond to the solid temperature at each spatial node). Figure 3 shows the time evolution of the temperatures and hydrogen mole fraction at the reactor exit, computed with the fiill-order ODE model and the reduced-order DAE model in the case of a 20% increase in the axial velocity of the gas mixture in the reforming side. Notice the fast initial transient exhibited by the stiff ODE model (the condition number of the linearized model is in the order of 10"^), which is not present in the evolution obtained using the DAE model. Both models capture the subsequent slow approach to steady state, with a very good agreement between the reduced-order and full models. 8 6 4 2

I 0

^ • • • ' • * •• • • • •

-2 4 •6

-10

-10

-10 log(Real)

-10

-10

Figure 2 Eigenvalues of the linearized reactor model (30-node Finite Difference discretization)

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M Baldea and P. Daoutidis

4. Conclusions We presented a generic dynamic analysis and model reduction framework for autothermal reactors. We demonstrated the presence of a two time scale dynamic behavior, and showed that the wall temperature evolves over a long time horizon, while the other variables evolve in a fast time scale. The derived low order DAE for the slow dynamics was shown, through simulations, to capture the evolution of the reactor very well, while eliminating the stiffness associated with the original model, and can thus be used for control purposes. The control of autothermal reactors is a topic that will be addressed in our future work. 0.5602

0.5602

-

0.56G1

0.5601

tS

0.5601

0.5601

0.56

^

1

2

1 time, s

2

3

0.56

5000

10000

15000

5000 10000 time, s

15000

1323.499

0) 3

S 1323.498 [ E


W 1323.497

Q^ 1323.496

0

Figure 3 ODE (solid) and DAE (dotted) simulations of the reactor dynamics

5. Acknowledgements Financial support from the National Science Foundation, grant CTS-0234440 is gratefully acknowledged. MB was partially supported through a University of Minnesota Doctoral Dissertation Fellowship. References Kolios, G., Frauhammer, J., Eigenberger, G. (2000) Chem. Eng. Sci., 55, 5945-5967. Gorgun, H., Arcak, M., Varigonda, S., Bortoff, S. (2005) Int. J. Hydr. Energ., 30,447-457 Vanden Bussche, K.M., Neophytides, S.N., Zolotarskii, I.A., Froment, G.F. (1993) Chem. Eng Sci., 48(19), 3335-3345 Kaisare, N.S., Lee, J.H., Fedorov, A.G. (2005) AIChE J., 51(8), 2254-2264 Gorbach, A., Eigenberger, G., Kolios, G. (2005) Ind. Eng. Chem. Res., 44, 2369-2381 Xu, J., Froment, G.F. (1989). AIChE J., 35 (1), 88-96. Trimm, D.L., & Lam, C.-W. (1980) Chem. Eng. Sci., 35, 1405-1413. van Sint Annaland, M., Scholts, H.A.R., Kuipers, J.A.M., van Swaaij, W.P.M. (2002). Chem. Eng. Sci., 57, 855-872.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Performance assessment and controller design for unknown systems based on gain and phase margins using modified relay feedback Jyh-Cheng Jeng,* Hsiao-Ping Huang Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan Abstract A systematic procedure for performance assessment and PI/PID controller design based on modified relay feedback test is proposed in this paper. It can estimate the gain and phase margins of unknown systems on-line to indicate the appropriateness of the controller parameters. When the retuning of controller is found necessary, the proposed modified relay feedback scheme can be applied to tune the PI/PID controller based on specifications of gain and phase margins. Performance assessment and controller design can be done simultaneously, which ensures a good performance of the control system. Keywords: performance assessment, relay feedback, gain margin, phase margin. 1. Introduction The PID controller is widely used in chemical process industries because of its simplicity and robustness to the modelling error. However, many control loops are still found to perform poorly. Therefore, regular performance assessment and controller retuning are necessary. Gain and phase margins have served as important measures of performance and robustness for the single-input-single-output (SISO) system. Because the calculation of gain and phase margins in traditional ways is very tedious, it is highly desirable to find a procedure for on-line monitoring of gain and phase margins. Recently, Ma and Zhu [1] proposed a performance assessment procedure based on modified relay feedback. Gain and phase margins are estimated by two relay tests. But their method may not give accurate results for processes with more complex dynamics such as process with righthalf-plane (RHP) zero and oscillatory modes. Controller designs to satisfy gain and phase margin specifications are well accepted in practice and in classical control. Astrom and Hagglund [2] used relay feedback for automatic tuning of PID controllers with specification on either gain margin or phase margin, but not both. Some approximate analytical PI/PID tuning formulas have been derived to achieve the specified gain and phase margins [3]. Most of them use simplified models such as FOPDT and SOPDT model. For processes with more complicated dynamics, the resulting control systems may not achieve userspecified gain and phase margins exactly. In this paper, a performance assessment procedure based on modified relay feedback test is proposed. It can on-line estimate the gain and phase margins for systems with both unknown process dynamics and controller parameters. The estimated ' Corresponding author. Tel.: 886-2-3366-3067; e-mail: [email protected]

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J.-C. Jeng and H.-P. Huang

results can be used to indicate the appropriateness of the controller parameters. When the retuning of controller is found necessary, a similar procedure can be applied to tune the PI/PID controller based on the user-specified gain and phase margins. 2. Modified Relay Feedback Structure The use of relay feedback for automatic tuning of PID controllers was first proposed by Astrom and Hagglund [2]. The block diagram of the standard relay feedback system is as shown in Fig. 1(a). The system generates a continues cycling with a period Pu if it has a phase lag of at least TT. The ultimate frequency from this relay feedback test is ^ = 27t/Pu and the ultimate gain can be approximately given by Ku = 4h/7ia, where h is the relay output magnitude and a is the amplitude of limit cycle. For the purpose of performance assessment, a modified relay feedback structure is proposed as shown in Fig. 1(b) where Gc, G, u^. and j ; are the controller, process, relay output, and process output, respectively. Moreover, a delay element, e~ , is embedded between the relay and the controller. Compared with the conventional relay feedback, the most important features of this modified structure are that the controller is always connected in line with the process and an additional delay is embedded. As a result, it can assess the performance of the closed-loop system by estimation of gain and phase margins on-line, as presented in the following section, to determine if a retuning of the controller is necessary.

(a)

(b)

Fig. 1 (a) Standard relay feedback system (b) Modified relay feedback system 3. Performance Assessment 3.1. Estimation of Gain Margin Consider the modified relay feedback system as shown in Fig. 1(b). For the estimation of gain margin, the delay A is set as zero. Let the loop transfer function be GLP(S) = Gc(s)G(s). The phase crossover frequency, OJ^, of GLP(S) can be calculated by OJ^ = 2n/Pp, where Pp is the period of the limit cycle. In addition, the amplitude of GLP(S) can be approximately calculated by G^p [jCOp) = KalAh [2]. However, the accuracy of such approximation is poor in some cases where the error may be as large as 20% [4]. For more accurate estimation, (^Lp(j(oA

can be computed based on

Fourier analysis as: (1) Therefore, the gain margin, Am, can be estimated as:

A=i/|<^.p(y^,)|

(2)

Performance Assessment and Controller Design for Unknown Systems

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3.2. Estimation of Phase Margin When the estimation of gain margin is finished, the delay A is then set as a nonzero value in order to extract the frequency information of GLP{S) at other frequencies for the estimation of phase margin. With a given value of A, assume that the system oscillates with a period of P and then we have the phase of the system as:

2irg{G,p{j(o)e-''] = 2x%{G,p{jQ))]-^0) = -n

(3)

where co = ITT/P. TO calculate the phase margin, the desired frequency is the gain crossover frequency, cOg, of GLP{S). Denote the desired value of A that makes the amplitude of GLP, at the corresponding oscillating frequency, equal unity as A ^ and the period of the limit cycle is Pg. In other words, the gain crossover frequency of Gipis) is equal to the phase crossover frequency of G^p{^s^e~^. In this case, Eq.(3) can be written as argJG^p (yty^ j | - A^ty^ =-7U where cOg = Ijt/Pg. Then, it follows that the phase margin, (j)^, can be estimated as:


(4)

To find the value of A j , an on-line iterative procedure as the following is presented. Starting from an initial guess A , the value of A is updated by A(-»=A<'>-/'>(|G,4y«('>)|-l)

(5)

where j ^ > 0 is the convergence rate and Y^^p \JC0 j is computed by \GW (M" )| = \GLP (M")e-^-^"''"

I=If

y{t)e-^'^"'dl\l\

f

uXt)e-''^"'dt\

(6)

Notice that we have A^ = 0 if ^^ = 1, and, in general, A^ increases as A^ and P increase. Thus, the value of A

is suggested as (^^ — l ) P / 6 . In addition, j'^^ is

chosen as:

/ o = (A<'> - A<-> ) / ( | G , , (jcd'^ )| -

\G,,

(y^<'-» )|)

(7)

which can make Eq.(5) have a quadratic convergence rate near the solution [2]. In each iteration, the value of A is set until the output generates two or three oscillating cycles and then switch it to the next value just like an on-line adaptive scheme. When Eq.(5) converges, the resulting value of A is taken as A^. 3.3. Extension to Multi-Loop Systems The proposed procedure can be extended to multi-loop control systems if the gain and phase margins are defined in the similar spirit as SISO system based on the effective open-loop process (BOP) [5]. The /-th BOP describes the effective transmission from the /-th input to the /-th output when all other loops are closed. With the formulation of BOP, the multi-loop control system can be considered as several equivalent SISO loops and the gain and phase margins of each equivalent loop can be estimated by sequentially using of the proposed modified relay feedback test.

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J.-C. JengandH.-P. Huang

4. Controller Tuning 4.1. Tuning of PI Controller Consider the PI controller of G^{s) = k^\\-\-\l{TjSy\

. Denote the user-

specified gain and phase margins as A^ and ^^, respectively. For a given value of A, the parameters, k^ and Tj, can be found to satisfy the specification of phase margin. In other words, they can be found such that the following two equations hold.

>A
(8)

|G..(7^J| = K(y«J^"'"^1 = l

(9)

However, the specification of gain margin may not be necessarily achieved by such obtained controller parameters. There exists a certain value of A which can make the gain margin of the resulting system meet its specification. Thus, an iterative procedure for tuning the PI controller using the modified relay feedback test is presented as follows: 1) Starting with a guessed value of A, i.e. A

.

2) Adjust Tj by the following equation:

rr)=rf-;^''(P<''-2;rA/<2>:)

(lO)

where P is the period of limit cycle in the /-th iteration. Eq.(8) holds when Eq.(lO) converges. 3) Adjust k^ by the following equation until it converges so that Eq.(9) holds,

kr=k^'-i^'i\G,,[M^^^^^^ where (O^^ ^InjP^^

and

|G^P(7W'^)|

(11) is computed by Eq.(6).

4) Set A = 0 and estimate A^ by Eq.(2). 5) Check if the estimated A^ equals A^ . If not, change the value of A by the following equation and go back to step 2) until A^ — A^ holds.

A<-)=A«-;^'->(4;->-^:)

(12)

The convergence rates, J^ , y^ ^^^ J ^ , are defined in the similar manner of Eq.(7). For FOPDT process, inserting a delay A in the relay feedback loop approximately results in an increase of 4A in the period of limit cycle. In order to improve the convergence of Eq.(lO), it is desirable that iTt^j^l

=P^(^P^ + 4 A ) . Thus, the

initial guess of A is suggested as A^^^ = / ^ / ^ 2 ; r / ^ ^ - 4 j 4.2. Tuning ofPID Controller For the tuning of PID controller, a similar procedure can be applied. The PID controller transfer fiinction is given as G^{s) = k^\\-\- ( I / T ^ ^ ) + T^SJ. The derivative

Performance Assessment and Controller Design for Unknown Systems

1505

time, T^ , is usually chosen as a fixed ratio of the integral time, Tj, as T^ = OCTj. Researchers have recommended that (X = 0.25 [2]. With this relation, the procedure for PI controller tuning presented in the previous section can be applied directly to tune the PID controller. If T^ is not chosen as a fixed ratio of Tj, then the extra degree of freedom can be used for achieving another performance requirement. 5. Simulation Examples 5.1. Example 1 Consider a control system with FOPDT process G(s) = e~ ji^s + l) and PI controller G^(5) = 0.616(l + l/(0.7655)) . Three different values of the process dead-time, ^ = 0.5, 1, 1.5, are used for simulation. The estimated gain and phase margins together with the values of A during iteration are shown in Table 1. The value of A converges after two iterations. The output responses during the estimation procedure are shown in Fig. 2, where period I is for estimating gain margin and periods II to IV are for estimating phase margin. For the case of ^ = 1.5 , the gain and phase margins are too small, so we retune the controller with the specifications chosen as A^ = 2 . 5 and ^^ = 54°. The results converge after two iterations of A (A^^^ = 2.239), and the PI controller is obtained as G,(5) = 0.424(1 + 1/(1.0085)) . The actual gain and phase margins of this resulting control system are A^ = 2.49 and (p^ = 53.2°, which are very close to the specified ones. The closed-loop responses before and after retuning are shown in Fig. 3. Table 1. Actual and estimated gain margin, pha se margin in example 1

0 0.5 1 1.5

A

0

4.64 2.11 1.33

61.5° 40.0° 18.6°

Ttn

Am 4.56 2.07 1.31

^(0)

^(1)

A^'\A,)

1.278 0.789 0.336

1.359 0.864 0.385

1.423 0.924 0.427

rm

60.7° 39.4° 18.2°

A^ , (f)^ : actual gain and phase margins

5.2. Example 2 Consider a process with RHP zero G{s) = {\-Ps)l{s

+ \f

and a PI controller

G^{s) = \\-\r 1/(25)), which are used for simulation in Ma and Zhu [1]. For the cases of J3 = I and /? = 0 . 1 , the actual gain and phase margins together with the estimated values by Ma and Zhu [1] and the proposed method are given in Table 2 for comparison. The proposed method can estimate the gain and phase margins more accurately. For the case of j3 = 1, retune the PI controller with the specifications given as A^=3

and (/f^ = 60°. These specifications are achieved after two iterations of A

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J.-C. Jeng and H.-P. Huang

0

10

20

30

40

50

60

70

80

Fig. 3. Closed-loop responses ( ^ = 1.5)

Fig. 2 The output re^nse during assessment (a) ^ = 0.5 ( b ) ^ = l ( c ) ^ = 1.5

(A*^^ = 4.961), and the PI controller is obtained as G^{s) = 0.32 (l + l / ( l . 5 2 4 5 ) ) . The actual gain and phase margins of this resulting control system are A^ = 3 . 1 2 and


^m

^m

1.28 3.49

20.0° 51.6°

Estimated [1]

A m

1.19 3.29

d rm

12.7° 41.8°

Estimated (proposed method) A^ cl>^ A,=A<^> 1.20 3.34

16.9° 51.8°

0.501 1.789

6. Conclusions A systematic procedure for performance assessment and controller design based on modified relay feedback test is proposed. The proposed method can estimate the gain and phase margins on-line for systems with both unknown process dynamics and controller parameters. The estimated results can be used to assess the performance of the closed-loop system. When the retuning of controller is found necessary, a similar procedure can be applied to tune the PI/PID controller based on the user-specified gain and phase margins. Simulation results have shown that the proposed method is effective for processes with different kinds of dynamics. References [1] M. D. Ma and X. J. Zhu, Performance Assessment and Controller Design Based on Modified Relay Feedback, Ind. Eng. Chem. Res., 44 (2005) 3538. [2] K. J. Astrom and T. Hagglund, Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins, Automatica, 20 (1984) 645. [3] W. K. Ho, C. C. Hang and L. S. Cao, Tuning of PID Controllers Based on Gain and Phase Margin Specifications, Automatica, 31 (1995) 497.

Performance Assessment and Controller Design for Unknown Systems

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[4] W. Li, E. Eskinat and W. L. Luyben, An Improved Autotune Identification Method. Ind. Eng. Chem. Res., 30 (1991) 1530. [5] H. P. Huang, J. C. Jeng, C. H. Chiang and W. Pan, A Direct Method for Multi-loop PI/PID Controller Design, J. Process Control, 13 (2003) 769.

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Graphical Modeling for the Safety Verification of Chemical Processes Jinkyung Kim, Younghee Lee and II Moon Department of Chemical Engineering, Yonsei University, Seodaemun-gu Shinchondong 134, Seoul 120-749, Korea Abstract This study focuses on verifying the safety using graphical modeUng and simulation for chemical processes logics. UPPAAL is used as an integrated tool for modeling, simulation and verification in this study. Most of chemical processes are highly automated and they are represented by complex network control logics. The safety analysis of these chemical processes is always difficult or sometimes impossible to verify the system perfectly because the system includes dynamic variables, various levels of control activities (safety interlocking, regulatory/discrete control and sequential control), numerous units, instruments and control software/hardware. All these must be considered simultaneously, therefore, this study presents an effective technique for the safety verification using a graphical modeling and simulation. Keywords: Safety verification. Graphical Modeling and Simulation, Chemical Processes, Model Checking, UPPAAL 1. Introduction As chemical processing systems are highly automated, the verification of their safety is becoming more difficult. The verification work must be complete. Most of chemical processes contain both discrete control logics and continuous dynamics, which are connected with networks and operated synchronously. All these must be verified at each stage of development (formal specification, design, implementation, construction, and maintenance). The changes of operating procedures make the verification work more difficult. When we design a chemical process and make an error free operation procedure of the process, it is necessary to verify the safety and operability of the process automatically and efficiently. In order to verify the chemical process automatically, the signal and phenomena exhibited by the control logics, various process variables, operating procedures, operator behavior and their networks have to be described in modeling part. UPPAAL is appropriate for graphical modeling for the chemical processes including nondeterministic variables, finite control logics and real time system because it uses timed automata, communications among channels and shared variables. The model checker in UPPAAL is based on the theory of timed automata and its modeling language offers additional features such as bounded integer variables and urgency. The query language for the specific properties and operability to be checked is a subset of CTL (computation tree logic). Timed-automata are finite state machines extended with clock variables for modeling and verification of real time system. Examples of other formalisms with the same purpose are timed Petri Nets, timed process algebras, and real time logics. A simplified version, namely timed safety automata is introduced to specify progress properties using

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J.K.Kimetal

local invariant conditions. Due to its simplicity, timed safety automata have been adopted in several verification tools for timed automata e.g. UPPAAL and Checkmate. CTL, expressing questions about the system being verified, is used as the query language in UPPAAL. CTL has been originally used in SMV (Symbolic Model Verifier), an automatic error finding system, and its powerful efficiency has been proved from a lot of applications for the system logic specifications. This study represents a novel safety verification method using graphical modeling and simulations of chemical processes and two kinds of batch process examples shows the effectiveness of this method.

System Model

No

Network of timed automata

Debugging information

Model Checker (Simulation + Verification)

Yes V ^ U e S I I O n Computation tree logic

Debugging information

Figure 1. Frame of the model checker UPPAAL

2. Modeling and Simulation UPPAAL consists of three main parts: a description language, a simulator and a model checker. The description language is a non-deterministic guarded command language with simple data (e.g. bounded integers, arrays, etc.). It serves as a modeling or design language to describe process behavior as networks of automata extended with time and process variables in chemical processes. The idea is to model a process using timedautomata, simulate it. Timed automata are a finite stat machine that is a graph containing a finite set of nodes or locations and a finite set of labeled edges with time. A system consists of a network of processes that are composed of locations. In UPPAAL, a system is modeled as a network of several such timed automata in parallel. The model is further extended with bounded discrete variables that are part of the state. A state of the system id defined by the locations of all automata, the clock constrains, and the values of the discrete variables. Every automaton may fire and edge separately or synchronize with another automaton, which leads to a new state. Transitions are used to change location and transitions between these locations define how the system behaves. To control when to fire a transition, it is possible to have a guard and synchronization. A guard is a condition on the variables and the clocks saying when the transition is enabled. The synchronization mechanism in UPPAAL is a hand-shaking synchronization: two processes take a transition at the same time, one will have a go\ And the other a go?, go being the synchronization channel. When taking a transition actions are possible: assignment of variables or reset of clocks. The clocks are the way to handle time in UPPAAL. The simulator is a validation tool which enables examination of possible dynamic executions of a system during early design stage and thus provides an inexpensive mean of fault detection prior to verification by model checker which covers the exhaustive dynamic behavior of the system. A graphical simulator which provides graphical visualization and recording of the possible dynamic behaviors of a system description, i.e. sequence of symbolic stated of the system. It is also used to visualize traces generated by the model checker.

Graphical Modeling for the Safety Verification of Chemical Processes

1511

3. Model Checking for Safety Verification The main purpose of a model checker is verifying the model with regard to a requirement specification. The model checking verification method is an alternative approach that has achieved significant results recently. Efficient algorithms are able to verify properties of extremely large systems. In these techniques, specifications are written as formulas in a proposition temporal logic and systems are represented by statetransition graph. The verification is accomplished by an efficient breadth first search procedure that views the transition system as a model for the logic, and determines if the specifications are satisfied by the model. There are several advantages to this approach. An important one is that the procedure is completely automatic. The model checker accepts a model description and specifications written as temporal logic formulas, and it determines if the formulas are true or not for that model. The model checker in UPPAAL is to check invariant and bounded-lively properties by exploring the state space of a system, i.e. reachability analysis in terms of symbolic states represented by constrains. Like the model, the requirement specification must be expressed in a formally well-defined and machine readable language. Several such logics exist in the scientific literature, and UPPAAL uses a simplified version of CTL (Computation Tree Logic). Like in CTL, the query language consists of path formula and state formula. State formula describe individual stated, whereas path formula quantify over paths or traces of the model. Path formula can be classified into reachability, safety and operability.

A[] q>: q> is invariant

£[] q>: q) holds at every state in some path

A<>
v ~>q): v leads to cp

£<> q>: q> potentially holds

Figure 2. Examples of computation trees satisfying five common CTL formulas.

4. Example 4.1. A tank with level controller A tank with high level and low level controller, two valves (inlet valve and outlet valve) illustrated in first case study. Volume Rates of inlet and outlet are 50 ml /s, high level sensor is to 500ml, and low level sensor is to 50ml. Process is operated twice (2 batches). 0. Initial state: tank is empty and all valves are closed. 1. Inlet valve is open. 2. Fill the tank.

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J.K. Kim et al

3. When level of the tank leads to high level, inlet valve is closed and outlet valve is open. 4. Empty the tank. 5. When level of the tank leads to low level, outlet valve is closed and inlet valve is open. 6. Return 2. (one more) 7. Drain and setup, (state 0.)

HL(500)

LL(50) V2 (outlet valve) '.50*.

4.1.1. Modeling and simulation

Figure 3. A tank with ievei controiier valve ^ |L

les«l ==0 8abat*==0

i==oas»8t*==2T"'"^^''^

= true,v3open:=t^^

v1 open;= false, v2Dperi:=fa!s8 ..i,i

bat* 1=2 .open: =

/ A open == true A * v2cperi == ti

P,«n ^ ^

^ ..

\ ^

^^)••,
/

v1open:= false, v2op8n := true

k —J ^\J

Figure 4. Simulation of a tank with level controller and two valves Two models are described for valve behavior and tank operation, including tank level variable, operation of filling and emptying and the number of batch. Senor signals are used as a synchronization connecting tank and valve. Two described models, therefore, is simulated at the same time and the result of simulation records dynamic variable (tank level and number of batch) and shows the behaviors sequence of the process description. 4.1.2. Specifications Queries in this case study follow; a. level == 500 ^ vlopen == false && v2open == true : This question means that inlet valve is closed and outlet valve is open as soon as tank level is up to high level. b. A[] !(vlopen == true && v2open == true) : This question means there is no path globally that inlet valve and outlet valve are open at the same time. c. A<> (50 < level < 500) && (vlopen == true or v2open == true) : This question means there is a state in every path that if tank level is between 50 and 500, one of two valves is open. statu s level==500 —> vlopen == false && v2open =^rue Property is satistied, A[] !(v1open == true && v2open == true) Property is satisfied. A<> (50 < level < 500) && (vlopen == true or v2open ==true) Property is satisfied.

Figure 5. Verification results of the example 4.1

Graphical Modeling for the Safety Verification of Chemical Processes 4.2. Distributed tank system This example is a distributed tank system which a material in main tank is delivered to three tanks with different volume rates of inlet. There are 4 valves and each tank has level controller. 0. All valve are closed and three sub tanks are empty. 1. Main valve is open 2. Three sub-valves are open 3. Fill three tanks 4. If some tank is up to high level, inlet valve of the tank is closed. 5. Three sub-valves are closed. 6. Main valve is closed.

1513

HL(3I

LL(0) majntank valve ^ V2

t

V3

-M - HL(600)-i ;

HL(600)

HM600)-n

;

ta^»

fp*1 LLL(0)

^

LL(0)

-LL(0)

J

Figure 6. Distributed system from main tank to tliree tanl<s

4.2.1. Modeling and simulation main^/alve ==true T21il!ing? maintartde\«l -= ta-!k2nlet

iniiankles«! == 3000 &a tank1 level == 0Mtank2level ==0 &&tank3levd ==0 mainvalvie :=true, rnairrtarklevel ;= 0

rnainv'ai\« ==true TSfiHing? aitntank level -~ tmk^nlet

h^ainvah/eClosed

main''/sil''.'e ==:ttue T1fillir>3? n ajntarfe ies«l -= tank 1 i niet

maintar<
\n open == tue Tin ill r^! tank1 le'-.«l += tank'l inlet

vtCols kTlfijII?

jfc.,^^ tank1 level < 600 T1 Filling \K^} Tlfillir-g! ^•^^- .^^ open: = true, tank1 level + = terrki Irtet

TlfiJ?

O

mainvalve == true && tank1 te\«l != 600

v1open:= false \ ••.•1 open : = true

tank1 level == 600 Titull! TiFull

Figure 7. Simulation of distributed tank system This model consists of level of main tank, main valve, three tanks and three valves. The variables for the level of main tank and the levels of each sub tank are calculated simultaneously. Four valves are operated according to the synchronization network. 4.2.2. Specifications Queries in this case study follow; a. A[] !(mainvalve==false && (vlopen==true or v2open= true or v3open==true)) : This question means there is no path globally that mainvalve is closed when one of tliree sub valve is at least open. b. A<> !((0 < tankllevel < 600) && vlopen == false) A<> !((0 < tank21evel < 600) && v2open == false) A<> !((0 < tank31evel < 600) && v3open == false)

1514

J.K.Kimetal.

: This question means there is no state in any path that although some sub tank level is not up to high level, the inlet valve is closed. Status AU Kmainvalve == false && (vtopen == true or v2open == true or v3open == true)) Pn,p>-rly r.. ^.nu-MQ. \A<> ! « 0 < fankllevel < 600) && viopen == false) ; P ' 0 p « r 1 y h •i.-^U-iued

\A<> ! « 0 < tankaeve! < 60)) && v2open == false) ipifiQ-^riK' ri

'•.:iti'-Jlf"i.

i A O !«0 < tankaevel < 600) && v^pen == false)

Figure 8. Verification results of example 4.2 5. C o n c l u s i o n s This study represents a novel method to verify the safety of chemical processes using graphical modeling and simulations. This method is applied to two batch processes and the safety and operability is verified. The graphical description of the process is model in system editor. The simulator provides visual sequence of symbolic states of the processes, calculates the dynamic variables, and records the possible behaviors of the process models at the same time. The model checker using CTL specifications verifies the safety and properties of the processes automatically. This study represents an effective and automatic technique for the safety verification using a graphical modeling and simulation. References I. Moon, 1994, Modeling PLCs for logic verification, IEEE Control Systems, Vol.14, No.2, pp. 53-59 J. Kim, M. Kim and I. Moon, 1999, Improved Search Algorithm for the Efficient Verification of Chemical Processes, Computers and Chemical Engineering, Vol. 23, SuppL, pp. S601-604 R. Alur and D. Dill, 1994, A Theory for Timed Automata, Theoretical Computer Science, Vol. 125, pp. 193-235 A. Furfaro and L. Nigro, 2003, Temporal verification of communicating real-time state machines using uppaal Industrial Technology, 2003 IEEE International Conference on, Vol.1, pp.399404 J. Kim and I. Moon, 2000, Synthesis of Safe Operation Procedure for Multi-purpose Bath Processes using SMV, Computers and Chemical Engineering, Vol. 24, pp.385-392 B. Justin, G. L. Kim, P. Paul, and Y. Wang, 2000, UPPAAL: A tool suite for automatic verification of real-time systems. Lecture notes in computer science. No. 1066, pp. 232-236 A. Bums, 1998, How to Verify a Safe Real-Time System: The Application of Model Checking and Timed Automata to the Production Cell Case Study, Real-time systems. Vol. 24, pp. 135151 S. Cha, H. Son, J. Yoo, E. Jee, and P. Seong, 2003, Systematic evaluation of fault trees using realtime model checker UPPAAL, Reliability engineering & system safety. Vol. 82, pp. 11-20 T. A. Henzinger and W. K. Peter, 1999, Discrete-time control for rectangular hybrid automata. Theoretical Computer Science, Vol. 221, Issues 1-2, pp. 369-392 G. L. Kim, P. Paul, and Y. Wang, 1995, Model-Checking for Real-Time Systems, Lecture Notes in Computer Science, Vol. 965, pp. 62-88 S. Yovine. Kronos, 1997, A Verification Tool for real-Time Systems, Journal of Software Tools for Technology Transfer, Vol.1, pp. 123-133 T. A. Henzinger, X. NicoUin, J. Sifakis, and S. Yovine, 1994, Symbolic model checking for realtime systems, Information and Computation, Vol. 111, pp. 193-244.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1515

Application of A Hybrid Control Approach to Highly Nonlinear Chemical Processes Yoshiyasu Sakakura^, Masaru Noda^, Hirokazu Nishitani ^, Yoshiyuki Yamashita^, Masatoshi Yoshida^, Sigeru Matsumoto ^ ^Graduate School of Information science, Nara Institute of Science and Technology, Takayama, Ikoma, Nara, 630-0192, Japan ^Department of Chemical Engineering, Tohoku University, Aramaki, Aoba, Sendai, Miyagi, 980-8579, Japan This paper proposes a new control approach for highly nonlinear chemical processes with operational mode changes. We combine a successive linearization and an MLD (mixed logical dynamical) formulation to transform an MPC (model predictive control) problem for a multimodal nonlinear dynamical system into an MIQP problem. We apply the proposed approach to the temperature control problem of CSTR (continuous stirred tank reactor), which is modeled as a bimodal nonlinear hybrid system. The simulation result shows a good control capability in the control of highly nonlinear hybrid systems. Key words: Process Control, Hybrid System, Model Predictive Control, MLD Formulation, CSTR, 1. INTRODUCTION In plant operations, operational modes change frequently due to product changes, emergency evacuations, and so on. The plant dynamics with operational mode changes are usually formulated as a nonlinear hybrid dynamical system. There are two problems to be overcome in nonlinear hybrid dynamical system control: nonlinearity of the process dynamics and logical rules in the hybrid dynamical system. Many approaches for nonlinear control have been proposed so far. One of the most effective is an MPC(model predictive control) using successive model linearization [2]. Shikanai et al. proposed a predictive control framework using successive linearization of a highly nonlinear process [4]. Many control approaches also have been proposed for hybrid dynamical systems. One has been proposed by Mhaskar et al. [3], where a Lyapunov-based predictive controller is designed for a each mode that allows for an explicit characterization of its stability region. Mode changes are then appropriately incorporated to the controller implementation. Another, and the most remarkable, approach was proposed by Bemporad et al [1], where a linear hybrid dynamical system is described as the MLD (mixed logical dynamical) model. This model is used in an MPC scheme. Since the method can be applied only to a linear discrete-time model, hnearization of the nonlinear process plays an important

1516

Y. Sakakura et al

role when the MLD model is used. In this study, we propose a new control method for nonUnear hybrid dynamical systems based on MLD formulation. We employ the modified linearization method proposed by Shikanai et al. [4] in a control scheme, and apply it to temperature control of a CSTR with operational mode changes. 2. METHOD 2.1. Problem description We consider a bimodal nonlinear hybrid system as given by Eqs. (1) and (2). X = fi{x,u) ^ = /2(^, u) y = Zx

if 5a: < 0 (mode 1) ii Sx >0 (mode 2) (2)

along with u G [t^min^'^max], l^t /i and /2 be nonlinear functions, where S and Z are coefficient matrices, and x and u denote state and input vector variables. Conditional inequalities in Eq. (1) indicate that the operational mode changes according to the state variables of the process. We can formulate a control problem of a bimodal nonlinear hybrid system as an MPC problem through the following two steps: 1. Discretization and linearization of nonlinear continuous ODEs in Eq. (1). 2. Formulate logical constraints in Eq. (1) as linear inequality constrains with binary variables. Through the above two steps, the control problem of a nonlinear hybrid system is transformed into an MILP problem, which is solved at every control period. 2.2. Discretization and linearization methods The nonlinear differential equations in Eq. (1) are discretized and linearized at every control period to transform a nonlinear hybrid system into an MLD model. In general, the conventional hnearized and discrete-time models of Eqs. (1) and (2) are given by Eqs. (3) and (4), jxk+j+i = Ai^Xk+j + Bi^Uk+j + Ci ]^Xk+j+i = A2uXk+j + B2uUk+j + C2 yk+j = Zxk^j

if Sxk+j < 0 (mode 1) if Sxk+j > 0 (mode 2)

(3) (4)

where k denotes the present time step, j{= {0,1, 2 • • • }) represents the number of steps from the present step, x^+j is the state vector variable, Uk+j is the input vector variable, Ann, Bnu ^rc coustaut matriccs, and Cn denote constant vectors in a control period. The subscript u indicates the conventional linearization method and n represents the mode of the system (1 or 2).

Application of a Hybrid Control Approach to Highly Nonlinear Chemical Processes

1517

In this study, we use the modified linearization method [4] in our control scheme. According to this method, Eq. (1) can be formulated as follows: ^fc+j+i = AiiXk+j + Ai2Xk+j-i + BiAuk+j

if Sxk+j < 0 (mode 1)

ock^j+i = A2iXk+j + A22Xk+j-i + B2Auk+j

if Sxk+j > 0 (mode 2)

where Auk+j = Uk+j — Uk+j-i- Because Eq. (5) is a finite difference model, the equation is calculated recursively to estimate state vector variables x at any time step j using both Xk and Xk-i as known values. We finally obtain a simple linear discrete prediction model in every control period as follows: \xk+j+i = ^ 1 + BiAuk+j

if Sxk+j < 0 (mode 1)

yxk+j+i = ^2 + B2Auk+j

if Sxk+j > 0 (mode 2)

where Ai = AnXk+j + Ai2Xk+j-i A2 = A2lXk+j

+

A22Xk+j-l

2.3. M L D formulation Bemporad et al. proposed a systematic approach for the control problems of hybrid dynamical systems [1]. According to the MLD formulation, Eq. (6) is formulated as follows: Xk+j+i = Al + Sk+j{A2 - Al) + {Bi + 6k+j{B2 -m

Sk+j + m<

Bi))Auk+j

S Xk+j < (M + e) 4 + j - s

(7) (8)

11 < Uk+j 7/;. , • < 7/__-_ ^max < < Umin

(9)

where Sk+j G {0,1} is the binary variable and £ is a small positive scalar. In addition, M and m denote the maximum and minimum values of Xk-\-j, respectively. Equation (9) denotes the constraint of the process inputs. Equation (7) represents mode 1 of Eq. (6) when 6k-j-j — 0. Logical rules in a hybrid dynamical system are transformed into inequalities Eqs. (8), which involve binary variables Sk-^j. As a result, the control problem of a nonlinear hybrid system given by Eqs. (1) and (2) is described as an MIQP problem as follows:

Mm^ _ J{k) = Yl Wy^+J - ^-f k+jWlu) + £

\\^^k^3\\\i)

subject to: Eqs. (4), (7), (8), and (9) where ^ref k+j denotes the reference trajectory, Q{j) and R{i) are weighting matrices. [H^,, Hp] is the prediction horizon, and [0, Hu — 1] represents the control horizon.

(10)

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Y. Sakakura et al - - ON/OFF Cooling System

Temperature Control System ^IN Q IN ^IN

Figure 1. Case study plant.

3. C A S E S T U D Y 3 . 1 . P r o c e s s description We will now examine a simulated non-isothermal CSTR as shown in Figure 1. The solvent and reactant A are supplied to the reactor and a first-order irreversible exothermal reaction producing B occurs. There are two coolant systems in this process: the temperature control system in Figure 1 is implemented to maintain the reactor temperature at a set point and the O N / O F F cooling system is operated depending on the reactor liquid level. When the level H \ m. (mode 2), the valve fully opens automatically. Temperature of the coolant flow Fi (Ti IN) is a manipulated variable to maintain the reactor temperature (T) at a set point (Tget)- Under these assumptions, the process dynamics can be described as follows: if 0 m < H{t) < 1 m

(mode 1)

dH ~ dAHCA dt dAHT dt dTi dt

k

A

= Fm{CA IN — CA) — AHkCA = FIN(TIN -T)

=

~

_

j.^)

(11)

fcoexp(^)

if 1 m < H{t) dH dt dAHCA dt dAHT dt dTi dt dT2 dt

+ ^^''(-AH)C. _ u ^ ^ T

(mode 2) A

= F I N ( C ^ IN - CA)

= F,^{T,^ -T)

-

AHWA

+ ^^M-AH)c. _ iM^^j,

_ y^) _ u^^j.

_ j.^^ (12)

= tmiN-T2) + =

fcoexp(^)

^^(T-T2)

Application of a Hybrid Control Approach to Highly Nonlinear Chemical Processes 1

Input Ti IN Output T Liquid level

360

r

'

360

Input Ti IN Output T Liquid level

1519 '

1.01 340

340 1

320

>

320

• '

0.99

300

>

300 280

280

0.98 C

10

20 30 Time[min]

40

5C

Figure 2. Control response of the proposed method.

C

10

20 30 Time[min]

40

50

Figure 3. Control response of the comparative approach.

All parameters in this model are summarized in notation. The MFC controller is constructed based on this model. In controller design, we use the following variables: y = {T}, u = { T U N } , X = {if, C A , T , Ti,T2}. It is assumed that all state variables are observed at every sampling period. 3.2. Results We applied the MIQP formulation represented by Eq. (10) to the problem. Control performance was simulated by using Mathematica 5.1. We changed the flow rate of the reactant (FIN) from 0.0366 m^/min to 0.0426 m^/min to cause mode shifts. In an MFC, we used the following reference trajectory Tref given by Eq. (13). Tref k+j = (1 - a')

T{k)

+ a^ Zet k+j

(13)

The set point Tget is 325 K and the MFC parameters are given as follows: Hy^ — 1, Ht = Hu = 5, Q{j) = 10, R(j) = 0.001, a = 0.8, Control interval = 1 min (see subsection 2.3). Figure 2 shows the control responses. The mode changes occur at time t = IS min (mode 1 -^ mode 2) and t = 41 min (mode 2 —> mode 1). In Figure 3, the control responses of MFC based on the conventional linearized model (Eq. (3)) are shown for comparison. Although the response shows overshoots, this system does provide good control capabilities, meaning an adequate sampling interval can give good control performance in nonlinear hybrid dynamical systems. 4. CONCLUSION We proposed a new control method for nonlinear hybrid systems. Modified successive linearization was used to transform each nonlinear continuous-time system into a linear discrete-time model. Hybrid hnear discrete-time models were then formulated as an MLD model and this model was employed in the MFC scheme. We applied this method to a highly nonlinear CSTR with operation mode changes. Result revealed that the MFC controller with modified linearization performed well in controlling nonlinear hybrid systems.

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Y. Sakakura et al

REFERENCES 1. Bemporad, A. & Morari, M. (1999). Control of systems integrating logic, dynamics, and constraints. Automatical 35^ 407-427 2. Lee, J.H. & Ricker, N.L. (1994). Extended Kalman filter based nonlinear model predictive control, Ind.Eng.Chem.Res., 33, 1530-1541 Mhaskar, P.,E1-Farra, N. H. , & Christofides, P. D. (2005) Predictive Control of Switched Nonlinear Systems with Scheduled Mode Transitions, IEEE Trans. Autom. Contr. , 50, 1670-1680. Shikanai, Y., Yamashita, Y., & Suzuki, M. (2002). Design of a robust model predictive controller. Proceedings of the 4^th conference of the automatic control association, 571-572 (in Japanese) NOTATION Parameters of a case study plant. description reactor temperature liquid level of the reactor concentration of reactant A CA temperature Ti,T2 feed temperature Tl IN feed temperature T2 IN feed ^IN cross-section of the reactor A heat transfer area Ahi heat transfer area Ah2 overall heat transfer coefficient UuU2 c constant specific heat of liquid mixture Op specific heat of heat transfer liquid Cpi,Cp2 density of reactor solution P density of thermal medium Pl^P2 flow late Fi flow late F2 concentration of reactant A (feed) CA IN feed temperature of reactor TIN frequency factor ko activation energy E AH heat of reaction gas constant R temperature of hot water Thot temperature of cold water ^cold setpoint of reactor temperature ^set Subscript 1 Temperature control system Subscript 2 O N / O F F cooling system

Variable ~T H

Value controlled state variable state variable state variable manipulated 278 3.66-4.26 3.14 6.28 3.14 5.11 X 10^ 0.04 3.14 X 10^ 4.18 X 10^ 800.9 997.9 0.5 5 8.24 294.4 1.18 X 10^ 69.8 69.8 8.3145 X 10^ 368 278 325

Unit K m kmol/m^ K K K lQ-2

j^^/^niYi

m^ m^ m^ J / m i n K m^ J/kgK J/kgK kg/m^ kg/m^ m^/min m^/min kmol/m*^ K 1/min MJ/kmol MJ/kmol J / K kmol K K K

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1521

Dynamic optimization of dead-end membrane filtration Bastiaan Blankert, Ben H.L. Betlem and Brian Roffel Faculty of science and technology, University ofTwente, P.O. Box 217, 7500 AE Enschede, The Netherlands 1. Abstract An operating strategy aimed at minimizing the energy consumption during the filtration phase of dead-end membrane filtration has been formulated. A method allowing fast calculation of trajectories is used to allow incorporation in a hierarchical optimization scheme. The optimal trajectory can be approximated closely by constant power filtration, which allows robust implementation of the optimal strategy. Under typical conditions, the relative saving in energy, is small compared to constant flux (0.1%) or constant pressure filtration (4.1%). Keywords: Dynamic optimization, dead-end filtration 2. Introduction Dead-end ultra filtration is applied in the purification of surface water to produce either process water or drinking water. Due to its high selectivity, economic scalability and low chemicals consumption, it is a promising technology in this field. However, the performance of membrane systems is often limited by fouling phenomena. Accumulation of retained particulates at the membrane surface increases the hydraulic resistance of the system. This increases the operating costs due to extra energy consumption and the necessity of periodic cleaning. Hence, dealing with membrane fouling is one of the main challenges in the application of this technology. Since the process settings are currently based on rules of thumb and pilot plant studies, it is believed that optimization will result in a reduction of the operational costs. Dead-end filtration is a cyclic process which consists of three phases. During the filtration phase clean water is produced and the membrane is subject to fouling. This is followed by the backflush phase, in which the flow is reversed in order to remove the fouling. After a number of alternating filtrations and backflushes, chemical cleaning is performed to remove "irreversible" fouling. The evolution of the fouling state during the sequence of filtrations and backwashes is illustrated in the left of fig. 1. This study is concerned with the sequence of alternating filtrations and backflushes. Since filtration and backflushing are fundamentally different, they need to be described by separate models. Both have the flux as control variable and the amount of fouling as state. Since the filtration and backflush phases alternate, the initial state and the cost of the final state are difficult to determine.

B. Blankert et al

1522

Therefore, a hierarchical structure with two layers is used. The top level coordinates the initial and final states of subsequent phases. It searches for a trajectory of initial and final conditions for which the total costs of the series of subsequent phases are minimal. The bottom level is concerned with reaching the final state at minimal costs. This is illustrated in the right of figure 1. A bottom up approach is followed to construct the hierarchical structure, starting with optimization of the filtration phase. This is a dynamic optimization problem which aims to minimize the energy consumption. As it is part of the hierarchical structure, a requirement on the final state and time must be satisfied. Furthermore, each iteration towards the total optimum at the top level involves a dynamic optimization at the bottom level. Hence, the optimal trajectory should be calculated fast. Cycle

Cycle w0,i "^T.!

^F,l

TF,i\

I

^0,1+1

Cs,,

.11

TB

Filtration J^t)

Time (t)

A

4

Backflush JB(t)

'r

T

Figure 1. Left: Semi continuous operation of dead-end filtration consists of consecutive filtration and backflush phases. Right: Hierarchical control structure which corresponds to the cyclic operation.

3. Theory 3.1. Model In dead-end filtration the fouling state (w) is proportional to the filtrated volume. The flux (J) is the control variable, which is also the production rate. The model parameters are given in table 1.

dw ~dt

= J

(1)

The driving force of the filtration process is the trans-membrane pressure, which is related to the flux and the hydraulic resistance of the system by Darcy's law:

AP = rjJRM7F

(2)

in which yp is the relative increase in the resistance due to fouHng, which can be given by (Blankert):

rF = i + — mvO(l + mvO rjfij) RM with O a correction factor for the geometry of the fiber, given by (Belfort):

(3)

Dynamic Optimization ofDead-End Membrane Filtration

1523

0 = - ^ l n l - ^ 2wx

\

(4)

r .

The relative increase in energy consumption due to the pump efficiency can be approximated by (Karasik):

v

f rp =

' /i,. /

(5)

4

Table 1. Model parameters and their values Specific cake resistance Compressibility Cake volume fraction Viscosity Membrane resistance Fiber radius Maximum pump pressure Maximum pump efficiency Flux at maximum efficiency

a

m' Pa^ Pas m-^ m Pa m/s

p X

ri RM

r p •*• m a x

VP,max

J.

1.00x10'' 5.00 X 10-^ 1.00x10-^ 1.01x10-^ 7.00 xlO^' 4.00 X 10-^ 1.33x10^ 0.50 4.16x10-^

3.2. Optimization The energy consumption per unit area is equal to the integral over the power per unit area (JAPyf), which can be given by:

Cp = \{vvp,m^RMrFrp'^^)^t

(6)

For this process the Hamiltonian can be given by:

H{w,

j,X)^Aj+mp,m.ArFrp'^'

(7)

In the Hamiltonian the adjoined state (X) is introduced. The first necessary condition for optimality states that the optimal flux minimizes the Hamiltonian, thus: (

J dYp

— = ^^+vnp,m.,RMrFrpJ\2 + YP V

J dfp ^ • + •

dJ

YF

=0

(8)

^J

This equation allows us to calculate the optimal flux as function of the state and the adjoined state. However, here it is used to eliminate X from eqn. 7. The result is the minimum value of the Hamiltonian as function of the flux and the state. ^

1^

Yp dJ

YF 3 /

(9)

This equation leads to two approaches which are discussed in the following paragraphs.

1524

B. Blankert et al

3.2.1. Simplified system One approach is simplification of the system. The effect of compressibility and pump efficiency are neglected (y5 = 0, j ^ = 1). In that case the right-hand term of eqn. 9 vanishes and the Hamiltonian is proportional to the power. Since the Hamiltonian is constant along an optimal trajectory, constant power filtration is optimal. From this consideration, constant gross power filtration is introduced as alternative for the dynamically optimal trajectory. The main advantage of this approach is that it can be implemented in a robust way. The power, which can be easily measured, can be kept constant by a feedback controller, which is part of a cascade control structure. The master controller uses the setpoint of the power to ensure the final condition (produced volume) is met. 3.2.2. Predefined trajectories The second approach also makes use of the consideration that for optimal trajectories the Hamiltonian is constant. With eqn. 9 the Hamiltonian is calculated for a large number of states and fluxes on a grid. The calculated points (J, w, H) are sorted in a table such that each row corresponds to a value of the state and each column corresponds with a value of the Hamiltonian. Each column contains a trajectory, which is optimal for some final condition. Since the state and the flux are known, the time and costs of each point in a column can be calculated and added to the table. This is done at the moment the model parameters are estimated. Then at each filtration phase, for a given final time and fmal state, the optimization problem is reduced to finding the correct row indices and column index. The row indices follow directly from the initial and final condition for the state. The column index follows fi*om the final time (duration) condition. It is equal to the difference between the initial and final column. The costs can be found in a similar way. 4. Results The optimal trajectories were calculated for model parameters shown in table 1. Fig. 2 shows these trajectories for WQ = Om, WT = 0.0375m and T = 1800s. The common operating strategies, constant flux (flow) and constant trans-membrane pressure (driving force), are shown in the figure as well. It can be seen that the constant gross power trajectory is close to the optimum. The constant flux trajectory is also close. The relative difference in costs between the optimal and suboptimal strategies are shown in table 2.

Table 2. Potential savings of reference strategies Complete model Final time (s) Final state (m) Constant flux Constant pressure Constant gross power

1800 3.75X IQ-^ 0.1 % 4.1 % < 0.1 %

3600 7.50X 10"^ 1.1 % 16.0 % <0.1%

Simplified model 1800 3.75X 10"^ 0.4 % 0.4 % 0

3600 7.50X 10"^ 1.0% 1.0% 0

Dynamic Optimization ofDead-End Membrane Filtration

1525

Constant gross power Optimal

Flux trajectories

500

1000 t[s]

Constant pressure Constant flux

Pressure trajectories

1500

2000

500

1000 t[s]

1500

2000

1500

2000

500

1000 t[s]

1500

2000

Yp trajectories

500

1000 t[s]

Figure 2. Top left: flux trajectories, top right: trans membrane pressure trajectories, bottom left: relative increase in resistance due to fouling, bottom right: relative increase of energy consumption due to pump efficiency.

5. Conclusion Constant flux and constant pressure filtration are equally expensive according to the simplified model. However, w^hen the pump efficiency, compressibility and cake volume are taken into account, constant pressure filtration consumes more energy than constant flux filtration. There is no significant difference (<0.1%) between constant gross pov^er filtration and the optimal trajectory. Hence, constant pov^er filtration can be a robust w^ay to implement the optimization. Under typical conditions, the relative saving in energy, is small compared to constant flux (0.1%) or constant pressure filtration (4.1%). Acknowledgements This v^ork is financially supported by: NWO/STW, Aquacare Europe, Hatenboer-Water, Norit Membrane Technology, Vitens Lab & procestechnology. References G. Belfort, R. H. Davis, Andrew L. Zydney, 1994, The behavior of suspensions and macromolecules in crossflow microfiltration, Joumal of Membrane Science 96, 1-58. I.J. Karasik, W.C. Krutzsch, W.H. Fraser, J.P Messina, 1976, Pump handbook, Mc Graw-Hill inc. B.Blankert, B.H.L. Betlem, B.Roffel, Dynamic optimization of a dead-end filtration trajectory: non-ideal cake filtration. In preparation

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1527

Combined Nonlinear Model Predictive Control and Moving Horizon Estimation for a Copolymerization Process M. Diehl^ P. Kuhl^ H.G. Bock% J.R Schloder^, B. Mahn^ and J. Kallrath^ " ^IWR, University of Heidelberg, INF 368, 69120 Heidelberg, Germany ^BASF AG, 67056 Ludwigshafen, Germany ^University of Florida, Gainesville, FL 32611, USA We implement and test a nonlinear model predictive control (NMPC) setup for first principles models, at the example of a copolymerization process. We address the important question of state and parameter estimation within a moving horizon estimator (MHE) framework. The employed "real-time iteration" algorithm for NMPC and MHE allows computation times 500 times faster than real-time. The MHE is able to estimate both states and unknown parameters well, and the overall scheme allows fast transients in grade changes. Finally, we test an "optimizing" control NMPC scheme which maximizes production while keeping the grade specifications. 1. I N T R O D U C T I O N Nonlinear model predictive control (NMPC) is often proposed for control of polymerization processes with frequent grade changes and strongly nonlinear behaviour and several industrial applications are reported, cf. [2,13]. The present study aims at demonstrating the practical potential of NMPC based on first principles models and their fast numerical solution, cf. [3,15]. We address in particular the important question of state and parameter estimation, by a moving horizon estimator (MHE). Our focus is on numerical methods for NMPC and MHE, and we present an efficient and novel scheme for MHE that is based on the multiple shooting method for state and parameter estimation in dynamic systems [5]. Due to its numerical stability even for seemingly ill-posed problems [10] this method is particularly well suited for MHE. The paper is organized as follows: In Section 2 the process and its model is described, Section 3 introduces the NMPC and Section 4 the MHE setup. Simulation results are presented in Section 5. 2. C O P O L Y M E R I Z A T I O N P R O C E S S A N D M O D E L We consider the copolymerization of two monomers, monomer A and monomer B, to a copolymer. The copolymer shall be produced in prespecified grades which are characterized by the mole fraction of monomer A in the product and the viscosity. Desired copolymer composition and viscosity depend on the market requirements. The process model has been developed by Bindlish in [4], to which we refer for all details concerning the model. The copolymerization takes place in a continuous stirred tank reactor (CSTR) with a downstream separator. As the reaction is exothermal, cooling has to be provided by a cooling jacket. In the feed stream not only monomers A and B are fed into the

1528

M.Diehletal

reactor, but also initiator (I), solvent (S), transfer agent (T), and inhibitor (Z). There are three manipulated variables (MV), the feed rate of monomer B (MVl), the feed rate of transfer agent T (MV2), and the coolant temperature Tj in the coohng jacket (MV3). All manipulated variables are constrained based on experience with an industrial plant. The controlled variables (CV) are the production rate (CVl), copolymer composition (CV2), and copolymer viscosity (CVS). Process model: The ordinary differential equation (ODE) model - that we will abbreviate by ± = f{x^u,p) - has the three MVs as inputs u, and contains 15 differential states x: educt concentrations in the reactor [ca Cb Q, Cg Ct Cz], reactor temperature Tr, copolymer compositions Aa and Ab, the first three moments of the molecular weight distribution of the dead copolymer, ^ Q , ^^, and ^25 ^^^ ^^^ separator concentrations Cas, Cbs, Css- Measurements are available of T^., Aa and Ab, Cas, Cbs, Cgs and of the three controlled quantities, ycv = h{x,u,p). We denote the vector of all measured variables together by ^m = hm{x,u,p). The parameter vector p summarizes parameters that are partly or completely unknown and need to be estimated online, along with the system state. In the numerical experiments shown later, two of the Arrhenius parameters (for polymer chain growth) are assumed to be uncertain. They need to be estimated online by the MHE, together with the system state. We denote estimated state and parameters at a given time ^0 by x{to) and p. The considered sampling time is 5 minutes. 3. R E A L - T I M E N O N L I N E A R M O D E L P R E D I C T I V E C O N T R O L Given the current state and parameter estimates x(to), p of the system at time to, NMPC computes a feedback control by solving an open-loop optimal control problem and applying the first controls to the real process for the duration of one sampling time (see [1] for an introduction to NMPC). The NMPC problem we address is: rto+T

min subject to

/

\\ycYit)-ycY\\l

+ 11^(^)111 dt

^(^^)_^(^^)^

(lb)

x{t) = f{x{t),u{t),p),

(Ic)

ycv{t) = h{x{t),u{t),p) c{x{t),u{t),u{t),p)

> 0,

(la)

(Id) te

[to,to + T].

(le)

Here, the last hue denotes control and path constraints like bounds. In the objective, ycv is the vector of desired reference values for the CV, and Q and R are diagonal weighting matrices, used for tuning the NMPC performance. Note that the MVs u are required to be continuous and that their rates of change u are penalized. This approach provides, under some assumptions on the estimation scheme, integral control [16]. For the stability theory of NMPC we refer to [12]. Real-time solution: The optimization problem (1) is numerically solved by the real-time iteration scheme [7] realized within the software package MUSCOD-II [11], an implementation of the direct multiple shooting method [6]. The real-time iteration scheme exploits the fact that in NMPC optimization a sequence of neighboring optimization problems has to be solved, which differ only by the estimated state x{to) and parameter vector p, and possibly by a change in the setpoint ycv- Solution information of the previous problem

Combined Nonlinear Model Predictive Control and Moving Horizon Estimation

1529

can be exploited for initialization of the following problem by an efficent initial value embedding strategy. This initialization procedure in conjunction with the direct multiple shooting method is so efficient for neighboring problems that it allows to perform only one single optimization iteration per optimization problem, without sacrificing much solution accuracy. For a more detailed description of the real-time iteration scheme and its convergence properties we refer to [7-9]. NMPC setup: The control horizon for the NMPC is chosen as T = 6600 min, which is long enough to come to steady state within all considered transients. The discretization of this horizon is not equidistant. The first intervals are smaller (in the range of minutes) than the last ones (in the range of several hours). The NMPC weighting matrices are Q = diag(10.0 50.0 20.0) (deviations of the CVs from their setpoints are scaled by 0.3, 0.6, and 38000, respectively) and R = diag(0.01 0.1 0.01). While the sampling time for the considered process is 5 minutes, computation of one control response, i.e. solution of each problem (1) within the real-time iteration scheme, took only a fraction of a second on a standard PC. 4. M O V I N G H O R I Z O N S T A T E E S T I M A T I O N In order t o estimate the current system state and parameters, we propose to use a moving horizon estimator (MHE) that is explicitly designed for estimation of both states and parameters. It is based on the measurements y^ at times ti, which are the past M sampling times to-M,---,^o- In contrast to standard MHE approaches, cf. [14], we do not consider state disturbances on the horizon, for which it is difficult to obtain variance estimates in practice. Instead, we regard a deterministic system model with Gaussian measurement noise only, as usual in offline parameter estimation [5]. It is easy to show that such an estimator delivers a maximum-likelihood estimate even for nonlinear models, if no initial regularization weight is used. In the practical MHE problem, however, we add an initial regularization term that can be regarded as a heuristic way to incorporate a-priori information. The MHE problem addressed in each sample is: 0

mm subject to

P - P

+

E \\y'n{ti)-yLrv

i=0-M

i(t) = f{x{t\u{tlpl

(2b)

ym{t) = h^{x{t),u{t),p), a^min < ^ ( ^ ) < :^max, Pmin
< Pmax-

(2a) (2c)

t G [ t o - M , ^o],

(2d) (2e)

The weighting matrix V is the inverse covariance matrix of yl^^ while the matrix P and vectors x and p express a-priori information. They are determined during the transition from one problem to the next by a discrete time extended Kalman filter (EKF) update. Within the EKF, an extra weighting matrix W is used that corresponds to white state noise in each transition. Note that P , x, p summarize past measurement information, which is downweighted by the assumption of state noise in the EKF. Real-time solution: The MHE problem (2) is parameterized by multiple shooting for parameter estimation [5]. The intervals are chosen as [t^,^^+i], and the online solution procedure follows the principle of the real-time iteration scheme, with one major difference:

1530

M Diehl et al CV1: production rate

0.4

iL 0.2 0.1

'^

1r~~^

K

MVI: feed rate monomer B

, \

:

20

I3Z1B

40

140

MV3: coolant temperature

160

380

-

360 g.

340 320

If 20

40

^ 80

100

"> 120

140

160

time [h]

Figure 1. CVs for nominal scenario, and Figure 2. MVs for nominal scenario; dashed desired setpoints (dashed). are the control bounds. during the transition from one problem to the next, a shift in the optimization variables is performed to account for the movement in time due to the fixed past control trajectory u{t). MHE setup: In the practical simulations, the samphng time for the measurements is 5 minutes. The MHE horizon is 20 minutes long and thus comprises five measurement samples, which proved sufficient to recover the states and, additionally, estimate the two reaction kinetic parameters p. The MHE weighting matrix is V = diag(y5s)~^. In the EKF W = diag(xss)~^ is used as weighting matrix for the state noise. The subscript ss denotes the steady state at grade C. 5. N U M E R I C A L S I M U L A T I O N S We present simulation results for a control scenario comprising several production grades: start in grade C (t = 0), setpoint change to grade B {t = 600 min), measured feed temperature disturbance from Tf = 353.0 K to Tf = 357.0 K (t = 1200 min), setpoint change to grade A {t = 3000 min), setpoint change to grade C {t = 6000 min), and back to grade A {t = 8000 min). The process in closed loop has been tested under different conditions: (i) nominal process model and no noise, and (ii) process model with two perturbed initial parameters and noisy measurements. The performance of the closed loop under conditions (i) is shown in Figures 1 and 2. In test (ii), both reaction parameters had been initialized largely wrong, one order of magnitude lower than their true values. They had to be determined by the estimator online, see Figure 5. In addition, measurement errors were introduced, in the form of zero-mean white noise with standard deviation 1% of the steady state value of grade C. The resulting closed loop simulation is shown in Figures 3 and 4 The overall control performance and duration of transients does not deteriorate much compared to the nominal scenario, even though the manipulated variables oscillate considerably. Optimizing NMPC: In a further test, we aim to maximize the production rate (CVl) while maintaining the grade specifications (CV2, CV3). Instead of the pure least squares objective (la) we use the cost rto+T

mm

-M-ycvi + Wvcvit) - ycvWq + Jto

Mt)\\l dt

(3)

Combined Nonlinear Model Predictive Control and Moving Horizon Estimation CV1: production rate 0.4 mw»^*imi»%' •9) ^

0.2 0.1

>

1

1531

MV1: feed rate monomer B

h JSN^_

M

Figure 3. CVs for noisy scenario: measure- Figure 4. MVs for noisy scenario; the ments (light) and estimates (solid). dashed line denotes the control bounds. Controlled Variables ^ 10= Reaction parameter I

Manipulated Variables

— Least Squares 1 - Economic |

r:

1 0-5 — 0.4 1^ 0.3

ta. 0.2

\l D

1000

2000

3000

4000

50

D

1000

2000

3000

4000

50

^

I 340

Figure 5. Estimates (light grey) vs. true Figure 6. Comparison of tracking and ecovalues (solid) for non measured quantities. nomic NMPC within a transient.

in problem (1) with factor /x := 10~^kg/min, and a very low weight within matrix Q on setpoint deviations in ycvi- Such a formulation leads to increased production rates that adapt to the current state and parameter estimate. A comparison of least-squares and "economic" objective is shown for one grade transition in detail in Figure 6. The production rate at both grades is increased by 7-12%, without deterioration of product specifications (CV2, CVS) or transition times. It is interesting to note that for MV2 the lower bound is touched, as the optimal operating point lies at the boundary of the feasible region. The lower bound on MV2 only becomes inactive during the transitions, where the additional degree of freedom is used to achieve a fast transient. Note that the resulting controller at the boundary of the feasible region has one sided gains and is highly nonlinear. Acknowledgements: The first four authors gratefully acknowledge financial support by the DFG under grant BO864/10-1.

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M.Diehletal

REFERENCES 1.

2.

3.

4. 5.

6.

7.

8.

9.

10. 11.

12. 13. 14.

15. 16.

F. Allgower, T. A. Badgwell, J. S. Qin, J. B. Rawlings, and S. J. Wright. Nonlinear Predictive Control and Moving Horizon Estimation - An Introductory Overview. In Paul M. Frank, editor, Advances in Control^ pages 391-449, Springer, 1999. R. D. Bartusiak. NMPC: A platform for optimal control of feed- or product-flexible manufacturing Proc. Assessment and Future Directions of NMPC, Freudenstadt-Lauterbad, Germany, pages 3-14, 2005. L.T. Biegler. Efficient solution of dynamic optimization and NMPC problems. In F. Allgdwer and A. Zheng, editors. Nonlinear Predictive Control, volume 26 of Progress in Systems Theory, pages 219-244, Basel, 2000. Birkhauser. Rahul Bindlish. Modelling and Control of Polymerization Processes. PhD-thesis, University of Wisconsin-Madison, 1999. H.G. Bock. Numerical treatment of inverse problems in chemical reaction kinetics. In K.H. Ebert, P. Deuflhard, and W. Jager, editors. Modelling of Chemical Reaction Systems, volume 18 of Springer Series in Chemical Physics, pages 102-125. Springer, Heidelberg, 1981. H. G. Bock and K.-J. Plitt. A multiple shooting algorithm for direct solution of optimal control problems. In Proceedings of the 9th IF AC World Congress, Budapest. Pergamon Press, 1984. M. Diehl, H.G. Bock, J.P. Schloder, R. Findeisen, Z. Nagy, and F. Allgower. Real-time optimization and nonlinear model predictive control of processes governed by differentialalgebraic equations. J. Proc. Contr., 12(4):577-585, 2002. M. Diehl, H. G. Bock, and J. P. Schloder. A real-time iteration scheme for nonlinear optimization in optimal feedback control. SI AM J. of Control and Optimization, 43(5), pages 1714-1736, 2005. M. Diehl, R. Findeisen, F. Allgower, H. G. Bock, and J. P. Schloder. Nominal stability of real-time iteration scheme for nonlinear model predictive control lEE Proceedings - Control Theory and Applications, 152(3), pages 296-308, 2005. J. Kallrath, H.G. Bock, and J.P. Schloder. Least squares parameter estimation in chaotic differential equations. Celestial Mechanics and Dynamical Astronomy, 56:353-371, 1993. D. B. Leineweber, I. Bauer, H. G. Bock, and J. P. Schloder. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects. Comp. & Chem. Eng., 27, pages 157-166, 2003. L. Magni and R. Scattolini. Stabilizing model predictive control of nonlinear continuous time systems. Annual Reviews in Control, 28, pages 1-11, 2004. S. J. Qin and T. A. Badgewell. A survey of industrial model predictive control technology Contr. Eng. Pract, 11, pages 733-764, 2003. C. V. Rao, J. B. Rawlings, and D. Q. Mayne. Constrained state estimation for nonlinear discrete-time systems: Stability and moving horizon approximations. IEEE Trans. Auto. Cont, 48(2):246-258, 2003. L.O. Santos. Multivariable Predictive Control of Nonlinear Chemical Processes. PhD thesis, Universidade de Coimbra, 2000. A. Zheng. Does nonlinear dynamic matrix control provide integral control? Comp. & Chem. Eng., 23, pages 1753-1756, 2000.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Automatic adjustment of data compression in process information management systems Frank Alsmeyer AixCAPE e.V., Intzestr. 1, 52072 Aachen, Germany Abstract Process information management systems (PIMS) use data compression methods where appropriate parameters must be set for each process variable. In this work, we describe algorithms to adjust these parameters automatically by analyzing raw process data. Our algorithms are based on wavelet analysis of the raw data and perform a conservative estimation of the noise variance. Based on these noise estimates, parameters for the compression algorithms are derived. The possible incorporation of prior information about the signal characteristics, like finite resolution of the AID converter, is discussed. Finally, we describe the validation of the proposed algorithms with real process data from the industrial members of the AixCAPE consortium. Keywords: data historian, compression algorithms, variance estimation, denoising, wavelets. 1. Introduction The compression methods in process information management systems (PIMS) are needed to store the masses of data measured in large-scale production processes. In practice, the quality of the compressed data is often unsatisfactory due to inadequate compression settings. This can lead to significant delays and high cost when production problems occur, because the compression settings must be corrected, and more data must be collected before the problem can be analysed and solved. The adverse effects of overcompression on data analysis have been examined in detail by Thomhill et al. (2004). The reason for inadequate compression settings is the high cost associated with the PIMS configuration, as it is mostly done manually. The piecewise linear compression methods in use today, like the swinging door algorithm (SDA; Bristol, 1990) or the modified Boxcar-Backs lope algorithm (MBBA; Aspentech, 2002) must be parameterized for each process variable separately. As a consequence, the parameters are often left at their default values, typically 1% of the maximum value. This is often larger than the actual noise variance in the data, resulting in loss of information. It has been shown that compression methods based on wavelets are superior to piecewise linear methods (Watson et al., 1998). These methods are adaptive to the signal itself and parameterization is, therefore, less critical than in the piecewise linear case. Unfortunately, these methods are not yet available in PIMS. As a pragmatic solution, we combine in this work the strengths of wavelet methods with the technical possibilities of current PIMS. We describe an algorithm to calculate adequate compression parameters for piecewise linear methods automatically. It is based on the analysis of uncompressed raw data in the wavelet domain and builds upon previous work on trend detection in process data (Flehmig, 2005). In the next section, we discuss possible configuration strategies by looking at the full signal path from sensor to data archive. Section 3 describes the algorithms used and

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discusses proper choice of wavelet bases and thresholding strategies. Finally, we validate the approach using real plant data from two commercial PIMS that implement the piecewise linear compression algorithms SDA and MBBA.

2. Strategy for automatic compression tuning Generally, we suggest using a conservative compression strategy as missing data can never be reconstructed again, and storage costs are constantly decreasing. To derive such a strategy, it is usefiil to consider the signal path from sensor to archive, as shown in Fig. 1. The electrical signal (1) at the sensor, usually linearized, is digitized in an analog/digital converter (2), resulting in a signal of finite resolution. It is not uncommon that this resolution, often not more than 12 or 16 bit, can be clearly seen as steps ("quantization") of the recorded signal (cf Figure 2a). The calibration model (3) that converts from the electrical to the underlying physical property is often linear, as in the case of a PT-100 temperature sensor, or only slightly nonlinear. The next two data transformations (4) and (5) depend on the PIMS at hand and on its configuration. Step (4) is used either as a preprocessing step to reduce the computational load of (5) by letting pass only those new values yi that differ significantly from the previous value yi.i (|yi - Yi-il < devpre)', or it is used alternatively to the piecewise linear compression step (5), which is, in this paper, either SDA or MBBA. These linear compression algorithms have in common that they keep only a subset of the original values. These values are chosen such that all discarded values lie within a specified band around the linear segments connecting the kept values. The acceptable deviation from the recorded linear segments dev is the essential compression parameter and the one that we derive below from the analysis of raw data alone. By raw data we mean the data entering step (4), because they are centrally accessible via the control system or the PIMS itself The idea is to analyze the data in an external configuration tool called Alanda, which then suggests compression parameters for steps (4) and (5). PIMS 1. linearized signal at sensor

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Figure 1: Signal path and interaction between PIMS and the configuration tool Alanda For the subsequent discussion, let A be the smallest step change entering (4), i.e. the visible data resolution, and a the standard deviation of the signal noise. We can always determine A, but a may be unaccessible if the resolution is too coarse. Figure 2 shows the effect of data resolution on the raw data. Both examples are fi'om real plants; details are not given for reasons of confidentiality. In Figure 2a, resolution dominates over process noise (o « A), creating a block-like structure when the measured data points are connected. The reason is that there are steady state periods where the measurement assumes one out of two or three adjacent values determined by the data resolution. Figure 2b shows a situation where process noise and resolution are

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comparable (a ~ A). Both situations may cause difficulties when a is to be determined from raw data. This is why we distinguish two types of data: Resolution and noise dominated data. The distinction is made using a heuristic: If more than half of the samples has the same value as its predecessor, a data set is considered resolution dominated. Resolution dominated data show a behavior as in Figure 2a or 2b. Wavelet denoising methods for noise estimation may not work as expected, but the resolution is a good starting point to suggest a reasonable compression parameter dev. Typically, one will avoid recording adjacent points within block-like steady states, i.e. dev should be larger than A, e.g. dev = 1.5A. We often observed block-like steady states with a greater width and, therefore, suggest dev = 2.5A. Note that integer multiples of A are not a good idea because unwanted samples may still be kept due to finite precision arithmetics. For noise dominated data, we can obtain a good estimate of a. How should dev be chosen in this case? Assuming a noisy steady state, where ideally we would want to record only the first and last sample, we can use standard statistical reasoning. We suggest dev = 2c which means that 95.5% of the samples will be discarded if we have white noise. The reasoning above applies to the piecewise linear compression step (5). For the precompression step (4), a reasonable parameterization is devpre = A/2. This discards only unchanged values and leaves the handling of the rest to linear compression (5) that is more effective.

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Figure 2: Influence of data resolution: a) resolution is dominating, block-like structure b) limiting case where noise level is similar to resolution

3. Algorithms and implementation The following algorithms have initially been implemented in Matlab and later made available in a standalone configuration tool called ALANDA. 3.1. Estimation of data resolution A The following algorithm is used to find the resolution A for a data vector yO of length m=l..M: 1. Sort the entries in yO (increasing values), resulting in y 1. 2. Remove the minimum and maximum entries in yl to avoid problems with data on bounds, resulting in y2 with length M2 < M. 3. Calculate the vector dy of absolute differences of y2, of length M2-1. 4. Remove all zero entries in dy, resulting in a vector dyl. 5. Use the minimum entry of dyl as A.

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Note that this algorithm will not work if there are not enough changes in the data. This can happen for small data sets or in the case of setpoint values. Our implementation checks if the data are adequate. 3.2. Estimation of standard deviation a using wavelets Wavelets are families of mathematical basis functions that are non-zero only on a finite interval. They have the ability to capture the essence of a signal y(t) in a small number of coefficients (Strang and Nguyen, 1996), by summing over translated and dilated basis functions. A signal y(t) is decomposed into

k

j=l

k

where Cj,k are approximation coefficients, dj,k detail coefficients, Oj k are the scaling and ^j,k the wavelet functions. J is the number of dyadic scales and j the scale index, and k is the translation index. For signal denoising or data compression, coefficients below a threshold t are set to zero. In practice, only very few non-zero coefficients are needed to achieve a good approximation of the original signal, which makes wavelets useful for data compression. A reasonable threshold can be inferred from the signal itself (Donoho and Johnstone, 1995) - called the Waveshrink method. There is one parameter - the decomposition level J - that must be adapted to the signal characteristics. A useful value is half of the maximum possible depth, yielding good results in practice (Nounou and Bakshi, 1999). Tona et al. (2005) have proposed an alternative method called Levashrink where J is also adapted to the signal itself In this work, we are using the biorthogonal spline wavelet of order 3.3 that has proven useful in previous applications on a wide range of industrial data sets (Alsmeyer, 2005). 4. Validation and Results 4.1. Implementation of original compression algorithms The original compression algorithms as implemented in the PIMS are needed to test the effectiveness of the tuning algorithms developed in this work. The algorithms were implemented in Matlab using the available documentation from the PIMS vendors. Then we used pairs of uncompressed snapshot data from the PIMS and the corresponding compressed data from the PIMS archive to validate our implementation. 4.1.1. Modified Boxcar-Backslope algorithm The Modified Boxcar-Backslope algorithm (MBBA) is described in the data base manual of InfoPlus.21 (AspenTech, 2002). The description is somewhat ambiguous, but we could reproduce the IP compression results after we introduced rounding of the time elapsed to the nearest fiill second. 4.1.2. Swinging Door algorithm The Swinging Door algorithm (SDA) is described in OSI's PI server reference manual (OSISoft, 2003). Note that the SDA here differs from the one in the original publication (Bristol, 1990). In the original SDA, a newly arriving point is not required to lie in the center of the parallelogram spanning the region of values that can be discarded. The algorithm in the PI system is, therefore, less efficient than the original one. In the PI system, there is an additional boxcar compression that is applied before the actual SDA (step 4 in Figure 1). We have implemented both compression steps.

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4.2. Validation with plant data We have used various pairs of compressed and uncompressed data sets from real plants to validate our approach for compression parameter adjustment. The data sets included all relevant types of measurement like temperature, pressure, flow, or level, as well as different categories like actual values, setpoints, and controlled variables. Below are two examples how existing compression settings can be corrected using our approach. Figure 3 shows a noise-dominated temperature measurement that is overcompressed when the current settings are used, with a mean absolute deviation MAD=1,24K The transient phenomenon after the step change on the left of the Figure is not well reproduced. With the corrected compression settings, the oscillations appear more naturally (MAD=0,28X), and it seems possible to use the reconstructed signal for tasks like controller performance assessment. The compression factor CF, which is the number of samples in the uncompressed signal divided by the number of retained samples, is 18 vs. 56 in the current scheme. 170 c 165

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Figure 3: Overcompression in a temperature measurement and automatic correction Figure 4 shows the reverse case of undercompression in a resolution-dominated pressure measurement. The current compression captures oscillations in blocks and in fact does not compresses at all (CF=1, MAD=OmZ?ar). After correction, the compression factor raises to CF=7.5, using less space for adequate reproduction with a small MAD=6,09wZ?ar. Generally, our approach yields compression settings that generate a reasonable signal when inspected carefiilly. In the data sets we examined, overcompression seemed to be somewhat more common than undercompression. In some cases, after correction, too many of the samples are retained, but this is by design: We definitely want to avoid overcompression as it is detrimental to data quality.

5. Conclusions We have developed an algorithm for automatic adjustment of compression parameters in process data archives (PIMS), based on the analysis of data resolution and noise in raw process measurements. We have validated this approach with extensive compressed and uncompressed data sets from real plants. The application of the automatic algorithm improves data quality from PIMS. In some cases the method has a tendency to

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undercompress, because it has been designed to yield conservative settings. This is not seen as a problem as storage costs will continue to decrease. Nevertheless, we are hoping to further improve the algorithms by incorporation of prior information about the process variables (e.g. setpoints). This is under current investigation.

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References Alsmeyer, F., 2005, Trend-based Treatment of Process Data: Application to Practical Problems, 7* World congress of chemical engineering, Glasgow, Scotland AspenTech, 2002, InfoPlus.2FM 5.0 Database User's Manual Bristol, E.H. 1990, Swinging door trending: Adaptive trend recording, in: ISA National Conference Proceedings, pp. 749-753 Cao, S. and R. R. PUiinehart, 1995, An efficient method for on-line identification of steady state, J. Proc. Cont., 5(6), pp. 363-374 Donoho, D. L. and I. M. Johnstone, 1995, J. Am. Stat. Assoc, 90, pp. 1200 Flehmig, F., 2005, Automatische Erkennung von Trends in Prozessgroi3en (automatic detection of trends in process quantities), PhD Thesis, RWTH Aachen Nounou, M. N. and B. R. Bakshi, 1999, AIChE J., 45(5), pp. 1041 OSISoft, Inc., 2003, PI Server Reference Guide, Version 3.4 Strang, G. and T. Nguyen, 1996, Wavelets and filter banks, Wellesley-Cambridge Press, Wellesney, MA, USA Thomhill, N. F., M. A. A. S. Choudhury and S. L. Shah, 2004, The impact of compression on data-driven process analyses, J. Proc. Cont., 14, pp. 389-398 Tona, R. V., A. Espu~na and L. Puigianer, 2005, Improving of wavelets filtering approaches. In: Proc. European Symposium on Computer Aided Process Engineering - 15 Watson, M. J., A. Liakopoulos, D. Brzakovic and C. Georgakis, 1998, A practical assessment of process data compression techniques, Ind. Eng. Chem. Res. 37, pp. 267-274

Acknowledgements Funding of this work by the AixCAPE member companies BASF, Bayer Technology Services and Degussa is gratefully appreciated.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Virtual Plant, New Paradigm for Future Production Management Hossam A.Gabbar, Kimitoshi Nishiyama, Ikeda Shingo, Teruo Ooto, Kazuhiko Suzuki Graduate School of Natural Science & Technology, Division of Industrial Innovation Sciences, Okayama University, 3-1-1 Tsushima-Naka, 700-853OOkayama, Japan Abstract Operator is a key player in plant operation. However, still operator working environment is limited to traditional interfaces and monitoring systems, which include actual plant, sensors, alarms, and other process and operation condition monitoring systems. Providing operator with virtual environment that integrates plant and process conditions in actual and virtual modes will support operator decisions in normal and abnormal situations. This research work discusses current limitations and proposes future generation virtual plant environment that enables operator to comprehend current plant and process condition and predict future states for safe and optimum operation. To achieve such target, process modeling is proposed to analyze operator activities in view of process design and operation practices. Keywords: virtual plant, plant operation environment, operator activity modeling. 1. Introduction Plant operation is a complex process where there are critical decisions and activities that need to be taken in timely and highly accurate manner. The system approach is widely used to improve plant operation starting from the concept, design, and operation engineering till the control and management stages. However, still there are major limitations in operator environment that requires further investigations and considerations. Chemical plant operation requires complete understanding and management of activities related to process chemistry, environmental impacts, product and production systems, etc. In addition, plant operation requires carrying out management activities related to maintenance, production scheduling, operation planning, human resources management, and financial matters. In such complex domain, operator requires extensive information in different forms to be able to perform the different tasks accurately and easily. Till to date, there are different solutions that have been proposed to address these needs, such as intelligent and integrated software systems, i.e. ERP, or by enabling new technologies such as multimedia and virtual realities. In current practices, operator is carrying out most of these activities efficiently using some automated tools via intelligent user interfaces, which provide operator with the required information. However, still accidents are happening due to operator errors, especially within the control room domain. In addition, there is always delay in production schedules, which causes costs and affects business reputation. In addition, there are other critical factors, such as those related to environment, recycling, energy management, and occupational safety, which are not supported by current operation management and interface systems.

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Process system modeling and engineering approaches are widely used to provide practical solutions that link process engineering practices with software / hardware, and technologies. Such system engineering approaches are based on process modeling where elements of plant operation are modeled in an integrated view so that operator environment can be reengineered and realized in practical manner, while establishing the links among different elements of operational environment. This research paper discusses current limitations of operational environment and their impact on plant safety. From such discussions, a proposed virtual plant framework, design, and its realization are illustrated, and system architecture of the proposed virtual plant environment is illustrated. In the following section, issues around current operational environment will be discussed. In section three, system approach is proposed to design target virtual environment, which is used to describe proposed virtual plant, as illustrated in section 4. 2. Current Operational Environment Currently, operator is carrying out different activities such as process monitoring, operation execution, maintenance tasks, and safety assessment activities. In addition, he is carrying other activities such as operation scheduling, planning, human resources management, scheduling, procurement, and financial. In such multidimensional environment, he is required to do job accurately, timely, optimally, and safely. Moreover, there is another constraint of increasing complexity of chemical plants. Most of these activities are carried out concurrently by operator with direct communication with operation engineers and management. In most of chemical and oil & gas production plants, operator works in control room and directly with equipment interaction in the plant process area, i.e. shop floor. In the control room, there are set of computers, DCS, monitors and other monitoring devices that reflect plant process condition. Usually, operator spend most of the time with DCS and HIS or human interface systems to monitor plant condition in terms of sensors and alarm. In most of production plants, plant and process condition monitoring systems are used to reflect real time condition for better operation decision and management. And for operation execution and monitoring, SFC, or sequential fiinction chart, is used to show the current step of plant operation. Operator needs to make mental work to link DCS values with SFC. In addition, operator carry out operation planning and scheduling activities in separate PC with a separate monitor based on received instructions of production planning and quality. From the incoming operation schedules, operation execution and control setup points will be defined and executed. Such switch among different computation environment and monitors might lead to errors, delay, and fiiistration. In addition, it might lead to exerting more efforts which has negative impact on physical and mental operator limits. In addition to these requirements, operator needs to carry out safety and environmental assessment regularly where some paper work is done on P&ID and engineering diagrams to write down list of possible hazards and causes, consequences, and safeguards for safe operation. Moreover, factors from waste and water treatment and energy management are considered as part of plant operation, which will ensure clean and cheap energy and less impact on environment. This caused additional work to be carried out by operator to

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reduce environmental impacts from energy utilization and water and waste treatment. To manage these activities, i.e. water, waste, and energy management, the different automated systems are usually executed in different workstations and interface systems. These limitations can be eliminated by using sophisticated virtual plant environment that augment the different monitoring, control, and interface systems. The proposed approach to achieve such virtual plant is via system approach, which is explained in the following section. 3. Process System Modeling Approach to Design Virtual Plant In view of current limitations of operator interface and support environment, this paper proposes a robust process system modeling approach that links the different automated systems, their information models, and operator activities and tasks, as shown in figure 1. The proposed process system approach is based on defining flowchart for the different activities that operator performs. These activities are classified in terms of process operation modes such as startup, shutdown, recovery, etc. In each mode, more detailed activities are developed such as in case of detected alarms or process deviations. In figure 1, operator activities are defined for a case when fault or abnormal situation is detected. In such case, operator validates sensor data, simulation data, operation history, and use incident / accident data to confirm possible and root causes and consequences. FDS or fault diagnosis system is proposed by Gabbar (2006) to do such task and provide list of possible fault propagation scenarios. For each activity, list of tasks are identified and associated information elements are listed along with the reference systems that owns / maintains such information elements, i.e. alarm data from DCS. Providing such list will enable the reengineering of operator interface systems and engineering of virtual plant environment with suitable visualization and interaction mechanism, such as 2D / 3D diagrams, voice, animation, textual, video, images, etc. In addition, human factors are analyzed for each task to provide suitable visualization and interaction mechanism. Human factors are analyzed using predefined list of main and detailed factors that are related to operation task, including physical and mental models (Jamieson, 2005). These factors are considered in each task / activity in process model diagrams, which provides input specification for virtual plant and operator interface system design. 4. Proposed Virtual Plant Virtual plant has set of key features that are proposed in view of system requirement and the reported limitations of current operator interface systems. One key feature is the synchronization between actual and simulation models where operator is required to perform real time operation by evaluating different operation scenarios using dynamic simulation. To overcome this requirement, integrated simulation environment is proposed where real time integration module is developed using VB. In this experiment dynamic simulator software is used to capture real time DCS data, which is stored in CSV or excel format and transferred every 20 seconds from DCS to simulator workstation. Such sensor data are compared with simulation data and used to tune simulation models and propose changes to operation control and execution. Detailed about this mechanism is explained in separate research report. Another feature of the target virtual plant environment is the use of latest technologies such as GIS or geographical information system, to capture and visualize plant and process condition in the corresponding geographical locations. Operator will be able to view high temperature locations in 2D / 3D in physical plant location for faster response and

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counteraction. In addition, CFD or computational fluid dynamics software is proposed to be integrated within virtual plant to visualize the inside of process equipments and facility locations. Task

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For example, gas distribution / emission can be visualized and simulated in case of leak in tank with hazardous materials. Synchronization between actual plant and simulation and CFD will help operator to have better understanding of what is going on inside tanks / facility locations. This will provide safer and faster response in case of emergency and recovery. One key feature of the proposed virtual plant is to provide operator with complete picture about plant and process conditions in terms of smell, sound, view, and textual data. This can be achieved by reflecting different sensor data to operator in suitable formal such as sound, image, video, and textual format.

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Figure 2. Technological Infrastructure Design for Integrated Virtual Plant Environment The main architecture of the target virtual plant is presented, which include integration with current DCS / PLC control systems to understand and interactively exchange operational data. Computer-aided modeling environment, or CAPE-ModE (Gabbar, 2002), is proposed as a hierarchical process modeling environment, which reflects process real time information in each model block / element for different hierarchical levels: process level (micro), production and supply chain level (macro). Operator will be able to visualize process data / domain knowledge such as accidents, risk, operation path, fault propagation scenario, etc. Operator can use different input devices such as keyboard, mouse, voice, etc. and different output media such as voice, image, video, text, etc. Operator will be able to switch among different modules and systems by simply switching among the different windows and interact with more than one system at a time. Using smart visualization controller will enable operator to perform smooth interaction in real time with different display windows. The proposed virtual plant is actually testing in experimental room in Okayama University using experimental plant that has been constructed for that purpose. DCS from Yokogawa (Centum 3000 small) is connected to the experimental plant and real interface system is tested from Hibino visualization provider. Evaluation version of CFD is used to reflect image of inside tank of the experimental process. Figure 3 shows system architecture

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where DCS interface modules are proposed to link real time data with different operational systems such as energy management, waste management, environmental assessment, fault diagnosis, ERP, etc. Windows from these systems are displayed within operator interface to enable operator to flexibly interact with the different system in real time basis for safer and efficient plant operation.

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Figure 3. Integrated Engineering Systems with Virtual Plant 5. Conclusion Operator is a key person in plant operation where he can ensure effective plant operation. However, current operator environments are suffering from major limitations in terms of smooth interaction with operator interfaces, comprehending plant / process conditions, and evaluating operation scenarios prior to execution. This paper presented new concept of virtual plant, which is used to enable operator to carry out daily work in easier and accurate manner. Process system modeling approach is proposed to analyze operator activities in view of process systems. Integrated technological infrastructure is proposed for effective operator interface system where human factors are considered in the design of each visualization mechanism and technology selection. The proposed virtual plant environment opened the door for new operation environment where actual and virtual plant can be synchronized for efficient real time plant operation.

References G.A. Jamieson, & K.J. Vicente (2005). Designing effective human-automation-plant interfaces: A control-theoretic perspective. Human Factors, vol. 47, pp. 12-34, 2005. H.A. Gabbar, 2006, FDS: Fault Diagnostic System, Technical Notes, Okayama University. http://syslab2.mech.okayama-u.ac.jp/staff^gabbar/fds.html. H.A. Gabbar, Y. Shimada, K. Suzuki, 2002, Computer-Aided Plant Enterprise Modeling Environment (CAPE-ModE) - Design Initiatives, Computers in Industry, 47 (2002), 25-37.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Adaptive monitoring statistics based on state space updating using canonical variate analysis Changkyu Lee,^ Sang Wook Choi,^ In-Beum Lee^ ^Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Hyoja-dong, Pohang, 790-784, Republic of Korea ^School of Environmental Science and Engineering, Pohang University of Science and Technology, San 31 Hyoja-dong, Pohang, 790-784, Republic of Korea Abstract Recently monitoring techniques using canonical variate analysis (CVA) based state space model are proposed by several researchers. They discussed that canonical variate analysis based approach is superior to dynamic principal component analysis (PCA) based that for fault detection and identification. To apply canonical variate analysis base approach to time-varying systems, we proposed adaptive monitoring method via CVA based state space model updating. Forgetting factor is employed to update current mean vector and correlation matrix and current state space model is recursively estimated using Cholesky factor updating scheme. Two state space model based monitoring indices are proposed to detect process abnormalities. One is state monitoring index and the other noise process monitoring index. They are formulated using state prediction matrix and noise extraction matrix which are derived from current state space model and the statistics of them are also recursively determined using the ;^ distribution. To adjust forgetting factors according to variation of process state, the forgetting factor updating criterions are introduced. The proposed algorithm is applied to the continuous stirred tank reactor under operation condition change. Application results provide the expectation that the proposed algorithm can be applied to practical time-varying or transient processes. Keywords: adaptive monitoring, state space model updating, canonical variate analysis 1. Introduction Multivariate statistical approaches, PCA and partial least squares (PLS), etc, have been developed to detect process abnormalities and identify the reasons for those in modem chemical processes. Conventional multivariate statistical approaches have been modified to be applied to these various types of processes, e.g. dynamic and nonlinear processes. This paper discusses on a monitoring scheme for a transient process and/or a time-vary system as an alternative modified approach. First the canonical variate (CV) based state space modeling is introduced and statistics of state space model based monitoring indices are discussed. And then the recursive modeling scheme is proposed using mean and correlation matrix updating. The recursive forms of state estimator and noise extractor are formulated using the recursive Cholesky factor updating. For more efficient updating, we employ the criterion of forgetting factor updating.

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2. CVA based monitoring 2.1. CVA based state space model A general state space model of vector autoregressive moving average (VARMA) process is gives as (Akio, 1990) x , + i = A x , + w „ w, = B v ,

^^^

where x^ e 9t^ is state vector; y^ E 9t^ observation vector; W^ stochastic disturbance; V^ measurement noise; A state matrix; C output matrix. The sub-matrices, A and C, can be calculated from the least square solutions. T

A

E(x,x/r

T

C

(2)

The CV which is a candidate of state vector is proposed by Akaike (1974). The estimation matrix of CV is calculated from the generalized singular value decomposition (SVD) of normalized Hankel matrix. Before the explanation for CV, the following stacked vectors of observations can be defined as

-

P, = [YM"^ f,-[y/

y,jr,

•• • y,,,_/f,

p, = [ y / •

E(p,p,') = 'L (3)

^(f,f/) = 5:^

• y,7r, Eip,p,') = t PP

where p^, f^, and p^ denote mean centered and scaled past, friture, and expanded past stacked vector of observations and L

» 21^ ? ^^^ ^

denote the correlation. And /

denotes the number of time lag or lead. The normalized Hankel matrix is factorized as follows.

E^-'-'i:^^!:^/'-' = u^s^v^^ - u^s^v^^

(4)

where 2 . denotes a joint correlation matrix defined as L . =£'(f^p/) • Using decomposed matrix, the CV as state vector can be calculated as x,«J,p,=vV2:,/-'p,

(5)

As alternative approach, the state vector is calculated using Cholesky factor of E

and

^#-

R/^E^R,/=u^s^v^^ x,«j,p,=\'\;'p, where R ^ and R

denote the Cholesky factors of L ^ and E

(6) , respectively, and

J^ means the ]i^ order state prediction matrix. £'(x^x^ ) in Eq.(2) is an identity matrix. Thus, using Eq.(6), Eq.(2) is represented as follows;

Adaptive Monitoring Statistics

1547

J;t^(P,+iP/)j/

(7)

£(y,p/)j; Using Eqs.(l) and (6), the total noise process of Eq.(l) is extracted as [J, e, =

V,

0]-[0 [I

[J,

AJJP, = a p , .

CJ,]

0]-[0 [I

AJJ-

(8)

CJJ

where I e W^'' and 0 € SR*^'' are the identity matrix and zero matrix, respectively, and E means the noise extraction matrix. From CV based state space model, two monitoring indices for state and noise of process can be formulated as follows;

n{n - k)

(9)

n{n -{k + p)) where S^^ = £"(6^6/) = S£'(p^p/)S^ and n andp denote the number of training samples and the dimension of the observations. 3. Recursive state space model based on CVA 3.1. mean, covariance, and correlation matrix updating Various adaptive monitoring schemes have been discussed by formulating the recursive structures of the mean, covariance, and correlation matrix. This paper employs exponential weighted moving average as a recursive form. (Wold, 1994; Liu et al., 2003 ) Recursive update forms of mean vector at time point t is represented as z^ = (1 - a^ )z^ + cr^z^_i

(10)

where Z^ and z^ denote the mean vector and a measured vector at time point t, respectively. And OC^ is a forgetting factor at time point t. Also, standard deviation to calculate the correlation matrix is updated as follows. D,=(l-;5j.diag{z,z/)+y3,D,-i

(11)

where D^ is a diagonal matrix whose elements are standard deviations a time point t, z^ =z^—z^, and /3^ is a forgetting factor to update the standard deviations. Using Eqs.(lO) and (11), the current correlation matrix can be recursively calculated as 2:, = ( I - A ) D ; " ^ Z , Z > - - + A 2 : M

(12)

where 21 ^ is a correlation matrix at time point t. 3.2. State prediction matrix and state space model updating Section 2 shows that CVA based state space model can be obtained using Cholesky factors of future, past, and expanded past correlation matrices. Cholesky factor updating

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C. Lee et al.

for a new measured vector has been developed by Pan and Plemmons. (1989) Recursive Cholesky and inverse Cholesky factor updating with the forgetting factor are given. (13) where (14) '= ;

t^J

'^M-./^M

a,=(l-ArA

R,-, x , = [ a „

«,.,

^,J

and ^t,o=l ^t,s=^^^^t,i+"'^t,s^ {s = l-",h) Using Eq.(13) and (14), the recursive form of normalized Hankel matrix can be obtained as follows. (15)

By applying SVD to Eq.(15), the current state prediction matrix, J^ ^, can be calculated. And the sub-matrices at time point t can be expressed as follows. A.

(16)

= E\ '^ppj

where

'^pp,t*'k,t

R =\W^'\ R' ,l = rR'*', R^^'^'l PP

*- PP,t

pp,t-i

I- pp,t

pp,t

and

R' „ R'*' e 91'^'*"^'''

J

pp,t^

PP,t ^

and

^*

^yit) ^yit-i) (^p{i+\yp Current sub-matrices, A^ and C^, can be recursively obtained pp,t^

pp,t

using current Cholesky factor of the correlation matrix of expanded past stacked vector. 3.3. Forgetting factor updating To update state prediction matrix and state submatrices effectively, updating criterion of forgetting factors, a^ and ^^ in Eqs.(10)-(12) can be considered as follows. (Choi et al., 2005) ^. = ^ . a x - K a x - ^ t n J [ l - e x p K ( | | Am,_, ||/|| Am„,, II)"}]

(17)

A = > f f t n a x - ( y » t n a x - > » n . i n ) [ l - e X p K ( | | A M , _ , ||/|| A M „ ^ , | | ) " } ]

(18)

where a^^, P^^^, a^^^, and /?^j^ are the maximum and minimum forgetting values, respectively, ^ and n are fiinction parameters. ||Am|| is the Euclidean vector norm of the difference between two consecutive mean vectors and ||AM|| is the Euclidean matrix norm of the difference between two consecutive correlation matrices. Here, ||Am„^J and || AM„^^ || are the averaged ||Ain|| and || AM || obtained by using historical data. Six variables in Eqs.(17) and (18) are user-defined values. (The default values of the function parameters are 6iinax=Aiax=0.99, ci;nin=Aiin=0.90, ^ 0.6931, and n=\.) The guideline of the user-defined values are introduced by Choi et al. (2005)

Adaptive Monitoring Statistics

1549

4. Adaptive monitoring statistics using CVA For a stationary process, multivariate variables which normally distributed follow the Hotelling's 7^ statistics and these depend on the degree of freedom related with sample number of training data. Under the assumption of multivariate normality of the observations, the 100(l-a)% control limit for Hotelling's 7^ is calculated by means of a 2^ approximation, i.e. Ty^ = zl(p)' which is generally used in quality control because of its simplicity. (Hall et al., 2000) Thus, adaptive multivariate statistics can be approximate as r/=x/x,-p/j,,/j,,,p,~j^(*,)

(19)

e, = p/s/2:,,_,"'S, p, ~zl(k, + P) where kt denotes current state order at sample time t and S^ indicates updated noise extraction matrix at sample time t defined as follows;

[J,,, 0]-[0 AA,J" [I QJ,,J

(20)

5. Application results The proposed is applied to the continuous stirred tank reactor (CSTR) simulator. (Choi et al., 2003) The Measured (Monitored) variables are 7;,, T,, C, C„ F„ Fa, F^, Q, and T in referred paper. Two thousand samples are generated as normal observations. First twelve hundred samples are obtained under stationary operating condition and the others are obtained after a set point of outlet temperature from 368K to 373K is changed. The monitoring results are shown in Fig. 1. (a)

CVA

c^«

" *

J 10=

fp|iip#i|iiji^^ 1500 Samples

Figure 1. Monitoring results, (a) r / (CVA), (b)r^/ (CVA), (c)r/ (Adaptive CVA), and (d) 7;./ (Adaptive CVA) In this simulation study, the number of time lag (or lead), /, retains 2 and the state order, k, is selected as that when the cumulative variance attains a value larger than 90%. The static CVA based model judges abnormal operating condition after the process change occurs (Figs. 1(a) and (b)), whereas the adaptive CVA based model adapts the

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H o t e l l i n g ' s T^s a n d their control limits, m a k i n g t h e current m o d e l valid. T h e update information for adaptive m o d e l is s h o w n in Fig.2. (c)

|¥MiVv<^'»HM>Wiv^>^

W 2000

2000

-T 1500

2000

Figure 2. Informations of Adaptive CVA (a) actual values and estimated mean of coolant flowrate, (b) actual value and estiamted mean of outlet temperature, (c)forgetting factor {oCt and fi^, and (d) state order Conclusively, the proposed adaptive approach is expected to be applied to transient processes or time-varying processes whose operating condition change could be predicted. As a future work, the criterion which distinguishes the process abnormalities and process transient state under normal condition is developed.

References M. Aoki, 1990, State Space Modeling of Time Series, 2^^ edition. Springer-Verlag, Berlin Heidelberg New York H. Akaike, 1974, Markovian representation of stochastic processes and its application to the analysis of autoregressive moving average processes. Ann. Inst. Statist. Math., 26, 363-387 S. Wold, 1994, Exponentially weighted moving principal components analysis and projections to latent structures, Chemometrics and Intelligent Laboratory Systems, 23, 149-161 X. Liu, T. Chen, S.M. Thornton, 2003, Eigenspace updating for non-stationary process and its appUcation to face recognition. Pattern Recognition, 36, 1945-1959 C.-T. Pan, R.J. Plemmons, 1989, Least squares modifications with inverse factorizations Parallel implications. Journal of Computational and Applied Mathematics, 27, 109-127 S. W. Choi, E. B. Martin, A. J. Morris, I.-B. Lee, 2005, Adaptive Multivariate Statistical Process Control for Monitoring Time-varying Processes, Journal of Process Control, accepted. P. Hall, D. Marshall, R. Martin, 2000, Merging and splitting Eigenspace models, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 1042-1049. S. W. Choi, C. K. Yoo, I.-B. Lee, 2003, Overall statistical monitoring of static and dynamic patterns. Industrial & Engineering Chemistry Research , 42, 108-117

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Scheduling of make and pack plants: a case study Cord-Ulrich Fundeling^, Norbert Trautmann^ ^Institute for Economic Theory and Operations Research, University ofKarlsruhe Kaiserstrafie 12, 76128 Karlsruhe, Germany ^Department of Business Administration, University of Bern Engehaldenstrafie 4, 3012 Bern, Switzerland Abstract We consider a case study published by Honkomp et al. (2000) that refers to the scheduling of a make and pack plant. Due to the problem size, the approaches known from literature for this kind of scheduling problem seem not to be capable of computing a feasible solution. We therefore present a new priority rule method where operations are scheduled successively taking into account the technological constraints. The order of the operations is determined by priority values assigned to groups of operations. Using this approach we are able to compute a feasible production schedule for the case study within less than one second of CPU time. To the best of our knowledge, no solution to this case study has been reported before. Keywords: case study, make and pack plant, scheduling, priority-rule based solution method 1. Introduction In the case study published by Honkomp et al. (2000) a make and pack plant is considered. A make and pack plant is a two-stage production plant where the two stages are linked by intermediate storage tanks with limited capacities. On the first stage (make stage), the production process takes place whereas on the second stage (pack stage), the products are packed into consumer packages for shipping. To produce and to package products, different operations have to be performed for which a set of parallel processing units (mixing tanks and packing lines) is available on each stage. Several technological constraints have to be respected: After the production process products have to be stored at least for a given quality release time. Processing units as well as the intermediate storage tanks have to be cleaned between the processing of different operations where the cleaning time depends on the products involved. In the case study, demand data of 10 weeks is given for the different combinations of products and consumer packages leading to a total of more than 650 operations per week to be scheduled. The problem consists in determining a production schedule for each week such that the demand is met, the technological constraints are respected, and the makespan is minimized. In literature, numerous models and solution procedures have been presented for a large variety of scheduling problems; for a review, we refer to Brucker (2004). There are only a few publications dealing explicitly with the scheduling of make and pack plants. Belarbi and Hindi (1992) develop a heuristic procedure for scheduling make and pack plants where intermediate storage capacity is limited. They illustrate their approach by solving two instances containing about 30 operations to be scheduled each. Ramudhin and Ratliff (1995) present a priority rule as well as a langrangian relaxation-based

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heuristic in order to solve a case study from food industries. Due to hygienic reasons no intermediate storage tanks have to be taken into account. Instances with up to 300 operations to be scheduled are solved to feasibility within at most 140 seconds. Mendez and Cerda (2002) develop a mixed-integer linear optimization model for the scheduling of make and pack plants. Intermediate storage tanks are assumed to have unlimited capacity. The largest instance that can be solved to feasibility within a few seconds of computing time contains 53 operations to be scheduled. Eventually, Gupta and Karimi (2003) state an alternative mixed-integer linear program. The capacity of intermediate storage tanks is again assumed to be unlimited. Due to the large number of binary variables 40 minutes of computing time are needed in order to solve an instance containing 90 operations to be scheduled. Since none of these solution approaches seems to be capable of solving problem instances of the size given in the case study we do not expect that the case study can be solved to optimality within reasonable time. We therefore present a new priority rule method where operations are scheduled one after the other taking into account the technological constraints. Priority values assigned to groups of operations determine the order in which operations are scheduled. Priority rule methods have been very successftil in finding near-optimal feasible solutions for various kinds of scheduling problems (cf. Franck et al., 2001).

2. Case study The data of the case study considered stems from the Procter & Gamble company (cf Honkomp et al., 2000). The make stage consists of 3 pre-mix as well as 6 final-mix tanks. Two final-mix tanks have been assigned to each pre-mix tank. 80 intermediate storage tanks are located between the make and the pack stage, the latter consisting of 7 packing lines. In total, 59 different final products can be produced and packed in various package types. In the case study, demand data is given for 203 combinations of final product and package type and varies unsystematically between 0 and 228,11 product units. A planning horizon of 10 weeks is considered. The production and packaging of an arbitrary final product runs off as follows (cf Figure 1): • Production (make stage). Raw material is converted into some intermediate product by executing an operation of type 01 on an appropriate pre-mix tank. The intermediate product is transferred (TF) to a final-mix tank immediately after the completion of this operation. By executing a second operation of type 02 on the final-mix tank, 10 units of some intermediate product together with some raw material are transformed into some final product. For some final products, no operation of type 01 needs to be executed. All operations are discontinuous, i.e., no material is added or removed during their execution, whereas the transfer of the intermediate product is a continuous process. Therefore, both the pre-mix as well as the final-mix tank are occupied during the transfer. Intermediate and final products are assigned to several product groups. Between the processing of products belonging to different product groups the pre-mix or final-mix tank, respectively, has to be cleaned. The processing times of the operations depend solely on the product in question but not on the pre-mix or final-mix tank. • Intermediate storage. After the termination of an operation of type 02 the final product is pumped out to one or two intermediate storage tanks (PO). The capacity of 6 of these tanks is 10 units whereas the remaining 74 tanks may contain only 5 units of some final product. No restriction is imposed on the choice of the tank. Different

Scheduling of Make and Pack Plants: A Case Study

1553

products, however, may not be stored in one tank at the same time. Final products can only be filled into an empty tank. Before being filled, each tank has to be cleaned. Filling, cleaning and unloading processes may not overlap. Each final product has to stay in some intermediate storage tank at least during a given quality release time t^^ depending on the product. 01

TF

Make stage

-> t TF

02

PO -#• t

10 + IS42

•••

1S80

> tQ^

•^

^t

03

Pack stage

- • t

"-> t Legend:

TpMxj

Pre-mix tank x

| ISxJ

Intermediate storage tank x

TRMX)

Finalmix tank x

PLx

Packing line x

• m

Cleaning or setup time

Figure 1: Production and packaging process at the make and pack plant • Packaging (pack stage). By executing operations of type 03 and 04, respectively, 5 units of some final product are packed on one of the packing lines. Not all package types can be handled on each packing line. The processing times for packaging depend on the packing lines. Final products are taken from the intermediate storage tanks and packed continuously. Between the packaging of final products belonging to different product groups or packed in different package types packing lines have to be cleaned or set up, respectively. An unlimited amount of each raw material is available. Capacities for the storage of packed final products are not considered. The total number of operations to be processed can be computed easily starting from the given demand quantities and varies between 679 and 917 per week. The problem tackled in this paper consists in assigning a processing unit as well as a starting time to each operation to be processed such that all technological constraints are met and the overall cycle time is minimized. Because the time horizon for processing the operations corresponding to each week is limited the problem of determining a feasible solution is already NP-hard. 3. Solution method The basic principle of the solution method is to schedule the operations to be processed successively taking into account the technological constraints. At the beginning of the

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scheduling process, priority values are assigned to each operation on the pack stage. An iteration of the solution method runs as follows: • In step 1, some operation of type 03 or 04 with the smallest priority value is scheduled during its latest execution time interval (packaging of 5 units of some final product). We thereby ensure that there is an intermediate storage tank Si whose allocation allows for storing 5 units of the final product to be packed for a sufficient time period. The fact that the final product has to be produced and stored in advance, however, is neglected. This constraint will be enforced in step 3 of the solution method. • In step 2, a further operation of type 03 or 04 with smallest priority value is scheduled during its latest execution time interval (packaging of another 5 units of the same final product). We again ensure that there is some intermediate storage tank S2 whose allocation allows for storing 5 units of the final product to be packed for a sufficient time period. Intermediate storage tanks Si and S2 may be different (two tanks with a capacity of 5 final product units) or identical (one tank with a capacity of 10 final product units). It has to be taken into account that the 10 final product units to be packed must be filled into tanks Si and S2 at the same time since they are produced as a whole. The material availability constraint is again ignored in step 2. • In step 3, two operations of types 01 and 02, respectively, are scheduled during its latest execution time interval (packaging of 10 units of the same final product). In steps 1 to 3, all technological constraints with the exception of the material availability constraint are taken into account. If in some step, no feasible execution time interval is found for some operation, the operations scheduled in the preceding steps are shifted to an earlier execution time interval. Steps 1 to 3 are repeated until the demand given for each final product is fiilfilled by processing the scheduled operations. The solution's overall cycle time depends critically on the order of scheduling und thus on the used priority rule. In unfavorable cases the overall cycle time may exceed the given planning horizon, i.e., no feasible solution is found. We make use of a multi-stage priority rule that applies different sorting criteria lexikographically to the operations to be scheduled where random numbers are used as tie-breakers. The following sorting criteria have been examined: • SWF (Smallest Wash-out Family): Operations are sorted according to different product groups. • SCF (Smallest Change-over Family): Operations are sorted according to different package types. • LMS (Least Mixing tanks Suitable): Operations are sorted according to the number of mixing tanks capable to produce the final product in demand. • LPS (Least Packing lines Suitable): Operations are sorted according to the number of packing lines capable to pack the desired package type.

4. Experimental performance analysis The priority rule method has been implemented in ANSI-C. In an experimental performance analysis we have examined which combinations of different sorting criteria lead to good feasible solutions for the case study described in section 2. The computational experiments have been performed on a PC with 2,8 GHz clock pulse and 256 MB RAM. A feasible solution has been found after 0,1 seconds on the average. For each out of the 10 weeks for which demand data is available, and for each out of the 64 possible combinations of sorting criteria, 10 runs of the solution method have been

Scheduling of Make and Pack Plants: A Case Study

1555

performed. As an example, Figure 2 shows the cycle times found by the 5 best as well as the 5 worst prioity rules for the demand data given for week 1. Significant differences exist between the solutions found by the different priority rules. The best schedules have been determined using sorting criteria LMS and SCF on stage 1 and 2 of the priority rule, respectively. For certain priority rules, however, the resulting schedules do not fulfill the given demand though there is enough production capacity. The results for weeks 2 to 10 are similar to those obtained for week 1.

__Planning "" horizon

LMS-SCF- LMS-SGF- LMS-SCF- LPS-SWF- LPS-SWF- SCF-LPS LPS-SWF SWF SWF-LPS LMS-SCF SCF-LMS

LPS-LMS

LPS-SCFLMS

LPS-SCF

Priority rule I ^ Minimum [1 Average B Maximum |

Figure 2: Overall cycle times found by the 5 best as well as the 5 worst priority rules (week 1) 5. Conclusions and Outlook In the present paper we have presented a new priority rule method for the scheduling of make and pack plants. Priorities are assigned to operations on the pack stage. Operations on the make stage are scheduled on demand by the pack stage. Hence, the main new characteristic of the solution approach is that priority values are assigned to groups of operations implicitly. Computational experiments have shown the efficiency of the method for a real-world case study with respect to computational times as well as solution quality. To the best of our knowledge no feasible solution to the case study has been reported before. One possible direction for future research consists in applying the solution method to generalized problem settings incorporating fiirther objective functions or constraints such as limited shelf life times. Furthermore, the performance of the solution approach should be analyzed for more complex process networks incorporating, e.g., more than two stages or convergent material flows. Moreover, it would be interesting to apply other more general priority rule approaches as well as commercial production scheduling software to the case study and to compare the solutions, if found, to the solutions determined by the presented priority rule method.

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References K. Belarbi, K.S. Hindi, 1992, Detailed scheduling for a class of two-stage intermittent manufacturing systems, Production Planning & Control, 3, 3 6 ^ 7 P. Brucker, 2004, Scheduling Algorithms (4* ed.), Berlin: Springer B. Franck, K. Neumann, C. Schwindt, 2001, Truncated branch-and-bound, schedule-construction, and schedule-improvement procedures for resource-constrained project scheduling, OR Spektrum, 23, 297-324 S. Gupta, LA. Karimi, 2003, Scheduling a two-stage multiproduct process with limited product shelf life in intermediate storage. Industrial Engineering & Chemistry Research, 42, 490-508 S.J. Honkomp, S. Lombardo, O. Rosen, J.F. Pekny, 2000, The curse of reality — why process scheduling optimization problems are difficult in practice. Computers and Chemical Engineering, 24, 323-328 C.A. Mendez, J. Cerda, 2002, An MILP-based approach to the short-term scheduling of makeand-pack continuous production plants, OR Spectrum, 24, 403^29 A. Ramudhin, H.D. Ratliff, 1995, Generating daily production schedules in process industries, HE Transactions, 27, 646-656

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Detection of abnormal alumina feed rate in aluminium electrolysis cells using state and parameter estimation. Kristin Hestetun and Morten Hovd'' "Norwegian University of Science and Technology, Department of Engineering Cybernetics, N-7491 Trondheim, Norway

Abstract The concentration of dissolved alumina in the electrolyte is one factor influencing the current efficiency in aluminium electrolysis cells. Too low concentration might lead to serious operational problems, like anode effect. In this paper measurements of the current through anodes in different parts of the cell are used to estimate states and parameters using an extended Kalman Filter. Faulty alumina feed rate causing abnormal alumina distribution might then be detected by examining how the expected model output differs from the state estimate from the Kalman Filter in combination with drift in the estimated parameter values. Keywords: Fault detection and diagnosis, state and parameter estimation, Extended Kalman filter 1. Introduction Process operation that was 'good enough' some years ago might no longer meet the required standards for economic and environmental performance. Fortunately, computer power and software have developed rapidly in the last decades introducing new possibilities for better monitoring and control of processes. Fault detection and diagnosis are the central components of abnormal event management (AEM) and have become a standard part of most process control systems. AEM deals with the timely detection, diagnosis and correction of abnormal process conditions or faults in a process (Venkatasubramanian, 2003). More precisely, a fault can be defined as an unpermitted deviation of at least one characteristic property or variable of the system (Isermann and Ball, 1996). Early detection, diagnosis and correction of process faults might prevent the process from entering an uncontrollable operating region thereby reducing production losses and increase performance. In aluminium electrolysis one of the largest expenses is the costs of electric energy and maximizing energy efficiency is an important control objective. One factor influencing this efficiency is the concentration of dissolved alumina in the electrolyte. Both too high and too low concentration of alumina will lead to serious operational problems. Anode effect is one such problem that occurs at low alumina concentration and is a major process disturbance. Besides giving low energy efficiency, anode effect severely disturbs the cell energy balance and produces significant amounts of environmentally harmful CFC-gases. Despite the importance of controlling the alumina concentration, tight control are difficult because of the lack of reliable online information. Key control variables like temperature and electrolyte concentration are usually not available, forcing most control strategies to rely heavily on measurements of cell pseudo

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K. Hestetun and M Hovd

resistance calculated from total line amperage and change in cell voltage. No cost efficient sensor for measuring alumina concentration in the electrolyte is presently available for industrial use. Beame (1999) states that the correlation between change in pseudo resistance and alumina concentration probably will continue to be used to control alumina concentration and alumina feed rate. On these premises, any additional measurements that can give information about what goes on in the electrolyte has potential for improving control of alumina concentration and reduce spatial variations within the cell. Due to improved sensing equipment and computer control, process information that was once considered of no practical use can now give additional information that can be used to optimize process performance. If the current through each anode, or the anode current distribution, is available, this might give valuable information about alumina distribution and consumption in the cell and thereby help prevent anode effects. Rye et al. (1998) showed a correlation between areas with low alumina concentration and anodes with less current when the alumina concentration was low. This paper continues the work started in a previous paper (Hestetun and Hovd, 2005) where this correlation is used to detect abnormal alumina concentration due to faulty alumina feed rate. Measurements of current through anodes in different sections of the cell are used in a state estimator (extended Kalman Filter) to estimate alumina concentration at different locations. The goal is to detect abnormal alumina feed rate early enough to prevent an oncoming anode effect. Hestetun and Hovd (2005) focused on using the state estimate to detect differences between expected and observed rates. This paper focus on what additional information can be gained from also utilising residuals generated based on variation in the estimated parameters values.

2. Short on aluminium electrolysis Liquid aluminium is produced by electrochemical reduction of alumina (AI2O3) as current passes through a high-temperature electrolyte in which alumina is dissolved. Liquid aluminium is formed at the metal/bath interface acting as the cathode while carbon-oxide gases are produced at the anodes, according to the main cell reaction Al ^O .{diss

) + —C{s)^

lAl

{I) + —CO 2 ( g )

^^^

A sketch of a modem prebake electrolysis cell is given in Fig. 1. 20-30 individual prebaked anodes are positioned in two rows and connected in parallel to the horizontal bus bars. The high amperage DC current enters the cell through the bus bars and distributes among the individual anodes. According to Ohm's law, more current will go where there is less resistance and how the current distributes between the anodes depends upon the resistance in the anode and in the inter-electrode gap. Since resistance in the electrolyte strongly depends on alumina concentration, measurements of the individual anode currents do give information about how alumina is distributed throughout the cell. Alumina is fed to the electrolyte through two feeders, one in each end of the cell. The electrolyte is covered by a layer of frozen electrolyte and alumina powder, so before each feeding operations a hole must be made in the top crust with a bar, allowing a controlled amount of alumina powder to be dropped into and dissolved in the electrolyte.

Detection of Abnormal Alumina Feed Rate in Aluminium Electrolysis

1559

Cells

3. Modeling, state and parameter estimation A simple model for estimation of alumina concentration at different locations in the cell has been developed. The resulting model is a non-linear state-space model with a number of unknown parameters: X = fix,u,0) (2) y = g{x ,w,6>) Both/and g are nonlinear functions, g calculates the total voltage drop for the cell. Several terms contribute to cell voltage drop, some of which depend on the alumina concentration. Equations for calculation oig are found from literature data (Solheim, 1998; Haupin, 1998)./is derived from the mass balance of dissolved and undissolved alumina and change in anode-cathode distance. For each control volume, the states are x = [\|/ c 6], where \|/ and c are the concentration of undissolved and dissolved alumina respectively, and 6 is the anode-cathode distance. The parameter vector 6 = [ki_^2 ki-^s kmtf Me] where ki^2 and k2-^3 are dispersion parameters describing transport of alumina between adjacent control volumes, k^tf is the mass transfer coefficient for dissolution of alumina and Mg is the mass of electrolyte in each control volume. The rate of dissolution of alumina is assumed to be proportional to both the concentration of undissolved alumina and the degree of under-saturation of alumina in the electrolyte. The rate of metal production (and alumina consumption) is given by Faraday's law and the cell current efficiency factor. The model inputs u = [Fl F2 ACD II .... I J are the feed rate of alumina for the two feeders, the change in anode-cathode distance and the anode currents through the n sections in the cell. A9

AlO

Bus bars A8

All

• F2 M

1

Alumina feeder

A?!

A12

A6

A13

A5

A14

A4

A15

A3

A16

Cmst breaker Side wall freez< /top cmst

Prebaked anodes Ml

Electrolyte

M Fia A2

A17

Al

A18

II

9 Feeder M Measurement point

Current collector bar

Fig. 1: Layout of prebake electrolysis cell used in the experiments. Placement of anodes (A), control volumes (V), Fig. 2: Sketch of prebake electrolysis cell, as seen feeders (F) and measurement points are from the shorter end. indicated. The objective of this work has been to detect abnormal situations rather than to accurately estimate concentration profile throughout the electrolyte. Earlier work (Jakobsen et al., 2001) dealt with the problem of estimating alumina concentration underneath each anode. However, to achieve good results with this model, reliable information about electrolyte flow pattern is required. This is in general not available during normal operation and to avoid the problem of the unknown flow pattern, the

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K. Hestetun and M. Hovd

number of control volumes has been reduced to three. This allows all mass transport of alumina between adjacent control volumes to be approximated as dispersion while the differences between different parts of the cell can still be detected. The state and parameter estimation is build around an extended Kalman Filter (EKF) representation implemented in Matlab. In general,

yk = gi^k^uj^,ej^)

+ w,

(3)

where Kk is the Kalman filter gain matrix. The parameter vector 6 and the initial states are unknown and must be estimated from experimental data. The program SENIT ModelFit was used to simultaneously determine initial values for states parameters by fitting the model response to measurement data collected from industrial electrolysis cells at Elkem Aluminium's smelter in Mosjoen, Norway. This program utilizes an algorithm based on a modified extended Kalman filter (Schei, 97) to estimate states and parameters in nonlinear state space models. An outer SQP-type optimization loop optimizes the initial values for the parameters and states by minimizing the difference between the measured and modeled output, here alumina concentration and total cell voltage drop for each control volume. 0"'' = min VA^i,,) (4) 1 ^ ^7V ( ^ 7 , A : ) ~

^i,k

= yTk^'

~\j~ 2^

-yi,k

\ ^ v,k^

v^

v,k

"•" ^ Al20^,k'^

Al jOj^

Al20i,k

)

'• = { v , ^ / 2 < ^ 3 }

Fig. 3 illustrates how the resulting model with optimal parameters and initial states (no Kalman filter update) fit the experimental data of measured alumina concentration. 4. Fault detection and diagnosis, results and comments. 4.1. Experimental data Faulty equipment, like a leak in one of the feeders or a malfunctioning bar, could mean that the amount of alumina that is actually dissolved in the electrolyte is considerably different from what is assumed by the process control system. If this goes on unnoticed it may lead to severe process disturbances. The experimental data collected from aluminium electrolysis cells at Elkem Aluminium ANS' smelter in Mosjoen, Norway contains data from periods with normal operations as well as data containing abnormal feed rate that eventually leads to anode effect. I. e., after a period with normal feed rate, the feed rate to the cell was reduced, either by closing one or both of the feeders in the cell (exp. 1 and 2 respectively) or by keeping the feed rate at a reduced rate too long (exp. 3). During the experiments alumina concentration was measured at six positions in the cell while the current through each individual anode were logged in addition to the standard measurements recorded in the process database. An overview of the layout of the cell can be seen in Fig. 2. 4.2. State and parameter estimation. Fig. 4 illustrates how measurements of anode current distribution in addition to standard measurements can be used in the EKF filter to indicate abnormal alumina distribution.

Detection of Abnormal Alumina Feed Rate in Aluminium Electrolysis Here (exp.l), the actual feed rate of the cell is only half of what is assumed (and recorded in the process database) since feeder Fl is actually closed down. Note that the measurements of alumina concentration are not used to update the estimates in the Kalman filter. These measurements are usually not available. Only measurements of cell voltage and anode current distribution in addition to assumed feed rate and change in anode cathode distance from the process database are used to produce the results in Fig. 4. The residual generated from the difference between the expected and estimated concentration (Fig. 4, third plot from the top) clearly indicates that something is wrong.

•.*..,-—-->.:.+

2.5

1561

Cells

•

•;+

-

2 Model output section 1

1.5 1

+

Measurement section 1

•*

'

'

+

'

^""~^'^"^^-~t Model output section 3 *

Measurement section 3

Time [sample index]

In previous work (Hestetun and Hovd, 2005) the model parameters were assumed to be Fig. 3: Measured alumina concentration constant. In reality the parameters represent compared to model output using optimal physical properties that vary with changing parameter values and initial states for exo. 1. process conditions. Even when the measured input rate is correct, the parameters might vary somewhat. For example will a changes in temperature most likely affect all parameters. Here the parameters are modeled as integrated white noise and estimated together with the states using the extended Kalman filter. When the model inputs are "faulty", parameters estimated "online" might also try to compensate for the faulty conditions. By monitoring the parameter behavior and detect out-of-normal variation, we can use the parameters in a fault detection and identification scheme in combination with the results indicated in Fig. 4. Fig. 5 and Fig. 6 show residuals generated from the Time [sample index] difference between the nominal and estimated parameters in exp. 1. The deviation in parameters can serve as a basis Fig. 4: Detection of abnormal feed rate using for detection and isolation of faults. The state and parameter estimation. From the top: change in parameter values in the last 200 Model response (no KF update) with faulty feed rate, state estimate from KF with faulty samples does clearly indicate an abnormal feed rate, difference in estimated and situation. The increase in mass of electrolyte modeled (No KF update) alumina feed rate, (Fig. 5) might indicate that there has been a and last assumed, faulty feed rate. temperature increase, which is consistent with both temperature measurements during the experiment and expected behavior prior

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K. Hestetun and M. Hovd

to an anode effect. In Fig. 6 the estimated diffusion parameters diverge in different directions. As the model believes that both Fl and F2 have the same, normal feed rate, the estimated diffusion parameters try to reflect the observed difference between the two sections of the cell. When an abnormal situation is detected, this additional information from the parameters might help to identify a likely cause of the fault. 5. Concluding remarks Even though much work remains in order to make this a robust fault detection and identification scheme, the observed results are interesting as they do give indication of what goes on inside the electrolyte. By considering both the parameters and the states when trying to identify an abnormal situation more information of the process can be obtained. It should be noted that the experimental data presented her all involve rather large manipulations compared to normal feed rate. Though we were able to detect the faulty feed rate with these data, data from normal operation containing more realistic abnormal situations should be analyzed. Other issues that should be adressed involve robust tuning of the Kalman filter matrices, how to ensure proper initial estimates and how to handle changes in anode-cathode distance.

References V. Venkatasubramanian, R. Rengaswamy, K. Yin, and S.N. Kavuri, 2003, A review of process fault detection and diagnosis Part I: Quantitative model-based methods. Computers & Chemical Engineering, 27, pp. 293-311. R. Isermann, and P. Ball, 1996, Trends in the application of model based fault detection and diagnosis of technical processes, In Proc. Of the 13^*^ IF AC World Congress, volume N, pages 1-12, Piscataway, NJ. G.P. Beame, 1999, The development of aluminium reduction cell process control. Journal of the Minerals, Metals and Materials Society, 51(5), pp-16-22. K.A. Rye, M. Konigsson, and I. Solberg, 1998, Current redistribution among individual anode carbons in a Hall-Heroult prebake cell at low alumina concentration, In: TMS Light Metals 1998, pp. 241-246. K. Hestetun, and M. Hovd, 2005, Detecting abnormal feed rate in aluminium electrolysis using extended Kalman filer, IFAC World Congress 2005, Prague. S.R. Jakobsen, K. Hestetun, M.Hovd and I.Solberg, 2001, Estimating alumina concentration distribution in aluminum electrolysis cells. In Proceeding of the 10* IFAC Symposium on Automation in Mining, Mineral and Metal Processing, pp. 253-258. W. Haupin, 1998, Interpreting the components of cell voltage. In: TMS Light Metals 1998, pp. 531-537. A. Solheim,1998, Reviderte aktivitetsdata for NaF, A1F3 and A1203, Technical Report SINTEF. T.S. Schei, 1997, Afinite-differencemethod for linearization in non-linear estimation algorithms, Automatica, 33(11), pp. 2053-2058.

/-J T.'>*

--^^at/

3.^!!!s-«<Sft;ij-if

,'/.v\ •'vAf

Time (sample intdexl

Fig. 5: Change in total mass of electrolyte during exp. 1.

Tim© (s^mpte iind#x|

Fig. 6: Change in dispersion parameter values, during exp. 1.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Thermodynamic diagram based estimation structure design for ternary distillation column Anna Pulis^ Carlos Fernandez^, Roberto Baratti^ Jesus Alvarez^ ^ Dipartimento di Ingegneria Chimica e del Materiali, Universitd di Cagliari, Piazza D'Armi, 09123 Cagliari, Italy ^ Departamento de Ingenieria de Procesos e Hidrdulica, Universidad Autonoma Metropolitana-Iztapalapa Apdo. 55534, 09340 Mexico, D.F., Mexico

Abstract In this work the ternary distillation column estimation problem is addressed, using an ethanol-tertbutanol-water pilot experimental column. The estimation problem is more difficult and less studied that the one of the binary case. The combination of a thermodynamic framework with results drawn in a previous structure-oriented geometric estimation study yields a procedure with criteria with physical meaning to jointly design the estimation structure and algorithm. The approach is illustrated and tested with an adjustable-structure geometric estimator implementation. Keywords: ternary distillation, estimation problem, geometric observer.

1. Introduction The need of designing or redesigning the separation process for better compromises between productivity, quality, safety and costs in conjunction with the availability of computing capability motivate the need of developing more systematic and reliable distillation column monitoring and control design methodologies, and this in turn justifies studies on the development of on-line estimation techniques to infer compositions on the basis of temperature measurements. The observability problems of binary and multicomponent distillation columns have been extensively studied and tested with a diversity of techniques. The related state of the art can be seen elsewhere [1-8], and here we circumscribe ourselves to mention that only a few studies have considered the multi-component case, that is considerably more difficult than the binary case, for the following reasons: (i) the relationship between temperature and composition is not uniquely defined, (ii) some colurmi trays may lie in stage zones with low temperature changes with compositions, and/or small composition changes, (iii) these features may change along a transient behavior, and (iv) there column model exhibits strong sensitivity with respect to the choice of the vapor-liquid equilibrium description. In a recent study [8], the multicomponent column estimator design problem was successfully addressed within an adjustable-structure geometric estimation framework [9]. However, the results are highly systems theory oriented and devoted from interpretation in the light of standard thermodynamic arguments, and this motivates the scope of the present work: the improvement of the ternary column estimator design by incorporating thermodynamic tools and arguments that are commonly used in the field of distillation column engineering. In this work, the ternary distillation column estimation problem is addressed, using an ethanol-tertbutanol-water pilot experimental ternary column. The combination of a thermodynamic fi'amework with results drawn in the above mentioned nonlinear geometric estimation study yields a combined structure algorithm estimation design in terms of concepts and criteria with

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A.Pulisetal

clear physical meaning. The approach is illustrated and tested with an implementation on the basis of experimental data. 2. Estimation problem Consider a continuous ternary distillation column with, feed rate F at composition Cp, distillate {D) and bottoms {B) at composition CD or CB, heat load Q (proportional to vapor flow rate V), and reflux flow rate R. Under the standard assumptions (liquid-vapor equilibrium at each stage, linear pressure drop along the column, quasi steady-state hydraulics and equimolar flow) the column behavior is described by the following equations [10, 11]: Stripping section (7
c. =(cf,c^)'

(la)

)J/7]-'(R + F)

(lb)

Feed tray (/ = np) c

=[R/tc

rip,

-VAv(c rip

) + F(c -c t

rip

n^

Enriching section {rip + 1 ^i
(Ic)

Top Tray (/ = A^ C^=[RAC^-VSV(C^)J/?]-'(R)

(Id)

Measurements y^=T^=fi(cJ,

se[ln^-lj

y^=T^=fi(cJ,

ee [n^^^,NJ

(le)

where ^c.=c.^^-c., ^M.,=^o=''(^.^' where

Av(c.) = v(c.)-v(c._^). cf+cf^c^=l

c.=(c^,c^)^rQ

v(c^) = (c^,cl)'

(If) (Ig)

the component molar fraction, ethanol and tertbutanol

respectively, in the i-th stage, js (or je) is the measured value of the temperature Ts (or Te) in the s (or e)-th stage (to be determined) of the stripping (enriching) section, v is the nonlinear (liquid-vapor equilibrium) two-entry vector frinction that determines the i-th composition pair ci in the vapor phase, fi is the nonlinear bubble point frinction that yields the temperature, and ^ is the tray hydraulics frinction that sets the exit molar flow rate from the i-th stage. Having as point of departure the results drawn from a previous study using an adjustable-structure estimation scheme, including numerically assessed observability measures, in this work the design procedure is frirther systematized and simplified by incorporating a thermodynamic framework to analyze the estimability property and to jointly design the estimation structure and algorithm, with structure meaning the choice of innovated and not innovated states and of sensor locations, and with algorithm

Thermodynamic Diagram Based Estimation Structure Design

1565

meaning the dynamic data processor that performs the estimation task. The approach is illustrated with a representative example and experimental data obtained from a pilot plant-scale column. 3. Dynamic behavior in the light of thermodynamic issues The ternary mixture water-ethanol-tertbutanol is separated in a continuous ternary distillation column. The column of 30 sieve trays, equipped by a shell and tube natural circulation thermosyphon and a total condenser, is located at the University of Padova (Italy). The feed enters in the 8-th stage, from the bottom, and temperature measurements are known in nine stages (0, 4, 8, 12, 16, 18, 22, 26, 30). The column has an important characteristic: the tray number is higher than necessary to have a distillate of high purity and works close to the azeotrope composition. Due to this fact, the temperature changes are significant only in the stripping section (low alcohols concentrations) while vanish in the enriching section (high alcohols concentrations) involving the presence of a flat region in the temperature surface, see Figure 1, where the temperature is almost constant ( r ~ 353K).

TTK)

Fig. 1 Temperature surface.

The selected experiment starts at steady state with low purity {R, F) = (3.486, 3.489) 10 "^ m^/s and higher vapor flow rate V = 1.431 gmol/s towards a high purity steady state (R, F) = (3.486, 3.489) 10 mVs by a step decrease of the vapor V= 0.963 gmol/s and a constant feed composition: c^ = 0.0979 and c^ = 0.0630. This is showed in the rip,

rip

Figure 2 where the distillation lines, at three different times, moves from low (c^ = 0, cl= 0, cl= 0.3, c^= 0.19) to high alcohol compositions (c^ = 0.019, c^ = 0.0043, c^= 0.43, c^= 0.31) jointly with the temperature gradient that originates in the high temperature peak (7 = 355K) and terminate close the depression of the boiling point surface. The borderline runs from the water-ethanol to the water-tertbutanol azeotrope and approximately follows the course of the valley in the boiling point surface. The objective of this work is to illustrate how to choose the observer structure (sensor location and innovated state) on the basis of the thermodynamic analysis of the system. For this purpose, let us write the corresponding geometric PI estimator [12] for the

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A. Pulis et ah

ternary column case of one measurement, located at stage i-th, and two innovated states (ethanol and tertbutanol) [8]:

'c.=f(dj._^,c.^^,u)j^i, w = kJy.-P(c.)],

u = (F,R,V,c.^^, e_=(ef,6^)',

n = (p,iy

(2b)

k =(2C-\-l)(0),co^)', k^ = co^

(2c)

- • — Omin -A— 70min O—133min

II Fig.2 Temperature level curves and distillation lines at three different times. where CE (or cr ) is the ethanol (or terbutanol) composition estimate at stage i-th, O is the observability matrix, w is the integral action state to eliminate the output mismatch, and kp is the vector of proportional gains, and k^ is the integral action gain, and co is the adjustable characteristic frequency of the prescribed output estimation error dynamics. In detailed notation, O is written as follows 0(c,c^_,,i^^^,u) =

p. (^,)

p. (^<) ddjc ,c „c .u)/dc

(3a)

ddjc.,c. „c. ,,u)/dc (3b)

The related singularity measure (5^"^) (the inverse of the minimum singular value {msv) of the observability matrix O) is given by ^ = Sf-' =[1/ msvO(c., C.J, c ._^, u)]

(4)

and the corresponding plot, at three different times, is shown in the Figure 3. From numerical assessments [8], we know that the ill conditioning of O grows with the separation degree, the closeness to azeotropic composition, and with the number of

Thermodynamic Diagram Based Estimation Structure Design

1567

innovated states. The question is now to look and interpret these features, related to the sensor location and the estimator functioning, in the light of composition and temperature profiles within a ternary thermodynamic diagram. By considering the distillate Hne shown in Figure 2 and equation (3), it is clear that being derivative of the temperature with respect the alcohols close to zero in the enriching section the singularity measure are higher than in the stripping section, where the derivatives assume a finite value. This suggest to locate the sensor in the stripping section and not to innovate the component in the enriching section.

^-'

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 stage

Fig.3 Singularity measure dependency at three different times. Moreover, high interactions between the entries of the observability matrix (presence of partial derivative of the equilibrium (v) and bubble {fi) point nonlinear function, stages close to azeotropic compositions), suggest to modify the observability matrix equation (3) in the following manner: drop off diagonal elements in order to improve conditioning. As a result, one obtains the passivated estimator: c=f(c,c

,c ,u) + 0 ^(c ,c ,c ,u)[mv + k [y

c.I =^ f,(c.,c. .c. .u) r I i-l 1+1 ^ w = k [y -p(c)],

-B(c)l

ji^i

(5a) (5b) (5c)

c =(c^:c^)

with diagonal matrix Op and observability measure Sf'.

«/V=

0

Pcfi)

S,=[l/msvO

(6JJ

(6)

Thus, the passivated estimator is underlined by a decentraHzed-coordinated innovated structure. Once the estimability structure has been designed on the basis of a thermodynamic assessment, the passivated geometric estimator was implemented to infer the effluent profile compositions, and the remarkable results are presented in the Figure 4.

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4. Conclusions The ternary distillation column estimation problem has been addressed, using an ethanol-tertbutanol-water pilot experimental column with experimental data as case example. The combination of thermodynamic concepts with the adjustable-structure geometric estimation technique yielded a procedure to jointly design the estimation structure and algorithm. Comparing with the EKF technique employed in previous ternary distillation studies [10], the proposed estimator yields similar results with a considerably simpler and easier to tune algorithm.

' I 'r

0 15 30 45 60 75 90 105 120135 time (min) Fig. 4 ethanol and tertbutanol profile composition estimated during the time:( — ) estimated, (•) off line experimental determinations. References [I] C. C. Yu and W. Luyben, Inst. Chem. Eng. Symp. Ser., No. 104 (1987). [2] L. Lang and D. Gilles, Comput. Chem. Eng., 14 (1990) 1297 - 1301. [3] E. Quintero-Marmol, W. Luyben and C. Georgakis, Ind. Eng. Chem. Res., 30 (1991) 1870 1880. [4] F. Deza, E. Busvelle, J. P. Gauthier, Chem. Eng. Sci., 47 (1992) 3935 - 3941. [5] R.M. Oisiovici and S. L. Cruz, Chem. Eng. Sci., 55 (2000) 4667 - 4680. [6] S. Tronci, F. Bezzo, M. Barolo, R. Baratti, Ind. Eng. Chem. Res., 44 (2005) 9884 - 9893. [7] W. Luyben, Ind. Eng. Chem. Res., 44 (2005) 7113-7119. [8] A. Pulis, C. Fernandez, R. Baratti, J. Alvarez, Geometric Estimation of Ternary Distillation Column, accepted to ADCHEM 2006. [9] T. Lopez and J. Alvarez, J. Process Control, 14 (2004) 99 - 109. [10] R. Baratti, A. Bertucco, A. Da Rold, M. Morbidelli, Chem. Eng. Sci., 53 (1998) 3601 - 3612. [II] S.Skogestad, Modeling, Identification and Control, 18 (1997) 177 - 217. [12] J. Alvarez and T. Lopez, AIChE J., 45 (1999) 107.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

O p t i m a l o p e r a t i o n of a m i x e d fluid c a s c a d e L N G p l a n t J0rgen Bauck Jensen^ & Sigurd Skogestad^* ^Department of Chemical Engineering, NTNU, Trondheim, Norway Studies on the operation of complex vapour compression cycles, like the one used for the production of liquefied natural gas (LNG), are not widely reported in the open literature. This is a bit surprising, considering the large amount of work that has been put into optimizing the design of such processes. It is important that the process is operated close to optimum to fully achieve the maximum performance in practice. There are possibilities for savings, both due to (a) identifying the optimal point of operation, and (b) selecting the controlled variables such that the optimal operation depends weakly on disturbances. In this paper we study the mixed fluid cascade (MFC) LNG process developed by The Statoil Linde Technology Alliance. We study the degrees of freedom and how to adjust these to achieve optimal steadystate operation. K e y w o r d s : Self-optimizing control, optimal operation, liquefied natural gas 1. I n t r o d u c t i o n Large amounts of natural gas (NG) are found at locations that makes it infeasible or not economical to transport it in gaseous state (in pipelines or as compressed NG) to the customers. The most economic way of transporting NG over long distances is to first produce liquefied natural gas (LNG) and then transport the LNG by ships. At atmospheric pressure LNG has approximately 600 times the density of gaseous NG. At atmospheric pressure LNG has a temperature of approximately -162°C, so the process of cooling and condensing the NG requires large amounts of energy. Several different process designs are used and they can be grouped roughly as follows: • Mixed refrigerant: The refrigerant composition is adjusted to match the cooling curve of NG. Some are designed with a separate pre-cooling cycle • Cascade process (pure fluid): Several refrigerant cycles are used to limit the mean temperature difference in the heat exchange • Mixed fluid cascade process: Energy eflaciency is further improved by using several mixed refrigerant cycles The process considered in this paper is the Mixed Fluid Linde Technology Alliance [1]. The MFC process has and the first cycle with two pressure levels. The steady-state model for this plant is implemented equations. Optimizing the plant takes in the order of and 512 MB RAM running GNU/Linux.

Cascade (MFC) process developed by The Statoil three different cycles, all with mixed refrigerant in gPROMS [6] resulting in approximately 14000 2 hours on a Pentium 4 computer with 2.8 GHz

2. P r o c e s s d e s c r i p t i o n A simpHfied flowsheet is given in Figure 1. For more details about the process consult [1], [2] and [3]. Nominal conditions: • Feed: NG enters with P = 6 1 . 5 bar and T = 1 1 ° C after pretreatment. The composition is: 88.8% methane, 5.7% ethane, 2.75% propane and 2.75% nitrogen. Nominal flow rate is 1 kmol/s * [email protected]

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J.B. Jensen and S. Skogestad

Figure 1. Simplified flowsheet of the MFC process. SUB - sub-cooHng cycle, LIQ - liquefaction cycle, P R E - pre-cooling cycle (two stages 1 and 2). N G I A to NG3 are the main heat exchangers. Degrees of freedom associated with variable active charge in each cycle are not shown

• Product: LNG is at P = 5 5 . 1 bar and T=-155°C • The refrigerants are a mix of nitrogen (N2), methane (Ci), ethane (C2) and propane (C3) and the compositions are used in optimization. • The refrigerant vapour to the compressors are super-heated 10°C • The refrigerants are cooled to 11° C in all sea water (SW) coolers (assumed maximum cooling) • Pressure drops are 0.5 bar in SW coolers, 0.5 bar for hot flows in main heat exchangers and 0.2 bar for cold refrigerant in main heat exchangers The SRK equation of state is used both for NO and the refrigerants. The heat exchangers are distributed models with constant heat transfer coefficients. The compressors are isentropic with 90% constant efficiencies. 3 . D e g r e e o f f r e e d o m analysis In this section we present a detailed degree of freedom analysis which is an important result of this work. In a single simple vapour compression cycle (e.g. a home refrigerator) there are two obvious manipulated inputs, namely the compressor and the valve^ In addition, there is a less obvious manipulated variable. This is the "active charge" in the cycle, which may be modified by introducing a unit (tank) with variable holdup [4]. The active charge may be changed by placing tanks at many different locations, but from a simple mass balance it may be verified that for each cycle one may have only one independent variable (tank) associated with the active charge. Thus for the cycles the number of manipulated variables are the number of compressors and valves plus one active charge for each cycle. Let us now look at the MFC process. ^In addition one might control flow of hot and cold fluid, but this is outside the cycle, so let us overlook that for now

Optimal Operation of a Mixed Fluid Cascade LNG Plant

1 b 71

3 . 1 . M a n i p u l a t e d variables ( M V ' s ) From the discussion above we find that there are in total 26 manipulated variables (degrees of freedom): • • • • • •

5 4 4 1 9 3

Compressor powers Ws,i Choke valve openings Zi SW flows in coolers NO flow (can also be considered a disturbance) Composition of three refrigerants active charges (one for each cycle)

3.2. Constraints during operation There are some constraints t h a t must be satisfied during operation. • • • • •

Super-heating: The vapour entering the compressors must be > 1 0 ° C super-heated ^rNG- NG Temperature out of NG3 must be <-155°C or colder Pressure: 2 b a r > P <60 bar NG temperature after N G I A and N G I B (not considered in this paper) Compressor outlet temperature (not considered in this paper)

3.3. Active constraints We are able to identify some constraints that will be active at optimum. In total there are 12 active constraints: • • • •

4 Super-heatings to be minimized (e.g. see [4]), that is ^Tsup,i—^^°C Excess cooling is costly so Tl^Q=-\^b°C Optimal with low pressure in cycles so Pi=2 bar (for all 3 cycles) Maximum cooling: Assume T = 1 1 ° C at 4 locations

at 4 locations

3.4. U n c o n s t r a i n e d d e g r e e s of f r e e d o m After using 12 of the 26 manipulated inputs to satisfy active constraints, we are left with 14 unconstrained degrees of freedom. In this work we consider the NG flow given from elsewhere (disturbance to the process), so we are left with 13 degrees of freedom in optimization. For a steady state analysis the pairing of inputs and outputs is insignificant, so say we are left with the following subset of the MV's: • 3 NG temperatures (after NGIA, N G I B and NG2) • Pm in SUB • 9 Refrigerant compositions In this paper we will not consider manipulating refrigerant composition in operation (only in the optimization), so of the 13 unconstrained degrees of freedom we are left with 4 during operation. 4. O p t i m i z a t i o n r e s u l t s In this section we are optimizing on the 13 degrees of freedom given above to locate the optimal operation of a given MFC LNG plant. The resulting temperature profiles for the four main heat exchangers are given in Figure 2. Some key values of the refrigerant cycles are given in Table 1 where the nomenclature is given in Figure 1. Some remarks: • The total shaft work is 10.896 MW • The optimal NG temperature out of N G I A , N G I B and NG2 is 255.9 K, 221.7 K and 196.1 K, respectively • In the true design there will separators at the high pressure side of the cycles, which has not been considered here. Further work will include an analysis of the eff'ect of this sub-optimal design • In SUB cycle the pressure ratios over the two compressor stages are far from equal (which is a rule of thumb for compression ratios). This is because the inlet temperature to the first stage (approximately -80° C) is much lower than inlet temperature to the second stage (11°C) • Nitrogen is present in SUB only to satisfy the minimum pressure of 2 bar

1572

J.B. Jensen and S. Skogestad

150 (a) NGIA

225 220

^

(b) N G I B

200

,

r-^

215 \

\
190 "s

"- . "^^>. \ ^ ^- :

NG • - • LIQ . . . SUB — LIQcold

180 170 160 150

a2oo

140

195

130

190

120 185

)

50

100 Position (c) NG2

1^)0

( (d) NG3

Figure 2. Temperature profiles

5. Control s t r u c t u r e d e s i g n In the section above we where able to identify the optimum for the process, but how should this optimum be implemented in practice? First we need to control the active constraints: • • • •

Degree of super-heating (4 locations): For this we may use the corresponding choke valve opening Pi is for each of the 3 cycles: For this we may use "active charge" (see discussion above) Maximum cooling in 4 SW coolers: SW flow at maximum LNG outlet temperature at -155°C: May use first compressor stage in SUB

The four remaining degrees of freedom should be used to control variables which have good self optimizing properties: "Self optimizing control is when we can achieve acceptable loss with constant setpoint values for the controlled variables (without the need to re-optimize when disturbances occur)" [5].

Optimal Operation of a Mixed Fluid Cascade LNG Plant

1573

Table 1 Optimal operation of a MFC process PREl Pi ^m

Ph

c, C2 C3 N2

Flow

w.

[Pa] [Pa] [Pa]

[%] [%] [%] [%] [mol/s] [MW]

PRE2

LIQ

SUB

2.00 6.45 15.03 15.03 0.00 0.00 37.70 37.70 62.30 62.30 0.00 0.00 464 685 1.2565 + 2.644

2.00

2.00 28.38 56.99 52.99 42.45 0.00 ' 4.55 627 3.780+1.086

6.45

-

20.58 4.02 82.96 13.02 0.00 390 2.128

SW(

Figure 3. Suggested control structure for the MFC process. SH are degree of super-heating controllers, P C and T C are pressure and temperature controllers respectively. Not shown: Three pressure controllers on the low pressure side using the active charge in each cycle

To evaluate the loss one needs to consider the effect of disturbances and implementation errors. A steady-state analysis is usually sufficient because the economics are primarily determined by the steadystate. Based on physical insight the following four variables may been suggested » Tout • ^NGIA ^ rpout

• Pm

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J.B. Jensen and S. Skogestad

A possible control structure with these four variables and the active constraints controlled is shown in Figure 3. However, note that the "pairings" of controlled and manipulated inputs are included primarily to illustrate that we have available degrees of freedom, as this does not matter for evaluating self-optimizing control at steady-state. It will be the subject of future work to compare this choice of controlled variables with one that follows from a systematic procedure. 6. Conclusion We have shown that the degrees of freedom in vapour compression cycles are equal to the number of compressors and valves plus one. The extra degree of freedom is related to the "active charge" in the system, and a tank with variable holdup should be included to gain this degree of freedom. A detailed degree of freedom analysis for the MFC process reveals that there are four unconstrained degrees of freedom in operation (not considering manipulating refrigerant compositions). To fully achieve the potentially high thermodynamic efficiency of the MFC process it is important that these four unconstrained degrees of freedom are utilized optimally. REFERENCES 1. W. A. Bach. Developments in the mixed fluid cascade process (MFCP) for LNG baseload plants. Reports on science and technology Linde, 63, 2002. 2. W. Forg, W. Bach, R. Stockmann, R. S. Heiersted, P. Paurola, and A. 0. Fredheim. A new LNG baseload process and manufacturing of the main heat exchanger. Reports on science and technology Linde, 61, 1999. 3. Statoil. Sn0hvit homepage, www.statoil.com/snohvit. 4. J. B. Jensen and S. Skogestad. Control and optimal operation of simple heat pump cycles. In European Symposium on Computer Aided Process Engineering (ESCAPE) 15, Barcelona^ 2005. 5. S. Skogestad. Plantwide control: the search for the self-optimizing control structure. J. Process Contr., 10(5):487-507, 2000. 6. h t t p : / /wiTV. psenterpr i s e . com/product s_gproms. html.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteHdes (Editors) © 2006 PubUshed by Elsevier B.V.

Multiplicity of steady states in an UOP FCC unit with high efficiency regenerator Joana L. Femandes,^ Carla I.C. Pinheiro,^ Nuno Oliveira,^ F. Ramoa Ribeiro^ ^CEBQ, Department of Chemical Engineering, Institute Superior Tecnico Av. Rovisco Pais, 1, 1049 - 001 Lisboa, Portugal ^GEPSI-PSE Group, Department of Chemical Engineering, University ofCoimbra Pinhal de Marrocos, 3030-290 Coimbra, Portugal Abstract Static bifurcation in a FCC unit is a problem that arises whenever studying the control of a FCC unit. The origin of this behavior is usually due to the exothermicity of the catalyst regeneration reactions and to important phenomena of backmixing in the regenerator. For this reason the geometrical and operational design of the regenerator unit plays an important role in the overall performance and dynamic stability of FCCs. Prior work has focused on model and control problems of different operating FCC units. However, none of these studies have considered a FCC unit with high efficiency regenerator. This paper presents an analysis of the static biftircation behavior of an UOP FCC unit with high efficiency regenerator. The results show that the high efficiency regenerator presents static bifurcation exhibiting multiple steady states, depending on the operating conditions. Keywords: Fluidized Catalytic Cracking; high efficiency regenerator; steady state multiplicity; nonlinear dynamics. 1. Introduction Fluidized Catalytic Cracking is an important refinery process, not just because of its economical relevance but also due to its complex nature and from a strong interaction between the reactor and regenerator units. The complexity of the FCC process originates a highly nonlinear system, quite interesting for control studies. A comprehensive analysis of its steady state behavior and detailed knowledge of the relationships between state and manipulated variables in the operating range is therefore essential for the design of the control systems for these processes. According to Arbel et al. (1995b) all FCCs should exhibit multiple steady states since they are autothermic reactors that require heating to start up the system. Another author (Elnashaie et al., 2004) refers the feedback effects resulting from complex nonlinear interactions between the regenerator and the reactor as the cause for static bifurcation. This author has investigated several industrial units and states that the behavior of the different industrial FCC configurations quantitatively differs, but qualitatively they are very similar. Even though configurations with riser type regenerator, such as the UOP high efficiency regenerator, as well as riser type reactor are expected to have bifurcation behavior albeit both parts of the unit as stand-alone, will not exhibit such behavior because they are in plug flow. In this paper the existence of static bifurcation behavior in a FCC unit with a riser type regenerator will be investigated.

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J.L. Fernandes et al

To minimize coke-on-regenerated catalyst, regenerators were designed with a long residence time. Thus by means of a long residence time and low regenerator temperature, coke-on-regenerated catalyst could be reduced to relatively low values. To reduce the time required for coke combustion and backmixing effects UOP has designed a high efficiency regenerator (see Fig. 1) that operates in a fast-fluidized flow followed by a plug flow as opposed to conventional back-mixed regenerators. The burning takes place in a fast-fluidized combustor. By conducting regeneration in this manner at high temperature, regeneration can be accomplished in a fraction of the time required in the back-mixed regenerators (Grace Davison, 1996). Combustoi" Regenerator Hue Gas

Dilute Hiase Dense Pliase

Fig. 1. Design of a high efficiency regenerator from UOP. (Couch et al., 2003) 2. High Efficiency Regenerator Model The FCC model used in this study was developed by Fernandes et al (2005a and 2005b) in what refers to the riser, stripper and standpipes models. The high efficiency regenerator was modeled in three parts: • combustor - lower part of the regenerator, where combustion air enters the unit and mixes with catalyst coming from the reactor (spent catalyst) and the regenerator vessel (recirculated regenerated catalyst); • lift or regenerator riser - takes the catalyst from the combustor to the upper regenerator vessel. • regenerator vessel - upper part of the regenerator where combustion gases disengage from catalyst and a dense phase of catalyst is formed. Most of the combustion reactions occur in the combustor and lift at normal operating conditions. The mixing of catalyst and gases occurs in the lower part of the combustor in the absence of reactions. The combustion kinetics considered in this model was previously studied at IFF by Vale (2002). The coke is considered to be composed mainly of carbon and hydrogen. Sulfur and nitrogen are also present but in small quantities. Therefore, it is only considered the combustion of carbon and hydrogen. Carbon combustion and hydrogen combustion can then be given by the following reactions:

" ' ^ p -

'

a+1

'co,t'

'

a+1

CO

(1)

Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator

lb'/'/

Where a is the intrinsic molar ratio between CO2/CO. /Z + I Q

3,gas-solid ^IH^O

(2)

The combustor and lift were modeled as a plug-flow reactor where combustion reactions occur and are considered to be in a pseudo steady state, since the residence time in the combustor and lift is much smaller than in the regenerator vessel. The regenerator vessel was modeled as a CSTR in dynamic state, with combustion reactions occurring in the dense phase. The equations for the regenerator vessel are the same as presented in Femandes et al. (2005a). The mass and energy balances for the combustor and lift are presented below: 2,1. Gaseous Species Molar Balance

dz

^,l(r/t;|.)+^.p,l(r>;,)

(3)

2.2. Carbon and Hydrogen Mass Balance dN

^=^^cPcj:(r;v;,) J

dz

(4)

2.3. Energy Balance dT 3^

Q.

^,S(r/A//f)+^,/.,Z(r;A//j)

T^iCp,

(5)

3. Simulation Results and Discussion The results shown in this paper where obtained through simulation by using a static version of the model presented. The operating conditions are the same as used by Femandes et al. (2005b). Combustion gases and coke heats of formation are the same as used by Han et al. (2001) and vaporization enthalpy is calculated through correlations found in Grace Davison (1996). The results presented in Figure 2 and Figure 3 where obtained by fixing all the inputs and changing respectively the catalyst-to-oil (COR ratio) and air-to-oil ratio (Air/Oil). As it can be seen the regenerator temperature goes through a maximum when going from low COR to high COR, which agrees with results presented by other authors (Han et al, 2001 an Arbel et al., 1995a). The air to-oil ratio has also the same effect on regenerator temperature; however the decrease in regenerator temperature after passing the maximum is very slightly. These effects are related to the quantities of coke formed and to the regime of combustion: partial or full combustion that is represented by the ratio CO/CO2. Partial combustion leads to lower temperatures and higher ratios of CO/CO2, since the oxidation of CO to CO2 occurs with low extension and is highly exothermic. At low COR the conversion is low since only small quantities of coke are formed due to low quantity of catalyst, in that way the quantity of coke to be burned is small and the temperature decreases due to a lower heat generation. At low air-to-oil ratio, since there

1578

J.L. Fernandes et al

is a deficient oxygen concentration, the combustion is more incomplete leading to lower temperatures. 0.7

800

0.6

750

- Air/Oil=0.7 -AirO=0.8

0.5

2 700 -I

O 0.4

t

S 0.3

650

0.2 600 0.1 550

0 4

6

8

COR (kg catatystl^g oil)

10

4

6

8

10

CORCkgcutab/stkgoil)

Fig. 2. Effect of COR ratio in regenerator temperature and partial or full combustion.

0.6

0.7

0.8

0.9

Air/Oil (kg air/kg oil)

0.7

0.8

0.9

Air;t»il (kg airikg oil)

Fig. 3. Effect of air/oil ratio in regenerator temperature and partial or full combustion. To investigate the existence of multiple steady states all the inputs were fixed and the heat balance was centered on the regenerator. Two simulations with different coke compositions with typical industrial hydrogen-to-carbon molar ratio (H/C) were made to investigate the existence of multiple steady states. By analyzing Figure 4 and Figure 5 it can be seen that there is always a cold steady state that is unconditionally stable and that resultsfi*omthe impossibility of preheating the feed sufficiently to achieve self-ignition. This cold steady state is physically impossible since it would result in the absence of feed vaporization, and only exists from a mathematical point of view. In Figure 4 three more steady states were obtained. The upper and lower ones are stable and the middle one unstable. All this three states appear at high regenerator temperatures, even though the lower one would take to smaller conversions not interesting from an economical point of view. Figure 5 shows that for higher hydrogen content on coke there is only one steady state at high temperatures besides the cold steady state. This results fi^om higher heat generation through the combustion of hydrogen. Different contents of hydrogen in coke are caused by feed composition and stripping efficiency, which depends on the stripper configuration, stripping steam flow and temperature.

Multiplicity of Steady States in an UOP FCC Unit with High Efficiency Regenerator 1579 The breakdown by sources (Figure 5) of the heat generated Hne shows that the extension of coke combustion in each section (combustor, lift and regenerator vessel) of the high efficiency regenerator varies significantly with temperature and in a nonlinear way which means that all the three components of the high efficiency regenerator contribute to the existence of multiple steady states. 2500

700

800

1000

Teinperatiiie |K)

Fig. 4. Heat generation and heat removal lines for coke combustion with H/C = 0.8. 2500 Hregenerator vessel

- H generated

J

• - - -Hcombustor

-Hremo\ed

^^"^ \

2000 - — - -Hlitl

^'

^'^

Hlost

m i f 1500 -

Hremoved

^'•''

•^

1

J J

^

1000 -

j.,,''^

'

/»'^'^"**-*w

J^'^ "'^ jT r

a» X 500 -

'

^^''^"^^^^ f^ j^s.

/ /

''

/

-- '

0 - r.^TT^-..^. 500

600

700

800

Temperature |K)

900

100C

*

y'

05

<^ 3

500

/

.y'^

\ . '^ A • . • - - -N

600

700

800

900

. . ^

1000

Temperature |K)

Fig. 5. Heat generation and heat removal lines for coke combustion with H/C =1.1 and breakdown of generated heat by sources.

4. Conclusions It is well established that the riser with its endothermic reactions in plug flow cannot be the cause of static bifurcation. However the highly exothermic catalyst regeneration reaction in the regenerator with several reactions occurring in series as an integrated and interactive part of the unit can cause this phenomenon. Several studies on the multiplicity of steady states in bubbling bed regenerators (Arbel et al., 1995b and Elnashaie et al. 2004) have shown the existence of more than one steady steady state, however none of these studies focused on a regenerator with a plug

1580

J.L. Fernandes et al

flow behavior without backmixing effects. Simulation results have shown that even the riser type regenerators such as the high efficiency regenerator from UOP, albeit having a flow regime near plug flow also exhibit multiple steady states depending on operating conditions. The knowledge of multiplicity steady states existence is quite important for future studies on control problems, since the unstable intermediate upper steady state may represent an interesting point for operation with good levels of conversion for lower regeneration temperatures. Nomenclature COR\ catalyst-to-oil ratio (kg catalyst/kg oil); Cp\ Specific heat capacity (Jkg'^K"^); F\ Mass flow rate (kg/s); iV/^: Molar flow rate of component / (mol/s); rf. Rate of the reaction j (molm'^s"^ or molkg'^s"^); t\ Time (s); T\ Temperature (K); V\ Volume (m^); W\ Inventory (kg); ¥{. Component / (in coke) content of catalyst (kg/kg); z: Axial coordinate (m); e: Volume fraction; p\ Density (kg/m^); a. Intrinsic CO2/CO molar ratio; Vp. Stoichiometric coefficient of component i with respect to the reaction j ; AH\ reaction heat (J/mol.s); i 2 Cross section (m^); gi gas; ct catalyst; §: solid References Arbel, A., Huang, Z., Rinard, I. H., & Shinnar, R. (1995a). Dynamic and Control of Fluidized Catalytic Crackers. 1. Modeling of the Current Generation of FCC s. Industrial & Engineering Chemistry Research, 34, 1228-1243 Arbel, A., Rinard, I. H., Shinnar, R., & Sapre, A. V. (1995b). Dynamic and Control of Fluidized Catalytic Crackers. 2. Multiple Steady States and Instabilities. Industrial & Engineering Chemistry Research, 34, 3014-3026 Couch, K. A., Seibert, K.D.,& Opdorp, P.V. (2003). Controlling FCC Yields and Emissions UOP Technology for a Changing Environment, in: NPRA Annual Meeting, National Petrochemical & Refiners Association, paper AM-04-45, San Antonio Elnashaie, S. S. E. H., Mohamed, N. F., & Kamal, M. (2004). Simulation and Static Bifurcation Behavior of Industrial FCC Units. Chemical Engineering Communications, 191, 813-831 Fernandes, J.L., Verstraete, J., Pinheiro, C.I.C, Oliveira, N., & Ribeiro, F.R. (2005a). Mechanistic Dynamic Modelling of an Industrial FCC Unit, in: Book and CD of Proceedings of the 15* European Symposium on Computer-Aided Process Engineering - ESCAPE 15, 589-594, Barcelona Fernandes, J.L., Pinheiro, C.I.C, Oliveira, N., & Ribeiro, F.R. (2005b). Modeling and Simulation of an Operating Industrial Fluidized Catalytic Cracking (FCC) Riser, in: Book of abstracts and Full Papers CD of the 4* Mercosur Congress on Process Systems Engineering - ENPROMER 2005, 38, Rio de Janeiro Grace Davison (1996). Guide to Fluid Catalytic Cracking, Part One. W.R. Grace & Co.-Conn. Han, I.-S., & Chung, C.-B. (2001). Dynamic Modeling and Simulation of a Fluidized Catalytic Cracking Process. Part II: Property Estimation and Simulation. Chemical Engineering Science, 56, 1973-1990 Vale, H. (2002). Development of a Simulator for a Complete R2R Catalytic Cracking Unit, IFP Report Acknowledgements The author Joana Fernandes thanks the financial support granted by the program POCTI - Formar e Qualificar from Funda9ao para a Ciencia e Tecnologia through the grant number SFRH/BD/12853/2003.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Pubhshed by Elsevier B.V.

On-Line Data Reconciliation and Parameter Estimation for an Industrial Polypropylene Reactor Diego Martinez Prata,^ Jose Carlos Pinto,^ Enrique Luis Lima^ ^Programa de Engenharia Quimica/COPPE/UFRJ, Cidade Universitdria, Rio de Janeiro- CP 68502, CEP 21945-970, Brasil

Abstract This work presents the implementation of a simple methodology for dynamic data reconciliation and simultaneous estimation of quality and productivity parameters using data from an industrial bulk Ziegler-Natta propylene polymerization process. A simple model of the real process, based on mass and energy balances, was developed including only the variables of greater significance for the desired parameter estimation problem. The resulting nonlinear dynamic optimization problem was solved using a sequential approach on a time moving window. Results have shown that including the energy balance increases information redundancy, leading to better estimations than the ones obtained when energy contribution are not considered. Keywords: Nonlinear Dynamic Data ReconciHation, Parameter Estimation, Real Polymerization Plant Data. 1. Introduction Advances in digital instrumentation techniques have provided the industries with very powerfiil tools that allow for on-line monitoring of several process variables. As a consequence mathematical models started to play an important role in modem process engineering techniques. In this promising scenario, it is possible to use a huge amount of reliable reconciled data for simultaneous online estimation of model parameters and non-measured variables. This information is fundamental for implementation of on-line optimization and automatic process control techniques. In this work a data reconciliation technique with simultaneous parameter estimation, applied to an industrial propylene polymerization process, is improved incorporating redundancy through a global energy balance.

Corresponding author: [email protected]

1581

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D.M. Praia et al

2. General Problem The data reconciliation and estimation problem is described as,

minj\y,m=t^\y{tj)-yj}v-U^-yj] ^

7=0 ^

s. t. dy{t) f dt .y{t) = 0 b[y{t)]=0 g\y{t)]>o where j)(r) represents the vector of estimated functions (reconciled measurements, model parameters and non-measured variables), yj are the discrete measured values, F i s a diagonal matrix of measurement variances,/is a vector of dynamic constraints, h and g are vectors of equality and inequality algebraic constraints, respectively. Several strategies have been proposed to solve similar constrained nonlinear dynamic optimization problems (Biegler and Grossman, 2004). In this work, a sequential strategy is applied to a time moving window. For every sampling time, the differential equation system and the resulting nonlinear optimization problem are solved sequentially using the measured data over the window, until convergence is reached. The solver ESTIMA (Noronha et al., 1993), which is based on a Gauss-Newton method, is employed here to solve the problem. The studied process consists of the bulk polymerization of propylene in a recycled single CSTR (LIPP-SHAC Shell Technology), using high-activity fourth generation Ziegler-Natta catalyst (TiCU/MgCh+PEEB+TEA) to produce polypropylene in liquid propylene (liquid pool or bulk reactor). A simplified model of this process was developed to be used as the dynamic constraints and is given in the appendix, with the corresponding nomenclature. The studied problem involves 15 input variables (m^, rricat, ^TEA> ^PEEB, i^bieed, rrirec, rrieA, rrieB, m^, Teu Terea TeA, TeB, T^u PH2 ), 7 output variablcs {Ca, nipoh XS, MI, L, T, T^o\ 8 initial conditions for each window {Peo, Pao, Cato, TEAQ, NO, XSo, Lo, To) and 2 process parameters {Kp, Q . The values of the parameters and operational conditions can be found in (Prata, 2005) and (Prata et al., 2005).

3. Results Six sets of real plant data were obtained for different production campaigns during distinct periods of time. Some of the obtained results are presented below.

On-Line Data Reconciliation and Parameter Estimation

1583

Figures 1 to 6 present results obtained from a data set recorded over a time interval of 7h (29h to 36h), with a sample time of 5 min, using a time window of2h. 3.0E+04

2.5E+04

•2.0E+04

oo°°^oOoo g£ 00 0

•-. . •••••^

-S^V

00

°°°

0° o*9»S»A 00

o o

° 0

' "8 0

1.5E+04

5.5E+04 5.0E+04 4.5E+04

[\

°0000<^

°

^ .,-.^'^.*^

1

Fig 2. mpoi: Measured (o), reconciled (•). 3.6E-04 3.4E-04

'"•••••$5^^;;;-.. 3.2E-04

***<

3.0E-04

'•4.0E+04 2.8E-04 3.5E+04 2.6E-04 3.0E+04

O

°^ ,

1.0E+04

Fig 1. Ca: Measured (o), reconciled (•), (+) reconciled without EB.

" ^

° '>^—,-„,

**t^

2.4E-04 2.2E-04

'•"-!*++':+•;.

-t;. -t++^+;;;-i+t*

+"+tttttttt+4:-

"•••%:t«ttttt

__, ___,

Fig 3. Kp! Estimated (+),error hmits (-).

Fig 4. C: Estimated (+),error limits (-).

Fig 5. MI: Measured (o), reconciled (•), (+) reconciled without EB.

Fig 6. T: Measured (o), reconciled (•).

^

Figures 1 and 2 show, respectively, the measured and reconciled data of the propane concentration in the recycle stream and of the produced polymer flowrate. For the propane concentration, reconciled data with and without the energy balance (EB) are compared, with much better results for the first case. The estimated values for Kp (a parameter associated with reaction rate), shown in Figure 3, follow the reduction of the polymer production as observed in Figure 2. It should be noted that this piece of information could be useful for plant operation, as it gives an indication of the process behavior (Kp is related to process productivity). Similar behavior can be identified in Figure 4, which represents estimated values of C, an important parameter that is linked to the average molecular weight of the product (see model equations).

1584

D.M. Praia et al

Figure 5 depicts the reconciled values of a final product quality parameter, M. These values are compared with the ones obtained when no energy balance is considered. Again, this comparison favors the results obtained using the energy balance. Finally Figure 6 shows the measured and reconciled data of the reactor temperature. It is important to note that the CPU time required for the calculations of each window is around 40s, a quite small value if compared to the sampling time of 5 min. This result indicates that a further increase on the number of estimated parameters, non-measured variables and routines to deal with gross error detection is undoubtedly possible. The computer used was a Pentium 4(3.0GHz) with 1024 MB memory.

4. Conclusions In this work, an estimation tehcnique for simultaneous on-line data reconciliation and parameter estimation has been implemented. An approach based on moving time window was used, and the resulting non-linear dynamic optimization problem was solved by a sequential strategy. The methodology was successfully applied to reconcile several sets of industrial data as well as to estimate non-measured parameters, like the ones associated to reaction productivity. It should be noted that the computational time required for the calculations on each sampling period was a small fi-action of such period, allowing for on-line implementation. As expected, results have indicated that an increase of redundancy, using energy balance information, results in better reconciled and estimated values.

References Biegler, L. T., Grossmann, I. E., 2004. Retrospective on optimization. Comput. Chem. Eng., 28, 1169. Noronha, F.B., Pinto, J.C, Monteiro, J.L., Lobao, M.W., Santos, T.J., 1993. Urn Pacote Computacional para Estima9ao de Parametros e Projeto de Experimentos, Technical Report, PEQ/COPPE/UFRJ. Prata, D. M., 2005. Reconcilia^ao de dados em um reator de polimeriza9ao. M.Sc. thesis, PEQ/COPPE/ UFRJ, Rio de Janeiro. Prata, D. M., Lima, E. L., Pinto, J. C, 2005. On-line quality and productivity monitoring in a polypropylene production process, 4* Mercosur Congress on Process Systems Enginnering, Mangaratiba, Brazil.

Acknowledgements The authors would like to thank CNPq and CAPES for financial support.

On-Line Data Reconciliation and Parameter Estimation

1585

Appendix The simplified model originally developed (Prata et. al, 2005) was improved incorporating a global energy balance, and is represented by the following equations, ^^^

n

^

dPa dt

Pe

(

%leed

Pa \Pa + Pe •m,bleed

dPol dt

=

^pol-^pol

dCat dt

•Cat

dTEA •= dt

d(XS) dt

\^\'^pol-^d-^^f

TEA Pe + Pa

mrj,,-a

dPEEB ' dt

dt

7?

^.w-a-^)|^j-'^,o/

(PEEB\ I^PPFTt

mpol

I Pol J

pol

Po7 Mi

[PEEB M:

dT_ ^ m^ • Cp^ jT) • {T, -T) + m,, • Cp, (T) • (?;,, -T) + j-AH) • R^, - £ dt [Pa • Cp^ (T) + Pe • Cp^ (T) + Pol • Cp^, (T)) rec ^ '"'bleed

ftpol ~ ^slurry '

slurry

^pol

_Pol ~ M Pa Ca = Pa + Pe

_K^CatPe ^pol

r+c-

Pe Pa + Pe

= PDM"^' Pa

=

M

1.00

M'--= PMPe-

=

pol

M = Pe + Pa + Pol

^po,

rac w

\~w

Pe

MT =

MI = Pol

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K,(M:Y

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V(L) =

r = 2.30010V[(L-16.5%).1.54410'] for K = 9.08810'[1.0610-'(Z) + 2.87 10-'(Z)'] for

Z.^---

m^'Cp^iT)

PEEB,

L>\6.5% L<16.5%

Mnin slurry

Nomenclature C: Ca. Cat: K,: K,: Ko,,K,s: L: M: fl^bleed, nirec'. meat, me, mpEEB,

niTEA-

rrieA, m^B'^poh

m-slurry^-

Mrl, M/'^: MJ": ML N: Pa, Pe, PEEB, Pol: PD: PH2: PMPe: Qe' Rpoh

T: TEA: TeA,

T

TeB-

T

•

Twi, Two'

K:' y^poh

XS: XS^: a: AH:

rX: Xe. pa , pe

Ppoh

kinetic constant for transfer to hydrogen propane concentration in the feed and recycle streams mass of catalyst in the reactor heat capacities of propane, propylene, polymer and water kinetic constant for homo-propagation catalyst deactivation constant parameter reactor volume total mass in the reactor liquid bleed and recycle flowrates input flowrates of catalyst, propylene (with propane traces), TEA and PEEB reflux flowrates of propylene from top condensers output polymer and slurry flowrates water flow rate of top heat exchanger instantaneous and cumulative number average molecular weights cumulative mass average molecular weight melting index of the final resin number of polymer mols masses of propane, propylene, PEEB and polymer in the reactor polydispersity hydrogen pressure in the reactor propylene molecular weight heat exchanged in the condenser rate of polymerization reactor temperature mass of TEA in the reactor temperatures in reflux flowrates of propylene from top condensers temperatures in input and recycle streams of propylene temperatures of input and output water streams from the top heat exchanger reactor volume propane concentration in the recycle stream polymer concentration in the slurry stream xylene extractable material of the final resin xylene extractable material reference value TEA recycle factor heat of reaction parameter parameter latent heat of vaporization of propylene densities of propane, propylene and polymer

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Optimal reactive scheduling of multipurpose, make-to-order industries Marta C. Gomes,^* Ana Paula Barbosa-Povoa,'' Augusto Q. Novais^ ^CESUR, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal ^CEG'IST, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal ^Department of Process Modelling and Simulation, Instituto Nacional de Engenharia, Tecnologia einovagao. Est. Pago do Lumiar, 1649-038 Lisboa, Portugal Abstract This work presents a new, generalized mixed-integer linear programming (MIL?) model for scheduling of multipurpose, make-to-order industries that accounts for the existence of recirculation (multiple visits to the same processing unit) and assembly to obtain final products. A reactive scheduling approach is built on the model and illustrated with an example, where different scenarios for the insertion of new orders in a predetermined schedule are optimally generated. Choosing the most suitable rescheduling policy requires balancing criteria of scheduling efficiency and stability.

1. Introduction Reactive scheduling has received far less attention in the scheduling literature than optimal or near-optimal production schedule generation, the development of which dates back to the 1950s. However, the ability to revise a schedule effectively in order to cope with unexpected events such as machine failures, processing time delays, arrival of new orders and unavailable material is as important as the scheduling problem itself in industrial environments, which are dynamic in nature. Accordingly, interest in reactive scheduling techniques intensified in the 1990s. In this paper, a new strategy for reactive scheduling of multipurpose, make-to-order industries is presented. It combines a MIL? model for batch scheduling and a reactive scheduling algorithm that shifts the scheduling horizon and solves the model to take account of changes in the system. The MIL? model is based on the continuous-time formulation proposed by Manne (1960) for job shop scheduling, recognized as having a good performance and still an object of fiirther developments in the scheduling literature. This was generalized to account for product assembly and recirculation, two features often present in real industrial contexts. The approach is applied to the problem of adapting a schedule to the arrival of new orders, a relevant and challenging issue in order-driven industries. By defining the subset of orders that may be rescheduled, leaving the others unchanged, different alternatives for inserting a set of new orders in an initial schedule can be generated and compared. To assist the user in choosing the most convenient one, scheduling solutions are characterized by measures of efficiency as well as measures of disruption of the previous solution. The approach is illustrated by a numerical example.

Corresponding author. E-mail: [email protected]

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2. P r o b l e m definition Given: • a multipurpose batch plant; • two sets of production orders, /; and I2, with different due dates; • an order may correspond to a single product or to several products that are combined in a final assembly operation; • different sequences of operations (processing routes) that each product follows; • recirculation in some processing routes (multiple visits to the same processing unit); • one processing unit assigned to each operation: products that share the same operation in their processing routes compete for this single resource; • sequence-independent setup times (included in the processing times); • unlimited intermediate storage; • earliness and tardiness costs for the orders; • costs of intermediate storage (for unfinished products); the problem modelled and solved in this paper consists in obtaining a production schedule for an initial set of orders /; (scheduling problem) and then adapting it to insert a set of new orders I2 arrived after production of the initial ones has started (reactive scheduling problem). Scheduling objectives are to finish each order as close as possible to the corresponding due date and to minimize storage of unfinished products in intermediate buffers.

3. Scheduling model The decision variables in Manne's model are the starting time of each job/task on each machine/processing unit (continuous variables) and the sequencing variables (binary variables) that establish precedences between tasks in each processing unit. The notion of global precedence is used instead of immediate precedence. Objective function is the maximum completion time or makespan. Manne's approach for sequencing jobs has been applied in the process industry context; recent examples are the work of Mendez and Cerda (2001), Mendez et al. (2002) and Harjunkoski and Grossmann (2002). Having Manne's model as a basis, the scheduling model developed in this paper additionally accounts for product recirculation, product combination to form an order and due dates for the orders. Indices, parameters, sets: / product J order s stage m processing unit Pim processing time of product / in unit m M(i) set of units m in the processing route of product / (except for the final operation). Variables: Xism Starting time of product / in stage s in processing unit m Yisi's'm equal to 1 if product / in stage s precedes (not necessarily immediately before) product / ' in stage s' in processing unit m\ 0 otherwise Dj delay/tardiness of ordery regarding the due date Aj advance/earliness of order7 regarding the due date.

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In this work, the stage index is used to identify multiple visits to the same processing unit (recirculation): ^ = 1 the first time a product is processed in a given unit, ^ = 2 in the second visit to that unit and so on. Accounting for recirculation implied the addition of a third index/dimension, besides those of products and processing units, to variables X and Y (that correspond to the ones in Manne's formulation). Variables D and A where added to the model following the approach of Zhu e Heady (2000) for the earliness and tardiness problem in a single-stage system of non-identical parallel units. Constraints: • Sequencing constraints impose that an operation in the processing route of a product can only start when the previous operation is finished (except for the first operation). • Disjunctive constraints, written in pairs, ensure that there is no overlap between operations of different products processed in the same unit. • Assembly constraints allow the final assembly operation of orders composed of different products to start only when all the corresponding products are finished. • Earliness and tardiness relations assign the delay and advance variables for every order. Objective Function: It is a weighted sum of earliness, tardiness and the time spent by unfinished products waiting for the next operation to be started; to be minimized:

Z = Y^ia.D.+p.A.) + Y.yXXr-XZ"'- Z pj j

i

(D

meM(i)

where o^ and y^ are the weights of order 7 tardiness and earliness, respectively, ;f is the weight of product / total waiting time in intermediate storage, XisJ^^''''^^ is the starting time of product i in the first operation of its processing route, XisJ^""^^ the starting time of the final operation of product / (an assembly operation in case it belongs to an order composed of different products) and the last term the summation of processing times of product /, except for the final operation in its processing route. 4. Reactive Scheduling Algorithm Based on the model described above, an algorithm was developed that inserts a set of new orders, arrived at a given instant (insertion point), in an existing schedule. It consists of the following steps: 1. 2.

Solve the scheduling model for the set of orders /;. Define: 2.1. The insertion point tj > 0 2.2. The set of new orders to insert, I2 2.3. The set of reschedulable old orders. 3. Fix the Yisrs'm variables for the old orders. 4. Fix the Xjsm variables for the non-reschedulable old orders. 5. Fix the Xjsm variables < t, for the reschedulable old orders. 6. Add constraint Xjsm > t, for the new orders. 7. Solve the resulting scheduling model for the set of orders /; u I2. The scheduling model is solved from scratch for set /; (old orders) and a first scheduling solution is obtained. To insert the set of new orders I2 in the schedule, old orders are divided into those that will be unchanged and those that may be rescheduled (non-

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reschedulable and reschedulable old orders). A new scheduling model is constructed by freezing the sequence of operations for the set of old orders: the values of the Yisi's'm variables are fixed (equal to the ones in the first scheduling solution). Operations for the non-reschedulable old orders are also fixed as well as those for the reschedulable old orders scheduled to start before the insertion point (the JG^m variables are equal to the ones in the first scheduling solution). Constraint Xism ^ tj is added for the new orders (operations cannot start before the insertion point) and the scheduling model is solved for sets /; and h. In short, to obtain the new schedule from the first one operations for the reschedulable old orders may be shifted in time, provided they start fi-om the insertion point onwards, and an operation for a new order may be placed between two operations for the old orders. This algorithm can be included in an iterative procedure that successively updates a schedule to incorporate incoming new orders. 5. Example The figure depicts a multipurpose plant where recirculation and assembly take place; a total of 15 processing units are present. Products may follow four processing routes (A,B,C,D) and two types of final products (orders) are obtained: one by combining a product from route A with a product from route C (assembly 1 in unit M14), the other by joining a product from route B with a product from route D (assembly 2 in unit M15). Routes B and C display recirculation: processing units M3, Mio and Mn are visited twice.

M7Hpr:("M^^

-<:>--TOTable 1 displays the orders to be scheduled, the corresponding due dates and the products that compose them. The initial set of orders /; contains orders Oi, O2, O3, O4 and O5 while the set of new orders to insert, I2 , contains orders O^ and O7, which arrive at instant 20 (insertion point //). Table 2 displays the processing times in each processing route, with processing units arranged in order. The processing time in a unit depends on the product type i.e. on the processing route that it follows. Costs in the objective function considered are: a, = 20 for tardiness; ^j = 1 for earliness and y, = 0.1 for work-in-process time (equal for all orders and products). Three alternatives or scenarios for inserting the set of new orders in the initial schedule were generated, by allowing none, two and all orders to be rescheduled from the insertion point onwards. In the intermediate scenario the old orders allowed to be rescheduled were O3 and O5. The MILP model was implemented in GAMS modelling system and solved to optimality with CPLEX version 8.1.0 on a 3 GHz Pentium 4 running Windows XP Professional. Table 3 summarizes the solutions obtained while table 4 shows model statistics and computational performance.

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Table 1 Order data Due date (days) 40 60 75 90 50 70 90

Order Ol 02 03 06 04 05 07

Processing Route A A A A B B B

Product 1 Dl p2 p3 Pll P7 p8

El3

Processing Route C C

Product 2 D4

P5 p6 pl2 P9 plO pl4

c c D D D

Table 2 Processing times (days) A

Route

^

g o o

OH

Ml M2 M3 M4 Ms M7 Mg

3 4 1 4 4 2 2

C M9 Mio(s=l) M u (s=l) M6 Mio(s=2) M u (s=2) M12

Mi4 ^Assembly 1)

B 5 7 6 3 5 4 3

Ml M3 (S=l) Ms Me M7 M3 (s=2) Mg

10

D 3 1 5 5 2 3 2

Mis (Assembly 2)

4 5 8 5 6 7

M3 M4 Mio Mu Mi3 M12 15

Table 3 Summary of results Number of Scenario reschedulable old orders 1 0 2 2 3 5

1202.5 1062.7 1062.6

Number of Tardiness Waiting time Old New -p^j^, Old New ^,^^^1 Orders Operations orders orders orders orders changed changed 27 33 60 15 10 25 0 0 30 23 53 18 9 27 2 15 30 23 53 22 4 26 4 22

Table 4 Model statistics and computational performance Number of variables Model Initial solution New order insertion

Number of

Single

Discrete

equations

CPU time (sec)

259 570

165 183

414 933

1.28 0.14-0.48

The initial solution and the one obtained in scenario 2 are depicted in the next page. Operations in each processing unit are represented by rectangles containing the start and finish times; waiting times between operations are also shown. Differences in the second solution compared with the first one, for the old orders, are shaded in grey. By comparing scenario 1 with scenarios 2 and 3 the conclusion can be drawn that allowing for rescheduling of the old orders results in a better insertion of the new orders (smaller values for tardiness and waiting time) at the cost of a slight deterioration of the same performance measures for the old orders; consequently there is a decrease in the optimal value of the objective function. Earliness is zero in all solutions. Scenario 2 is only slightly worse than scenario 3 (equal tardiness; one more unit in total waiting time) but changes in the initial schedule are smaller: 15 operations are changed in products p3, p6, p8 and plO which correspond to 2 orders (03 and 05) while scenario 3 displays changes in 22 operations in products p2, p3, p6, p7 and plO

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affecting 4 orders (02, 03, 04 and 05 ). Thus, scenario 2 is more favourable for new order insertion than scenario 3 since changes to the initial schedule are more restricted. Initial solution: Route/ Product A p1 P2 p3

M1

Wait

M3

Wait

M4

Wait

iM5

Wait

M7

Wait

M8

W

Assembly 1 Tardiness

0

16

20

0

20

21

0

21

25

0

25

29

0

29

31

0

31

33

0

~33

42

45

0

45

49

0

50

54

0

54

58

0

58

60

0

60

62

0

62

72

55

0

55

59

0

50 60

0

52

49 59

0

60

64

0

64

68

0

68

70

0

70

72

0

72

82

Wait

IM3 ( s s l )

Wait

Wait

IM5

iM6

Wait

M7

Wait

M3 {S''!)

Wait

M8

Wait

16

19

0

19

20

0

20

25

0

25

30

1

31

33

0

33

36

0

36

38

33

36

0

36

37

0

37

42

0

42

47

1

48

50

0

50

53

0

53

55

Wait

M9

p4 p5

Wait

16

IMI

C

M2

13

0

5

0

29 37

34 42 M3

IVI10(s=1) Wait

M i l (s=1) Wait

M6

21

47

50

0

55

58

0

0

12

18

0

18

0

34

41

0

41

47

0

0

42

49

0

49

55

0

M4

Wait

Wait

M10

M10 (s=2) Wait

0

12

Wait

Wait 21

5

Mil

M i l (s=2) Wait 30

0

55

59

0

63

67

0

26

0

26

50

55

0

58

63

0

Wait

M13

Wait

4

8

0

8

13

0

13

21

0

21

26

0

26

32

12

16

0

16

21

5

26

34

2

36

41

0

41

47

25 54

25

29

\m m 88

\m

0

M12

M12 30

43^

Assembly 2

>

40

55

55

70

Wait 33

0

59

62

0

67

70

2

Wait

33

40

47

54

29 58

31 60

31 60

\n: 92

•'n\

\n94

33 62 •m\ 96

36 53 78

38 55 80

Insertion of new orders (scenario 2): 13

Assembly 1 Tardiness

M4

Ml

duct

16

16 45

20 49

179m

m\ 83

pi P2 p3

42

45

is

m 1

p11

76

79

P7 p8

16

19

33

36

36

20 37

pi 3

58

61

61

62

p4

0

5

p5

29

Wait

M9

M3(s=1) 19

Wait

20 49

50

j '12.

84

Wait

M5 20 37 62

5

12

0

12

34

0

34

41

0

42

49

68

75

p6

37

42

pi 2

63

68

0

p9 plO p14

4 12 41

M3

Wait 8 16 45

0 0 0

M4 8 16 45

Walt 25 42 67

58

fti 92

88

Wait 25 42

30 47

67

72

1

Wait 33

0

l i i i liliii i M I i i i i 1 Walt

M6

M7 31 73

75

0

M10 (s=2) Wait

94

M3 (s=2) 33 50

36 53

75

78

Wait

Assembly 2 Tardiness

M i l (s=2) Walt

M12

l-H 55

0

18

21

0

21

26

0

26

30

0

30

33

41

0

47

50

0

50

55

0

55

59

0

59

62

0

0

49

55

0

55

58

0

81

0

81

84

n 93

n

e

75

«i n 89 93

0

0

96

0 5 5

Wait

M10 13 26 55

21 34 63

Mil

Walt

0

21

0

u #

»

63

$ n u 0 84 89

26 68

0 0 1

M13 26

0 Wait

32

""W"""m' 69

75

1 0 0

0

70

Wait

18 47

Wait 13 21 50

54

1 ^^

M i l (s=1) Wait

0 0

si 1

83

M10(s=1) Wait

21 50

21

0

0

Ml2 33

40

if

".75 m 82 m' \

6. Conclusions and future work An MILP continuous-time formulation for scheduUng of multipurpose, make-to-order industries with recirculation and assembly is presented, and a Reactive Scheduling Algorithm proposed to address the problem of inserting new orders in a previously determined schedule. The usefulness of the approach as a decision support tool for scheduling is illustrated by creating different scenarios for new order insertion in a medium size example. In the future, the model should be extended to the case where multiple units are available per operation. The improvements of Liao and You (1992) to the model of Manne (lower and upper bound calculations) should be considered to increase model performance when applied to larger problems. Other situations should be modelled like temporary unavailability of processing units and changes in order specifications (due date, demand) and order canceling, which are relevant in the make-to-order sector.

References 1. 2. 3. 4. 5. 6.

Harjunkoski, I. and Grossmann, I.E. (2002), Comp. & Chem. Eng., 26, 1533-1552. Liao, C.-J. and You, C.-T. (1992), J. Operational Research Soc, 43, 1047-1054. Manne, A. S. (I960), Operations Research, 8, 219-223. Mendez, C.A. and Cerda, J. (2002), Comp. & Chem. Eng., 26, 687-695. Mendez, C.A., Henning, G.P. and Cerda, J. (2001), Comp.&Chem.Eng, 25, 701-711. Zhu, Z. and Heady, R.B. (2000), Comp. & Industrial Eng, 38, 297-305.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Chapter 1

Optimal steady-state transitions under constrained predictive control David K Lanf and Christopher L.E. Swartz^ "^Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton L8S 4L7, Canada There has been an increasing focus in recent years on the design and operation of chemical process plants that need to respond rapidly to frequently changing market demands. We consider in this paper the transition from one steady-state operating point to another, with constrained predictive control in use as the regulatory control system. An automation structure is proposed in which the optimal transitions are implemented through specification of appropriate set-point trajectories determined at an upper level. A strategy for computing optimal reference trajectories is presented that takes into account the dynamics of the underlying closed-loop control system. Its effectiveness for determining optimal transitions is illustrated through example problems. 1. Introduction There are increasing economic incentives for demand driven operation with product diversification in the chemical industry, which requires flexible operation in responsive plants [1]. In particular, the ability to optimize transitions between product grades in response to changes in demand while satisfying operational, safety and product quality constraints, is a key component to maximizing economic performance in a competitive market. Backx et al. [1] advocate the use of model-based regulatory control in conjunction with dynamic real-time optimization for such intentionally dynamic, market-driven operations. McAuley and MacGregor [2] consider optimal grade transitions in a gas phase polyethylene reactor. They compute optimal input trajectories as the solution of a dynamic optimization problem that minimizes an economic based objective function. Feedback control on quality variables is not included, which the authors demonstrate could lead to suboptimal policies and offset in the presence of plant/model mismatch or disturbances. Chatzidoukas et al. [3] pose an optimization problem for optimal grade transitions in which input trajectories, a control structure and controller tuning parameters are simultaneously computed. Integer variables are used for the control structure selection, resulting in a mixed-integer dynamic optimization (MIDO) problem. Multi-loop PI control is used to control four output variables, with the density and melt index (product quality variables) controlled via open-loop optimal control.

D.K. Lam and C.L.E. Swartz

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In this paper, we consider optimal steady-state transitions where the product quality variables are controlled via constrained model predictive control (MPC). Optimal setpoint trajectories are computed, taking into account the dynamics of the underlying closed-loop control system. This is illustrated in Figure 1 as an additional layer in the standard process automation hierarchy [4,5], in which optimal set-point trajectories are determined to move the plant to a target steady-state determined at the real-time optimization (RTO) level. This results in a multi-level optimization problem which we solve using an interior point solution approach. In the sequel, we present the mathematical formulation and illustrate its application through two example problems.

C V measurements and disturbance estimates

Real time optimizer Determines optimal economic setpoint targets subject to slow disturbances, input and output constraints, and a steaj^-state process model CV targets

Dynamic optimizer CV setpoints

Determines the optimal and feasible setpoint trajectory subject to input and output constraints, constrained MPC and a dynamic process model

Model predictive controller Minimization of control objective function subject to fast disturbances, input and output constraints and dynamic process model Optimal MVs

Process Under mvestigation for steady-state transitions mmmM

Figure 1. Proposed optimization and control hierarchy for steady-state transitions. 2. Mathematical formulation The steady-state transition problem that computes optimal set-point trajectories subject to constraints on the closed-loop response of a system controlled using constrained MPC may be posed as follows:

min 0(y(*),u(*))

(1)

S.t.

ymin < y(k) < Ym:

(2)

Umin < U ( ^ ) < U„

(3)

rmin
(4)

Optimal Steady-State Transitions Under Constrained Predictive Control

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x^(^+l) = f(x,(^),u(^))

(5)

y{k) = g{xjk))

(6)

u(^) = \i{n(k-\\y{k\r{k)\

k = 0,...,K

(7)

In the above, (2) and (3) are constraints on the closed-loop output and input trajectories, and (5) and (6) comprise the dynamic plant model. The plant inputs in (7) are generated through the solution of an MPC open-loop optimal control problem. We consider quadratic dynamic matrix control (QDMC) [6] with input constraints. The inclusion of hard output constraints within the MPC formulation could result in closed-loop instability [7], and are therefore applied here in the outer optimization problem through constraints (2). The MPC calculation at each time step may be formulated as a quadratic program (QP) of the form, min x^Hx+g^x

(8)

X

s.t.

^x = b x>0,

(9) (10)

where the time index k has been omitted for clarity of notation. We reformulate the multi-level optimization problem by replacing the QP subproblems given by (8)-(10) by their Karush-Kuhn-Tucker (KKT) conditions, Hx-A^\+g-yv == 0

(11)

Ax = h

(12)

WiXi = 0

(13)

(w,x) > 0,

(14)

which transforms the multi-level optimization problem into a single mathematical programming problem with complementarity constraints (MPCC). We then solve the resulting system using an interior point approach in which the complementarity constraints are relaxed as w^X/ < Sju with ju driven to zero as the algorithm iterates toward the solution. The implementation in IPOPT-C [8] was found to be highly effective for problems of this type [9], and was used in this work.

D.K. Lam and C.L.E. Swartz

1596 3. Application

3.1. Example 1 - SISO System A single-input single-output system comprising three continuously stirred tank reactors in series, based on a problem in [10], was considered. It is desired to change the target mole fraction of component ^ in the product stream, XAS, from 3% to 4%, with an output constraint on product quality ofxAs < 4%. The output, XA3, is controlled by manipulating the valve position of an inlet stream, with the plant transfer function given by

As)=

0.039 v(.)(5^ + 1)^

(15)

The initial steady-state valve position was taken to be v = 20% open. The system was controlled using constrained model predictive control with the manipulated variables constrained to [0,80] % open, executed every 2 minutes with a prediction horizon of 30 and an input horizon of 10. The initial tuning with an output to input move suppression weighting ratio of 100:1 resulted in a 7% overshoot relative to the set-point change. Detuning the controller results in a feasible response, but with an increased settling time. However, a feasible transition can be achieved while maintaining the more aggressive controller tuning by using the reference trajectory optimization formulation given in the previous section. Figure 2 shows the set-point trajectory and resulting closed-loop response using a transition objective function comprised of the sum of squared deviations between the output and set-point target at each time step over the simulation horizon. 4.2

'

'

^

'

100

'

4.0 ^ £

3.8 • i

1 3-6

£ 2 3.4 3.2 3.0

j

/

/

/

"'i iy

C

'

_

80

-

^

60

-

1. 40

-

/

-i i .i i

'

-

/

/ /

-

/ 10

20

30 Time (min)

40

,

50

60

>

-

20 0

[]

10

20

30 Time (min)

40

50

6

Figure 2. Optimal reference trajectory and corresponding closed-loop response.

3.2. Example 2 - MIMO System The multi-input multi-output application considered involves a styrene polymerization reaction process based on [11] and [12], and shown in Figure 3. The nonlinear system was approximated by a linear transfer function model, which is used in this study. The number average molecular weight (NAMW) and reactor temperature (7) are controlled by manipulating the initiator flow rate (Qt) and the coolant flow rate (Qc), consistent with the control structure in [11]. The objective in our study is to determine an optimal set-point trajectory for a change in NAMW, with set-

Optimal Steady-State Transitions Under Constrained Predictive Control

1597

point tracking implemented through constrained MPC. The following constraints are imposed, a G [0,150], e,G [0,500],

NAMW^

[50,80],

T E [323,324]

(16)

with only the input constraints applied at the MPC level.

> Effluent Q (Ml [I] T

Figure 3. Styrene polymerization in continuously stirred tank reactor. The response using set-point trajectory optimization is shown in Figure 4, where a weighted sum of squared deviations of the outputs from their targets is minimized, subject to path constraints on the closed-loop response. Direct application of the single target set-point to the MPC controller (with the same tuning) without reference trajectory optimization results in a temperature constraint violation. 4. Conclusions The cost of transitions between steady-state operating points becomes a significant factor in the economics of process operations that are required to respond to rapidly changing market demands. In this paper, a mathematical formulation for reference trajectory optimization with consideration of the closed-loop dynamics of constrained model predictive control was presented. The supervisory controller aims to achieve feasible and optimal operation, while enabling the regulatory controller to remain unaltered both in structure and tuning, thus retaining capability for disturbance rejection. The application of the methodology was illustrated on two example problems with linear dynamics. Extensions currently under development include application to nonlinear systems, explicit incorporation of economics in the reference trajectory optimization problem, and the use of feedback at the supervisory level to compensate for plant/model mismatch.

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20

30 Time (h)

Swartz

40

Figure 4. Optimal reference trajectory and corresponding closed-loop response.

References 1. T. Backx, O. Bosgra and W. Marquardt, IF AC Symposium Advanced Control of Chemical Processes, Vol. 1, 2000, pp. 249-260. 2. K.B. McAuley and J.F. MacGregor, AIChE J. 38 (1992) 1564. 3. C. Chatzidoukas, J.D. Perkins, E.N. Pistikopoulos and C. Kiparissides, Chem. Eng. Sci. 58 (2003)3643. 4. T.E. Marlin and A.N. Hrymak, In: J.C. Kantor, C.E. Garcia and B. Camahan (Eds.), Fifth International Conference on Chemical Process Control, AIChE Symposium Series Vol. 97, 1997, pp. 156-164. 5. S.J. Qin and T.A. Badgwell, Control Engineering Practice 11 (2003) 733. 6. C.E. Garcia and M. Morshedi, Chem. Eng. Commun. 46 (1986) 73. 7. E. Zafiriou and A.L. Marchal, AIChE J. 37 (1991) 1550. 8. A.U. Raghunathan and L.T. Biegler, Comp. Chem. Eng. 27 (2003) 1381. 9. R. Baker and C.L.E. Swartz, AIChE Annual Meeting, Cincinnati, 2005. 10. T.E. Marlin, Process control: Designing processes and control systems for dynamic performance 2"^ edition, McGraw-Hill Inc., New York, 2000. 11. B.R. Maner, F.J. Doyle, B.A. Ogunnaike and R.K. Pearson, Automatica 32 (1996) 1285. 12. P.M. Hidalgo and C.B. Brosilow, Comp. Chem. Eng. 14 (1990) 481.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Optimal Configuration of Artificial Neural Networks Vivek Dua Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London JVCIE 7JE, United Kingdom Abstract In this work a mixed-integer programming approach for training the network is presented. This approach reHes on modelhng the existence or non-existence of nodes by introducing 0-1 binary variables. The interconnection between the nodes and the layers is also similarly modelled. This results in a mixed integer program where the objective is not only the minimization of the error but also the number of nodes. The key advantage of this approach is that a reduced number of nodes and a much simplified network are obtained, for a similar performance in terms of minimization of the error. Keywords: Training of Artificial Neural Networks, Optimization, Mixed Integer Programming

1. Introduction Artificial Neural Networks (ANN) are mathematical models that mimic simple biological nervous systems and have been extensively used in many applications including system identification and model reduction (Prasad and Bequette, 2003), control of chemical processes (Hussain and Kershenbaum, 2000; Hussain et al., 2003) and fault detection and diagnosis (Rengaswamy and Venkatasubramanian, 2000). A network consists of a number of layers and each layer consists of a number of neurodes where neurodes are considered to be similar to neurons. Each neurode receives a number of inputs each of which is weighted, summed and then modified by an internal transfer function to give an output. This output then becomes either an input to another neurode or the result itself. A typical architecture of ANN which consists of interconnected neurodes is shown in Figure 1 where neurodes are denoted by circles. There is an input layer which receives the data and an output layer which gives the response of the network to the data. The transformation of the input data to the output response is facilitated via an extra neurode, known as bias, and one or more hidden layers. The output of an ANN is determined by the architecture of the network, internal transfer functions of the neurodes and the weights on the nodes. For a fixed architecture and internal transfer functions, the error between the output response of the network and the correct output is minimised by iteratively computing the weights and biases. This is known as training the network. Shang and Wah (1996) presented a global optimization technique for training the network. Teng and Wah (1996) presented cascade-correlation learning and population-based learning mechanisms for reducing the number of hidden units for the case when there are binary outputs and Kim and Park (1995) proposed an expand-and-tmncate learning algorithm that guarantees convergence for any binary-tobinary mapping and automatically determines the number of nodes required in the hidden layer. In this paper a mixed-integer programming approach for training the network is presented, where the existence or non-existence of the nodes is modeled by

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introducing 0-1 binary variables and the objectives are to minimize the number of nodes and minimize the error between the ANN prediction and desired output. This can result in reduction in the computational time required by the ANN to compute the output for the given inputs. The rest of the paper is organized as follows: the next section presents the proposed approach, an illustrative example is given in section 3 and concluding remarks are presented in section 4.

•

OUTPUT

BIAS

Input Layer

Hidden Layer

Output Layer

Figure 1. Artificial Neural Network

2. Mixed-Integer Programming Approach for ANN Let Xi denote the input values to the network where / = l,...,A^jc combinations of these inputs gives the activation variables:

and Nn linear

i=\

where A^„ is the number of nodes in the hidden layer, the superscript 1 denotes the index of the hidden layer, Wji are the weights and bj the biases. These activation variables are then transformed non-linearly to provide: h)=i2ir)h{a)) where /?| is the output of the first hidden layer; note that nonlinear transformations other than tanh are also used in the literature, /zj becomes the input to next hidden layer such that:

optimal Configuration of Artificial Neural Networks

1601

where a^ denote the activation variables of the second hidden layer which are also transformed nonlinearly and then become the input to third hidden layer. Similarly the activation variables, a^^, for Nh, the last hidden layer, are given by:

af^='zwJ/hf^-'+bf^

j = l,...,N„

i=\

which are transformed nonlinearly to obtain:

hf' = tmh(af') which are then combined to provide the outputs:

u,='zWk,hf^+B,

k = l,...,N,

where No is the number of outputs and Wki and B^ are the weights and biases respectively. Let w^ denote the desired output, the training of the network can then be formulated as the following optimization problem:

mm

E=

a,b,w,n,W ,B,u

Tiuk-uj,) j^^i

subject to equations described earlier, where E is the error function. In this formulation it is assumed that number of hidden layers and the number of nodes in each of the layers is given. The configuration of the network can be optimized so that the sum of the number of the layers and nodes is minimized and yet E is within a pre-specified tolerance. This can be achieved by introducing 0-1 binary variables as follows: -M'y]
y = l,...,iV„ ,/ = !,..., iV,

where Mis a large positive number, y^j is the vector of 0-1 binary variables (Floudas, 1995, p239). Note that if jy = 1 then a^ is unrestricted whereas if y^j = 0 then a^ = 0 which implies thaty^*^ node of the f^ hidden layer does not exist. Optimization of neural network configuration can then be formulated as follows: NhN„

mmZXj/ y

i=\j=i

subject to E less than or equal to the pre-specified tolerance, equations for activation variables and the logical conditions representing the existence or non-existence of the nodes. Some simplifications can be introduced in this formulation, such as, if a hidden layer does not exist then the outputs from that layer are equal to the inputs to that layer.

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Note this is mixed-integer nonlinear program which can be solved by using outerapproximation algorithm (Viswanathan and Grossmann, 1990); for the case when no nonlinearities are present then a mixed-integer linear program is obtained which, for example, can be solved by using branch and bound algorithms. These mixed-integer programs aim at minimizing the number of nodes and similarly the problem can be formulated so as to minimize the number of interconnections between the nodes by introducing the following constraints: -M'd],<w),<M-S),

7 = 1,...,7V,

for w^ji, where Sj^ are the 0-1 binary variables, introducing similar constraints for b, W and B and minimizing the summation of 0-1 binary variables. The key advantage of the formulations presented in this work are that a reduced configuration of the network is obtained which will result in reduced computational effort for computing the outputs for given inputs. Another feature is that these formulations can be augmented by logical conditions which emanate from the engineering understanding of the system under consideration, for example, correlations such as "if an input is greater than or equal to a certain value then an output should be less than or equal to some given value". This can not generally be guaranteed by formulating the problem in the traditional setting where the objective is to minimize E. The next section presents an example to illustrate the main ideas of the proposed formulation.

3. Example: 3-bit Binary-to-Binary Mapping Function Consider a binary-to-binary mapping fiinction where the inputs, X/, and the outputs, Uk, are 0-1 binary numbers. The activation variables, QJ '', take a value of I if aj^ > 0 and 0 otherwise. The values of Uk are similarly decided. Consider a function of three inputs such that if inputs are {000, 010, Oil, 111} then the output is 1; if the inputs are {001, 100, 110} then the output is 0; if the input is {101} then we are not concerned what the output is. Kim et al. (1998) reported a solution for this problem which is given in Table 1 and Figure 2 where the numbers on the interconnections are the weights and the numbers in the circles are the biases; the interconnections with zero weight are not shown; the problem was formulated such that the weights and the biases are integer variables. Table 1. Input-Output Mapping for 3-bit Binary Function (Kim et al., 1998) Xi

000,010,011 001,100, n o 111

u 1 0 1

ai

Cl2

1 0 0

1 1 0

u 1 0 1

In this work the problem was formulated so as to minimize the number of interconnections for one hidden layer such that the output of the network is equal to the desired output and the weights and biases are integer variables; this was modeled by

Optimal Configuration of Artificial Neural

Xi

Networks

X2

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X3

Figure 2. The configuration of a three-layer network for 3-bit example (Kim et al., 1998)

Xi

X2

Figure 3. Optimal Network Configuration for 3-bit Example

X3

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V. Dua

expressing weights and biases in terms of binary variables (Floudas, 1995, p i l l ) . The lower bounds of -6 and 0 and upper bounds of 9 and 7 were considered on weights and biases respectively. This resulted in a mixed-integer linear program which was solved by using GAMS/CPLEX (Brooke et al., 1998). The results are shown in Figure 3, a reduced configuration with one interconnection less than that reported by Kim et al. (1998) is obtained.

4. Concluding Remarks A mixed-integer programming approach for optimizing the configuration of neural networks has been presented. The basic idea is to model the existence of nodes and interconnections by introducing 0-1 binary variables and minimizing these binary variables. This results in a reduced configuration that still satisfies the error criteria specified by the user. The reduced configuration would take less computational effort for computing the outputs for the given inputs. Although not discussed in this paper, it is recognized that more than one network configuration may be optimal and they can be obtained by recursively introducing integer cuts. Another important issue is that logical conditions relating certain inputs and outputs can also be incorporated in the proposed formulations.

References A. Brooke, D. Kendrick, A. Meeraus and R. Raman, 1998, GAMS: a user's guide, GAMS development corporation, Washington. C.A. Floudas, 1995, Nonlinear and mixed-integer optimization, Oxford University Press, New York. M.A. Hussain and L.S. Kershenbaum, 2000, Implementation of inverse-model-based control strategy using neural networks on a partially simulated exothermic reactor, Trasanctions of the Institution of Chemical Engineers (Part A), 78, 299. M.A. Hussain, C. Ng, N. Aziz and I.M. Mujtaba, 2003, Neural network techniques and applications in chemical process control systems. Intelligent systems techniques and applications, 5, 326-362, CRC Press. J.H. Kim and S.-W. Park, 1995, The geometrical learning of binary neural networks, IEEE Trasanctions on Neural Networks, 6(1), 237-247. J.W. Kim, S.-W. Park, H. Oh and Y. Han, 1998, Synthesis of three-layer threshold networks, in Algorithms and Architectures, C.T. Leondes (Editor), Academic Press, San Diego. V. Prasad and B.W. Bequette, 2003, Nonlinear system identification and model reduction using artificial neural networks. Computers and Chemical Engineering, 27, 1741-1754. R. Rengaswamy and V. Venkatasubramanian, 2000, A fast training neural network and its updation for incipient fault detection and diagnosis. Computers and Chemical Engineering, 24(2-7), 431-437. Y. Shang and B.W. Wah, 1996, Global optimization for neural netwrok training, IEEE Computer, 29(3), 45-54. C.-C. Teng and B.W. Wah, 1996, Automated learning for reducing the configuration of a feedforward neural network, IEEE Transactions on Neural Networks, 7(5), 1072-1085. J. Viswanathan and I.E. Grossmann, 1990, A combined penalty function and outer-approximation method for MINLP optimization. Computers and Chemical Engineering, 14, 769.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Diagnosis of oscillations in process control loops Yoshiyuki Yamashita ^ ^Department of Chemical Engineering, Tohoku University, 6-6-07 Aramaki Aoba, Sendai 980-8579, Japan Abstract Valve stiction is the most common cause of the problem on control loops in process industry. To improve the productivity and quality, industrial engineers demand a tool to facilitate loop monitoring. This paper present a new algorithm to detect valve stiction for diagnosis of oscillation in control loops. The method is for the level control loops and based on a statistical analysis in phase plane by using controller output and level signals, those are available for all the level control loops. The usefulness of the method are successfully demonstrated on a simulation data set and several industrial data sets. Keywords: fault diagnosis, process monitoring, control performance 1. INTRODUCTION Among many control loops in a process plant, quite a few loops have oscillatory behavior. Notwithstanding, it is usually too time consuming and sometimes too difficult to maintain all the control loops in proper working order. Therefore, to improve the productivity and quality, industrial engineers demand a tool to facilitate loop monitoring. These oscillations can be caused by improper controller tuning, external disturbances, and so-called stiction in a control valve. For example, more than 20% of all control loops in paper mills reportedly oscillate because of valve stiction. Detection of oscillations in process control loops have been investigated by many researchers. The next step of the loop monitoring is to indicate likely causes of the oscillations. Several methods have been reported to identify the causes. When the valve position or the corresponding flow rate is available, plot of controller output v.s. valve position or corresponding flow rate represents characteristic pattern for oscillatory loops caused by valve stiction and therefore this pattern can be used for the identification of the cause [1]. Unfortunately the valve position or the corresponding flow rate is not often measured in real plant. Therefore, a method is highly required to detect valve stiction without using this information. Several methods have been proposed for the detection of valve stiction without requiring the valve position. Horch and Isaksson developed a detection method based on the probabiUty density function of the second derivative of the process output [2]. Singhal and Salsbury proposed a simple method to check if the shape of the process output signal is similar to

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Controller Output

(b) Figure 1. IVpical plots for a valve stiction loop

\ -\9

^ (a)

(b)

Figure 2. Typical plots for a non-stiction loop

sinusoidal or not [4] . Rossi and Scali used square sum of the differences between typical oscillatory patterns and the observed data [5]. In this study, a method is proposed to detect stiction in a control loop by using only information of controller input and output. As the result of investigation on the behavior of the controller input and output in polar coordinate, the distribution of the sampling points was found to be lopsided for the oscillation caused by valve stiction, although it is almost symmetrical for the loop with bad tuning. This characteristics can be well displayed in an angle histogram of the controller input and output. Moreover an index to detect stiction is developed based on the skewness of the histogram. The method is illustrated with several level control loops of simulation and industrial plant. 2. Method 2.1. Observation Atfirst,oscillating data in level control loops are visually inspected. If the valve position or correspondingflowrateis available, it is relatively easy to find valve stiction because the plot of controller output and the valve position shows typical parallelogram shape as shown in Fig. 1(a). Automated method for detection of stiction based on these variables is also proposed [1]. Figure 2 shows plots of another oscillatory control loop, which does not include stiction. In this loop, corresponding input-output plots does not show parallelogram shape (Fig 2(b)). However, valve position or correspondingflowrateis not always measured in the industrial plant. Even if the valve position is not available, controller output and controlled measurement variable are always available. Comparing the Figures 1(b) and 2(b), it may be difficult to find typical characteristics for stiction in a plot between controller output and level signal. To find characteristic features for stiction, other plots are investigated. Let the gravity

Diagnosis of Oscillations in Process Control Loops

^

1607

'•^•^.

Si

- 4 - 3 - 2 - 1 0 1 2 3 4

(a) stiction

(b) non-stiction

Figure 3. Plots converted in polar coordinate

center of the controller-output and the level plots be the new origin, the original plots can be converted into polar coordinate. Figure 3 shows the polar plots of the two loops. By comparing these twofigures,distributions of the sampling points is found to be lopsided for the oscillation caused by valve stiction, although it was almost symmetrical for the loop with bad tuning. This characteristics can be well displayed in an angle histogram of the controller input and output. Fast motion in the slip jump probably causes this uneven distribution of stiction pattern. 2.2. Algorithm Based on the above mentioned observation, a method to identify stiction is developed. The following steps shows the details of the process. 1. Normalization: Let A?i and A^ be each time differences of controller output {u) and tank level {y) during the sampling period. Both of the values are normalized between zero and one by using their mean and standard deviation. 2. Polar conversion: Plots in /S.u v.s. Ay plane are converted in polar coordinate r v.s. 6 as r = v^Au2~+~Ay2, i9 = tan"^(A2//A'?x)

(1) (2)

3. Histogram: To clarify the distribution of the sampling points in polar coordinate, number of samplings are counted for each interval of 6, which is divided into 100 intervals. If the histogram is symmetric, the loop oscillate because of poor tuning. If it is asymmetric, the loop will have stiction. The same information can also be visually shown in rose plot, which is a polar plot showing the distribution of values grouped according to their numeric range. Rose plot is useful to find asymmetrical distribution by human observation. Since the trajectory in polar coordinate is periodic in the 6 direction it is possible to fold into half plane and get 0: if(9>0, e if(9<0.

(3)

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4. Stiction Index: To represent an asymmetrical distribution in the half plane polar histogram, an index based on the skewness is defined:

h=

1

N-l

(4)

where (f)i is the count of each class interval, 0 and a are the mean and the standard deviation of (t)i. If this index shows relatively large value, the loop will have valve stiction. 3. Case Studies 3.1. Simulation Data The method first applied to simulation data sets, which are the closed loop response of a PI controller combined with a stiction model [3]. Two types of simulation data were prepared corresponding to the behavior of the normal and stiction. Input-output behavior of the data are shown in Figure 4. For these two data sets, histogram in polar coordinates were calculated as Figure 5. Although both of the figures may seem similar, by concentrating attention on both sides of the peaks, one can find that the stiction case shows higher asymmetry than the normal case. Calculated stiction index for normal case is 0.195 and that for stiction case is 0.401. This result shows that this index have larger value for stiction case than normal case. It indicates the possibility of the usage of this index for detecting a stiction.

- 2 - 1 0 1 Controller Output

Controller Output

(a) normal (b) stiction Figure 4. Input and output of the controller (simulation data)

(a) normal (b) stiction Figure 5. Histogram in the polar coordinate (simulation data)

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Diagnosis of Oscillations in Process Control Loops

Controller Output

(a) Case 1 (a) Case 2 (a) Case 3 (b) Case 4 Figure 6. Controller output v.s. Level of the industrial data sets

1,0

-0.5

0.0

0.5

e/7t

(a) Case 1

1.0

-1.0

-0.5

0,0

9/re

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

e/jt

(b) Case 2 (c) Case 3 Figure 7. Polar plots of industrial data sets

-1.0

-0.5

0.0

0.5

1.0

e/7c

(d) Case 4

3.2. Industrial Data This section presents an evaluation of the method using four industrial data sets [1], Each of the data sets is of level control loop and has 1440 samples at 1-min intervals. Figure 6 shows plots of controller output and controlled level signals. All the data are normalized divided by its standard deviation, respectively. In this figure, only the Case 1 is of stiction loop. Case 2 and 3 oscillate because of bad tuning. Case 4 is considered to be normal. Figure 7 shows the plots in polar coordinate. Figure 8 shows histogram of the polar plots. This figure is also represented in the angle histogram as shown in Fig 9. These two histograms represent the same information in different forms. Case 1 shows two peaks in this histogram and these shapes are clearly asymmetrical, which is the typical characteristics for valve stiction. Case 2 did not show sharp peaks, which are not for stiction. The histogram for Case 3 data did not show clear characteristics but the corresponding angle histogram shows symmetrical shape and indicates that the loop does not have stiction. The histogram for Case 4 shows relatively clear peaks but their shapes are symmetrical, which are not for typical stiction characteristics. From these results, calculated values of the proposed stiction index are summarised in Table 1. The index showed largest value for the stiction loop. It indicates that this stiction index can be used as an measure to detect stiction. Table 1 Stiction Index for the industrial data sets Data Set Case 1 (Stiction) 0.374 Case 2 (Bad tuning) 0.050 Case 3 (Bad tuning) 0.271 Case 4 (Normal) 0.124

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I •2 10

(a) Case 1

0.01

o

0.00

< -0.01 -0.02 0.00 0.02 AController Output

(a) Case 1

(d) Case 4

(b) Case 2 (c) Case 3 Figure 8. Histograms for industrial data sets

^

^

'

0.02

Jl# 5

^MK

<

^ % .

-0.01 0.00 0.01 AController Output

0.00

-0.02 -0.02 0.00 0.02 AController Output

(b) Case 2 (c) Case 3 Figure 9. Angle histograms of industrial data sets

w w

•

•

-0.02 0.00 0.02 AController Output

(d) Case 4

4. Conclusion To identify the root cause of an oscillating level control loop, polar plots of controller output and level signal are investigated. Based on the visual inspection of the plots, a method for diagnosing valve stiction for level control loop was developed. The method is based on a statistical analysis of the polar plots. The proposed index showed good performance in detecting stiction on a simulation data set and several industrial data sets. Acknowledgements The author acknowledge to Mr. Toshio Miura for his help in preparing the manuscript. Mr. H. Kugemoto is acknowledged for providing the industrial data set. Dr. M. Kano and Mr. H. Maruta are acknowledged for the contribution of simulation data.

REFERENCES [1] [2] [3] [4] [5] [6]

Y. Yamashita, Control Engineering Practice, 14 (2006) 503. A. Horch and A.J. Isaksson, European Control Conference, Porto, Portugal (2001) 1327. M. Kano, M. Maruta, K. Shimizu and H. Kugemoto, DYC0PS7, Cambridge, USA (2004). A. Singhal and T.I. Salsbury, J. Process Control, 15 (2005) 371. M. Rossi and C. ScaH, J. Process Control, 15 (2005) 505. M.A.A.S. Choudhury, N.F. Thomhill and S.L. Shah, Control Engineering Practice, 13 (2005)641.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Advances and Future Directions in Morphology Monitoring and Control of Organic Crystals Grown from Solution Xue Z Wang, Kevin J Roberts, Jorge Calderon De Anda Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, The University of Leeds, Leeds LS2 9JT, UK Abstract Organic crystals grown from solution are known to exhibit multiple morphology and habits which are of great importance to the end use property of the product such as the bioavailability and down stream processing such as in filtration and drying. The crystal morphology can also dictate other quality measures such as the size. This paper reviews recent developments in on-line crystal morphology measurement and control using online imaging and image analysis by reference to a case study of cooling crystallization of (L)-glutamic acid in a batch reactor. On-line imaging was found to be able to capture with high fidelity crystal shape and polymorphic transitions in real-time. The images were analyzed using a new multi-scale image analysis method to extract the crystals from the image background, and to calculate shape descriptors which were then used for shape recognition and deriving new monitoring charts showing the ratios of different polymorphs in real-time. Preliminary study on estimating crystal growth rates and kinetics parameters for different facets for needle shaped crystals was also presented. Finally a framework integrating morphology modeling, multi-dimensional population balance and computational fluid dynamics, with on-line 3D imaging and image analysis is presented which provides the basis for model predictive automatic control of the morphology of growing crystals. Keywords: crystal morphology and shape measurement and control, model predictive control, morphology modeling, imaging, image analysis, population balance 1. Introduction High value-added speciality chemicals such as pharmaceuticals are often manufactured in batch crystallization processes. The shape, size and polymorphic form are properties of great importance to crystalline drug products. It is known that certain crystal morphological forms and habits have been related to difficulties in dissolution rate, process hydrodynamics, solid-liquid separations, drug tableting, storage and handling, or in milling and grinding. Although there has been a large amount of research work on on-line measurement of other quality measures such as the size and concentration using various spectroscopy techniques including ultrasound, infrared, near infrared, and Uv spectroscopy. X-ray diffraction and Raman spectrometer(Braatz et al. 2002), the literature on monitoring crystal morphology is scarce, and so does that on crystal morphology control. This paper presents recent advances towards developing an enabling technique for real-time measurement and manipulation of the morphology of growing crystals through integrating on-line imaging, image analysis and morphology modeling, and

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proposes a framework and highlights challenges and ftiture research needs for model predictive control of crystals morphology. 2. Morphology Measurement: Imaging and Image Analysis Laser diffraction techniques were investigated previously for the recognition of nonspherical particles with only limited success mainly due to the difficulty in obtaining a single-particle pattern in mixtures(Ma et al. 2001; Yamamoto et al. 2002). Use of attenuation acoustic spectroscopy, Raman spectrometer, NIR and X-ray diffraction techniques though can detect polymorphs, but they cannot give detailed quantitative shape information. Several recent studies have demonstrated the effectiveness of using on-line imaging as an instrument for monitoring the shape of the crystals (Calderon De Anda et al. 2005c; Calderon De Anda et Fig. 1 The on-line imaging system mounted al. 2005d; Calderon De Anda et al. on a 5 litre batch reactor 2005a; Patience and Rawlings 2001; Wilkinson 2000). In our work, we investigated the use of an on-line imaging instrument developed by researchers of GlaxoSmithKline to monitor the onset as well as polymorph transitions during cooling crystallization Fig 2 Polymotph transition for L-glutamic acid captured in real-time, of L-glutamic acid from a form (left) to (3 form. The right figure shows mixed a and [3 (Calderon De Anda et indicating transition being taking place al. 2005d), developed an effective multi-scale image analysis technique for segmentation of the crystals from the complex background of image frames (Calderon De Anda et al. 2005a), and subsequently derived shape descriptors, classification 1 ~~~~~ ^ techniques and novel process monitoring charts (Calderon De Anda et al. 2005c). Figure 1 shows the on-line imaging system mounted on the outside wall of a 5 liter batch reactor which is able to take maximum Fig 3 crystal images were effectively extracted from 30 images per second of the pixel the background of an imageframeobtained by the GSK imagine system resolution of 480 x 640. Figure 2 shows polymorphic transition was captured in real-time during the cooling crystallization of L-glutamic acid.

Advances and Future Directions in Morphology Monitoring

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On-line images of slurries with particles suspended in a solution pose much greater challenges to image analysis methods than images of particles obtained with off-line equipment. The major challenges lie in the fact that the slurries in a stirred reactor are in continuous motion, and that the variation of distances from the camera lens of particles captured in a snapshot makes some particles rather vague compared to others. In addition, the light effect and temporal changes of hydrodynamics within the reactor may lead to varied intensity in the image background. Fig 4 crystal images were effectively extracted As a result a multi-step multiscale from the background of an image frame obtained approach was developed which by the Lasentec PVM imaging system. proved to be effective in extracting objects from the image background for images obtained by the GSK on-line microscopy system, as well as for the Lasentec's PVM probe (Barrett and Glennon 2002), as demonstrated by Figures 3 and 4. 3. Shape Descriptors, Shape Classification and Monitoring Charts Following image segmentation, the next step is to classify the shapes. Shape descriptors should be invariant under rotation, translation and size scaling. It is also important to exclude descriptors that do not represent shape. After comparing various descriptors, we found that Fourier descriptors based on spectral transforms provide robust performance, accuracy and compact features and low computational complexity, therefore were used

20 30 time (min)

50

400

600

800

1000 12001400

Process evolution

(b) (a) Fig 5 (a) crystal polymorphic transition charts in number percentage for the polymorphs a and p. Each point represents the previous ten minutes; (b) evolution of roundness values for shapes detected as a and 3 forms indicating transition of a to 3 for L-glutamic acid crystallization in our study (Calderon De Anda et al. 2005c). The descriptors were then used by an unsupervised neural network algorithm, ART2 (adaptive resonance theory) to classify the crystals. Based on the results, advanced new monitoring charts were developed, as shown in Figures 5 (a) and (b). 4. Faceted Crystal Growth Rates and Kinetics Given that the fimdamental process of crystal growth and its associated kinetic control is surface controlled, the use of a single scalar parameter, particle size, usually defined

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as a volume equivalent diameter, i.e. based on a spherical assumption of particle shape can be misleading for a number of practical crystallization systems, notably pharmaceutical products, 200 -| where facetted particles H Length 180 • Supeisaturation • B ? -L defined by non-unity aspect • Turbidity ^ • ratios (Calderon De Anda et al. 160 ^ Temperature J * • • 2005b). Hence, measurement B 140 of the growth rate for each • •^B 1 120 • individual crystal surface in ^1 m real-time and within 100 • i processing reactors could s 80open the way for the >% 60 development of more • effective processes and , ^ ^ L 40concomitant product quality Time (min) control. Using on-line Fig 6 Crystal length evolution, plotted against imaging and image analysis, supersaturation, temperature and turbidity. Each point a preliminary study was represents the average of previous 60 seconds containing conducted on the estimation 300 images of the growth rates of needle-shaped crystals in two dimensions for p-form L-glutamic acid in cooling crystallization under a cooling rate of O.lO^C/min (Calderon De Anda et al. 2005b). The length and width of each needle-shaped crystal were measured every 60 seconds, ranging from 100 to nearly 200 |Lim in length and from 30 to 45 |Lim in width, and the values were used to estimate growth rates on both directions (Fig. 6). The growth rate in length was found to be 4 to 6 times greater than for the width. The (101) plane was found to be the fastest growing surface of the morphology studied and an attempt was also made to estimate its growthkinetics parameters from measurements of length, whilst it was harder to estimate kinetics from measurements of width for other crystal faces. In the temperature range between 68.34^C to 67.5 l^C, the length growth rate is estimated as between 2.440x10'^ ~ 2.995x10"^ m/s, while the growth rate for the width is between 0.558x10"^ ^ 0.502x10" ^ m/s. The capability to measure crystal growth rates in different directions could be used to estimate the parameters associated with growth kinetics in multi-dimensional directions. If a semi-empirical kinetic model is used, R = kG"", k ^1.761x10'' m / s , an d /; «2.61. It was assumed for [3 L-glutamic acid, the growth rate in length is very close to the growth rate of the faces {101}.

5. Model Predictive Control of Crystals Morphology 5.1. Model Predicted Crystal Morphology Control The recently developed imaging and image analysis technique for real-time measurement of crystal morphology makes inroads into automatic control of the morphology of crystals grown from solution. Since once an unfavorable polymorph or morphology is formed, to reverse the process e.g. through heating to dissolve and regrowing the crystals may not be a preferred option in industrial operation, advanced control based on shape predictive models is therefore attractive. Since morphology prediction is formulated for a single crystal, in order to consider the whole population of crystals in a reactor to give a statistical measure of the shape distribution, morphology

Advances and Future Directions in Morphology Monitoring

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modeling should be integrated with population balance (PB) modeling (as indicated in the next sub-section, this gap has yet to be filled), and very importantly, PB has to be conducted based on multi-dimensions, instead of being based on the On-line 3D Imaging assumption of an equivalent spherical Image Analysis shape with a single dimension of the I diameter to represent the size Shape Recognition distribution. Multi-dimensional PB Multi-dimensional modeling depends on the number of Faceted Growth Rates Population and Faceted Population facets of a crystal, each with its own Balance Modelling Balance growth rates. Some researchers (Briesen 2006; Puel et al. 2003) have experimented the idea of two dimensional PB modeling for needle shaped crystals. In early part of this Morphology paper we showed results of measuring Control Based on the growth rates in two dimensions for Predictive Models needle-shape crystals. Not only being Fig 7 The necessary components for developing restricted to two dimensions, work so model predictive control strategy for crystal far has also assumed that there is only morphology one polymorph in a reactor. If more than one polymorphs exist at the same time in the reactor, the problem clearly becomes more complicated because the transition between two polymorphs also needs to be taken into account. Fig. 7 shows a conceptual framework highlighting the necessary components for developing a model predictive control strategy of crystal morphology. Key future research needs are highlighted below. 5.2. Shape Measurement Although proved to be effective, so far the measurement of shape using on-line imaging has been restricted to giving 2-D images. Li et al (Li et al. 2005) proposed to use a camera model to construct the 3-D shape from 2-D images, there are still a few obstacles to be overcome before it can become practical. We are experimenting the use of two video cameras to simultaneously image the same object so that the 3-D shape can be fully constructed. 5.3. Morphology modeling Traditionally, morphology prediction of crystals had almost assumed that the crystals were growing in vacuum without (a) (b) adequately considering process Fig. 7 (a) traditional modelling and control: based on a spherical assumption, e.g. operational conditions such as cooling growth rate in m/s of diameter; (b) new rates, supersaturation, solvents and modelling and control: based on impurity or additives etc though these multidimensional model, e.g. growth rates, factors are known to affect the polymorph m/s for each surface. and morphology. Progress are being made in improving the morphology models to take into account these factors and the work has been reviewed (Clydesdale et al. 1996; Gadewar and Doherty 2004; Liu and Bennema 1996; Liu et al. 1995; Winn and Doherty 2000). Success has been achieved on predicting the effect of solvent on the shape of several organic crystal systems. Despite

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the progress, there is still some way to go before robust models available that can fully take into account process conditions to give confident prediction of crystals shape. This does not mean however, this represents the weakest link that will jeopardize the whole framework for model predictive morphology control research. In fact, on-line shape measurement techniques could contribute to the development of morphology prediction models since they can be used for model validation. 53.1. Integration with CFD The effect of mixing on crystal growth has long been recognized and computational fluid dynamics (CFD) has been integrated into PB modeling (though only for one-dimension based on the spherical assumption) (Kulikov et al. 2005) so that PB modeling can be conducted in different zones with varied mixing characteristics. A major challenge is computational speed. Research on either model reduction or data compression is needed in order to be able to effectively use the CFD data into crystal morphology model predictive control.

6. Final Remarks The recent developments in on-line shape measurement as well as in crystal morphology prediction and multi-dimensional population balance modeling opens the way for developing model-predictive control of the morphology as well as size of crystals grown form solution. This will need integration of on-line real-time 3-D shape measurement, multi-scale modeling of morphology, multi-dimensional PB modeling and CFD. Future research needs towards this goal are highlighted.

Acknowledgements Financial supports from EPSRC for the Shape project (EP/C009541) and for the Chemicals Behaving Badly project (GR/R43860), and from Malvern Instruments Ltd for the Vision and IntelliSense projects are greatly appreciated. The first author thanks Malvern for sponsoring his Readership in Intelligent Measurement and Control. References Barrett, P., Glennon, B., 2002, Chem. Eng. Res. Des., 80, 799-805. Braatz, R. D., Fujiwara, M., Ma, D. L., Togkalidou, T., Tafti, D. K., 2002, Int. J. Modem Physics B, 16, 346-353. Briesen, H., 2006, Chem. Eng. Sci., 61, 104-112. Calderon De Anda, J., Wang, X. Z., Roberts, K. J., 2005a, Chem. Eng. Sci., 60, 1053-1065. Calderon De Anda, J., Wang, X. Z., Roberts, K. J., 2005b, J. Phar. Sci. Calderon De Anda, J., Wang, X. Z., Lai, X., Roberts, K. J., 2005c, J. Proce. Cont., 15, 785-797. Calderon De Anda, J., Wang, X. Z., Lai, X., Roberts, K. J., Jennings, K. H., Wilkinson, M. J., Watson, D., Roberts, D., 2005d, AIChE J., 51, 1406-1414. Clydesdale, G., Roberts, K. J., Docherty, R, 1996, J. Cryst. Growth, 166, 78-83. Gadewar, S. B., Doherty, M. F., 2004, J. Cryst. Growth, 267, 239-250. Kulikov, v., Briesen, H., Marquardt, W., 2005, Chem. Eng. Res. Des., 83, 706-717. Li, R. F., Thomson, G. B., White, G., Calderon De Anda, J., Wang, X. Z., Roberts, K. J., AIChE J, in press, 2006. Liu, X. Y., Bennema, P., 1996, J. Cryst. Growth, 166, 117-123. Liu, X. Y., Boek, E. S., Briels, W. J., Bennema, P., 1995, Nature, 374, 342-345. Ma, Z. H., Merkus, H. G., Scarlett, B., 2001, Powder Techno!., 118, 180-187. Patience, D. B., Rawlings, J. B., 2001, AIChE J., 47, 2125-2130. Puel, F., Fevotte, G., Klein, J. P., 2003, Chem. Eng. Sci., 58, 3715-3727. Wilkinson, M. J., Jennings, K. H., Hardy, M., 2000, Microscopy Microanal., 6, 996-997. Winn, D., Doherty, M. F., 2000, AIChE J., 46, 1348-1367. Yamamoto, H., Matsuyama, T., Wada, M., 2002, Powder Technol., 122, 205-211.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Molecular Weight Control in Acrylonitrile Polymerization with Neural Network Based Controllers Atasoy I, Yuceer M., Berber R. Department of Chemical Engineering, Faculty of Engineering, Ankara University, Tandogan, 06100 Ankara, Turkey

Abstract Acrylic fiber is commercially produced by free radical polymerization, initiated by a redox system. Industrial production of polyacrylonitrile is a variant of aqueous dispersion polymerization, which takes place in homogenous phase under isothermal conditions with perfect mixing. The fact that the kinetics is a lot more complicated than that of ordinary polymerization systems makes the problem of controlling molecular weight a difficult one. On the other hand, abundant data is being gathered in industrial polymerization systems, and this information makes the neural network based controllers a good candidate for a difficult control problem. In this work, neural network based control of continuous acrylonitrile polymerization is studied, based on our previously developed new rigorous dynamic model for the polymerization of acrylonitrile. Two typical neural network controllers are investigated: model predictive control and NARMA-L2 control. These controllers are representative of the variety of common ways in which multilayer networks are used in control systems. As with most neural controllers, they are based on standard linear control architectures. The concentration of bisulfite fed to the reactor as the manipulated variable and weight average molecular weight of the polymer as an output function are used in control studies. The results present a comparison of two common neural network controllers, and indicate that the model predictive controller requires larger computational time. Furthermore, the model predictive controller involves difficulties in determining the values for the weighting factor and the prediction horizons. The NARMA-L2 controller requires minimal online computation. Keywords: Acrylonitrile polymerization, NN predictive control, NARMA-L2 control. 1. Introduction Neural networks have been applied successfully in the identification and control of dynamic systems (Hagan et al., 2002). Rather than attempt to survey the many ways in which multilayer networks have been used in control systems, we concentrated on two typical neural network controllers: model predictive control (Narendra et al., 1997), NARMA-L2 control (Narendra et al., 1990). These controllers are representative of the variety of common ways in which multilayer networks are used in control systems. As with most neural controllers, they are based on standard linear control architectures. There are typically two steps involved when using neural networks for control: system identification and control design. In the system identification stage, we develop a neural network model of the plant that we want to control. In the control design stage, we use the neural network plant model to design (or train) the controller. In each of the two

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control architectures described in this paper, the system identification stage is identical. The control design stage, however, is different for each architecture. 2. NN predictive control The aim of the predictive control strategy using neural predictors is two step: (a) to estimate the future output of the plant: The controller then calculates the control input that will optimize plant performance over a specified future time horizon. (h) to minimize a cost function based on the error between the predicted output of the processes and the reference trajectory. The cost function, which may be different from case to case, is minimized in order to obtain the optimum control input that is applied to the nonlinear plant. In most of the predictive control algorithms a quadratic form is utilized for the cost function. 2.1. System identification The first stage of model predictive control and NARMA-L2 control are to train a neural network to represent the forward dynamics of the plant. The prediction error between the plant output and the neural network output is used as the neural network training signal. 2.2. Predictive control This controller uses a neural network model to predict future plant responses to potential control signals. An optimization algorithm then computes the control signals that optimize future plant performance. The neural network plant model is trained offline, in batch form, using any of the training algorithms. The controller, however, requires a significant amount of on-line computation, since an optimization algorithm is performed at each sample time to compute the optimal control input. The model predictive control method is based on the receding horizon technique. This controller uses a neural network model to predict future plant responses to potential control signals. The neural network plant model is trained offline, in batch form, using any of the training algorithms. The predictions are used by a numerical optimization program to determine the control signal that minimizes the following performance criterion over the specified horizon:

^=i(>'.(^+y)-7j^+7)f+/'i:(«'(^+7-i)-«'(^+;-2F j=N,

0)

7=11

where Ni; N2 and Nu define the horizons over which the tracking error and the control increments are evaluated. The u' variable is the tentative control signal, yr is the desired response and ym is the network model response. The value of p determines the contribution that the sum of the squares of the control increments has on the performance index. Figure 1 shows block diagram that illustrates the model predictive control process. The controller consists of the neural network plant model and the optimization block. The optimization block determines the values of u' that minimize J; and then the optimal u is input to the plant.

Molecular Weight Control in Acrylonitrile Polymerization

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"A

Coiitix>Ilei' I

yr Optirrdzatioti

Nesuml Network Model

Plant

Figure 1. NN predictive control 3. NARMA-L2 control The controller is simply a rearrangement of the neural network plant model, which is trained offline, in batch form. The only online computation is a forward pass through the neural network controller. The main idea of this type of control is to transform nonlinear system dynamics into linear dynamics by canceling the nonlinearities. As with the model predictive control, the first step in using NARMA-L2 control is to identify the system to be controlled. The nonlinear autoregressive moving average model is used to represent the system: Kk + d) = f[y{k\y{k - \\..,,y{k -n + l),...,i/(A: - m +1)] + g{y(k\y(k - l)v..X^ - « + l),w(A: - \\..M{k -m + \)lu(k)

^2)

where d < 2. Using the NARMA-L2 model, the controller is obtained of the form: U(k + 1):

where

yr(k + d)-f[Y,U] g[Y,U]

Y=[y(k),...,y(k-n + l)] U = [u(k),u(k-l),..,u(k-n + l)]

(3)

(4)

This model is in companion form, where the next controller input u(k) is not contained inside the nonlinearity. Figure 2 is a block diagram of the NARMA-L2 controller.

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1 Rediel«n.ce 1 lodel 1

-'

1 ^ -{- ^

Contra

+ J Iter

.1

i

^AC^

Sir 1 / 1 1s 1 i

—

>U-

^ IL

1 T 1 1 1^ 1L 1

Figure 2. NARMA-L2 controller. 4. Mathematical model The dynamic model comprises 9 state variables, namely the concentrations of monomer, sulfate radical, bisulfite radical, bisiilfite ion, persulfate ion, active radical of one monomer with sulfate end group, active radical of one monomer with sulfite end group, active radical of n monomers with sulfate end group, active radical of n monomers with sulfite end group. The mathematical model developed was based on the following simplifying assumptions: (i) Polymerization takes place in homogenous phase under isothermal conditions with perfect mixing and all species are soluble in the solvent, (ii) Reactivities of sulfate and bisulfite radicals are the same, (iii) All radicals propagate, terminate and transfer with the same velocity constant, (iv) Branching reactions do not take place, (v) Equilibrium is established after 5 to 6 dwell time, (vi) Addition and propagation steps proceed with the same velocity constant, (vii) All reactions are of second order, (viii) All radicals propagate with the same velocity. Addition and growing rate of monomers to the polymer chain were taken to be equal. As our dynamic studies indicated that 6 of those variables were not changing considerably, we assumed that they would hold constant in control studies, and we were thus able to obtain a simplified model comprising of 3 differential equations coupled with the outputfiinctionas follows:

^=-t,M^(Iso,-]+ko;])*ML-M at

V — OC

u

(5)

u

(6)

at Probability of growth;

\-a

6

6

Molecular Weight Control in Acrylonitrile Polymerization ^._

KM

_

1621

K{^] (o)

z

The average (number and weight) molecular weights of the pol5niier are defined by, _ r„ =

2kp[M] p

_

:(l + 2y)

'""

M =W *m

;

kp[M][3 + 2y] z(l + y)'

M =W *r

^p^ (10)

where y and z are ^AHSO;^^

z = (2k,k,[Fe-]M-])>^

^^^^

The manipulated variable was the concentration of bisulfite fed to the reactor. 5. Results and Conclusions A neural network model of the plant was constructed using data from the simulation model. A multilayer perceptron neural network with 3 neurons in the hidden layer. The Acrylonitrile Polymerization model is solved MATLAB environment using variable order ODE solver. The model generated input data considered of 10 000 sample points. It was divided into three subsets, namely; training set (5 000 sample points), validation (2 500 sample points) and testing (2 500 sample points). Then through the optimization routine, the model predictive controller provides a control action to the system depending on the predictive horizon and the control weighting factor. A model predictive controller with a longer prediction horizon and a small control-weighting factor provides good performance in terms of reduced error. The controller employs an online optimization algorithm, which requires more computation than the NARMA-L2 control. The performance of the NARMA-L2 controller depends on the identification of the system with a neural network. An accurate neural network model of the system provides good results in terms of set point tracking, hence reduced error values. However, as there is no factor to adjust the control weighting, use of a limiter on the control action appears to be necessary. The performances are hence reduced but the control signal variation is also reduced. The controller is a rearrangement of the plant model, and requires minimal online computation. Table 1 shows the final NN-Predictive Controller and NARMA-L2 Controller parameters for acrylonitrile polymerization control application. The performances of neural network controllers are highly dependent on the accuracy of the plant identification. For our applications, we typically collect training data while applying random inputs which consist of a series of pulses of random amplitude and duration. The duration and amplitude of the pulses must be chosen carefully to produce accurate identification. If the identification is poor, then the resulting control system may fail. Controller performance tends to fail in either steady-state operation, or transient operation, or both. When steady-state performance is poor, it is useful to increase the duration of the input pulses. Unfortunately, within a training data set, if we

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have too much data in steady-state conditions, the training data may not be representative of typical plant behaviour. This is due to the fact that the input and output signals do not adequately cover the region that is going to be controlled. This will result in poor transient performance. Figure 3 shows the performance of controllers for parameters given in Table 1. Table 1. Parameters for NN-MPC and NARMA-L2 Parameters

NN-MPC

NARMA-L2

Size of hidden layers

3

3

Training function Training epochs

Levenberg-Marquardt 1000

1000

Cost horizon (N2)

7

-

Control horizon (NJ

2

-

Minimization routine

Minimization with backtracking

-

Control weighting factor (p)

0.001

-

Search parameter (a)

0.01

-

1200

1000

800

—\

r^ 1

1

1

r-

600

• NARMA-L2 • Setpoint • NN-MPC 5 time, s

7

8

9

10

xio"^

Figure 3. Response of molecular weight controllers to set point changes References T. M. Hagan., H. B. Demuth and O. D. Jesus, An introduction to the use of neural networks in control systems. Int. J. Robust and Nonlinear Control, 2002; 12, 959-985. K. S. Narendra, S. Mukhopadhyay, Adaptive control using neural networks and approximate models, IEEE Transactions on Neural Networks, 1997, 8, 475-485. K. S. Narendra, K. Parthasarathy, Identification and control of dynamical systems using neural networks, IEEE Transactions on Neural Networks, 1990; 1, 4-27.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A New Approach to Chance Constrained Process Optimization and Control under Time-dependent Uncertainties Harvey Arellano-Garcia*, Tilman Barz, Walter Martini, Gunter Wozny Department of Process Dynamics and Operation, Berlin University of Technology, Sekr. KWT-9, Str. Des 17. Juni 135, Berlin 10623, Germany

Abstract In this work, a novel approach to solving nonlinear chance-constrained dynamic optimization problems under time-dependent uncertainties is proposed. The approach considers a nonlinear relation between the uncertain input and the constrained output variables. In fact, the approach is relevant to all cases when uncertainty can be described by any kind of joint correlated multivariate distribution function. The essential challenge lies in the efficient computation of the probabilities of holding the constraints, as well as their gradients. However, the main novelties of this approach are that nonlinear chance constrained dynamic optimization can now also be realized efficiently even for those cases where no monotonic relation between uncertain input and constrained output exists. This is necessary, particularly, for those process systems where the decision variables are critical to the question of whether there is monotony or not. Furthermore, novel efficient algorithms are proposed to consider dynamic random variables. Thus, the solution of the problem has the feature of prediction, robustness and being closed-loop. The performance of the proposed approach will be demonstrated through application to the optimal operation and control of a high pressure column embedded in a heat integrated column system. In addition, a novel chance constrained nonlinear MFC scheme is introduced to show the efficiency and potential of the chance constrained approach for online optimization and control under uncertainty. Keywords: Time-dependent uncertainty, chance constraints, dynamic optimization, NMPC.

1. Introduction Optimization and control under uncertainty is deemed to be of ftindamental significance in several discipline and application areas. In dynamic processes, in particular, there are parameters which are usually uncertain, but may have a large impact on the targets like the objective value and the constrained outputs. Thus, explicit consideration of the stochastic property of the uncertainties in the optimization approach is necessary for robust process operation and control. Uncertainty and variability are inherent characteristics of any process system. Moreover, measurements often contain random errors that invalidate the process model used for optimization and control. This implies that neither the magnitude nor the sign of the error can be predicted with certainty. However, the uncertainties considered are continuous variables, not results of discrete events. This means that there is infinity of possible "discrete" values for the events associated with continuous time-dependent variables. The only possible way these weaknesses can be characterized is by use of probability distributions. To whom correspondence should be addressed at [email protected]

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to

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tf

to

tf

Figure 1. Constant und time-dependent uncertainty The stochastic distribution of the uncertain variables may have different forms. The values of mean and variance are usually available. Uncertain variables can be constant or time-dependent in the future horizon (Fig. 1). They are, however, undetermined before their realization. Moreover, usually only a subset of variables can be measured. The unmeasured variables are, however, open loop but should be constrained under uncertain disturbances. 2. Chance Constrained Approach In this work, chance constrained programming for process optimization and control under uncertainty is proposed. The stochastic property of the uncertainties is included in the problem formulation such that the output constraints are to be complied with a predefined confidence (probability) level. The resulting problem is then transformed to an equivalent deterministic NLP problem. Here, the basic idea is to map the probabilistic constrained output region back to a bounded region of the uncertain inputs. Hence, the output probabilities and, simultaneously, their gradients can be calculated through multivariate integration of the density function of the uncertain inputs by collocation in finite elements. Recently, a new promising optimization fi-amework for dynamic systems under uncertainty was introduced for the off-line optimization under probabilistic constraints and successfully applied to a large-scale nonlinear dynamic chemical process where the monotony of the constrained output to at least one uncertain input is utilized (Arellano-Garcia et al. 2003, 2004). However, this approach can only be used, if the required monotony exists. In this contribution, we extend the chance constrained approach to allow for such stochastic dynamic optimization problems where no monotone relation between constrained output and any uncertain input variable can be guaranteed. Moreover, the novel approach explicitly considers time-dependent uncertainties. The entire optimization framework also involves efficient algorithms for the computation of the required (mapping) reverse projection and is relevant to all cases when the uncertainties can be described by any kind of joint correlated multivariate distribution function.

superior layor

multivariate integration

.,_^ (dhrnamic solvir

^ ^ ^ e f . er Figure 2. Chance constrained optimization framework.

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The proposed approach uses a two-stage computation framework (Fig. 2). The upper stage is a superior optimizer following the sequential strategy. Inside the simulation layer there is a two-layer structure to compute the probabilistic constraints. One is the superior layer, where the probabilities and their gradients are finally calculated by multivariate integration. The inferior sub-layer is the key to the computation of the chance constraints with non-monotonous relation. In case of strict monotony the optimization step in the sub-layer becomes unnecessary. 3. An Approach to Time-dependent Uncertainty In this work, a dynamic process with 7V7 time-dependent output state variables y(t), NU time-dependent control variables u(t), and A/O time-dependent uncertain parameters ^(t) is considered. The probability Pr of complying with a certain restriction yi^ which corresponds to the output state variable y^^ at every time point t during the process operation is to be calculated and formulated by the following expression: Pr{/^(/,^.,f)<X^ VfE [/„/,]}

(1)

In order to transform the infinite number of time points to a finite number of representing values, the entire time horizon is divided into several short time intervals where both the control variables and the uncertain variables are piecewise constant. Due to the monotony between the restricted output y^^ and at least one uncertain parameter, the value of this uncertain parameter ^^^, which corresponds to the bound of the constrained output yi^, can be calculated for every time interval JT according to the following equation: ^'l^MM= f(^w^^JTA^'"^^NMM-xy[]

^ith NMM = JTxN^

(2)

However, for NT time intervals, the probability of complying with the constraint for all time intervals can be computed by multivariate integration of a probability density function over all uncertain parameters as follows: psp

pp

Pr= j . . . j j J... J j p{^x.^^^,^NMM)d^NMM'-d^x

(3)

with NMM = NTxN^ Each integration bound of the uncertain parameter (^^^ corresponds to the bound of the constrained output yi^ within the corresponding interval. All the other uncertain parameters will be integrated over their entire space. Since only few discretization points are required for the integration over a relatively large integration space with an acceptable accuracy, the orthogonal collocation method on finite elements has been proved to be very efficient. Thus, in this work we propose a calculation scheme where the first uncertain parameter ^i is discretized in the first interval. Then, for each resulting collocation point, a value for the second uncertain parameter ^2^^ can be obtained which exactly corresponds to the bound of the constrained output yi^ within this interval and, thus, forming the bound for the second integration layer. Over the new derived integration space, the second uncertain parameter can be discretized. Thus, for instance, in case of two dynamic random variables, NK^ collocation points result for the first interval. This procedure will then be repeated until the next-to-last integration layer.

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first interval. This procedure will then be repeated until the next-to-last integration layer.

Figure 3. Solution strategy for time-dependent uncertainties. Consequently, the last time interval is reached. This approach leads to the computation (tree) structure described in Fig. 3. Since the values below the integration bound are only used for calculations in the following integrals, only one value which corresponds to the bound of the constrained output is required for the last uncertain parameter of the last time interval. Thus, the probability of complying with the restrictions of the last interval under given values of the other uncertain variables, and all control variables will be obtained. This value can also be seen as a part of the probability density function of the next-to-last integration layer. The integration along this layer leads to the probability calculation concerning this layer. This procedure will be carried out up to the most superior integration layer and, thus, we finally obtain the originally wanted probability for fulfilling the constraints of the entire time horizon. Furthermore, to solve the NLP-problem with a standard NLP solver such as SQP, gradients of the objective function and the constraints w.r.t. the control parameters u(t) are additionally required. These are computed simultaneously based on the proposed solution strategy in Fig. 3. 4. Optimal Process Operation under Chance Constraints Model-based optimization has widely been used to develope operation points and operating trajectories for industrial processes. The task of process control systems has been to mantain these predetermined operating points, or follow the given operating trajectory. However, optimization and control have generally been considered individually. The major drawback of performing the two issues separately is the discrepancy in treating process disturbances. Furthermore, the true process optimum lies on the boundary of the feasible region defined by the active constraint or equipment capacities (e.g. maximum allowable pressure, temperature, etc.). Due to the uncertainty in the parameters and the measurement errors, the process optimum and the set-point would be infeasible (e.g. if the pressure of the plant swings). Thus, usually a back-off from the active constraints in the optimization is introduced such that the region around the set-point is moved inside the feasible region of the process to ensure a feasible operation as closely to the true optimum as possible. This can lead to a conservative operation with a much higher product purity than required. Our main concern is that there are variables which are often monitoring for the sake of safety but not close-loop controlled. They should, however, be constrained under uncertain disturbances. This

A New Approach to Chance Constrained Process Optimization and Control

1627

leads to a nonlinear optimization problem under chance constraints. To follow these setpoints, the manipulated variables have to be varied corresponding to the realized disturbances. 1 0.998 ^

0.4 0.6 X-Acetonitrile

126

0.8

0.996

127 128 129 Temperature "C

0.59

0.6

0.61 0.62 0.63 0.64

D i s t i l l a t e X Acetonitnle

Figure 4. a) y-x diagram; b) operating set-point; c) constrained output variable The disturbance variables are described with stochastic distributions which can be achieved based on historical data. The novel approach is applied to the optimal operation and control of one column embedded in a coupled two-pressure column system for the separation of an azeotropic mixture (acetonitrile/water). The operating point is defined by the distillate x^^ and bottom product jcf^ specifications, cooling outlet temperature limitations T^^ , as well as the maximum pressure of the considered high-pressure column Ptop (Fig. 4a). The expected disturbances and implementation errors concern the maximal allowable system pressure, the sensitive tray in the stripping column section r/^^' as well as the feed flow rate and its dynamically changing composition x^eed' However, the values of the setpoints and controls are adjusted so that the target area will be modified according to the changing disturbances. The objective fiinction is defined by the minimization of the total energy in the considered time horizon. However, the uncertain parameters also have an impact on the objective function. The usual way is to reformulate it to its expected value (Darlington et al., 1999). On the other hand, for practical application, it is more convenient to assure a certain reliability of the realization of the calculated objective value. This can be achieved by minimizing an upper bound p, and the compliance of it, can be guaranteed with a certain reliability by formulating an additional chance constraint. Thus, the entire stochastic optimization problem will be formulated as follows with / as the probability levels: mm s.t. model equations, direct adjusted state variables: rpOUt

rpOUt

,

1

^CW ~ ^CW,ref "*" ^ 7 |_

/ rpOUt

\

yCW

jj '

indirect adjusted constrained output variables: Prjjc^ >x^ ^ ^ \^Ac

]>

— -^Ac,spec\ —

1 »

originally replaced objective fiinction as chance constraint uncertainties:

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and finally regulatory deviation variance. The formulation of individual pre-defined probability limits of complying with the restrictions incorporates the issue of feasibility and the contemplation of trade-off between profitability and reliability. However, the computation results demonstrate the efficieny of the proposed approach (Fig. 4c).

5. Robust Nonlinear MPC Under Chance Constraints Predictive control is well developed in terms of handling constraints and bounded uncertainty but there is still a need for a efficient framework addressing problems involving stochastic objectives and constraints (Batina et al., 2002). However, since the prediction of future process outputs within an NMPC moving horizon is based on a process model involving the effects of manipulated inputs and disturbances on process outputs, the compliance with constraints on process outputs is more challenging than these on process inputs. Furthermore, as the model involves uncertainty, process output predictions are also uncertain. This leads to output constraints violation by the closeloop system, even though predicted outputs over the moving horizon might have been properly constrained. Thus, a robust predictice control strategy is proposed, namely a nonlinear MPC scheme where the output constraints are to be held with a predefined probability with respect to the entire horizon. Due to the property of the moving horizon approach the control strategy can be extended to on-line optimization under uncertainty. Here, different confidence levels can be assigned to different time periods within the moving horizon by using single chance constraints. Consequently, a decreasing factor, i.e., a lower confidence level for the future periods in the horizon can be introduced. The outcomes of the application to a semibatch reactor with safety restrictions will show the potential of the new approach. 6. Concluding remarks In this work, a new chance constrained approach is proposed. The developed optimization framework demonstrates to be promising to address optimization and control problems under uncertainties. Furthermore, novel efficient algorithms have been integrated to consider time-dependent uncertainties. The solution strategy has been applied to the optimal operation of a high pressure column. The solution provides a robust operation strategy in the future time horizon. Moreover, the relationship between the probability levels and the corresponding values of the objective function can be used for a suitable trade-off decision between profitability and robustness. Tuning the value of / is also an issue of the relation between feasibility and profitability. In addition, a novel robust NMPC scheme will be introduced for the online optimization of a semibatch reactor under hard constraints.

References Arellano-Garcia H., Martini W., Wendt M., Li P., Wozny G., 2003, Chance Constrained Batch Distillation Process Optimization under Uncertainty. In: I. E. Grossmann, C. M. McDonald (Eds.): FOCAPO, pp. 609-612. Arellano-Garcia H., Martini W., Wendt M., Wozny G., 2004, A New Optimization Framework for Dynamic Systems under Uncertainty, In: A. Barbarosa-Povoa, H. Matos (Eds.): Computer Aided Process Engineering -14, Elsevier, 2004, 553-558.

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Batina, I., Stoorvogel, A.A., Weiland, S., 2002, Optimal control of linear, stochastic systems with state and input constraints, Proc. IEEE Conf. Decision & Control, 1564-1569. Darlington J., Pantelides C.C, Rustem B., Tanyi, B.A., 1999, An algorithm for constrained nonlinear optimization under uncertainty. Automatica, 35, 217.

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A Lab-on-a-Chip Simulation Framework A. J. Pfeiffer^, X. He^ T. Mukherjee^ and S. Hauan^* ^Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh PA 15213 ^Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh PA 15213 A Lab-on-a-chip (LoC) miniaturizes and integrates chemical synthesis and analysis capabilities onto a microchip structure. Since LoCs are both physically and chemically complex, a fast, accurate simulation methodology is required for design. Here we present an efficient, configurable LoC simulation framework that can be readily incorporated within design automation and synthesis methods. We compare our simulator to a general computational fluid dynamics (CFD) tool using a LoC design benchmark. Our simulator results in less than 5% error and over 3 orders of magnitude speedup. We demonstrate the efficiency of our approach by redesigning an experimentally generated design from the Uterature [1]. 1. INTRODUCTION A Lab-on-a-Chip (LoC) is essentially a miniaturized microchip implementation of an analytical chemistry laboratory. LoCs are typically fabricated in glass or plastic and are about the size of a credit card. They have been used in the life-science and biomedical industries for applications in genomics, drug discovery, point-of-care analysis and in-vivo diagnostics because they are fast, accurate, readily automatable and inexpensive to fabricate. Microscale unit operations such as mixing, reaction and separation can be constructed entirely on-chip [2]. However, the widespread use of LoC technology has been hindered by the lack of adequate design tools. LoC design combines complex physiochemical phenomena with challenging chip layout and channel interconnectivity issues. The chemistry that takes place during chip operation, as well as the chip layout and manufacturing process must be understood so that the appropriate design trade-offs and constraints are considered. Channel geometry, and the system's channel topology have been shown to contribute a great deal to the overall performance of the final LoC design [3]. Current LoC design practices employ laboratory experiments or iterative computational fluid dynamics (CFD) simulations [4] which are both time consuming and difficult to automate. Simulators using reduced order models [5] have been created, but these tools require CFD pre-solves to extract model parameters. Fast, accurate models have been implemented in a commercial circuit simulator [6]. However, black-box simulators are difficult to integrate within automated design tools. We are developing Computer-Aided Design (CAD) tools to address the complex nature of LoC design by combining electrical circuit simulation [7] and chemical process simulation * Corresponding author email: [email protected]

A J. Pfeiffer et al

1632 Sample: [Ag*. Ag]

Sample: [Ag*, Ag]

Figure 1. Four phases of LoC operation: (a) Sample and reagent are introduced (shaded area) (b) Steady-state loading phase in which fluids are mixed (Jl) and reacted (Jl - J2). Flow direction shown with dotted arrows, (c) Transient injection phase where a narrow band of material is injected into the separation channel, (d) Fluids are separated electrophoretically into unique species bands.

strategies [8] with fast, accurate, models [9-11] derived from the coupled partial differential equations that describe LoC physiochemistry. We present an efficient, configurable, systemlevel simulation framework for channel-based microfluidic LoCs that can be incorporated into optimization and synthesis tools. 1.1. Basic LoC Operation Figure 1 illustrates the operation of a canonical LoC-based immunoassay [12] where we wish to determine the presence and concentration of a desirable antigen Ag* in a mixture of undesirable antigen Ag. First, the mixture of Ag*, Ag and an appropriate antibody Ab are input into the sample and reagent wells respectively (Fig. 1(a)). In the second phase (Fig. 1(b)), voltages VI and V2 are applied while V4 is grounded. This voltage drop generates an electric field resulting in theflowof sample and reagent to the sample waste well in a continuous fashion. The mixture of Ag* and Ag is contacted with Ab at junction Jl. Between Jl and J2, Ag* and Ab react according to Ag* + Ab v^ Ag*Ab. At J2, the fluid stream is compressed or pinched by applying voltages V3 and V5 to aid in downstream separation. We refer to this collective set of operations as the loading phase. We call the third phase the injection phase (Fig. 1(c)). Here, the voltages are instantaneously switched such that a narrow band of material is injected into the separation channel by applying voltage V3, grounding V5 and setting VI, V2 and V4 to draw back the excess material. Finally, in the time-dependent separation phase (Fig. 1(d)), the species within the injected band undergo electrophoresis and separate into unique species bands as they travel toward the buffer waste well. The antigen-antibody complex Ag*Ab can then be differentiated from the excess reactants. We use this canonical example as our LoC benchmark because it illustrates many of the complexities common in most LoC applications. Fluid mixing, reaction, injection and separation are all integrated into a single, multi-function LoC. The loading phase, a steady-state process, is followed by the injection phase, a transient process, after a discrete voltage switching action.

A Lab-on-a-Chip Simulation Framework

Function: Well Mixer Splitter

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Type:

•

Reactor Separator l\/lixer Injector

H D •r+

(a)

Figure 2. The LoC simulation process: (a) Partial library of LoC units, (b) Channel topology constructed from the library of units, (c) Resistor network representation of the channel topology, (d) Directed Acyclic Graph representation of the channel topology.

We have developed a system-level LoC simulator that accounts for these complexities by combining Kirchoffian network analysis and topological sorting from electrical circuit simulation [7] with the sequential-modular structure of process flowsheet simulation [8]. Our simulator employs fast, accurate physiochemical models [9-11] and allows us to simulate complex designs in only seconds. 2. SIMULATION FRAMEWORK In our approach, we construct LoC channel topologies from a library of microfluidic unit operations. Figure 2(a) illustrates part of our current library of unit operations. A unit is defined in terms of its function or physiochemical process, and in terms of its type or physical geometry. Units communicate using four standard interface objects. The flow = {FC, a'^, Cmax, t} object contains phenomenological information for each chemical species in the system. It is updated by each unit and passed to the appropriate downstream unit, flow contains an array of Fourier coefficients FC that describe species' concentration distributions during the loading phase, and species' band-shapes during the separation phase . flow also contains the variance (j^ or band broadness, the peak concentration Cmax, and the cumulative transit time t for each species exiting the current unit. We also pass a geom object which contains the length L, width uj, and depth d of a. unit, a props object which contains the diffusivity Z), and mobility // of each species, and a buffer object which contains the electrical and thermal conductivity (A and hi respectively) and the concentration Cb of the background electrolyte or buffer solution. The information in flow can be used to calculate a separation performance metric known as resolution R between any two species in the system. Although ^ = 1.5 corresponds to baseline resolution, our simulator also accurately predicts the higher values commonly reported in the literature [12]. We use R as one way to assess overall design quality. In Fig. 2(b) we construct the channel topology of our benchmark. Our simulation approach involves translating a given channel topology into a resistor network (RN) and a directed acycUc graph (DAG). The RN and DAG for the topology shown in Fig. 2(b) are shown in Fig. 2(c) and Fig. 2(d) respectively. The mapping between the DAG and the RN is as follows: nodes with

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Table 1 Comparison between FEMLAB and simulator for the channel topology shown in Fig. 2(b). FEMLAB Simulator % Difference Ag* Ag*Ab Ag* Ag*Ab Ag* Ag*Ab a2 4.9 • 10^ 6.4 . 10^ 5.1-10^ 6.3-10^ 4.1 1.6 (M) 0.83 1.12 0.81 1.08 2.4 3.6 24.1 24.0 0.4 R {sec.) CPW ~ 7200.0 - 1.5 4800 X speedup t 2GHz CPU, 1GB RAM degree one in the DAG correspond to nodes of degree one in the RN, degree two nodes in the DAG correspond to a resistor with nodes at either end in the RN, and nodes with degree greater than two in the DAG correspond to a set of resistors equal to the number of incident edges in the DAG with nodes at each resistor endpoint and a single internal node (eg. node 13 in 2(c)). The RN can be used to calculate the flows and potentials for the loading phase (iioad and vioad) and the injection phase (iinj and Vinj) as shown in Eq. 1. R

Bo\

Bl

0 ;

l^iioad\ ^ fyCioad\

f

\vioad)

\Bl

V 0

; '

R

B, 0 l-Hnj ^ yCinj\ 0 ) \VinjJ \ 0 J

... ^^

Here R is the resistance matrix, which in our work is the electrical resistance of each channel section. It is straight forward to include static pressure effects in R. B is the edge-node incidence matrix of the RN. BQ is the matrix formed after removing the columns that correspond to specified nodes from B so that linear independence is maintained [7]. The computational overhead of constructing these matrices is only incurred once for a proposed topology and thus enables an efficient search of applied potentials vcioad and vcinj. In the final step, the DAG is processed in topological order and each unit behaves in a sequential-modular or signal-flow fashion [8]. The flows and potentials calculated from Eq. 1 are passed to the appropriate units as required. This approach works well because LoCs do not contain recycle loops, and a stream-tearing approach [8] is not required. Since the models within each unit are highly nonlinear, our approach avoids the computational expense of simulating the system using a simultaneous approach. Further, our simulator does not require a priori initialization as is often the case with Newton-type solvers. By decomposing the simulation into a simultaneous linear part and a sequential nonlinear part, we can simulate the entire system in the time required for approximately one Newton iteration. 2.1. Example Results To test the accuracy and performance of our simulator, we compare it against an equivalent FEMLAB [13] simulation. In Table. 1 we compare the simulation results generated by FEMLAB and our simulator for our 9 unit benchmark (Fig. 2(b)). Our simulator and FEMLAB are in good agreement for the variance cr^, peak concentration Cmax^ and resolution R between Ag* and Ag*Ab. However, our simulator is over three orders of magnitude faster and thus enables iterative design and optimization. We can simulate designs with 50 units in under 3 seconds, but have been unable to simulate designs of the same complexity in FEMLAB. Fig.3(a) is a LoC from the literature [1] designed to perform a competitive immunoassay where labeled and unlabeled TheophylUne, Th* and Th, compete for antibody as shown in the

A Lab-on-a-Chip

Buffer

CD

(1:

Simulation

111

Tb*

C5]

®

Framework

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I sample waste

^—• detector \ buffer waste Sample VvaMc

1.97cm

(b)

(a)

Figure 3. Competitive immunoassay design comparison: (a) Design from literature with a 7.6cm X 7.6cm footprint [1]. (b) Simulator-based design with a 1.97cm x 2.61cm footprint.

Table 2 Comparison between literature design and simulation-based design.

a^ Sep. time R Area

(/im^)

{sec.) {cw?)

Literature [1,6] Th* Th*Ab 3.0 • 10^ 6.7 • 10^ 29.1 18.6 24.0 57.8

Double T Th* Th*Ab 2.6 • 10^ 6.0 • 10^ 2.7 1.7 27.0 5.1

Cross Th* Th*Ab 2.0 • 10^ 5.4 . 10^ 2.7 1.7 29.4 5.1

following reactions: Th* + Ab # Th*Ab and Th + Ab ^ ThAb. This multi-function chip incorporates mixing, reaction, injection and separation and reportedly fits within a 7.6cm x 7.6cm glass microchip. We can simulate this chip in approximately 2 seconds and our results are in good agreement with a previously published analysis [6]. Fig. 3(b) shows our new design of this chip. It achieves better performance than the original chip, and requires approximately 11 times less area. The design in Fig. 3(b) maintains the same reactor length and voltages as the original. We are also able to quickly evaluate the influence that interchanging particular units in the design has on the overall system performance. Table 2 shows a comparison between the original chip, which uses a double T injector (see Fig. 2(a)), and our new design, where we investigate both a double T and a cross injector. We are able to investigate design alternatives that would be prohibitively time consuming using CFD and experimental approaches. 3. Conclusion We have demonstrated that our simulator is capable of efficiently and accurately evaluating complex LoC systems. The ability to embed a modular simulation tool within our CAD method-

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ologies has been a key to the success of our earher work. We have explored design automation approaches for capillary electrophoresis (CE) LoCs using a distributed agent-based computing approach [14] and a tailored strategy based on Very Large Scale Integration (VLSI) circuit design and standard optimization algorithms [15]. As our library of LoC unit operations grows, we are able to address more complex LoC systems. The simulator presented here allows us to extend our design automation approaches to multi-function LoC systems like the immunoassay LoC benchmark discussed previously. Acknowledgments This research effort is sponsored by the Defense Advanced Research Projects Agency (DARPA) and U. S. Air Force Research Laboratory under agreement number F30602-01-2-0987 and by the National Science Foundation (NSF) under award CCR-0325344. The authors would like to thank members of the S YNBIOS YS group at Carnegie Mellon University.

REFERENCES 1. N.H. Chiem and DJ. Harrison. Microchip systems for immunoassay: an integrated inmiunoreactor with electrophoretic separation for serum theophylline determination. Clin. Chem., 44(3):591-598,1998. 2. A. Auroux, D. lossifidis, D.R. Reyes, and A Manz. Micro Total Analysis Systems. 2. Analytical standard operations and applications. Anal. Chem., 74:2637-2652, June 2002. 3. A.J. Pfeiffer, T. Mukherjee, and S. Hauan. Design and optimization of compact microscale electrophoretic separation systems. Ind. Eng. Chem. Res., 43:3539-3553,2004. 4. O. Geschke, H. Klank, and R Tellemann. Microsystem Engineering of Lab-on-a-Chip Devices. Wiley-VCH, 2004. 5. T. Korsmeyer, J. Zeng, and K. Geiner. Design tools for BioMEMS. In Design Automation Conference (DAC '04), pages 622-627,2004. 6. Y. Wang, R. Magargle, Q. Lin, J. Hoburg, and T. Mukherjee. System-oriented modeling and simulation of biofluidic lab-on-a-chip. In TRANSDUCERS '05, pages 1280-1283,2005. 7. T.L. Pillage, R.A. Rohrer, and C. Visweswariah. Electronic circuit and system simulation methods. McGraw Hill, 1998. 8. L.T. Biegler, I.E. Grossmann, and A.W Westerberg. Systematic Methods of Chemical Process Design. Prentice Hall, Upper Saddle River, NJ 07458,1997. 9. Y. Wang, Q. Lin, and T. Mukherjee. System-oriented dispersion models of general-shaped electrophoresis microchannels. Lab-on-a-chip, 4:453^63,2004. 10. Y. Wang, Q. Lin, and T. Mukherjee. A model for complex electrokinetic passive micromixers. RSC Lab-ona-Chip, 5(8):877-887,2005. 11. R. Magargle, J.F. Hoburg, and T. Mukherjee. Microfluidic injector models based on neural networks. In NanoTech (MSM '05), pages 616-619,2005. 12. WS.B. Yeung, G.A. Luo, Q.G. Wang, and J.R Ou. Capillary electrophoresis-based immunoassay. Journal of Chromatography B, pages 217-228,2003. 13. The Mathworks Inc. FEMLAB - Finite Element Modelling LABoratory version 3.2. http://www.femlab.com. 14. A.J. Pfeiffer, J.D. Siirola, and S. Hauan. Optimal design of microscale separation systems using distributed agents. In Foundations of Computer-Aided Process Design (FOCAPD '04), pages 381-384,2004. 15. A.J. Pfeiffer, T. Mukherjee, and S. Hauan. Simultaneous design and placement of multiplexed chemical processing systems on microchips. In International Conference on Computer-Aided Design (ICCAD '04), pages 229-236. 2004.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Two level control of the sequence fed batch continuous hybridoma bioreactor Irina D. Ofiteru, Alexandru Woinaroschy, Vasile Lavric Chemical Engineering Department, University Politehnica of Bucharest, Polizu 1-7, 011061 Bucharest, Romania

In the present study a recirculation system for monoclonal antibodies production, operated consecutively fed batch and continuously, was modelled and subjected to optimal control. The optimization procedure uses a two level approach: one regarding the overall process, and two inner ones, concerning the fed batch and continuous steps. The best switch time between the two operating modes was calculated, together with the best control variable profile for each section. Keywords: hybridoma, two-level optimization, optimal control, genetic algorithms 1. Introduction The monoclonal antibodies (MAbs) are produced in the last years in large quantities and still the potential growth of the market is more than 25% per year. There is a high demand for these products (used especially against cancer) but the current production costs are very high (by factors of 20 to 200 times per gram) compared with the ones from classical chemical synthesis (Sommerfeld and Strube, 2005). The main efforts are now focused on cutting down the operating costs. This can be achieved using the large potential for optimizing the processes; not only its upstream (cells, culture medium, supplements) and downstream (separation and purification) sections, but also the bioreactor itself. For small product concentrations in the fermentation step, the cost distribution between upstream and downstream is approximately 50-50%. Once the product concentration rises, both the overall and the upstream costs decrease (with more than 20%)), although the proportion of downstream costs in the overall costs increases (Sommerfeld and Strube, 2005).

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In the upstream section, one of the pertinent challenges is reducing the formation of large amounts of secondary metabolic products, which have inhibitory effects on cell growth and production rates. It is therefore important to maintain the cells in a physiological state characterized by a minimum production of waste metabolites and a maximum production of valuable products. This goal implies the development of an optimal nutrients supplying strategy which modifies the growth medium in such a way that the cells alter their metabolism to produce as much MAbs as possible, with minimal waste. Optimization studies have been made both for fed batch (Dhir et al., 2000; Sarkar and Modak, 2003; Woinaroschy et al., 2004) and continuous processes (Ofiteru et al., 2005). Our previous studies (Woinaroschy et al., 2004; Ofiteru et al., 2005) in optimal control of the hybridoma bioreactor treated separately the fed batch and the continuous operating modes, pointing out their benefits and drawbacks. The results suggested a combined approach, namely using the sequence fed batch - continuous. In the present study, a recirculation system composed of a perfectly mixed bioreactor operated consecutively fed batch and continuously, respectively, a cell separator, a mixer and a purge was modelled then subjected to optimal control. The optimization procedure uses a two level approach. The outer optimization, searching for the proper time switch between fed batch and continuous operation modes, is based upon the direct search algorithm of Luus and Jaakola. Basically, in the feasible region of parameters takes place a random search in discs with given radius having as centres the proposed values for these parameters. As a result, a possibly new optimum point could be reached, the centres of discs being moved accordingly. After each jump, a contraction of the discs' radii is imposed, till either the minimum value for the objective function is found or the allowed number of iterations is reached. The inner optimization procedure is based upon genetic algorithms, which are applied for the optimal glutamine set point computation for the fed batch operating mode(Woinaroschy et al., 2004) and the determination of the inlet flow profile in time for the continuous mode, both aiming to maximize the MAb production (Ofiteru et al., 2005). 2. Mathematical model The representation of the process is given in Figure 1, together with the main notations. Since the concentration of the cells is rather low, the recirculation fraction, a, was set to 0.15, while the purge fraction, p, to 0.005. For the first operating stage, which is fed batch, there is no recirculation, and that the process is formed only by the reactor, together with the feeding. The Nielsen kinetic model (Ryszczuc and Emborg, 1997) was used, such as in the aforementioned studies for the optimal control of the fed batch, respectively continuous bioreactors. This kinetic is a one-compartment model assuming amino acids as a limiting factor and saturated glucose metabolism. The cells produce monoclonal antibody (P), lactate (L), ammonia (M) and alanine (A)

Two Level Control of the Sequence Fed Batch — Continuous Hybridoma Bioreactor

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using a medium which has glucose (S) and glutamine (Q) as substrates. The production rate for MAb was modified (Ofiteru et al., 2003) to be computed the same way the specific growth rate is. For the detailed mathematical model used for each of the stages of the process, see Woinaroschy et al. (2004) and Ofiteru et al. (2005), respectively. aDv,Nv,r,N:D^S,OAI-rM,P

Figure 1. Sketch of the process, together with the main notations used in the mathematical model

2.1. The objective function The objective function should encode the search for the maximum MAb production through an optimum switch time between fed batch and continuous operation, an optimum glutamine set-point profile for the fed batch stage and an optimum flowrate profile for the continuous stage. The overall operating period for the system is fixed. Since we have a two level optimization problem, a specific objective function has to be used for each level. For the fed batch stage, as there is no dead cells removal, optimum glutamine set-point profile should be sought such as to keep their concentration as low as possible, besides the maximum MAb production. After some preliminary studies, two objective functions were employed (subject to minimization): Fobj^j,

=~

P ^ FB

P ^ FR

V ^ FB

V

.^ K

(1)

(2)

' Fl

The objective function described by eq, (1) takes into account the dead cells concentration. But it was found that the overall performance of the process was lowered by the request of keeping this concentration small, so that the objective function was simplified (see eq. (2)). The results obtained will be compared and the best compromise between keeping the dead cells concentration low and increase the overall performance of the process will be selected. For the continuous stage, the objective function should encode the search for the maximum MAb production through an optimum flowrate profile, Dv(t), for a given operating period:

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^Cont

(3) For the whole process, we seek for the maximum MAb production, so the natural choice of the objective function for the second level is: ^^^Process

= ^FB ' ^FB + Pcont

f/^\

3. Results and discussions 3.1. Solving procedure The fixed-time two-level optimization problem had to be solved off-line, due to the long computation times, using Luus and Jaakola algorithm to calculate the proper time switch between fed batch and continuous operating modes and genetic algorithms to optimize each individual sequence. The process starts as fed batch, with glucose and glutamine as substrates, until the proposed switch time is reached. After that, the continuous operation begins, starting from the cells' and metabolites' concentrations achieved in the fed batch period. For the fed batch step the command variable is the glutamine set-point (Woinaroschy et al., 2004). The command time profile is encoded into a chromosome, every glutamine set-point corresponding to a time stage being represented as a gene. For the continuous step the command variable is the inlet flow. The operating period was divided into the same number of intervals as the fed batch time and the same encoding procedure was applied (Ofiteru et al., 2005). In order to test the method's convergence, three replicas were done for both cases: with or without dead cells concentration in the objective function. Each run started fi-om the same conditions as in our previous studies. Also, some supplemental runs were done with higher values of substrates concentration in the inlet flow. 3.2. Discussion of results The maximum MAb obtained using the objective function without dead cells was 4385 mg, while the maximum MAb obtained considering them was 3686 mg. Clearly, this difference is the result of the higher concentration (respectively mass) of MAb after the fed batch step: 10.408 mg/1 (1766 mg) for the former case, 8.356 mg/1 (1067 mg) for the later case, respectively. Another significant difference is between time switches: 562.312 h for the former case, respectively 441 h for the later. The request of keeping the dead cells concentration optimally low, while maximizing the living cells concentration, slows down the MAb synthesis process. This is inherent, since the cells' death is the results, partly, of adversely medium conditions, which in turn favour the valu-

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able MAb production. As is already know, the glutamine decomposes spontaneously forming ammonia, which is toxic for the cells. This influences the command profile for the fed batch step (glutamine set point), which can vary in the interval 0.3 - 1.0 g/1. When the objective function does not take into account the dead cells concentration, the glutamine set point has values constantly higher then 0.6 g/1, with one exception (see Figure 2a). When the dead cells concentration enters the objective fUnction, the variation is more abrupt and the minimum allowed value (0.3 g/1) is reached several times. Comparing the results, in terms of productivity gain (expressed in mg MAb/h), we observed that the growing rate for the dead cells' concentration, 1.27 (the ratio between the final concentrations when using eq. (2) and (1) as objective function) is lower than productivity rate increase, 1.3, showing that the constraint imposed by eq. (1) are somehow relaxed. A side effect when using eq. (1) as objective function is the length of the fed batch process, lower than for the other case. It should be mentioned that the fed batch step is much longer than in the previous cases, while the concentration of the dead cells is rather higher (Woinaroschy et al. 2004; Ofiteru et al., 2003), irrespective of the objective function used. Definitely, this proves that the system's optimality does not imply the optimal behaviour of its parts. Command continue

Figure 2. The main state variable profiles for the system, a) objective function (1) and (3); b) objective function (2) and (3). Ny/No - viable/dead cells, cP/mP - MAb concentration/ mass

During the continuous period of the system, the dead cells concentration decreases constantly, due to the beneficial effect of the purge (see Figure 2, on the RHS of the dotted line). Around 800 h system time, the MAb concentration reaches a pseudo-stationary value which is maintained till the end of the process and, concomitantly, the living cells concentration starts to grow exponentially. This is supported by the high values of the inlet flow, which reaches constantly the maximum value allowed, thus ensuring a high level of substrates concentrations. One side effect of these high inflows is an increased ammonia concentration (data not shown), the only by-product (and in the same time the most deleterious one for the cells) which rises above the MAb concentration. Analysing the results obtained, we observed the low values for the valuable product concentration and concluded that some run-tests with higher glucose and glutamine concentration in the inlet flow should be done, so we increased

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them five times. Several interesting differences compared with the "normal" situation were noticed (no figure is presented, due to the lack of space). First, the switching time between the fed batch and continuous steps has closer values: 257.8 h for the objective function (1), respectively 277.3 h for the objective function (2). Second, the amount of MAb obtained is significantly higher, 10400 mg for the former case and respectively 9629 mg, for the later. Third, the dead cells concentration does not decrease after switching to the continuous step, but keeps increasing, its growth rate surpassing the viable cells concentration's one. This effect is more pronounced when using the objective function (1), although its influence upon the system's performance is indirect. This can explain why the inlet flow is kept to smaller values in this case. 4. Conclusions The analysis of a recirculation system (Figure 1) was done, first modelling it, and then searching for its optimal control. Two approaches were used, one more conservative, that considers maintaining the dead cells concentration at lowest level, the other more relaxed, taking into consideration only the highest MAb production. Although it could seem surprising, the relaxed objective function gave better results, in terms of MAb production, but at the cost of higher byproducts concentrations. A final selection of the variable command profile would not be proper without considering also the performance of the downstream separation processes. Based only upon the MAb production and byproducts outlet concentrations and without a sound economic analysis, we concluded that the advantage of using higher substrate concentrations in the feed (a bigger amount of MAb produced) is counter-balanced by the increase in dead cells and by-products concentration. But, if the market price for the MAb is high enough, this could be the recommended way of increasing the productivity. References Dhir, S., K.J. Morrow, R.R. Rhinehart, T. Wiesner, 2000. Dynamic optimization of hybridoma in a fed - batch bioreactor. Biotechnol. Bioeng. 67, 197 - 205. Ofiteru, I.D., Lavric, V.and A. Woinaroschy, A., 2003. Sensitivity analysis of the fed-batch animal cell bioreactor. Chemical Engineering Transactions 3 (3), 1845 - 1850. Ofiteru, I.D., Woinaroschy, A., Lavric, V., 2005. Optimal control of a continuous perfectly mixed hybridoma bioreactor. ESCAPE 15*, May 29 - June 1, Barcelona, Spain Ryszczuc, A., Emborg, C , 1997. Evaluation of mammalian fed - batch cultivation by two different models. Bioprocess Eng. 16, 185 - 191. Sarkar, D. and J.M. Modak, 2003. Optimization of fed-batch bioreactors using genetic algorithms. Chem. Eng. Sci. 58, 2284 - 2296. Sommerfeld, S., J. Strube, 2005. Challenges in biotechnology production - generic processes and process optimization for monoclonal antibodies. Chem. Eng. and processing 44, 1123 - 1137. Woinaroschy, A., Ofiteru, I.D., V. Lavric, 2004. Time-free schedule optimization of an animal cell fed-batch bioreactor. ESCAPE 14*, 16-19 May, Lisbon.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Optimal Delivery of Chemotherapeutic Agents in Cancer Pinky Dua^, Vivek Dua^, Efstratios N. Pistikopoulos^ ^Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom ^Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London JVCIE 7JE, United Kingdom Abstract In this paper, derivation of the optimal chemotherapy schedule is formulated and solved as a dynamic optimization problem. For this purpose two models representing the tumour growth kinetics are considered. The dynamic optimization problem for the first model, which is cell cycle non-specific, takes into account multiple time characteristics, drug resistance and toxicity. The discontinuity in the model is formulated by introducing integer variables. For the second model, which is cell cycle specific, the tumour growth is modelled via two compartments: proliferating and resting compartment. Keywords: Drug Delivery Systems, Cancer, Chemotherapy, Optimal Control, Mixed Integer Dynamic Optimization 1. Introduction Cancer is a collective term that describes a group of diseases characterized by uncontrolled and unregulated growth of cells leading to invasion of surrounding tissues and spreading to the parts of the body that are distant from the site of origin. There are around 200 types of cancer and cancers of lungs, breast, bowel and prostrate are the most common ones. There are three main stages in the process of carcinogenesis: initiation, promotion and progression. The normal cell changes to an initiated cell and then to cancer differentiated cell and finally invades and spreads to the surrounding cells. The simplest mathematical model describes the entire cell cycle as a uniform entity, where all the cells contained in a tumour are of the same type. The cell cycle nonspecific models consist of one compartment so that the effect of the anticancer agents is same on all the cells. However these models fail to describe the action of cycle specific drugs due to their over-simplified nature. The more detailed multi-compartment models (cell cycle specific models) are considered for this purpose. Here the cell cycle is divided into compartments depending on the types of cells that are affected by the drug. Chemotherapy is one of the most commonly used treatments for cancer that uses anticancer or cytotoxic drugs to destroy or kill cancer cells. The suitability of chemotherapy and the choice of drugs depend on many factors including the type of cancer, the location of the origin of the cancer in the body, how mature the cancer cells are and whether the cancer cells have spread to the other parts of the body. Chemotherapy targets dividing cells which does not only include cancer cells but any

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normal cells that are dividing such as the hair producing cells, cells that line mouth and digestive system and those in the bone marrow and the skin. Harrold and Parker (2004) recently proposed a mixed integer linear programming approach for the derivation of optimal chemotherapy schedule. In this paper, optimal chemotherapy schedules are derived with the objective of minimizing the final number of tumour cells at the end of the treatment. The rest of the paper is orgainzed as follows: in Section 2 optimal cancer chemotherapy schedule is derived for a cell cycle non-specific model whereas a cell cycle specific model is considered in Section 3; concluding remarks are presented in Section 4. 2. Optimal Control for Cell Cycle non-Specific Model A pharmacokinetic/pharmacodynamic model given in Martin (1992) is used for the derivation of the schedule. The objective is to obtain the drug dosage over a fixed period of time so as to minimize the number of cancer cells at the end of the period subject to constraints on toxicity and resistance of the drugs. The optimal control problem is formulated as follows: min J(w) = -z{T) s.t. z(t) = -h{t) + k{v{t)-Vth)y

z(0) = l n [ ^ ^ i){t) = u{t) - yv{t) t;(0) = i;o=0 0
forall/e[0,r]

T

\v{s)ds < v^^^ 0

z(21)>ln(200) z(42) > ln(400) z(63) > ln(800) vit)>yVth,

forall/e[0,r]

v(t) - Vfh < yM,

for all t e [0, T]

where u = [wi,....,w„]^e9t" is the vector of the rate of delivery of drug, z(t) is the nondimensional tumour size, yt is a growth parameter, k is the proportion of the tumour cells killed per unit time per unit drug concentration, v(t) is the concentration of the anticancer drug at time t, Vth is the therapeutic concentration of the drug, >; is a 0-1 binary variable, 0 is the plateau population or the carrying capacity of the tumour, A^o is the initial tumour cell population, Vmax is the maximum amount of drug in the plasma, Vcum is the cumulative toxicity and Mis a large positive number. Note that in this formulation the binary variable, y, is introduced to model the discontinuity so that y takes the value 0, if 0 < tXO ^ '^th, or the value 1, if v{t) > Vth and the following transformation is introduced to make the model tractable and tumour size dimensionless (Dua, 2005):

Optimal Delivery of Chemotherapeutic Agents in Cancer

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where A^ is the number of tumour cells. The data for the parameters in the model is taken from Martin (1992). This problem is a mixed-integer dynamic optimization problem (Bansal et al, 2003) and is solved using gPROMS (gPROMS, 2003) by taking the control interval of one day and a time period of 84 days. The profiles of the optimal chemotherapy and tumour growth are shown in Figures 1 and 2 and are consistent with those reported in the open literature on optimal control strategies for cancer chemotherapy.

CD 3

O

^

m (0

45 40 35 30 25 20 15 10 5 0

*mnmmnmmmmn%

0

20

10

30

40

50

60

70

80

Time (days) Figure 1 Optimal chemotherapy protocol for the cell cycle non specific model

1.2E+10 \-.--*^^ 1.0E+10 i H^ •

•« 8.0E+09

•

o 6.0E+09 0)

•

1 4.0E+09 3

^

2.0E+09

\

i HF+nA

0

1

t

1

i

10

20

30

40

1 ^WiiiiiifctA^AA^

50

60

A^^^^^^^.>^A.>^^.>

70

80

90

Time (days) Figure 2 Predicted tumour growth using the optimal treatment protocol Initially no drug is delivered until the time approaches 21 days, the first interval of the muhiple characteristic time constraints, z(21) > ln(200), which are introduced to model

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drug resistance. This can be attributed to the fact that an initial high dose may not necessarily result in an overall decrease in the tumour size at the end of the time period as compared to the minimum tumour size obtained by the optimal chemotherapy protocol, while the constraints on toxicity and drug resistance are satisfied. Moreover, the drugs are given in large doses intermittently rather than in small doses continuously. The tumour follows the Gompertz growth until the first injection of the drug that results in a reduction of 70% of the initial tumour size. High intensity chemotherapy, from time 40 days till the end of the therapy, results in an increase in the rate of tumour reduction. The value of the objective function at the end of the treatment is given by 7.5x10"^ cancer cells. As a conclusion, the optimal way to reduce the tumour size was to apply high intensity therapy towards the end of the chemotherapy period. The optimal chemotherapy protocol shown in Figure 1 produces a 99.9% reduction of the initial tumour size.

3. Optimal Control for Cell Cycle Specific Model The model of Panetta and Adam (1995) describes the administration of the anticancer drug in the case of cell cycle specific chemotherapy. In this model, the effect of the drug depends only on the duration of the injection and not on the amount of the drug that is injected. For this reason, this model is modified so as to relate the effect of the drug with the rate of delivery of the drug (Dua, 2005) and minimization of the final tumour population is formulated as the following optimal control problem: mmJ{u) = P{T) + Q{T) u{t)

s.t. P{t) = {a-m-

n)P(t) + bQ(t) - git)P(tl

P(0) = PQ

Q(t) = mP(t)-bQ(tl

Q(0) = Qo

y(t) = ^(01 1 - ^ 1 - g(t)y(tl

y{0) = yo

v(t) = u(t)-l^i(t% g(t) = k^Vi(t) y^^
t;i(0) = i;o for all/G [0,7]

where P and Q represent the cycling and non-cycling tumour cell mass respectively, a is the cycling cell growth rate, m is the rate at which cycling cells become non-cycling, n is the natural decay of cycling cells, b is the rate at which non-cycling cells become cycling, y is the number of normal cells, ^is the growth rate, K is the carrying capacity of normal tissue, g is the effect of drug on the cells, Vi is the concentration of the drug in the body, u is the rate of the delivery of the drug, ki is the parameter for the kill rate and ;^is the parameter for the decay of the drug. The parameters for this model are taken from Panetta and Adam (1995). The problem was solved by using gPROMS by considering a time horizon of 60 days and control interval of one day. The optimal chemotherapy schedule derived is shown in Figure 3. Initially, the drug is delivered at a high dose, followed by a low intensity therapy for ahnost the rest of the treatment period. Finally, a large amount of drug is applied in the last two days. The objective fimction represents the size of the tumour, including both the proliferating (cycling) and resting (non-cycling) compartments. The

Optimal Delivery of Chemotherapeutic Agents in Cancer

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evolution of number of cancer cells during the therapy period is illustrated in Figure 4. After the optimal drug protocol is applied to the patient, the predicted number of cancer cells drops from 10^^ cells to 4x10^^ cells, which corresponds to 60% reduction of the initial tumour size. The final reduction in the number of cancer cells is less when compared to the results obtained for the cell cycle non-specific case. Nevertheless, the cell cycle specific model includes the constraint that places strict limits to the acceptable number of normal cells, i.e. j^min = 10^ As shown in Figure 5, for almost the entire treatment period, the level of normal cells is near to its lower bound. It is obvious that the first high dose of the drug had an enormous effect on normal cells, which did not allow further application of high doses, apart from that in the last two days.

35 3

30 25

= ^ 15 Q

o

" 10

5 0

^•* **••*••••

10

»>>*•••••»•

20

30

••••••^•»»

40

50

60

50

60

Time (days) Figure 3 Optimal chemotherapy protocol for the cell cycle specific model

1.2E+12 1.0E+12 i^ _2 1) 8.0E+11 O

0 6.0E+11 o 1 4.0E+11 ^ 2.0E+11 O.OE+00

10

20

30

40

Time (days) Figure 4 Predicted tumour growth using the optimal treatment protocol - cycling and non-cycling cells

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1.2E+09 (A

r

1.0E+09 1 •

® 8.0E+08

' •

«^ f 6.0E+08

•

£ 4.0E+08 3

2

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^^^^-'H-.^ ' '^**t>t>#H###t#»##»#»»##»M###t^<Mf

n OF+nn 0

10

20

30

40

50

f^

60

Time (days) Figure 5 Predicted growth of the normal cell population using the optimal treatment protocol

4. Concluding Remarks In this paper, two models for the derivation of optimal chemotherapy schedules were considered. The first model involves the use of cell cycle non-specific agents. The tumour is modelled as one compartment and the drug is assumed to affect all the cancer cells in the same way. Toxicity to normal cells is controlled with constraints that limit drug concentration during therapy, as well as the total cumulative toxicity at the end of the treatment. The problem is formulated and solved as a mixed integer dynamic optimization problem. The optimal chemotherapy protocol derived in this case kept the initial administration of the drug at a low level, while most of the drug was delivered towards the end of the therapy. The second model describes the effects of the cell cycle specific drugs. The tumour is divided into two compartments; the cycling and the noncycling compartments. Toxicity is monitored through the differential equation that describes the effect of the drug on normal cell population. The problem is formulated and solved as a dynamic optimization problem. The optimal chemotherapy indicates that the anticancer agent should be administered in high doses at the beginning and at the end of the treatment period. The current work involves formulating these problems such that the objective is not only to minimize the tumour size at the end of a given time period but also to optimize the time period for the therapy. References V. Bansal, V. Sakizlis, R. Ross, J,D. Perkins and E.N. Pistikopoulos, 2003, New algorithms for mixed integer dynamic optimization, Computers and Chemical Engineering, 27, 647-668 P. Dua, 2005, Model based and parametric control for drug delivery systems, PhD Thesis, Imperial College London, University of London gPROMS, 2003, Introductory User's Guide, Release 2.2, Process Systems Enterprise Limited, London, U. K. J.M. Harrold and R.S. Parker, 2004, An MILP approach to cancer chemotherapy dose regime design, Proc. American Control Conference, 969-974 R.B. Martin, 1992, Optimal control drug scheduling of cancer chemotherapy, Automatica, 28, 1113-1123 J.C. Panetta and J. Adam, 1995, A mathematical model of cycle-specific chemotherapy. Mathematical and Computer Modelling, 22, 67-82

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Dissipative particle dynamics simulation of ibuprofen molecules distribution in the matrix of solid lipid microparticles (SLM) *

Chunxia Long, Lijuan Zhang, Yu Qian School of Chemical Engineering, South China University of Technology, Guangzhou 510640, P. R. China

Abstract: Dissipative particle dynamics (DPD) simulation technique is used to simulate the distribution of ibuprofen molecules in the drug carrier of solid lipid microparticles (SLM). For the SLM with tristearin as its carrier material, ibuprofen molecules are adsorbed on the surface of the carrier and a shellspherical structure is formed, while for the SLM made from glyceryl behenate, ibuprofen molecules distribute in the outer area of the carrier matrix. The release performance of the SLM is predicted basing on the simulated microstructure. A better sustained release of ibuprofen is expected in the glyceryl behenate SLM. Keywords: dissipative particle dynamics, solid lipid microparticles, drug release, ibuprofen, structure and properties 1. Introduction Drug carriers in submicron size are attracting increasing attention in recent years. Their micro-sizes enable them possess distinctive properties. Solid lipid microparticles (SLM) are newly developed colloidal drug carriers made from fatty acid, glyceride, fatty alcohol and solid wax as matrixes with high melting points (Gasco, 1993; Mehnert et al., 2001). They exhibit better biocompatibility and more efficient controlled and sustained drug release performance. SLM are complex multi-phase systems and their properties are determined not only by their composition but also by their microstructures. The distribution of drug molecules in a carrier is an essential characteristic of the microstructure, thus directly determine their stability and release performance. For a drug carrier in submicron size, it is difficult to observe its drug

Corresponding author: E-mail: [email protected]. Phone and Fax: +86(20)87113046.

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distribution by experiment. Computer simulation provides us a convenient approach to investigate its drug distribution. Dissipative particle dynamics (DPD), proposed by Hoogerbrugge and Koelman (1992), is a mesoscopic simulation technique to simulate complex fluid dynamical phenomena. Its simulation strategy is regarding clusters of atoms or fluid packets as fluid particles or beads that move according to the Newton's equations of motion (Hoogerbrugge et al., 1992). The method has been successfully applied to study the behavior of surfactants at the water/oil interface (Dong et al., 2004), microphase separation of block co-polymers (Groot et a l , 1998) and polymer melts (Zhang et a l , 2000). However, DPD method used in a drug delivery system is seldom reported. In this work, DPD simulation technique is applied to explore the distribution of drug molecules in the matrix of SLM, in which tristearin and glyceryl behenate are chosen as the carrier materials, ibuprofen as the model drug, and polyvinyl alcohol (PVA) as the stabilizer, respectively. Ibuprofen is a nonsteroidal antiinflammatory drug and has a low solubility and short half-life. Embedding ibuprofen in the solid matrix of SLM can sustain its release and improve its therapeutic efficacy. A release performance prediction is made on the base of the simulated microstructure. 2. Calculation of the simulation parameters In DPD simulations, a set of beads move according to the Newton's equations of motion. Each bead is subject to three non-bonded forces from its neighbours: a conservative force {Ftf), a. random force ( F / ) , and a dissipative force (Fjf). In addition, beads connected in a molecule experience a spring force (Fjj ) due to bonded neighbours (Espanol et al., 1995).

da CH-

I

db

-CHCH2^/ ^^3

I

I

dc

\^CH—COOH

^ = ^

I CH3

l^^^^gcrP-Itd^M^^^^ CH—0-6-CH2-(CH2)i5CH3 (!:H2-0-H!!!—CH2—(CH2)i5CH3

(a)

(b)

Figure 1. The molecular structures of ibuprofen(a) and tristearin(b).

In DPD simulations, coarse-grained approach is used where one DPD bead represents a group of atoms or a liquid volume. The coarse-graining procedure of the molecules in our investigation is shown as follows. A water molecule and a monomer of PVA are represented with a single bead denoted by w and /?, respectively. The molecular structure of ibuprofen is shown in Figure 1(a). Each part separated with dashed lines is represented with a single bead named da, db and dc, respectively. The molecular structure of tristearin is shown in Figure

Dissipative Particle Dynamics Simulation oflhuprofen Molecules Distribution

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1(b). The group in the dashed line box is represented with a bead / The hydrocarbon chain are represented with several beads named a. Glyceryl behenate is a mixture of mono-, di- and triglycerides of behenic acid (C22). The coarse-grained molecular structure of the triglyceride is similar with that of tristearin. For the mono- and diglycerides, the group with a hydroxyl is presented with a bead named h. Since the coarse-grained model of each molecule has been developed, the repulsion parameters between each bead pair are calculated. The repulsion parameters between beads of the same type are calculated according to equation (1) for all types (Groot et al., 1997):

a,=—k,T P

(1)

Here the compressibility of pure fluid is chosen as p=3, which is close to that of water (Groot et al., 1997), thus a.. = 25k^T. As Hoogerbrugge and Koelman did (1992), we choose the conservative interaction potentials kBT=\, The repulse parameters between different types of beads are obtained with equation (2) (Groot et al., 1997):

^,-^.+3.27;^

(2)

where X/y is the Flory-Huggins parameter, which is obtained from two approaches of experiment and molecular simulation. Flory-Huggins parameter x are calculated from solubility parameters using the formula (3) (Mailti et al., 2004): X,={S,-d.fVIRT

(3)

where 6^ and dj are the solubility parameters of / andy, respectively, while Fis the molar volume of the bead. The solubility parameter is calculated using Discover and Amorphous Cell program in the software Materials Studio (Accelrys) with the COMPASS force field. The calculated interaction parameters in DPD simulation are listed in Table 1. Table 1. The interaction parameters used in DPD simulation. parameter aij

w

a

f

da db dc P h

w 25 102.12 55.55 118.17 83.83 57.11 54.81 39.03

a

/

da

db

dc

P

h

25 28.40 25.00 28.58 39.13 34.20 44.30

25 29.06 25.00 28.83 26.79 32.91

25 29.25 41.37 35.87 47.89

25 29 26.88 33.38

25 25.24 26.12

25 26.95

25

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C Long et al

The simulation is performed in a 20x20x20 box (p=3) with a periodic boundary condition in all three directions. The beads of the same molecule are connected by a harmonic spring and the spring constant C= 4 (Greet et al., 1998). Each DPD simulation run 10000 steps with a time step of 0.05ns. In the simulation system, the relative amount of PVA, carrier material, ibuprofen and water are 3, 8, 4 and 85, respectively. Glyceryl behenate is a mixture of mono-, di- and triglycerides with a molar fraction of 15%, 45% and 40%). All the components are considered into the simulation system. 3. Results and discussion 3.L Simulation results of ibuprofen distribution in SLM Shown in Figure 2 are the snap shots of phase separation of tristearin SLM during the simulation. To display the molecular arrangement of the drug carrier clearly, water molecules are not shown here. As seen in Figure 2(a), all the components are mixed together at the beginning. After 2000 simulation steps, the phase separation is obvious. The system reach equilibrium after 5000 steps, and no obvious changes are observed for the system with increasing simulation time. It is seen in Figure 2(c), in the final equilibrium state, PVA molecules enwrap on the surface of the drug carrier and serve as the stabilizer. The simulation is carried out on a PC with Intel® Pentium® 4 CPU 3.0GHz and spends about 10 hours.

(a) 100 steps

(b) 2000 steps

(c) 10000 steps

Figure 2. The phase separation of tristearin SLM at different simulation steps ( ^ * PVA, ^ carrier material, • • • non-polar bead of ibuprofen, • • polar bead of ibuprofen).

To reveal how ibuprofen molecules distribute in the carrier, section views of SLM are taken in Figure 3 without PVA. It is shown in Figure 3(a), in the tristearin SLM, ibuprofen molecules are adsorbed on the surface of the carrier with their polar groups towards the water phase and non-polar groups to the carrier, and a shell-spherical structure is formed. However, in the glyceryl behenate SLM, shown in Figure 3(b), ibuprofen molecules are not adsorbed on

Dissipative Particle Dynamics Simulation oflbuprofen

Molecules Distribution

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the surface of the carrier, but distribute in the outer area of the carrier matrix. This is because the mono- and diglycerides of glyceryl behenate possess hydrophilic hydroxyls which can form hydrogen-bonds with the carboxyls of ibuprofen molecules. Hence, in the glyceryl behenate SLM, the hydrophobic groups of ibuprofen molecules remain in the body of the carrier with their carboxyls at the oil/water interface along with the hydroxyls of glyceryl behenate molecules.

(a) tristearin SLM

(b) glyceryl behenate SLM

Figure 3. The section views of the SLM (©= carrier material, € ^ non-polar bead of ibuprofen, #™ polar bead of ibuprofen).

3.2. Prediction of the SLM release performance From the simulation results above, the release performance of SLM can be predicted. A sustained release of ibuprofen dispersed in the SLM achieves in comparison with ibuprofen crystal. However, ibuprofen molecules do not distribute evenly in the whole matrix. Instead, they are adsorbed on the surface or locate at the outer area of the carrier. This will most probably induce an initial burst release for both tristearin SLM and glyceryl behenate SLM. However, the extents of their burst release are different. In the tristearin SLM, ibuprofen molecules are adsorbed on the surface of the carrier and a shellspherical structure forms. The resistance of drug release is the stabilizer layer. On the other hand, in the glyceryl behenate SLM drug molecules distribute in the matrix of the carrier. During the release process, drug molecules diffuse through the carrier matrix and then pass through the stabilizer layer. The drug release rate is thus much reduced. A better sustained release of ibuprofen will be obtained in the glyceryl behenate SLM. A property prediction of a drug carrier is made from its microstructure, which is helpfiil in designing a drug carrier and optimizing its property.

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4. Conclusions The DPD simulation technique is used to model the SLM with tristearin and glyceryl behenate as carrier materials. The simulation results indicate that, for the tristearin SLM, ibuprofen molecules are adsorbed on the surface of the drug carrier and a shell-spherical structure is formed, while ibuprofen molecules mainly distribute in the outer area of the matrix of the glyceryl behenate SLM. The release performance of the SLM is predicted from the simulated microstructures. Simulation results presented in this work are interesting and consistent with the experimental results. The corresponding in vitro release performance of the SLM is also investigated, which will be demonstrated in another paper. The DPD method enables us to have an insight into the microstructure of a drug carrier and predict its property. In this way, a desired property of the drug carrier can be obtained by optimizing its microstructure, which would hopefully accelerate the new product design and reduce the cost and time. Acknowledgments Financial support from the National Natural Science Foundation of China (No. 20536020, No. 20476033), the China Excellent Young Scientist Fund (No. 20225620) and Guangdong Province Science Fund (No. 04020121) are gratefiilly acknowledged. References Dong, F.L., Y. Li and P. Zhang, 2004. Mesoscopic simulation study on the orientation of surfactants adsorbed at the Hquid/Uquid interface. Chemical Physics Letters, 399, 215-219. Espanol, P. and P. Warren, 1995. Statistical mechanics of dissipative particle dynamics, Europhysics Letters, 30(4), 191-196. Gasco, M.R., 1993. Method for producing solid lipid microspheres having a marrow size distribution, US Patent No. 188837. Groot, R.D. and P.B. Warren, 1997. Dissipative particle dynamics:Bridging the gap between atomistic and mesoscopic simulation, Journal of Chemical Physics, 107(11), 4423-4435. Groot, R.D. and T.J. Madden, 1998. Dynamic simulation of diblock copolymer microphase separation. Journal of Chemical Physics, 108 (20), 8713-8724. Hoogerbrugge, P.J. and J.M.V.A. Koelman, 1992. Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhysics Letters, 19(3), 155-160. Mailti, A. and S. McGrother, 2004. Bead-bead interaction parameters in dissipative particle dynamics: relation to bead-size, solubility parameter, and surface tension. Journal of Chemical Physics, 120(3), 1594-1601. Mehnert, W. and K. Mader, 2001. Solid lipid nanoparticles production, characterization and applications, Advanced Drug Delivery Reviews, 42, 165-196. Zhang, K. and C.W. Manke, 2000. Simulation of diblock copolymer melts by Dissipative Particle Dynamics, Computer Physics Communications, 129, 275-281.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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An Integrative Systems Biology Approach for Analyzing Liver Hypermetabolism Eric Yang^, Francois Berthiaume^, Martin L.Yarmush^'^, loannis P. Androulakis^'' ^Biomedical Engineering Department, Rutgers University, Piscataway, NJ, USA ^Center for Engineering in Medicine, Harvard Medical School, Boston, MA, USA Abstract Severe injury and other disease conditions (cancer, diabetes) lead to a hypermetabolic state featuring increased turnover of proteins, fatty acids and carbohydrates. Prolonged hypermetabolism is accompanied by progressive organ failure and serious complications. Understanding the biochemical underpinnings of the hypermetabolic response will lead to a better understanding of the changes at the systemic level and ultimately provide clues for limiting its harmful consequences. We examine integrated bioinformatics approaches combining novel modeling and computational techniques, animal injury models, and gene expression measurements for analyzing hypermetabolic states. We demonstrate how integrative systems biology developments upgrade the information content of diverse biological data and ultimately advance our fundamental biological understanding Keywords: hypermetabolism, gene expression, regulation. 1. Introduction Severe injury, such as trauma, bums and other common disease conditions such as cancer and diabetes, leads to a hypermetabolic state which features increased turnover of proteins, fatty acids and carbohydrates in the whole body. Prolonged hypermetabolism is typically accompanied by a severe loss of lean body mass and progressive organ failure, and predisposes patients to complications such as infections and multiple organ dysfunction syndrome (MODS), which threatens recovery and survival (Evans and Smithies 1999). An important player in systemic hypermetabolism is the liver, which is the organ that is responsible for detoxification processes and maintaining circulating levels of important metabolites, such as glucose, ketone bodies, amino acids, and many plasma proteins including clotting factors. Thus, understanding the biochemical underpinnings of the hypermetabolic response of the liver may lead to a better understanding of the changes at the systemic level and ultimately provide clues for limiting the harmful, if not lethal, consequences of hypermetabolism. Metabolic changes following severe injury, such as bums, have been studied using biochemical and physiological techniques (Lee, Berthiaume et al. 2000) and many laboratories have undertaken organ-specific studies that involve the administration of a thermal injury to an animal followed by perfusion of the organ for determining net rates of production and uptake of a large number of metabolites. This systems-based approach has been used to describe changes in metabolite levels and establish a ' Corresponding author: [email protected]

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metabolic basis for the liver response to bum injury (Chen, Fei et al. 2003). However, these techniques are limited by the number of metabolic reactions that can be studied simultaneously and hence lack the ability to comprehensively describe a disease state. Furthermore, these studies do not address the underlying mechanisms that drive hypermetabolic changes, since the observed metabolic alterations are only one component of the overall liver response, and other aspects such as changes in liver gene expression following bum injury have not yet been fully characterized. Furthermore, understanding the genetic basis of the disease state may provide clues for the development of improved therapeutics. Therefore, in addition to characterizing the metabolic states by collecting metabolic profile data, the incorporation of gene expression profiles would generate a more complete picture and enhance our understanding of the mechanisms that control the onset, maintenance, and resolution of the hypermetabolic response. Although numerous studies have been previously undertaken to assess global gene expression pattems in cancer and other malignancies, (Alon, Barkai et al. 1999; Golub, Slonim et al. 1999), very few studies have used transcriptional profiling in the context of inflammation and sepsis, (Kaminski, AUard et al. 2000; Chinnaiyan, Huber-Lang et al. 2001). In order to elucidate the transcriptional characteristics of hypermetabolism following severe injury, various animal models have been proposed to induce and quantify in vivo the appropriate gene expression response (Chinnaiyan, Huber-Lang et al. 2001; Vemula, Berthiaume et al. 2004). However, very few studies have focused on global expression analysis for characterizing transient alterations in the context of severe trauma on the liver. In this paper we address issues related to the development of integrative systems biology approaches in terms of characterizing temporal events in transcriptional profiles. Gene expression dynamics of inflammation-stimulated hepatocytes and the potential for genetic reprogramming of the underlying gene expression is evaluated. We propose novel unsupervised computational approaches, based on proximity preserving hashing, for the identification of critical motifs of gene expression and demonstrate how an integrative framework combining transcriptional data, bioinformatics tools and regulatory models can be put together to quantify the observations and initiate the process of hypothesis generation in a systematic way. 2. Characterization of Temporal Events in Transcriptional Profiles 2.1. Experimental system. We present our analysis in the context of the experiments described in (Vemula, Berthiaume et al. 2004). Animal studies were performed with male Sprague-Dawley rats. Rats were anesthetized, and a cutaneous 1>^^ degree bum injury over -20% of the rat's total body surface area was induced by scalding for 10 s. Animals were sacrificed at 5 time points (0, 1, 4, 8, and 24h), total RNA harvested from livers was isolated and was hybridized to a U34A GeneChip that had 8,799 probes represented on each chip. The data are available at http://www.ncbi.nlm.nih.gov/geo (accession number GSE802). 2.2. Classification of temporal expression patterns. In order to characterize the expression waveforms we follow the basic formalism for time-series analysis presented in (Keogh, Lin et al. 2005). The Symbolic Aggregate approximation (SAX) of time series is based on the premise of transforming a time series into a corresponding sequence of symbols. The normalized time series are piecewise averaged thus transforming the time series to a lower dimensional space. An equiprobable discretization technique is applied to transform the series to a sequence of symbols. The symbolic representation makes it possible to further analyze the time

An Integrative Systems Biology Approach for Analyzing Liver Hypermetabolism

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Figure 1: Motif distribution and characteristic average motifs series in an order to uniquely characterize the overall dynamic response of each transcriptional profile. In order to assign a unique identifier to each profile, we explore the concept of hashing where the sequence of symbols is mapped onto a single integer w

h(c, w,a) = 1 + y [ord(c )-l]xa'^"^ where a is the size of the alphabet, w is length of the word, and c is the "letter" sequence to which the expression profile is assigned to, where w is a time window within the duration of the experiment. Genes with similar normalized expression profiles "hash" to similar motif values. As a result, we generate a distribution of such motif values and identify (i) overpopulated motifs, (ii) genes sharing similar motif values, i.e., similar expression profiles. The basic ideas were recently presented in (Androulakis, Vitolo et al. 2005). Figure 1 depicts the results for the bum data in terms of the distribution of motifs values and characteristic, average, profiles for the most abundant expression motifs. 2.3. Definition of transcriptional state and selection of informative expression patterns. The cellular system at the beginning of the experiment is assumed to be at some state of equilibrium. Once an external disturbance is introduced the system starts to deviate from its original homeostatic state. Excursions from this distribution are the result of the transcriptional changes in response to the external stimulus. Thus, we define the transcriptional state as the distribution of gene expression levels at any given time from the distribution at the control state. With an informative subset of genes, we should see time points where the transcriptional state is both non-Gaussian, and deviating significantly from their initial set point before the perturbation. We use the Kolmogorov-Smimov statistic to determine just how far from the basis curve (distribution at time t=0 hr) our subset deviates during the course of the experiment (Rassokhin and Agrafiotis 2000) with the assumption that maximally informative genes will be the ones whose transcriptional activity is being maximally modified. This statistic allows us to compute a metric that defines just how different the two distributions of expression levels are. Given the quantitative metric for establishing a deviation from the control transcriptional state we have implemented a rudimentary selection mechanism by which we incorporate peaks from the distribution in an attempt to identify the ensemble of profiles that maximized the deviation from the original distribution of expression values while maximizing the maximum point deviation relative to the distribution at time t=0, that ismaxmax|F(Yi(t))-F(Yi(0))|. Motifs t

l
'

(peaks) are sequentially introduced as long as the KS metric increases. Figure 2 depicts the deviation from the t=0 distribution for the informative genes belonging to six most

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Figure 2: Temporal evolution of transcriptional state indicative of a two-wave process and KS metric comparison between informative and uninformative important motifs. As shown in Figure 2, the expression at any time past the initial time is significantly non-Gaussian for the genes in the informative motifs (Figure 1). In fact, compared to the distribution of time zero, there is a significant difference in the shape of the curve characterizing the transcriptional state. Since the genes that we have selected have shown a significant difference, it therefore follows that the genes are reacting to outside perturbations. Had a non-informative subset of genes been chosen, we would expect to see less of a difference between the curves. Therefore from our transcriptional state definition, it becomes possible to define a metric that specifies just how significant the subset of genes is. The lower right part is a plot of the objective as a function of time. We thus identified the minimum number of expression signatures whose presence is critical for reproducing the observed transcriptional response in its entirety. It is also important to realize, that consistent with previous theoretical analyses of the system (Jayaraman, Yarmush et al. 2000) we also observe a potential genetic reprogramming due to a network of interacting transcription factors giving rise to two waves of gene expression alteration. This is clearly observed in the temporal evolution of the transcriptional state as well as the plot of the objective over time (lower right part of Figure 2) indicating an initial deviation from the original state, return to it by t=4hr and an eventual irreversible deviation by t=24h. 2.4. Regulatory, functional and metabolic characterization of the informative genes. In order to identify the underlying structure of a potential regulatory network, we assembled the set of transcription factors for all the genes involved in the six maximally informative motifs by making use of TRAFAC (Jegga, Sherwood et al. 2002), which runs the Genomatix Matlnspector analysis suite in the background. Once the necessary transcription factors have been extracted from TRAFAC, the conditional probability, p(TF|Clusteri), of a transcription factor lying in a specific cluster was computed. In support of our clustering, we identify a very high degree of clustering of transcription factors specific to each one of the identified motifs, quantified via the conditional probability and block-diagonal structure of the corresponding matrix. Figure 3 left panel, hence, verifying the co-regulation of genes belonging to similar motifs. The observed deviations for cluster 6 are more indicative of the lack of available data for vertebrate animals than anything else. A gene ontology search via the Rat Genome

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^

Figure 3: Conditional probabilities of motif-specific transcription factors and ontologies Database (http://rgd.mcw.edu/) was run to characterize the functionality of the genes belonging to the critical motifs. Significant gene ontologies that were common within each identified motif include the ubiquitin dependent protein catabolism, which is a major mechanism involved in muscle wasting which is characteristic of the hypermetabolic state that results after serious bum injuries, as well as pathways related to the activation of 15-hydroxyprostaglandin dehydrogenase involved in the inflammatory response. IL-1 and IL-6, as well as many other inflammatory cytokines were not expressed, possibly due to the early time points (up to 24 h) examined. As a result, none of these define significant events. Functional characterization was also obtained through the analysis of the gene ontology (GO) annotations associated with the Affymetrix RG-U34A array. Conditional probabilities of each GO verified the strong bias of our clusters to assemble genes sharing similar fimctionalities. Figure 3 right panel. In addition to transcriptional profiling (Vemula, Berthiaume et al. 2004) provide a number of changes in metabolic processes as a result of the thermal injury. Significant similarities between these observations and our selections were identified: (i) fatty acid metabolism increases after bum injury, cluster 4 most highly correlated with FA metabolism follows similar temporal pattems; (ii) cholesterol biosynthesis, exhibiting strong down-regulation following an upregulated spike, was found to be most abundant in genes of cluster 6; (iii) ATP biochemical measurements indicate a upward spike followed by decreased levels and cluster 6 was most highly associated electron transport and follow closely ATP dynamics. Finally, we employed Network Component Analysis (Liao, Boscolo et al. 2003; Kao, Yang et al. 2004) as a basis for reconstmcting signals present in gene expression profiles and quantifying the strength of the connections between genes and transcription factors as well the levels of the corresponding TF activities. A total of 133 transcription factors were identified and their activities quantified. A sample is depicted in Figure 4. Among the leading TFs, known inflammation-specific TFs are strongly modulated (e.g., HSF, IRF, cAMP, etc.) 3. Summary We presented a novel approach for the analysis of temporal expression data. We demonstrated how to integrate gene expression analyses techniques, gene selection algorithms and functional and regulatory characterization tools in order to analyze complex physiological responses. The framework was demonstrated in the context of a post-injury hypermetabolic rat liver.

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1660 PAX-2/5/8 binding sites

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4. References Alon, U., N. Barkai, et al. (1999). Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays. Proc Natl Acad Sci U S A 96(12): 6745-50. Androulakis, I. P., J. Vitolo, et al. (2005). Selecting maximally informative genes to enable temporal expression profiling analysis. Proceedings of Foundations of Systems Biology in Engineering, Santa Barbara, CA. Chen, C. L., Z. Fei, et al. (2003). Metabolic fate of extrahepatic arginine in liver after bum injury." Metabolism 52(10): 1232-9. Chinnaiyan, A. M., M. Huber-Lang, et al. (2001). "Molecular signatures of sepsis: multiorgan gene expression profiles of systemic inflammation. Am J Pathol 159(4): 1199-209. Evans, T. W. and M. Smithies (1999). ABC of intensive care: organ dysfunction. Bmj 318(7198): 1606-9. Golub, T. R., D. K. Slonim, et al. (1999). Molecular classification of cancer: class discovery and class prediction by gene expression monitoring. Science 286(5439): 531-7. Jayaraman, A., M. L. Yarmush, et al. (2000). Dynamics of gene expression in rat hepatocytes under stress. Metab Eng 2(3): 239-51. Jegga, A. G., S. P. Sherwood, et al. (2002). Detection and visualization of compositionally similar cis-regulatory element clusters in orthologous and coordinately controlled genes" Genome Res 12(9): 1408-17. Kaminski, N., J. D. Allard, et al. (2000). Global analysis of gene expression in pulmonary fibrosis reveals distinct programs regulating lung inflammation and fibrosis. Proc Natl Acad Sci U S A 97(4): 1778-83. Kao, K. C, Y. L. Yang, et al. (2004). Transcriptome-based determination of multiple transcription regulator activities in Escherichia coli by using network component analysis. Proc Natl Acad Sci U S A 101(2): 641-6. Keogh, E., J. Lin, et al. (2005). HOT SAX: Efficiently finding the most unusual time series subsequences. 5th IEEE International Conference on Data Mining. Lee, K., F. Berthiaume, et al. (2000). Metabolic flux analysis of postbum hepatic hypermetabolism. Metab Eng 2(4): 312-27. Liao, J. C, R. Boscolo, et al. (2003). Network component analysis: reconstruction of regulatory signals in biological systems. Proc Natl Acad Sci U S A 100(26): 15522-7. Rassokhin, D. N. and D. K. Agrafiotis (2000). Kolmogorov-Smimov statistic and its application in library design. J Mol Graph Model 18(4-5): 368-82. Vemula, M., F. Berthiaume, et al. (2004). Expression profiling analysis of the metabolic and inflammatory changes following bum injury in rats. Physiol Genomics 18(1): 87-98.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Solid Fuel Decomposition Modelling for the Design of Biomass Gasification Systems David Brown,^* Tetsuo Fuchino,^ Fran9ois Marechal ^ "^Tokyo Institute of Technology, SI Bldg, #363, 2-12-1 Ookayama, Meguro Tokyo Japan ^Ecole Polytechnique Federale de Lausanne, LENI-ISE-STI-EPFL, Station 9, CH-1015, Lausanne Switzerland Abstract A novel equilibrium reaction modelling approach is proposed for the efficient design of biomass gasifiers. Fuels and chars are defined as pseudo species with properties derived from their ultimate analyses; tars as a subset of known molecular species and their distribution determined by equilibrium calculations. Non-equilibrium behaviour for gas, tar, and char formation is explained by reaction temperature differences for a complete set of stoichiometric equations. A nonlinear regression, with an artificial neural network (NN), relates changes in temperature differences to fuel composition and operational variables. This first principles approach, illustrated with fluidised bed reactor data, improves the accuracy of equilibrium calculations, and reduces the amount of required data by preventing the NN from learning atomic and heat balances. Keywords: biomass, tar, equilibrium temperature differences, neural networks. 1. Introduction Biomass gasification is of interest for combined heat and power production (CHP), be it for onsite conversion and cogeneration, or for the production of synthetic fuels such as methane (e.g. Duret et al, 2005). A major impediment to CHP remains equipment fouling problems related, namely, to the production of a wide array of condensable species commonly termed tar. In this respect, reaction modelling is a challenging task, because of the difficulty of identifying and quantifying heavier products, and the lack of thermophysical properties for a variety of feedstocks, chars, and tars. In addition, the quantity and the nature of tars vary according to several factors such as reactor temperature and pressure (Shafizadeh, 1982; Evans and Milne, 1987), the nature and ratio of oxidising gases (Kinoshita et al, 1994), the nature of biomass feedstocks (Fagbemi et al, 2001) especially the catalytic effect of their inorganic content (Shafizadeh, 1982), and even the types of gasifiers (Buekens and Schoeters; 1985). These issues have an impact on choices between design alternatives that affect the overall efficiency of process design. Indeed, it is important to reliably estimate tar distributions to determine product condensation points in order to design the appropriate contaminant removal configuration. To meet this end, we are developing a simple modelling approach that broadens the applicability of chemical equilibrium calculations. The approach relies on equilibrium reaction temperature difference parameters, derived from a database of standardised fuel analyses and quantified product compositions deduced from pilot gasifier measurements obtained under different operating conditions. These parameters serve as an estimator of kinetic, catalytic, and fluid dynamic effects that are not explained by equilibrium modelling. * Author to whom correspondence should be addressed: [email protected]

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2. Description and illustration of modelling approach The approach is illustrated with air gasification data from pilot circulating fluidised bed reactors of similar dimensions operated at atmospheric pressure (Garcia-Ibanez et al, 2004; van der Drift et al, 2001). The operational variables of the database are feedstock compositions, gasifying air equivalence ratios (ER), and reaction temperatures (T). Upon verifying mass balances, the data of van der Drift et al (2001) (set B) was kept for modelling, and the data of Garcia-Ibanez et al (2004) (set A) for model validation. 2.1. Generating additional compositional data from a non-stoichiometric model As it would be costly if not impossible to separately quantify each tar specie, in the first step of the approach, quantity and elemental composition measurements of the total tars produced, are used to calculate a distribution for a subset of unmeasured cyclic species that typically result fi-om biomass gasification (Evans and Milne, 1987). This is done with a non-stoichiometric (NS) equilibrium model, i.e. under the constraint of the reaction mixture's elemental composition (Smith and Missen, 1982). The assumption is that experimental measurements have precedence over chemical equilibrium. The Gibbs free energy of the system is minimised by SQP optimisation under individual measurement constraints for major species, and the total elemental stoichiometry of tar (and other lumped) species. Another assumption, to avoid convergence problems that arise when considering phase equilibrium, and that is acceptable at low and atmospheric pressures (Evans and Milne, 1987), is that the tar species form entirely in the gas phase. The molecular species are listed in Table 1. Species of sets A and B are on the first line, and species of which the formation was modelled on the following lines. There are two types of lumped species in set B: tars and remaining hydrocarbons (HC). The species in italics were finally withdrawn after preliminary NS calculations, because their formation is not thermodynamically favoured. The model enables estimating distributions of a considerably large subset of additional tar (24), and HC (10) species, that would otherwise not appear after equilibrium calculations. Feedstocks and chars are modelled as pseudo species with unit carbon formulas determined from their dry ash free ultimate analyses; standard enthalpies of formation by use of Thornton's (1917) rule with the constant of Patel and Erickson (1981); standard entropies of formation from the correlation of Battley and Stone (2000); and sohd specific heat capacities by adapting a modified partition function (Merrick, 1983) to a Kopp's rule. Table 1. Species quantified in references and species added for equilibrium calculations A & B

N2 H2 H2O CO2 CO CH4 C2H4 C2H6

B

C6H6 C7H8 CgHio^"^^* NH3 H2S HCl (2) . u „ rC4H8^' .T4 (6)^ rC4H1 .u,(^) HCN COS SO2 CI2 Remaining HCs: C2H2 CsHe^"^ rCsHg

Added light gases & tars

Tars: styrene, indene, ethyltoluene^^\ methylnaphthalene^^\ naphthalene, Cj2Hjo^^^^, pyrene, fluoranthene, furfural, furfur alcohol, phenol, cresol^^\ guaiacol, dimethylphenol^^\ pyridine, picoline^^\ quinoline^'^\ quinaldine

Notes. ^"Wmber of isomers specs * "xylene" in B, added ethylbenzene %iphenyl & acenaphthene

2.2. Formulation of a stoichiometric model ft)r gasification Secondly, as in Duret et al (2005), a stoichiometry is defined, and fitted to the product distribution by letting reaction equilibrium temperatures vary from the measured gasification temperature. A complete stoichiometry is generated by writing the molecule and element formula matrix in its reduced row echelon form (Smith and Missen, 1982), A=(Ic Z), from which N=(-Z IF)^ is deduced

(1)

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Where C is usually the number of atomic elements, and F the number of independent stoichiometric equations (usually the difference between species and elements). The temperature differences are therefore deduced by considering that the compositions resulting from the NS model are constant and by solving the non linear problem,

'dG^

= Z n A ( r + Arr.,P,n)=0

V7 = U , F

Eq. (2) implies that each stoichiometric equation is generated with the same molecular specie for each element. In our example, there are 47 independent equations derived from the six constitutive species (i.e. C, H, O, N, S, CI) of the system. We have considered using either one of the two following general equation formulations analogous to H2O -Eq. (3)- or CO2 -Eq. (4)- gasification, C„H„OpN^Sp,+(n-p)H,0^nCO + f^ + n - p - ^ q - r - ^ V + q N H 3 + r H , S + tHCl U 2 1) ^^j C„H,OpN,S,Cl, +(n-p)CO, ^(2n-p>::0 + f ^ - | q - r - ^ l H , +qNH3 +rH,S + tHCl (4) ATrj ranges and averages obtained with both stoichiometric formulations, for the twelve samples of set B (NH3 formation, the water gas shift and methane reforming reactions, naphthalene, anthracene and char formation) are indicated in Table 2. Duret et al (2005) reported similar values for the average ATrj of the shift and methane reforming reactions (40 and -224 K respectively). However, results indicate that the spread of the shift reaction is high. Also, most ATrj spreads are larger for Eq. (3) than for Eq. (4). Table 2. ATrj averages and ranges & linear correlation coefficient p-values for certain reactions React. mnAT mxAT avAT H

C N O S CI FC V hm As T ER Eq. (3) stoichiometrv NH3 -657 -482 -565 1. 0. 16. 1. 51. 32. 66. 0. 50. 0. 21. 5. Shift -120 939 159 1. 1. 31. 3. 87. 40. 86. 1. 5. 1. 97. 0. Refo. -367 -230 -281 34. 39. 2. 55. 34. 5. 53. 39. 44. 32. 6. 3. Naph. -494 -383 -428 43. 48. 3. 61. 38. 3. 62. 46. 41. 39. 5. 4. Anth. -440 -299 -378 95. 79. 12. 74. 66. 40. 12. 91. 54. 90. 41. 48. Char -650 -568 -605 69. 79. 2. 84. 27. 1. 68. 70. 70. 65. 1. 13. Eq. (4) stoichiometry (same stoichiometry as above for NH3 and water gas shift reactions) Refo. -299 -213 -248 67. 74. 3. 88. 13. 6. 46. 74. 92. 65. 0. 19. Naph. -386 -331 -359 81. 77. 10. 74. 12. 4. 61. 81. 56. 87. 0. 61. Anth. -371 -249 -312 46. 64. 38. 31. 56. 70. 12. 36. 77. 49. 35. 64. Char -536 -480 -509 56. 49. 12. 56. 12. 3. 74. 58. 38. 61. 0. 92. Notes, mn: minimum; mx: maximum; av: average; FC:fixedcarbon; VM: volatile matter; hm: humidity; ER: equivalence ratio. Units. Temperature differences [K], p-values [%] 2.3. Nature of correlation between temperature differences and independent variables The ATrj are strongly dependent of fiiel composition and operating condition variables. However, the validity of the temperature difference model relies on the assumption that

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certain ATrj parameters are not affected by changes in processing conditions. Hence, sample correlation coefficients between each ATrj and input variables have been computed to determine whether any uncorrelated variables exist. Linear and nonlinear correlations (e.g. logarithms, exponentials, and inverses) of input variables have been tested. The p-values of the linear correlation coefficients are also in Table 2. This Fisher distribution variable is the probability of randomly obtaining a correlation as large as the one observed. T and ER (both independent of fuel properties) are of particular interest. At a 10% significance level (single digit percentages in bold), it appears that, • The ATrj are independent of T for the shift and ammonia (2NH3 4-> 3H2 + N2) reactions, and of ER for most other equations of Eq. (4). (i.e. CO2) stoichiometry. • Fixed carbon is uncorrelated to all reactions; humidity, volatile matter and ash are correlated to the NH3 and shift reactions, as is the ash content to tar formation. • The ATrj are strongly correlated to major elements {C H 0 } for the NH3 and shift reactions, minor elements {N CI} for most other reactions; and weakly to N for NH3 formation, and {CHS} for HC, tars, and char reactions (exponential of inverse test). 2.4. Modelling the temperature difference using artificial neural networks The ATrj represent a relationship between several operational variables, that was not physically modelled, but approximated instead by a nonlinear regression. Multilayer feed forward artificial neural network (NN) models have been used to represent the variation of each ATrj as a fiinction of the operational variables. Having established that strongly correlated variables are not the same for each reaction, a fiilly connected two layer NN is defined for each reaction. Each NN has a number of hidden sigmoid nodes that vary in fiinction of the number of inputs, and a single linear node as the output. As suggested by Sarle, (1994) direct input/output layer connections are added to account for the lower order effects noted in Table 2. The problem formulation is.

mmY,[^Tr.^-^fr.}j s

(5)

with. mh

^%

mf

I/=!

W

=^- +£w^J l + exp b,+Y,Wfl,x^

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The problem is solved with standard backpropagation of errors to the hidden layer. Incomplete target vectors are assigned a null error for unmeasured target values. 2.5. Network training and validation NNs can be estimators of arbitrary square-integrable fiinctions (White, 1990), however their major drawback is the high dimensionality of their weight space, which implies the risk of obtaining poor interpolations between training points. Generally speaking, large data samples, i.e. a number a least superior to the number of weights and biases, are needed to obtain good interpolation properties (termed generalisation). With the twelve observations of set B, as indicted in Table 3. (in bold), there would be fewer parameters than observations only in single input networks, or with two variables and a single hidden node per variable. Obviously a larger data sample would be preferable, but it can be costly in practice to generate a sample of several hundred or thousands of entries.

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4 6 Number of iterations

4 6 Number of iterations

Figure 1. Impact of input vars. on shift reaction

Figure 2. Size of hidden layer (Anthracene)

Preliminary validation tests and results from the literature allow reducing the number of input variables and of hidden nodes per input. Fixed carbon (FC) and volatile matter (VM) can be considered as dependent variables because the FC ratio is proportional to both the H/C and O/C ratios (van Krevelen, 1950; Jenkins et al, 1998). We used the preliminary correlation analysis (Table 2.) to further reduce the number of model parameters. For instance, Fig. 1. shows that, for the shift reaction, in concordance with p-value tests, error minimisation is more difficult with apparently uncorrelated inputs (but even more without ER). Fig. 2. indicates that to correlate the (Eq. (4)) set of ATrj of anthracene, there can be less than two hidden nodes per input, but that a single node is insufficient. The results obtained from the training set B have been assessed with another set of data (set A). The quality of generalisation for certain reactions is given in Table 4. as the relative error between modeled and calculated ATrj. Generalisation is particularly poor for the shift reaction while it is better for char and HC reactions. Table 3. Number of parameters (w & b) per input variables and hidden nodes per input. h nodes/vars.

1

2

1

4

5

6

7

8

9

10

11

12

1 2 3

5 8 11

11 19 27

19 34 49

29 53 77

41 76 111

55 103 151

71 134 197

89 169 249

109 208 307

131 251 371

155 298 441

181 349 517

Table 4. Errors on calculatec (Eq. (4) stoichiometry) and interpolated (100000 iterations) ATrj h/vars 3 3 6 6

Reaction

Input variables

NH3 Shift Shift Reform. Char

{C, H, O, hum, ash, ER} {C, H, O, hum, ash, ER} All 12 input variables 10 vars. (all but FC&VM) 10 vars. (all but FC&VM) 3

av B 0.000%

A.1 -17%

A.2 -17%

A.3 -22%

A.4 -19%

0.77% 0.000% 0.020% 0.002%

-150% -71% 29% 14%

-79% -125% 34%

-58% -131% 15%

-50% -123%

8.1%

-0.1%

29% 6.4%

3. Conclusion and recommendations A reaction model has been developed for the rapid computation of product compositions of biomass gasification. An NS equilibrium model based on total tar measurements, is first applied to estimate the distribution of tar species. The product distribution is then formulated as a stoichiometric equilibrium model with reaction equilibrium temperature

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differences. Although certain temperature differences appear to be uncorrected to independent variables such as T and ER, other temperature differences are strongly correlated to these variables. Since there is no clear evidence of any single or characteristic relationship between operational variables and temperature differences, the use of an appropriately designed N N appears as a solution to parameterise the reaction temperature differences, even with a data sample of limited size. Future improvements to generalisation could include adding data samples and/or additional constraints to improve the smoothness of the NN regression, e.g. weight decay (Krogh and Hertz, 1991), and using prediction intervals (De Veaux et al, 1998). Nomenclature ajk or A: number of atomic elements k in molecular specie i, and element molecule formula matrix b: network biases C: rank of formula matrix (usually C = M, the number of atomic elements) F: number of stoichiometric degrees of freedom (of linearly independent equations) G: Gibbs function [kJ/kmol]; I: identity matrix ni or n: quantity of molecular specie i (or of all species) at equilibrium [kmol] N: number of molecular species; N: complete stoichiometric matrix m: number of terms in a sum; M: number of elements P: reaction pressure [kPa]; R: gas constant [kJ/kmol-K]; T: reaction temperature [K] ATrji temperature difference between equilibrium and actual composition for reaction j [K] tar & tars: reconciled measurement of tar concentration and subset of tar species Xf! input variable f to network; w: network weight Z: matrix of dimension C x F when the only compositional constraint are element abundances Greek letters |x: chemical potential [kJ/kmol]; v: stoichiometric coefficient; L,y extent of reaction j Indexes f: input variables to network; i: molecular species; I: isomers h: hidden nodes; j : chemical reactions; k: atomic elements; s: observations

References E.H. Battley and J. R. Stone, 2000. Thermochimica Acta., 349, 153. A.G. Buekens and J.G. Schoeters 1985. in Fundamentals of Thermochemical Biomass Conversion, R.P Overend, T.A Milne and L.K Mudge eds., Elsevier Applied Science (Ch. 35). R.D. De Veaux, J. Schumi, J. Schweinsberg, L.H. Ungar, 1998. Technometrics., 40, 273. A. Duret, C. Friedli and F. Marechal, 2005. Journal of Cleaner Production., 13-15, 1434. R.J. Evans and T.A. Milne, 1987. Energy & Fuels., 1, 123. L. Fagbemi, L. Khezami, R. Capart, 2001. Applied Energy., 69,293. P. Garcia-lbanez, A. Cabanillas and J.M. Sanchez, 2004. Biomass & Bioenergy., 27, 183. B.M. Jenkins, L.L. Baxter, T.R. Miles Jr. and T.R Miles, 1998. Fuel Proc. Technol., 54, 17. CM. Kinoshita, Y. Wang, J.Zhou, 1994. J. Analytical and Applied Pyrolysis., 29, 169. A. Krogh and J.A. Hertz, 1991. Neural Information Processing Systems., 4, 950. D. Merrick, 1983. Fuel., 62, 535. S.A. Patel and L. E. Erickson, 1981. Biotechnology & Bioengineering., 23, 2051. W.S. Sarle, 1994. Proc. 19th annual SAS users group international conference., 1538. F. Shafizadeh, 1982. Journal of Analytical and Applied Pyrolysis., 3, 283. W.R. Smith and R. Missen, 1982. Chemical Reaction Analysis: Theory and Algorithms., USA.: John Wiley and Sons (Ch. 2 & Ch. 6) W.M. Thornton, 1917. Phil Mag., 33, 196. A. van der Drift, J. van Doom and J.W. Vermeulen, 2001. Biomass & Bioenergy., 20, 45. D.W.van Krevelen, 1950. Fuel., 29, 269. H. White, 1990. Neural Networks., 3, 535. Acknowledgements Funding provided by the Ministry of Education, Culture, Sports, Science and Technology of Japan is gratefully acknowledged.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Hybrid metabolic flux analysis/data-driven modelling of bioprocesses A. Teixeira", C.M.L. Alves", P. M. Alves^ M. J. T. Carrondo"'^ R. Oliveira^ "^ FCT/UNL, Laboratorio de Engenharia Bioquimica, P-2825 Monte da Caparica, Portugal ^IBET/ITQB, Apartado 12, P-2781-901 Oeiras, Portugal Abstract This work proposes a hybrid modeUng method that combines data-driven modeHng, metaboUc flux analysis, and bioreactor transport phenomena models. The main objective was to understand the time evolution of metabolism in fedbatch cultivations and to identify favorable conditions for product formation. The overall metabolic network is simplified using the elementary flux modes method. The hybrid modeling scheme was implemented in HYBMOD, an inhouse developed software. Preliminary simulation studies confirmed that HYBMOD could identify the "true" kinetic fianctions of elementary flux modes provided that a given set of state variables are measured. The method was applied to analyze the data of recombinant BHK-21 cultures. Keywords: Metabolic Flux Analysis, Elementary Flux Modes, Hybrid Modelling, BHK21 culture. 1. Introduction The main metabolic pathways of many biological systems with industrial interest are today well known. In principle, classical bioreactor dynamic optimization schemes could profit by the incorporation of this knowledge. Under the balanced growth condition, networks of metabolic reactions are reflected in large stoichiometric matrices and flux kinetics that must obey the constraint that net formation of intracellular components is zero. The overall reactions network may be simplified to a smaller size network using the elementary flux modes method [1]. Elementary flux modes (EFM) are the simplest paths within a metabolic network that connect substrates with endproducts. Each flux mode lumps all the reactions in the particular path, and is governed by the rate-limiting step. The reaction mechanism within the biological system may thus be viewed as a collection of EFMs. Bioreactor dynamical models can be established from material balance equations of EFM input/output compounds in a standard fashion [2]. The main problem of this method is the establishment of reliable and accurate kinetics of EFM, which should obviously reflect the "true" intracellular modulation mechanisms. Provided that a given minimum set of state variables is experimentally

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available, the identification of kinetic functions from measured data becomes possible. In this work, we studied the identification of elementary flux mode kinetics with artificial neural networks (ANN). Thus the approach proposed is a hybrid modeling method that combines data-driven modeling (in our case ANNs), metabolic flux analysis, and bioreactor transport phenomena models. This framework was applied to a recombinant baby hamster kidney BHK-21 cell line expressing the fusion glycoprotein IgGl-IL2 for cancer therapy [3]. The main objective was to understand the time evolution of metabolism in fedbatch cultivations and to identify favourable conditions for product formation. BHK-21 elementary flux modes were obtained with the FluxAnalyser software [4]. The hybrid modelling scheme was implemented in HYBMOD, an in-house developed software. Preliminary simulation studies confirmed that HYBMOD could identify the "true" kinetic functions of elementary flux modes. The application of the same method to experimental data provided the metabolic flux distribution over the time. In future studies, bioreactor dynamic optimization supported by the hybrid model will be performed. 2. Proposed method 2.1. Elementary flux modes The first step is to analyze the metabolic network structure of the biological system under study. The objective is to extract only the relevant knowledge to be used for bioreactor performance optimization. Here we use a metabolic flux analysis (MFA) method called "elementary flux modes". Elementary flux modes are the simplest paths within a metabolic network that connect substrates with end-products [1], thus they define the minimum set ofn species that must be considered for modeling and how they are connected in a simplified reaction mechanism. If a given system has m elementary flux modes, then the result of this analysis is a nxm stoichiometric matrix K. 2.2. General bioreactor hybrid model Once a reaction mechanism has been established using the EFM method, the next problem is to identify the kinetics of EFM from data. Here we adopted the hybrid model structure represented in Figure 1, where the main hypothesis is that reaction kinetics of EFM are partially known. This model structure can be formulated mathematically by the following two equations: - - = r(c,w)-Dc + u at r(c,w) = K((pj(c)xpj(c,w)).^^^ ^^

(la) (lb)

with c a vector of n state variables, r a vector of n volumetric reaction rates, K a nxm coefficients matrix obtained from the elementary flux modes analysis, ^(c) are m known kinetic ftinctions established from mechanistic and/or empirical knowledge, /?Xc,w) are m unknown kinetic functions, D is the dilution rate, u is a vector ofn volumetric input rates (control inputs).

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Biological system Bioreactor system

Mechanistic/empiric kinetics w

Unknown kinetics: w nonparametric model

^ Material balance equations

Kinetic model Figure 1. General hybrid model for bioprocesses Functions /7y(c,w) are to be identified form data using nonparametric modeling techniques with w a vector of parameters that must be estimated from data. A 3-layered backpropagation neural network with sigmoidal activation function was used:

p(c,w) = P^ax^(w2^(WiC + bi) + b2)

(2)

with Pmax a vector of scaling factors with dim(pn,ax)='w, Wi, bi, W2, b2 are parameter matrices associated with connections between the nodes of the network, w is a vectored form of Wi, bi, W2, b2 and s(.) the sigmoid activation function defined as follows: s(x) = -

—

1 + e ""

(3)

^ ^

2.3. Identification of unknown kinetics The parameters vector w must be estimated from measured data of control inputs and process state over time: {Z)(t), u(t), c(t)}. For the identification of w, a least squares criteria of residuals in concentrations was adopted. The least squares problem was solved using the quasi-Newton algorithm with conjugate gradient with line search (MATLAB^^ optimization toolbox). This optimizer requires analytical residuals gradients, which were computed using the sensitivities method (see [5] for details) 3. Case study: recombinant BHK-21 culture 3.1. Process description A BHK-21 A cell line (a subclone of ATCC accesion CCL-10) expressing the fusion glycoprotein IgGl-IL2 was used. The experiments were carried out in serum free and protein free medium (SMIF6, Life Technology, Glasgow, UK). The batch culture was set up in a 2 1 reactor volume and the fed-batch cultures were set up at 3 different volume scales (2, 8 and 24 1). Sparger aeration was

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employed. Dissolved oxygen concentration was set at 15% of air saturation. Agitation rate used was 60 rpm; pH was set as 7.2 and controlled through the addition of CO2. Analysis techniques are described elsewhere [3]. Experimental data of viable cells cconcentration and six extracellular species (glucose, glutamine, lactate, ammonia, alanine and desired product) was collected. 3.2. Elementary flux modes The metabolic network considered takes into account the most relevant pathways involving the two main nutrients (glucose and glutamine) whithin the central metabolism of BHK cells, which are glycolysis, glutaminolysis, TCA cycle and nucleotides synthesis. The FluxAnalyzer software [4] was used to determine the EFM of BHK metabolic network. The information that must be provided to the algorithm is (i) all metabolites of the system including information whether they are internal or external and (ii) all reactions including information whether reactions are reversible or irreversible. There are seven EFMs describing the BHK metabolic network. Each one is a collection of reaction steps. The hypothesis of balanced growth allows the elimination of the intermediate metabolites resulting in a simplified set of reactions connecting extracellular substrates (glucose and glutamine) with end-products (lactate, ammonia, alanine, carbon dioxide, purine and pyrimidine). Furthermore, some assumptions concerning the fluxes were made based on literature, resulting in five EFMs. The following stoichiometric matrix was obtained |l-kd ri [ 1 0

X2 ra XA rs rigo 0 0 0 0 0

0 - 1 - 1 0 0 - 2

0

Glc

0

0 0

-1 -1 -5

0

Gin

K= 0

2 0

0

0

0

0

Lac

0

0 0

1

2

2

0

Amm

0

1

0

0

0

Ala

0 0

0

0

0

1

IgG

0 0 0

(4^

Note that K also accounts for cell growth and product formation as completely independent fluxes since the stoichiometry of theses reactions is not available. 3.3. Identification of EFM kinetics Off-line measurements of the seven state variables from five experiments were used for model training and validation. The neural network had three inputs: glucose and glutamine, the main limiting nutrients and ammonia, the main toxic by-product. The output vector was formed by the following seven specific kinetics: |i-kd, ri, r2, rs, r4, rs, rigo- The structure of the artificial neural network was selected by trial-and-error. The criterion to stop the parameter estimation algorithm was the minimum modeling error for the validation data set. The best result was obtained with a single hidden layer and five hidden nodes. Figure 2 presents the hybrid modeling results for one of the training and

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one of the validation sets. A relevant result is the fact that the hybrid model was able to describe simultaneously all five experiments with high accuracy. The flux distribution identified by the hybrid model for one of the fed-batch cultures is shown in figure 3. 2.5

100 200 time(h)

300 0

50

100 150 time(h)

200

Figure 2. Hybrid model results for a training data set (a) and a validation data set (b).

0.07

0.05

^

0.03

0.01

-0.01

^

130

160

190

time (h) -0.03

Figure 3. Elementary flux modes identified by the hybrid model

220 > .

250

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4. Conclusions Analyzing the intracellular flux distribution over the time course of the bioreaction, conclusions can be taken concerning cellular control mechanisms, which in this case are consistent with published works of other mammalian cell lines. The agreement between theory and model predictions confirms that the proposed framework can produce valid conclusions. The proposed hybrid modelling approach, which integrates knowledge from measured data, metabolic flux analysis, and bioreactor transport phenomena, will be used to develop optimal control strategies for fed-batch bioprocesses.

Acknowledgements The authors acknowledge the financial support provided by the Funda^ao para a Ciencia e Tecnologia through project POCTI/BIO/57927/2004 and PhD grant SFRH/BD/13712/2003.

References [1] S. Schuster, D.A. Fell, T. Dandekar. (2000) A general definition of metabolic pathways usefiil for systematic organization and analysis of complex metabolic networks. Nature Biotech., 18: 326-332. [2] A. Provost and G. Bastin. (2004) Dynamic metabolic modeling under balanced growth condition. J, Process Control, 14: 717-728. [3] A. Teixeira, A.E. Cunha, J.J. Clemente, J.L. Moreira, H.J. Cruz, P.M. Alves, M.J.T. Carrondo, R. Oliveira. (2005) Modelling and optimization of a recombinant BHK-21 cultivation process using hybrid grey-box systems. J. Biotechnol. 118: 290-303. [4] S. Klamt, J. Stelling, M. Ginkel, E.D. Gilles. (2003) FluxAnalyser: exploring structure, pathways, and flux distributions in metabolic networks on interactive flux maps. Bioinformatics, 19(2):261-269. [5] R. Oliveira. (2004) Combining first principles modelling and artificial neural networks: a general framework. Comp. Chem. Engn. 28, 55-766.

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Rotavirus-Like Particle Production: Simulation of Protein Production and Particle Assembly Antonio Roldao , Helena L. A. Vieira , Manuel J.T. Carrondo ' , Paula M. a

u

Alves and R. Oliveira

""IBET/ITQB, Apartado 12, P-2781-901 Oeiras, Portugal * FCT/UNL, Laboratorio de Engenharia Bioquimica, P-2825 Monte da Caparica, Portugal Abstract In this work we study the production of rotavirus virus-like particles (VLP) using the baculovirus expression vector system (BEVS). A model-based optimization of the infection strategy was performed in order to maximize the production of correctly assembled VLPs. A structured mathematical model describing the relevant intracellular processes (baculovirus adsorption and trafficking, DNA replication and gene expression) was employed. Some intracellular processes take several hours for completion and may be regarded as pure time delays. A modified 4^V5^^ order RungeKutta solver was employed to integrate the ODEs system with pure time delays. A coinfection program using different combinations of multiplicity of infection (MOI) for each gene {vp2, vp6 and vp7) was investigated. The best results were obtained for MOI combinations of 2 (vp2)H-5 (vp6)+8 {vp7) or 5+2+8. It was also concluded that viral protein 7 (VP7) was the limiting component for VLPs assembly. This study highlights the usefulness of mathematical modeling in the design of improved infection strategies for VLPs production. Keywords: Rotavirus, VLP, modeling, simulation, assembly. 1. Introduction In the last decades. Rotavirus disease (RVD) emerged worldwide to become the primer cause of severe gastrointestinal illness in children, with an incidence estimated at 111 million episodes and a total of 440.000 deaths per year in children with less than 5 years of age [1]. Rotavirus, a non-enveloped virus can be mimicked for vaccine purposes by a triple-layered concentric VLP protein structure: the innermost layer composed by 60 dimers of VP2 (102.7 kDa) [2]; the middle shell formed by 260 trimers of VP6 (44.9 kDa) [3] and the third, outer layer composed by 780 monomers of glycoprotein VP7 (37.2 kDa) [3]. The viral proteins VP2, VP6 and VP7 (and the corresponding VLP assembly) may be effectively produced in Spodoptera frugiperda Sf-9 cells by infecting with recombinant baculovirus containing the three genes of interest, bvp2, bvp6 and bvp7. The process main steps involved in VLPs production are summarized schematically in Fig. 1 for a co-infection strategy using three monocistronic baculovirus vectors, each one expressing the bvp2, bvp6 and bvp7 genes individually. Step 1 encloses four main events: i) the binding of extracellular baculovirus containing the

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genes of interest to the plasmatic membrane and their entry by adsorptive endocytosis [4]; ii) the release of viruses' nucleocapsids (containing the viral genes) to the cytoplasmic space due to the pH decrease inside the endosomme; iii) the nucleocapsids migration to the cell nucleus and iv) the nucleocapsids binding to the nucleus membrane (thereby inserting the genes into the cell nucleus) that triggers the viral DNA (vDNA) replication. The vDNA transcription is the main event in Step 2. In the following step, Step 3, the corresponding mRNA leaves the nucleus and migrates to ribosomes where the coded VPs are synthesized. The synthesized proteins, VP2, VP6 and VP7, assemble into triple layered VLP in Step 4 according to mechanisms that are not yet fully known. Finally, Step 5 consists in the VLPs release to the extracellular medium. The main aim of this study is to optimize the VLPs production using a detailed structured mathematical model describing all five steps presented in Fig. 1. A co-infection program, in which Sf-9 cells were infected which three monocistronic baculovirus vectors (each one expressing the vp2, vp6 and vp7 genes individually) was investigated. 2. Mathematical model The mathematical model used in this work (Table 1) has been previously calibrated/validated by our group with original data [5]. A brief description of the model is provided below. A set of simplifying assumptions were taken: i) MOIs grater than two correspond to synchronized virus infection [6]; ii) infection kinetics are independent of the expression vector; iii) negligible virus budding and release [6-8] and also iv) VLPs assembly is a fast process ruled by the underlying stoichiometry. Plasmatic Membrane Nuclear Membrane mRNA

Baculovirus DNA

Protein VP7 Protein VP2 Protein VP6

VP7-Q

TrimersofVP6 Monomers of VP7 Monomers of VP6 VLP Uncompleted double layered 2/6VLP

Figure 1. VLPs production steps for co-infection strategy using three monocistronic baculovirus vectors each one expressing vp2, vp6 and vp7 genes individually.

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Depletion of extracellular virus due to binding to insect cells surface is expressed by Eq. (1), where Vj represents the extracellular concentration of virus7 (copiesDNAj.ml"^), with subscript index 7=2,6,7 corresponding to vp2, vp6 and vp7 genes respectively, ka the adsorption rate equal to 1.3x10'^ ml.cell^min'^ [6] and Ni the concentration of infected cells (celLml'^). The infected cells equation has two terms (Eq. (2)): the first, accounts for the increase in infected cells concentration due to binding of baculovirus to uninfected cells while the second, represents the cells death rate (Eq. (4)). Table 1. Mathematical Model Equations -^ = -kNy. dt

Eq.(l)

a I J

—L = kNV.-k,N, ^f

a u t

Eq. (2) a I

'-^-Wr^.r'^u

Eq.(3)

kd=kdx+k^^{TOl-r,)

Eq. (4)

— = 1^ A V.{t-r^ r)^k^^.,,

.

^

J

fo..At) = [Tt~

.(0

Eq. (5)

DNAJ

'-'^'-^'^

^ = ksRNA jI^NA Y fRNA Jit) - knmA jKNA j dt dRNA dVP ^=kypjRNAj(t-Typj)fypjit)N^ dt

f^.,,

Eq.(6) Eq. (7)

Eq. (8)

The uninfected cell population, Nu (celLml"^), in Eq. (3) has two terms: the first term is the "conversion" of uninfected cells into infected cells due to virus binding and the second term is the intrinsic cell death rate. The intracellular dynamics of vDNA, DNAf""" (DNA.ceir^), is given by Eq. (5). The first term characterizes the transport of genes from extracellular virus into the cell nucleus, the trafficking efficiency and the virus trafficking time. The second term reports the vDNA replication kinetics, which is assumed to be of Michaelis-Menten type. The term^^A^^j (t) accounts for the intrinsic metabolic decay due to infection (Eq. (6)). The transcription of vDNA is defined by first order kinetics in Eq. (7), with RNAj (RNA.cell"^) the concentration of mRNA from gene j \ ksRNAj (h'^) the first order kinetic constant and koRNAj the first order mRNA degradation rate (h'^). The fmAjit) represents the progressive loss of capacity to synthesize mRNA, here similar to Eq. (6). Concerning the VP synthesis, previous models [9,10] have used zero order kinetics to predict/simulate VPs production. The zero-order kinetic constant has been defined empirically as a function of number of viruses infecting the cells. In this study, intracellular mRNA is calculated and therefore

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a more mechanistic relationship between VP synthesis kinetics and mRNA concentration is possible. Therefore, the protein synthesis, VPj (jigj.mr^) was defined by first order kinetics on mRNA (Eq. 8), with kypj the first order kinetic constant (|ig.RNA" \h"^), Tvpj the time delay accounting for glycosilation in protein VP7, equal to 0.5h [11] and/^p/O the linear decay function of VP synthesis capacity, analogous to Eq. (6) for DNA. The mathematical model has pure time delays associated due to protein glycosilation and virus trafficking inside the cell. Therefore, a modified 4^/5 order Runge-Kutta solver, in which all intermediate integration results are stored, was used to integrate the ODE system with pure time delays. All simulations were performed in MATLAB™^ (The MathWorks, Inc., US, 1994-2006).

3. Infection strategy optimization A co-infection program in which the insect cells were infected with three monocistronic baculovirus vectors (each one expressing the vp2, vp6 and vp7 genes individually) was evaluated through simulations. VLPs production was calculated based on total VP2, VP^ and VP7 synthesized and the VLPs stoichiometric composition. The theoretical mass stoichiometric ratio VP2:VP6:VP7 of correctly assembled particle is known to be 1:2.8:2.4 (w/w). The co-infection program investigated is given in Table 2. The individual MOIs varied (either 2, 5 or 8) while the total MOI was kept constant to be 15 (thus guaranteeing that the level of DNA polymerase is similar to that of the experiments used to validate the model). Table 2. Co-infection program MOI = 5 {vp2) MOI = 5 {vp6); MOI = 5 (vpT)

Prog. (1)

MOI = 2 (vp2) MOI = 8 {vp6); MOI = 5 {vp7)

Prog. (2)

MOI = 2 (vp2) MOI = 5 {vp6); MOI = 8 (vp7)

Prog. (3)

MOI = 5 {vp2) MOI = 8 {vp6); MOI = 2{yp7)

Prog. (4)

MOI = 5 {vp2) MOI = 2 {vp6); MOI = 8 (vp7)

Prog. (5)

MOI = 8 {vp2) MOI = 2 {vp6); MOI = 5 (vp7)

Prog. (6)

MOI = 8 (vp2) MOI = 5 {vp6); MOI = 2 (vp7)

Prog. (7)

Figure 2 shows predicted protein profiles for the different infection strategies. As expected, higher MOIs promote higher expression levels of the corresponding viral protein. However, higher intracellular concentration of a particular VP does not necessarily imply higher VLPs yield. Viral proteins must be synthesized at correct stoichiometric ratios. Figure 3 shows the total production of correctly assembled VLPs for the different infection strategies. The higher VLPs yield was obtained for the co-infection strategies where the MOI of VP7 was higher, i.e., the combination MOI = 2, 5 and 8 for vp2, vp6 and vp7 respectively and also MOI = 5, 2 and 8. The results obtained clearly indicate

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that VP7 is the limiting VP in the process of particle assembly. Therefore, infection strategies favoring the formation of VP7 seem to be the key for VLPs production optimization.

(>n -

i

80 • 70 •

/ 1 /

605040 -

solo •

/ / / /

10 • 0• 48

72

96

/ .

. 48

Time (hpi)

.

.

72

96

B 120

144

Time (hpi)

48

72 96 Time (hpi)

Figure 2. Simulations of intracellular VP production in co-infection strategy. Pattem for VP2 (A): the full line (—) represents Prog. (1), the dot line ( ) represent Prog. (2-3), the dash line ( ) indicates Prog. (4-5) and dot and dash line (—-) Prog. (6-7). Pattem for VPg (B): the fiill line (—) represents Prog. (1), the dot line ( ) represent Prog. (5-6), the dash line ( ) indicates Prog. (3-7) and dot and dash line (—-) Prog. (2-4). Pattem for VP7 (C): the full line (—) represents Prog. (1), the dot line ( ) represent Prog. (4-7), the dash line ( ) indicates Prog. (2-6) and dot and dash line (—) Prog. (3-5). 4. Conclusions and future work The production of Rotavirus-like particles (VLPs) adopting a co-infection scheme was assessed using the baculovirus expression vector system (BEVS). A detailed structured model accounting for intracellular processes such as vDNA replication and transcription, usually not taken into consideration in modeling this type of systems [9,10,12,13], was used. The model allowed investigating the effect of MOI on each VLPs production step and on its final VP and VLP yield. The best resuhs were obtained for MOI combinations of 2 (vp2)+5 (vp6)+S (vpT) or 5+2+8. It was also concluded that viral protein 7 (VP7) was the limiting component for VLPs assembly.

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et al

Figure 3. VLPs content simulations for each co-infection strategy reported in table 1.

This study highlights the usefulness of this mathematical model in the design of improved infection strategies for VLPs production. In future studies, the model will be improved in order to consider infection statistics, which is known to follow a Poisson distribution. Bibliography [I] [2] [3] [4]

[5]

[6]

[7]

[8] [9]

[10] [II] [12] [13]

U.D. Parashar, E.G. Hummelman, J.S. Bresee, M.A. Miller and R.I. Glass, 2003, Global illness and deaths caused by rotavirus disease in children, Emerg Infect Dis, 9, 5, 565. M. Labbe, A. Charpilienne, S.E. Crawford, M.K. Estes and J. Cohen, 1991, Expression of rotavirus VP2 produces empty corelike particles, J Virol, 65, 6, 2946. B.V. Prasad, G.J. Wang, J.P. Clerx and W. Chiu, 1988, Three-dimensional structure of rotavirus, J Mol Biol, 199,2,269. L.E. Volkman and P.A. Goldsmith, 1985, Mechanism of neutralization of budded Autographa Califomica Nuclear Polyhedrosis Virus by a monoclonal antibody: Inhibition of entry by adsorptive endocytosis. Virology, 143, 185. A. Roldao, H.L.A. Vieira, M.J.T. Carrondo, P.M. Alves and R. Oliveira, 2006, Intracellular dynamics in Rotavirus-like particles production: Evaluation of multigene and monocistronic infection strategies. Submitted, K.U. Dee and M.L. Shuler, 1997, A mathematical model of the trafficking of acid-dependent enveloped viruses: Application to the binding, uptake, and nuclear accumulation of baculovirus, Biotechnol Bioeng, 54, 5, 468. J.F. Power, S. Reid, K.M. Radford, P.P. Greenfield and L.K. Nielsen, 1994, Modeling and optimization of the baculovirus expression vector system in batch suspension culture, Biotechnol Bioeng, 44, 6, 710. M. Rosinski, S. Reid and L.K. Nielsen, 2002, Kinetics of baculovirus replication and release using real-time quantitative polymerase chain reaction, Biotechnol Bioeng, 77, 4,476. Y.C. Hu and W.E. Bentley, 2001, Effect of MOI ratio on the composition and yield of chimeric infectious bursal disease virus-like particles by baculovirus co-infection: deterministic predictions and experimental results, Biotechniques, 75,1, 104. Y.-C. Hu and W.E. Bentley, 2000, A kinetic and statistical-thermodynamic model for baculovirus infection and virus-like particle assembly in suspended insect cells, Chem Eng Sci, 55, 3991. H. Lodish, D. Baltimore and J. Darnell (eds.), Molecular Cell Biology, Scientific American Books, Inc., New York, USA, 1986. G. Enden, Y.H. Zhang and J.C. Merchuk, 2005, A model of the dynamics of insect cell infection at low multiplicity of infection, J Theor Biol, 237, 3, 257. B. Jiang, M.K. Estes, C. Barone, V. Bamiak, CM. O'Neal, A. Ottaiano, H.P. Madore and M.E. Conner, 1999, Heterotypic protection from rotavirus infection in mice vaccinated with virus-like particles. Vaccine, 17, 7-8, 1005.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Prediction of Secondary Structures of Proteins Using a Two-Stage Method Metin Turkay and Ozlem Yilmaz and Fadime Uney Yuksektepe College of Engineering, Kog University, Rumelifeneri Yolu, Sariyer, 34450 Istanbul, TURKEY Abstract Protein structure determination and prediction has been a focal research subject in life sciences due to the importance of protein structure in understanding the biological and chemical activities in any organism. The experimental methods used to determine the structures of proteins demand sophisticated equipment and time. In order to overcome the shortcomings of the experimental methods, a host of algorithms aimed at predicting the location of secondary structure elements using statistical and computational methods are developed. However, prediction accuracies of these methods rarely exceeded 70%. In this paper a novel two-stage method to predict the location of secondary structure elements in a protein using the primary structure data only is presented. In the first stage of the proposed method, folding type of a protein is determined using a novel classification model for multi-class problems. The second stage of the method utilizes data available in the Protein Data Bank and determines the possible location of secondary structure elements in a probabilistic search algorithm. It is shown that the average accuracy of the predictions increased to 74.1%. Keywords: Protein Structure, Data Classification, Mixed-Integer Linear Programming 1. Introduction Proteins are large molecules indispensable for existence and proper functioning of biological organisms. Proteins are used in structure of cells, which are main constituents of larger formations like tissues and organs. Bones, muscles, skin and hair of organisms are made basically up of proteins. Besides their necessity for structure, they are also required for proper functioning and regulation of organisms such as enzymes, hormones, antibodies. Understanding functions of proteins is crucial for discovery of drugs to treat various diseases and disorders. A protein molecule is the chain(s) of amino acids also called residues. A typical protein contains 200 - 300 amino acids but this may increase up to approximately 30,000 in a single chain. There are 4 basic structural phases in proteins: primary structure, secondary structure, tertiary structure and quaternary structure. The primary structure is the sequence of amino acids that make up the protein. The secondary structure of a segment of polypeptide chain is the local spatial arrangement of its main-chain atoms without regard to the conformation of its side chains or to its relationship with other segments. This is the shape formed by amino acid sequences due to interactions between different parts of molecules. There are mainly three types of secondary structural shapes: a-helices, P-sheets and other structures connecting these such as loops, turns or coils. Alpha-helices are spiral strings formed by hydrogen bonds between CO and NH groups in residues Beta-sheets are plain strands formed by stretched polypeptide backbone. Connecting structures do not have regular shapes; they connect a-helices and P-sheets to each other. Turns enable parts of polypeptide chain to

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fold onto itself reversing the direction of the polypeptide chain to form its threedimensional shape. Proteins are classified according to their secondary structure content, considering ahelices and P-sheets. Levitt and Chothia ^^^ were the first to propose such a classification with four basic types. "All-a" proteins consist almost entirely (at least 90%) of ahelices. "All-p" proteins are composed mostly of P-sheets (at least 90%) in their secondary structures. There are two intermediate classes which have mixed a-helices and P-sheets. "a/P" proteins have approximately alternating, mainly parallel segments of a-helices and P-sheets. The last class, "a+P" has mixture of all-a and all-P regions, mostly in an antiparallel fashion.^^^ Due to bottlenecks in experimental methods to determine protein structures, computational approaches to predict protein structures are developed. All structure prediction methods basically rely on the idea that there is a correlation between residue sequence and structure. Most methods to predict protein structure from residue sequence utilize information on known protein structures. Databases are formed and examined for relationships between amino acid sequence and protein structure. First predictions were made in 1970s with a few dozen structures available. Currently structures of about 33,500 (as of November 2005) proteins are identified that means vast amount of data supporting more reliable predictions with better accuracy is available. Protein structures are stored in and accessible from Protein Data Bank.^^^ Among different computational methods developed to predict protein structures, the most successfiil ones include neural network models, database search tools, multiple sequence alignment, local sequence alignment, threading, hidden Markov model-based prediction, nearest neighbor methods, molecular dynamic simulation, and approaches combining different prediction methods. Neural networks are parallel, distributed information processing structures and the method tries to solve the problem by training the network.^"^"^^ The most successfiil ones are Copenhagen^^^ PSI-BLAST^^^, PHD^^'^^ and SSpro^^^l The multiple sequence alignment method aligns each sequence such that one base in a sequence corresponds to bases in the other sequences to reveal the similarity of genetic code, evolutionary history, and common biological fiinctions of the strings. Consensus is one of the latest approaches utilizing this method with significant performance.^^ ^^ The local sequence alignment approach utilizes local pair-wise alignment of the sequence to be predicted and the most significant method developed with this approach is named PREDATOR.^^^^ Threading maps the unknown structure to the most similar known sequence.^^^^ Hidden Markov Model-Based Prediction of Secondary Structure (HMMSTR) considers similarity of unknown protein to segments of known structures.^^"*^ The nearest neighbor methods operate by matching segments of the sequence with segments within a database of known structures, and making a prediction based on the observed secondary structures of the best matches.^^^"^^^ There are two significant approaches: combination of GOR Algorithm and Multiple Sequence Alignment Method and combination of Nearest-Neighbor Algorithms and Multiple Sequence Alignment Method. The combination of GOR algorithm and multiple sequence alignment method^^^^ starts with selection of a set of proteins (12 proteins) with well-determined structures, none of which belonging to or having identity to any of proteins in databank of GOR program. Next, multiple sequence alignment of these proteins is carried out and the results are the inputs for GOR algorithm. A scoring system considering sequence-similarity matrix, local structural environment scoring scheme and N and C-terminal positions of secondary structure types is utilized with a restricted database of a small subset of

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proteins that are similar in the combination of nearest-neighbor algorithms and multiple sequence alignment method. Nearest-neighbor algorithmic part is followed by multiple sequence alignments J^^^ The comparison of various methods for predicting secondary structure of globular proteins in general are tested on 195 proteins and three state per residue performances (Q3: helix, sheet or other), and Qs are measuredJ^^^ Prediction accuracies of these methods are given in Table 1. Table 1: Average three-state accuracy indices calculated for six prediction algorithms based on 396proteins^^^l IVfpflinH

n.(o/^\

PHn[6'9]

71 Qsn

NNSSpt^^l

71.400

DsrJ^^^

68.4n

PRKDATORt^^l

68.602

ZPRFD^^^^

59.637

Consensus^^^^

72.707

In this paper, a two-stage algorithm for the secondary structure prediction of proteins is presented. The algorithm has a probabilistic approach utilizing data on all structurally identified proteins having the same folding type with the new unknown protein. The first stage in the method is determination of class of unknown protein. This is accomplished by solving a mixed integer linear program (MILP) problem with 100% accuracy. The objective of the first stage is to reveal some of the uncertainties in the protein structure by determining the folding type accurately. The second stage involves decomposition of the amino acid sequence to overlapping sequential groups of 3 to 7 residues. A local database is formed for each folding type by extracting structural data from PDB files. After matrix of frequency of occurrences of each sequential group of new residue chain is generated, probabilities of being in an ahelix, a P-sheet or a connecting structure are calculated for each residue. The structure with maximum probability is accepted as the structure. 2. The Two-Stage Method The two-stage algorithm decomposes the secondary structure prediction problem into two steps: first the overall folding type of the protein is predicted, and then the secondary structure is predicted using the refined statistical data from the first step. 2.1. Prediction of Folding Type The overall folding type of a protein depends on amino acid composition.^^"^^ Several methods are developed to exploit this theory in the prediction of folding type of proteins.^^^"^^^ These methods use statistical analysis and separate multi-dimensional amino acid composition data into several folding types. The prediction of protein folding type is a typical multi-class data classification problem. Classification of multidimensional data plays an important role in the decision determining main characteristics of a set. Support vector machines is a data mining method to classify data into different groups.^^^^ Although this method can be efficient in classifying data into two groups, it is inaccurate and inefficient when the data needs to be classified into more than two sets. Mixed-integer programming allows the use of hyper boxes for

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defining boundaries of the sets that include all or some of the points in that set. Therefore, the efficiency and accuracy of multi-class data classification can be improved significantly compared to traditional methodsJ^^'^^^ Another approach is to define piecewise linear functions to separate the data that belongs to different classes from each otherP^^ The main differences between these three approaches are illustrated in Figure 1.

;

•

i

•

»

1

: - i

.-

• • •• 3_ "^*

• . \ * \

*

A

••

i

AN

• ^

:

m

A

. A

• • ••

(a) Hyperlanes

(b) Hyper-boxes

(c) Piecewtse-Linear Functions

Figure 1. Three approaches to multi-class data classification problems: (a) support vector machines, (b) MILP using hyper-boxes, (c) piecewise-linear functions. The protein folding type prediction problem is considered in two parts: training and testing. The objective of the training part is to determine the characteristics of the proteins that belong to a certain class and differentiate them from the data points that belong to other classes. After the distinguishing characteristics of the classes are determined, then the effectiveness of the prediction must be tested. The prediction accuracies with different methods for the data set given in are summarized in Table 2. Table 2: Average prediction accuracies with different methods for the folding type problem. Method SVD^^*J NN'^'l SVMP'J

ccf^^i Hyper-boxes'^^'^ Piecewise-Linear Functions''"^

al-a 66.7% 68.6% 74.3% 84.3% 87.5% 100%

al-3 90.1% 85.2% 82% 82% 85.7% 100%

a+3 81% 86.4% 87.7% 81.5% 91.3% 100%

a/B 66.7% 56.9% 72.3% 67.7% 50% 100%

Overall 81% 74.7% 79.4% 79.1% 83.3% 100%

2.2. Prediction of the Secondary Structure The basis for the algorithm is searching segments of its residue sequence in pool of known protein structures and predicting structure for each residue on the basis of frequency of occurrence. To determine the number of residues in each segment to be searched, two facts are considered: the segment should be long enough to have a legitimate reason to search considering interactions and bonds formed between amino acids to shape their structures. Every structure is searched in the relevant database whose dimensions were stated in previous section. The algorithm considers 3 to 7residues-long segments of this chain in the database as illustrated on a sample primary sequence in Figure 2.

Prediction of Secondary Structures of Proteins Using a Two-Stage Method

a

r\

V

1

a v|

akv

II o y ci 1

raq

hsy aqhs

akvr

akvraq akvraqh

l\

1 III

qkl

aft

O

1

ttqkl hsyaft

1

msr

II aty

fhn

qkim

yaft qhsya

akvra

L \f

1683

naty

srfh

naty.

msrfh fhnaty

qklmsr syaftqk

aty...

Figure 2. Representation of overlapping 3,4,5,6, and 7 residue segments. Then, the probability for a particular residue to be in a secondary element for each residue segment is calculated according to its folding type. (1) where Pyk represents the probability of residue / being of structure type j in A:-residue segments, ryk is the total number of occurrences of residue i in structure type j in kresidue segments, and ttk is the total number of occurrences of residue / in A:-residue segments. Then, the ultimate probability, Qy, for residue / to be in structure type j is calculated as follows. (2)

Qv=l>kP,k

The weights, Wk, are determined for each folding type using the data available in SCOP^^^^ database with least squares regression.^^^^ The three ultimate probabilities calculated using Eq. (2) are compared and the secondary structure type that has the highest ultimate probability is selected as the secondary structure of the residue. The results of the algorithm are given in Table 3. Table 3: Results of secondary structure prediction. Training Set

#ofnroteins # of stnictiires

All-a All-P

o/p a+p Total

2000 2766 3375 2866 11007

41139 67373 100255 62935 271702

Test Set

# of nroteins

419 579 707 601 2306

Qs (%)

# of residues

203302 313844 505693 279355 1302194

80.5 72.4 71.9 75.5 74.1

3. Conclusions A novel two-stage method to predict the secondary structure of proteins is presented in this paper. The objective of the first stage is to reveal some of the uncertainties in the protein structure by determining the folding type accurately. The second stage involves decomposition of the amino acid sequence to overlapping sequential groups of 3 to 7 residues and calculation of probability for a particular residue to be in a secondary

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structure. It is shown that the novel two-stage method performs better compared to the state-of-the-art general methods for globular proteins. References [I] Levitt,M, and Chotia,C. (1976) Structural patterns in globular proteins. Nature, 261, 552-558. [2] Mount, D. W. (2001), Bioinformatics: Sequence & Genome Analysis, Cold Spring Harbor Laboratory Press, Woodbury, New York. [3] Protein Data Bank, htttp://www.pdb.org/. [4] Bohr, H., Bohr, J., Brunak, S., Cotterill, R. M J., Fredholm, H., Lautrup, B., and Petersen, S. B. (1990), A novel approach to prediction of the 3-dimensional structures of protein backbones by neural networks, FEBS Letters 261, 43-46. [5] Cai, Y. D., Liu, X. J., Xu, X. B., and Chou, K. C. (2002), Artificial neural network method for predicting protein secondary structure content. Computers and Chemistry 26, 347-350. [6]Rost, B. (2001) Review: Protein secondary structure prediction continues to rise. Journal of Structural Biology 134, 204-218. [7] Petersen, T. N., Lundegaard, C , Nielsen, M., Bohr, H., Bohr, J., Brunak, S., Gippert, G. P., and Lund, O. (2000), Prediction of protein secondary structure at 80% accuracy. Proteins 41, 17-20. [8] Altschul, S., Madden, T., Shaffer, A., Zhang, J., Zhang, Z., Miller, W., and Lipman, D. (1997) Gapped Blast and PSI-Blast: A new generation of protein database search programs. Nucleic Acids Res. 25, 3389-3402. [9] Rost, B., and Sander, C. (1993), Prediciton of protein secondary structure at better than 70% accuracy, J. Mol. Biol. 232, 584-599. [10] Baldi, P., Brunak, S., Frasconi, P., Soda, G., and PoUastrt, G. (1999) Exploiting the past and the future in protein secondary structure prediction, Bioinformatics 15, 937-946. [II] Cuff, J. A., Barton, G. (1999), Evaluation and improvement of multiple sequence methods for protein secondary structure prediction. Proteins 34, 508-519. [12] Frishman, D., and Argos, P. (1997), Seventy-five percent accuracy in secondary structure prediction, PROTEINS: Structure, Function and Genetics 27, 329-335. [13] Thiele, R., Zimmer, R., and Lengauer, T. (1999), Protein threading by recursive dynamic programming. Journal of Molecular Biology 290, 757-779. [14] Bystroff, C , Thorsson, V., and Baker, D. (2000), HMMSTR: A hidden Markov model for local sequence - structure correlations in proteins, J. Mol. Biol. 301, 173-190. [15] Sen, S. (2003), Statistical analysis of pair-wise compatibility of spatially nearest neighbor and adjacent residues in a-helix and p-strands: Application to a minimal model for secondary structure prediction. Biophysical Chemistry 103, 35-49. [16] Westhead, D. R., and Thornton, J. M. (1998), Protein structure prediction, Current Opinion in Biotechnology 9, 383-389. [17] Yi, T. M., and Lander, E. S. (1993), Protein Secondary Structure Prediction Using Nearestneighbor Methods, Journal of Molecular Biology 232, 1117-1129. [18] Salamov, A. A., and Soloveyev, V. V. (1997) Protein secondary structure prediction using local alignments, J. Mol. Biol. 268, 31-36. [19] Kloczkovski, A., Ting, K.-L., Jemigan, R.L., and Gamier, J. (2002), Protein secondary structure prediction based on the GOR algorithm incorporating multiple sequence alignment information. Polymer 43, 441-449. [20] Salamov, A. A., and Soloveyev, V. V. (1995), Prediction of Protein Secondary Structure by Combining Nearest-neighbor Algorithms and Multiple Sequence Alignments, J. Mol. Biol. 247,11-15. [21] SCOP, http://scop.mrc-lmb.cam.ac.uk/ [22] King, R. D., and Stemberg, M. J. E. (1996), Identification and application of the concepts important for accurate and reliable protein secondary structure prediction. Protein Science 5, 2298-2310. [23] Zvelebil, M. J. J. M., Barton, G. J., Taylor, W. R., and Stemberg, M. J. E. (1987), Prediction of protein secondary stmcture and active sites using the alignment of homologous sequences. Journal of Molecular Biology 195, 957-961. [24] Nakashima, H., Nishikawa, K., and Ooi, T. (1986), J. Biochem. 99, 152-162. [25] Chou, K.C. (1995), Does the folding type of a protein depend on its amino acid composition?, FEBS Letters 363, 127-131. [26] Bahar, L, Atilgan, A.R., Jemigan, R.L., and Erman, B. (1997), Understanding the Recognition of Protein Stmctural Classes by Amino Acid Composition, Proteins: Stmcture, Function, and Genetics 29,172-185.

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[27] Cai, Y.D., Liu, X.J., Xu, X.B., and Zhou, G.P. (2001), Support Vector Machines for predicting protein structural class, BMC Bioinformatics 2, 3. [28] Uney, F., and Turkay, M. (2005)"A Mixed-Integer Programming Approach to Multi-Class Data Classification Problem", European Journal of Operational Research, in print. [29] Turkay, M., Uney, F. and Yilmaz, O. (2005), Prediction of Folding Type of Proteins Using Mixed-Integer Linear Programming, Computer-Aided Chem. Eng., vol 20A: ESCAPE-15, L. Puigjaner and A. Espuna (Eds.), 523-528, Elsevier, Amsterdam. [30] Turkay, M., Bagirov, A. and Uney, F. (2005), Prediction of Folding Type of Proteins Using Piecewise-Linear Functions, manuscript under preparation. [31] Cai, Y.D., Zhou, G.P. (2000), Prediction of protein structural classes by neural network, Biochemie, 82, 783-785. [32] Chou, K.C., Liu, W.M., Maggiora, G.M., Zhang, C.T. (1998), Prediction and classification of domain structural classes, ProteinsL Structure, Function, and Genetics, 31, 97-103. [33] Yilmaz, O, (2003), A Tw^o-stage mathematical programming algorithm for predicting secondary structures of proteins, MS Thesis, Koc University, Istanbul, Turkey.

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Reconstruction of Transcriptional Regulatory Networks via Integer Linear Programming Joao M. S. Natali, Jose M. Pinto Polytechnic University, 6 Metrotech Center, Brooklyn, NY 11201, USA

Abstract Much effort has been recently dedicated to the identification of genome-wide transcription regulatory networks by means of comprehensive high-throughput experiments that are capable of capturing the systemic behavior of the transcription coordination phenomenon. The present work comprises the development of Linear Programming (LP) and Integer Linear Programming (ILP) approaches to model and analyze the gene regulatory network of Saccharomyces cerevisiae, centered on a logic inference based representation of regulatory events and on a direct evaluation of experimental quantitative results. Our models are based in a simple representation of regulatory logic. Initial results show coherence to published data and improvements on the logical representation of regulatory events are currently under development. Keywords: Integer Linear Programming, Transcriptional Regulatory Networks, Optimization, Logical Modeling. 1. Introduction The concept that proteic molecules are produced via the initial transcription of the genetic code into mRNA strands and the following translation of this mRNA into a sequence of amino acids - which is often referred to as the Central Dogma of Biology has been widely known and accepted by the scientific community for decades. However, the mechanisms underlying the regulation of these processes, which would ultimately explain why proteins are produced in such differing quantities under diverse metabolic conditions, are far from being fully understood. The transcription of a gene relies, among many factors, on the activities of a class of enzymes called RNA-polymerases. The binding of the RNA polymerase to the genetic code may depend on the existence of other chromatin binding proteins and complexes, known as transcription factors (or simply TF's), which can aid or obstruct the enzyme's binding and fiirther genetic transcription. Therefore, the affinities and activities of transcription factors are key elements of the cell's transcription regulation. In recent years significant effort has been put into deciphering transcription regulatory elements and regulatory networks on a genomic scale. This attempt has been founded on the insight that the information that can be extracted from the establishment of a coordinated network of regulatory interactions may reach far broader scopes of understanding than the usual recognition of individual regulatory elements alone. One of the most successful experiments aiming at the identification of a genome-wide regulatory network was carried out by Richard Young's group (Lee et al., 2002) based upon the ideal eukaryotic microorganism Saccharomyces cerevisiae. These authors relied on Chromatin-Immunoprecipitation (ChIP) and Microarrays techniques to identify all gene promoter regions that were physically bounded by a comprehensive set of transcription factors. The outcome from this approach was an array of p-values that provided information on the likelihood of each of the promoter regions from the whole

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studied genome to be bound (and, thus, potentially regulated) by each of the transcription factors considered. Additionally, a number of authors undertook the effort of defining mathematical methods and computational procedures that would be capable of providing information on gene regulatory networks in a in silico manner. Gupta et al. (2005) used linear and non-linear dynamical models of mRNA production and gene expression to obtain regulatory patterns from a set of expression profiles. Hasty et al. (2001) provided a comprehensive review on computational studies of gene regulatory networks. Using a procedure based on statistical analysis and relying on the results from Lee et al. (2002) and on the vast amount of expression data currently available, Gao et al. (2003) proposed the confrontation of the information extracted fi-om transcription factor occupancy data and gene expression data to obtain a compendious set of TF-Gene interactions that represent a concise and coherent regulatory network. For that purpose, however, a number of simplifications based on statistical calculations were performed. These simplifications are justifiably expected not to exert great influence on the general outcome of the method; however, they significantly limit the reproducibility of the results while inserting information that is not exclusively provenient of the biological phenomenon studied.

2. Proposed Approaches We propose two approaches for the Regulatory Network Reconstruction problem, one based on Linear Programming (LP) and another on Integer Linear Programming (ILP). The formulated problems involve the modeling of regulatory events and the automated decision making regarding which of the interactions between transcription factors and intergenic regions, previously pointed by genomic location analysis (Lee et al., 2002), are indeed relevant to the global regulation of transcription in yeast and, therefore, belong to its regulatory network. 2.1. Linear Programming/Minimum Cost Network Flow Model The proposed LP formulation is a Minimum Cost Network Flow (MCNF) model in which supply nodes are set to represent transcription factors, demand nodes are regarded as genes, and arcs represent the paths through which regulatory signals flow. The model is based on the representation of regulatory elements and signalling pathways as a network comprised of a bipartite graph and input/output flows. A graphical representation of such network is presented in figure 1. F(TFxRG)

I flows: M flows:

signaling strength required for each transcription factor in an experiment, log-ratios from microarray experiments - measure of the signaling flow required by each regulated gene

Figure 1: Regulatory interaction network in LP model.. The model is defined in the following manner: let

TF = {l,2,...,nj.p] be the set of

transcription factors, and i?G = {7,2,...,«^^} the assumed set of regulated genes. The bipartite graph that represents the interconnections between transcription factors and

Reconstruction of Transcriptional Regulatory Networks

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genes is given byF = (F^,£'^), where V^ =TFuRG and Ep=TFxRG. The network model is based on the straightforward concept of flow balance around each defined node, in which microarray experiment results are used as a measure of the signal intensity of positive regulation required by each gene and, thus, determine the overall intensity and units of the global flow through the network. This signal flow is distributed to every gene from each transcription factor through the network F. A cost parameter C. \/(i,j)s [TF,RG) is further associated with each unitary flow through arcs in F, and is defined as the p-value for the existence of an interaction found by Lee at al. (2002). Finally, a demand for flux Mj V/'E RG is assigned to each gene in the network, given by the base-2 logarithm of the ratio of scanned luminescence intensity between the test and control media in each run of the microarrays experiments. The problem is, then, formulated as a MCNF problem (Ahuja et al., 1993). Only the datafi-oma single microarray experiment is considered in the definition of the problem. Therefore, the comparison between different solutions using dissimilar expression data is important for the interpretation of the results. The resulting optimization problem is as follows:

min Z=XZQy-^.y ieTF jeRG

s.t. j F,_j Vi e TF, Vy s RG \ Y, F,j >Mr,Y,F,^>0;0< [

ieTF

f;., < f"

(1)

jeRG

2.2. Integer Linear Programming/Logic Inference Based Representation Let EX = {l,2,...,n^] be the set of experiments used as input data, and TF and RG be defined as previously. Furthermore, we define Xj,^ e [True,False] V/'G RG,\/ke EX as a Boolean variable which is true if and only if gene J is expressed in experiment k, and Y.j^ V/e TF,\/ke EX as a Boolean variable true if and only if a transcription factor / is produced in an experiment k. Moreover, we define two sets of binary variables representing the topological characteristics of the reconstructed transcriptional regulatory network. Let Sp. j \/[i,j)e [TF,RG) be a set of Boolean variables which are true if and only if the transcription factor i activates the transcription of gene y, and, concordantly, Sn.j \/[i,j)e [TF.RG) which are true if and only if the transcription factor /• represses the transcription of geney. Using this formalism, the logical relationship between the variables that represent the regulatory network topology and the inferences that connect TF's and genes can be posed as follows: SP,J

—iSp. J A —tSn. J

No Regulation

y{i,j)e{TF,RG)

(2)

In disjunction (2), L^(Xjj^,Y.A represents a set of logical propositions that describe relationships between the expression of a gene and the binary activity of a transcription factor under activation interactions, and I^ (z^^, 1^ ^) , similarly, for repression.

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The present model is based on the simple logical relationships between the activities of transcription factors and the genes that are regulated by them as below: Sp,j

\/{i,j,k)s{TF,RG,EX)

and

(3)

which can be converted to integer constraints (Raman and Grossmann, 1991). The OF is the maximization of the existence of activation interactions, weighted by the log ratio values of each interactions shown below, where R is the set of log ratio values for the interactions between each pair of transcription factors and genes found by Lee et al. (2002). The optimization model can be defined by: max

Z=i;Z^..y(%v+'5"M)

(4)

ieTF jeRG

sJ. Y.,-X^,>{l-Sp,j)

yieTFyjeRG,keEX

Y,,+Xjj^-l<[l-Sn.j)

yieTFyjeRG,kGEX

(5) (6)

jeRGkeEX

Sp,j + Sn.j <1

V/ G TF, yj G RG

(8)

^M.^,..%y.^«/,y e {OJ} V/e TFyje RG,ke EX (9) The above formulation results in a relaxed problem, provided that no constraints on Y are defined. To address this issue, and considering that the activation and repression events are only connected by the objective fimction and by the mutually exclusiveness constraint, the problem was divided into two subproblems shown below. max Z = X Z ^ u ( % , ; ) max Z = Z Z ^ / j ( % y ) ieTF JeRG

S.L (5), (7), (P) Y.,=0 yiGTF,kGEX

KJ-^^ sx. {6\ (7), (9) Y.,=l yiGTF,kGEX

(10)

The a priori definition of the transcription factors activities is based on the logical nature of the model. Considering the activation problem, the postulation of no TF activity implies that all genes which are positively regulated be non-expressed. This generates a trade-off with the OF, which seeks the maximization of regulatory interactions and the imposition of a lower bound on genetic expression. The same argument is used to justify the repression model. It is important to note, however, that using the simple logic proposed, alongside the restrictions on the values of Y, both models are reduced to the same set of constraints and, thus, to the same formulation.

3. Results and Discussion 3.1. Case Study: Yeast Transcription Network The system employed in this study is the transcription network from S. cerevisiae, comprised of the microorganism's entire genome (6270 genes) and a set of 113 transcription factors, chosen due to the availability of the chromatin binding data obtained by Lee et al. (2002). Genome-wide binding data consist of two-dimensional arrays containing all the base 2 log ratios of the scanned intensities of colored tags from the ChIP concentrated solution and the control solutions for all considered transcription factors. Microarray experiment results were obtained from the Saccharomyces Genome Database\ Expression profiles from four different cultivation conditions were used: evolution in limited glucose; ^ http://www.yeastgenome.org/

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diauxic shift; cell cycle phases S/G2/M (elutriation); and cell cycle phases M/Gl (afactor release). 3.2. MCNF Results Computations based on the linear model were carried for each set of expression data obtained. The optimal flow distribution from transcription factors to genes was obtained and positive interactions were considered as active arcs in the network. The input flow from each TF - equal to the sum of outbound flows in each TF node - was plotted for each experimental data. Moreover, a search for motifs was carried and the results were compared to the motifs found by Lee et al. (2002). (a)

""

(b) Results

Number 3f Regulatory Motifs Found . Multi- Regulator Single Multi Comp. Chain Input Input 3 188 90 49 81 4 35 42 13 27 1 23 40 15 33 0 29 34 21 30 2 32 36 19 38

Autoreg Feedfwd Lee et al. (2002) Expl Exp2 Exp3 Exp4

10 2 1 3 3

Transcriptton Factoi

Figure 2: (a) Flux intensityfromeach transcription factor considered and for each genetic expression datasets. (b) Number of regulatory motifs found for each experimental dataset and published resultsfromlocation analysis. Figure 2 shows coherent results regarding assignment of transcription factors to genes when different microarray results are used. Moreover, the set of regulatory motifs found is seen to be smaller than the ones found at Lee et al. (2002). This result was expected provided that we are attempting to contrast such results to expression experiments to obtain a more concise set of interactions. 3.3. ILP Results The ILP model was solved simultaneously for the four sets of experimental data selected. The results shown in figure 3 refer to the behavior of the activation model, since, as discussed, the logical inference used was not sufficient to provide a distinction between both models.

ILsp.l

4000

TFs

MXLo

Figure 3: Total number of interactions: (a)fromeach transcription factor taken for different lower bounds on gene expression; (b) for each transcription factorfromliterature, obtained by location analysis, (c)fromthe entire solution space, as a function of the lower bound on expression. Figure 3 a shows that increasing the requirement for expression, the number of regulatory interactions observed suffer considerable drop. This can also be seen by the results displayed in figure 3 c, which shows that the overall number of interactions also

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decreases in a similar fashion. However, it can be observed that this trend is not followed with the same intensity by all the TF's, which is closely related to the greater requirement for expression by some genes relative to others, given by the log-ratio intensity in the parameter M. It is also observed that transcription factors that are associated with a larger number of interactions have a tendency to maintain these interactions for small decreases in the value of M^"^, whereas less connected transcription factor present a stronger dependence on this parameter. Results exhibited in table 1 corroborate the observed tendency of obtaining a reduced set of regulatory motifs, in comparison to published results from location analysis. Figure 3b illustrates that the obtained results share a good correlation with the ones found by location analysis (Lee et al., 2002). Table 1: Regulatory motifs found in literature and those obtainedfromthe ILP model. Results Lee et al. (2002) ILP (MY^^= 1000) ILP (M\^^ = 6500)

Autoregulation 10 6 3

Number of Regulatory Motifs Found Feedforward MultiRegulator Loop Component Chain 49 3 188 36 2 102 19 0 56

Single Input 90 76 29

Multi Input 81 64 25

4. Conclusions Two mathematical programming models for the reconstruction of franscriptional regulatory networks were proposed. The MCNF model represented a simple approach whose simplicity and the impossibility of incorporating multiple expression datasets limits its applicability. Results, nonetheless, show good coherence with information available in literature and a relatively high consistency with the expected behavior of the system. The second proposed model is initially formulated in disjunctive form and transformed into an Integer Linear Program. It provides a framework capable of incorporating sophisticated logical relationships between transcription factors and regulated genes that are able to describe complex regulatory relationships. The model was evaluated with a simple relational logic, which carried restrictive simplifications, and the obtained results, regardless of such complexity reductions, presented good agreement with published results and with the physics of the problem.

References R.K. Ahuja, T.L. Magnanti and J.B. Orlin, 1993, Network Flows, Prentice-Hall, New Jersey F. Gao, B.C. Foat and H.J. Bussemaker, 2004, Defining Transcriptional Networks Through integrative Modelling of mRNA Expression and Transcription Factor Binding Data, BMC Bioinf.,Vol. 5,No. 31 A. Gupta, J.D. Vamer and CD. Maranas, 2005 , Large-Scale Inference of the Transcriptional Regulation oiBacillus subtilis, Comp & Chem. Eng., Vol. 29, pp. 565-576. J. Hasty, D. McMillen, F. Isaacs and J.J. Collins, 2001, Computational Studies of Gene Regulatory Networks: In Numero Molecular Biology, Nature Reviews, Vol. 2, pp. 268-279 T.I. Lee et al., 2002, Transcriptional Regulatory Networks in Saccharomyces cerevisiae, Science, Vol. 298, pp. 799-804 R. Raman and I.E. Grossmann, 1991, Relation Between MILP Modelling and Logical Inference for Chemical Process Design, Comp. and Chem. Eng., Vol. 15, No. 2, pp. 73-84

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Systematic Design of Drug Delivery Therapies Michalis Xenos, Libin Zhang, MahadevaBharath R. Somayaji Srinivasa Kondapalli and Andreas A. Linninger* Dept. of Chemical Engineering & Bioengineering, University of Ilhnois at Chicago, 851 S. Morgan St., Chicago, IL 60607, USA. *email: [email protected] Abstract This paper presents an engineering approach for optimal drug delivery into the human brain. The hierarchical design procedure addresses three major challenges: (i) physiologically consistent geometric models of the brain anatomy (ii) discovery of unknown transport and metabolic reaction rates of therapeutic drugs by problem inversion and (iii) a quantitative method for determining parameters of invasive drug injection policies. The proposed interdisciplinary approach integrates medical imaging and diagnosis with systems biology and engineering optimization in order to better quantify transport and reaction phenomena in the brain in-vivo. It will enhance the knowledge gained from clinical data by combining advanced imaging techniques with large scale optimization of distributed systems. The new procedure will allow physicians and scientists to design and optimize invasive drug delivery techniques systematically based on in-vivo drug distribution data and rigorous first principles models. 1.

Introduction More than eighty million people in the world are affected by neurodegenerative diseases of the central nervous system (CNS) such as Parkinson's, Alzheimer's and Huntington's disease (NIH, 2005). Development of more efficient therapies for diseases of the CNS is hampered by difficulties related to administering therapeutic drugs to the affected areas deep inside the brain. Many of the larger proteins with promising pharmacological potency in-vitro are often unable to pass the blood brain barrier (BBB), thus never reaching the cells they are targeted to remedy. Direct injection of the drugs via catheters into the region of interest in the brain can effectively by-pass the BBB and the penetration depth of the drug can be enhanced by convection (Morrison et al, 1994). This technique imposes a convective bulk flow field in the extracellular space of the porous brain tissues in order to carry the drug deeper into the brain. This convectionenhanced drug delivery adds another design option to the problem. The properties of new drugs are usually studied by animal experiments. The interpretation of animal drug distribution data inside the three-dimensional tissues of rats, rabbits or sheep is not amenable to simplistic lumped or one-dimensional approaches. Moreover, the enormous size differences between the human and animal brains make the scaling of drug delivery options a non-trivial task. Therefore, there exists a need to create a systematic datadriven approach for the effective design of invasive drug delivery therapies that takes into account the geometric, physiological and biochemical complexity of brain-drug interactions. Novel analytical imaging techniques such as MRI, fimctional MRI, Difftision Tensor Imaging (DTI), Computer Tomography (CT) and Positron Emission Tomography (PET) have improved medical diagnosis. However, the existing imaging technologies are serving physicians in their diagnosis only in a qualitative sense. Quantitative information such as transport or metabolic reaction properties is typically not extracted from these images. There appears to be a gap between high quality of

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imaging data and their use in quantitative analysis. In order to address the open challenges in accurately predicting the drug distribution, its metabolic reaction rates and drug clearance through the blood, a new computational approach integrating the clinical imaging data with first principles transport equations is presented. The proposed design methodology aims at overcoming three major challenges. The first one is to accurately reconstruct the three-dimensional complex brain geometry. A process to seamlessly integrate imaging data with state-of-the-art geometric image reconstruction tools is presented. The second one concerns the lack of exact data for physical and chemical properties such as drug diffusivity and metabolic uptake. A new computational method for accurate determination of transport and reaction properties from images with novel mathematical programming techniques will be introduced. The efficiency of this new transport and kinetic inversion problem (TKIP) for the recovery of unknown diffusivity and metabolic reaction rates from histological or imaging data will be demonstrated. The quantitative information from TKIP will help design optimal drug delivery policies whose parameters include: optimal placement and orientation of the injection catheter, catheter dimensions, and the number of drug release openings. Figure 1 depicts the proposed methodology of the systematic drug design approach. Outline. Section 2 describes the accurate reconstruction of the anatomically consistent brain geometry. Section 3 introduces the TKIP approach for the unknown parameter estimation and section 4 outlines the quantification of drug distribution and provides best options for drug delivery polices into the human brain. 2.

Capturing the Complex Brain Geometry In this section we present our computer assisted brain analysis for accurate geometric reconstruction of the brain's inner structure using histological data shown in Figure 2- left obtained from literature (Warner, 2001). The gyrated surfaces of the cortex, the segmented extensions and physiological differences render the brain geometry complex. The magnitude of these variations differs in each subject and thus leads to the development of a patient-specific approach for better design of drug delivery policies. Quantification of transport processes requires accurate reconstruction of the complex brain geometry. Figure 2-right displays the reconstructed twodimensional computational mesh and the magnification in lower half shows the finely Figure 1. Schematic of the computer resolved computational grid near the catheter assisted diagnosis approach. tip. This two-dimensional reconstructed brain geometry composed of volumes and boundary surfaces has been extracted using Mimics (Materialise, 2005). The subsequent grid generation step segregates the geometric objects using a computational mesh with well-defined mathematical properties (Bohm et al, 2000). The computational grid composed of a fine mesh of polygons serves as the input to a detailed computational analysis of drug transport and metabolic equations to Figure 2. The detailed 2D brain geometry (right) extractedfromhistology (left). MR Imaging/Histological data Direct experimental measurements

Reconstruction tools

I Capturing the anatomic ( complexity of the brain

Grid Generation

TKIP/Optimization

rates fbi t t i * spaclflc drug

Quantitative analysis and Design of Drug Delivery • OpUmal cattMtor Design •Prediction of Drug m

Mathematical modeling 'of systematic dmg delivery based on first principles

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quantify drug distribution in the brain (Linninger et al, 2005). Before predicting the drug distribution, key transport and reaction properties of the drug have to be known. A methodology for estimating these unknown transport and metaboHc reaction parameters is given in the following section. 3.

Discovery of unknown transport properties and metabolic rates Our proposed computational method will accurately quantify unknown transport and drug properties by interpreting concentration profiles obtained by advanced imaging techniques. We propose to adjust the unknown transport and kinetic properties so that the measured concentration field observed in the image and the model predictions are perfectly aligned. Experimental data from clinical images We solve with mathematical programming a large X Construct the geometry of scale transport and kinetic inversion problem for the computational domain unknown parameter set. Since the concentration and flow data are varying with time and position, these experimental Initialize the unknown parameters measurements are often obtained indirectly and J Solve the transport & kinetic discretized ' '"' equations and response surface contain inherent errors. Under the assumption of normal distribution of errors, maximum likelihood Calculate the Newton and Steepest descent direction estimation reduces to a least squares problem as given Update the unknown parameters in Eq. (1). The TKIP has the least square error in the objective function. The transport and reaction of the drug and the bulk fluid momentum equations are Output the optimal parameters constraints. Figure 3: Flow diagram for TKIP of imaging data. Convergence?;

miny/(P) = {0(x,t,P)-e(x,t)f

F'^

{(/>(x,t,P)-e(x,t))

(1)

s.t. Convection-Diffusion equation (2) dt Du _ dp + pV^u + pg + S Transport of the bulk fluid (3) Dt ~ dx In problem (l)-(3), 0(x,/) represents the imaging data in space (x^) and as a fiinction of time. (/>(xJ,P) represents the predicted concentration fields of the drug as a function of the unknown transport and reaction properties, P = {r(x),A:(x)} as given in Eq (2). The term R in Eq (2) represents the metabolic uptake of the drug. The diffusivity, r(x) as well as the kinetic rates may be a function of the spatial region of the brain. For example, the diffusion coefficient is known to vary along the orientation of the axons in the white matter, thus qualifying the porous white matter as anisotropic medium. Eq. (3) represents the momentum equations for flow through the porous brain parenchyma and S is the additional pressure drop incurred (Dullien, 1979). The TKIP approach is composed of three steps: First, discretization of the distributed system obtained from the reconstructed computational grid using discretization schemes such as the finite volume approach (Patankar, 1980). For accommodating complex geometry like the brain, the generalized curvilinear coordinate transformation and unstructured computational meshes are used (Fletcher, 1991; Date, 2005). Second, response surfaces are needed in the optimization algorithm for an efficient and robust evaluation of the first derivative as given in Eq. (4). Finally, we

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have developed inexact trust regions methods to solve large scale inversion problem (Zhang and Linninger, 2005; Conn et. al., 2000). With the use of response surface approximation and the trust region radius y, problem (l)-(3) is converted into subproblem (5)-(6). ^(x,^P + AP) = ^(x,/,P) + - ^ A P + 0(AP^) dP

(4)

min ij/{P^ + AP) = (^(x,t,P^)-\-^'AP-0(x,0)^

(5)

Ap

dP

s.t. \\AP\\
& & » P 4b. Table 1 shows the estimated parameters obtained from TKIP. 0. llcmls-^ Drug Diffiisivity (D) This demonstrates the ability of the 0.0042 s Drug Clearance (K^ ) TKIP approach to effectively obtain 0.0009 s Drug Conversion Rate ( A:^*) the unknown transport properties along with the metabolic reaction rates of the drug molecules within the highly segmented areas of the human brain. 4.

Quantitative analysis and design of drug administration policies The final stage of the systematic therapy design aims at accurately predicting the achievable treatment volumes based on the transport and reaction properties discovered with TKIP. The efficacy of the drug administration measured in terms of the achievable treatment volume that is defined as the brain region dosed with a drug Figure 5. Schematic of catheter design, concentration above the therapeutic threshold. Other considerations include metabolic uptake, blood clearance, and toxicity limits of the drug. The drug delivery policy determines the exact location of the catheter, its stereotaxic orientation, the number and design of catheter outlet ports. Figure 5 displays different catheter hole alignments (output ports) as well as a 3D representation of the brain parenchyma with the inserted

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catheter. In addition, the infusion rate, drug concentration and infiision pressure trajectory will have to be determined. 4.1 Prediction of Treatment Volume for Injection in Thalamus We used the finite volume method assessing the transport and reaction properties obtained by TKIP to quantify drug distribution (Vd) as a function of the catheter placement. The procedure is illustrated using a two-dimensional coronal cut of a human brain with dimensions of 15 cm x 13 cm Table 2. Parameters used X 0.1 cm with a total volume of 19.5 cc Parameter Value (Figure 2). The catheter used in the (p- porosity GM 0.2 computer aided therapy development is a WM 0.28 10-^^ m^ single port catheter. It is assumed that k - permeability GM WM x:6*10-^^m^ caudate nucleus is the target and the y: 2.5*10"^^ m^ therapeutic threshold is fixed at 50% of p - Inertial Factor GM 8.32*10^^ m-^ the initial concentration. The drug of WM x:5*10'^^m-^ interest is Nerve Growth Factor (NGF) y: 2.1*10^^ m-^ which is a large molecular weight |x - bulk viscosity 0.8937*10-^ Pa.s chemotherapeutic agent with a molecular GM: Gray Matter, WM: White Matter weight of 27000 Kg/Kmol (Reisfeld et al, 1995). Therefore, invasive drug infusion is the only feasible option. Parameter values for the computer simulation are given in Table 2. For treating the cells at caudate nucleus the plausible injection sites included injection near the thalamus, putamen, internal capsule and the lateral ventricle.

1 week

4 weeks 2 weeks 3 weeks Figure 6. Evolution of Drug Distribution over time. The brain was modelled as an anisotropic heterogeneous porous medium (Morrison et al, 1994) in which the qualitative and quantitative patterns of Vd strongly depends on the catheter placement. Figure 6 shows the therapeutic patterns of the drug distribution for injection in the thalamic nucleus over a four-week period. With white matter anisotropy as a constraint, the drug distributes asymmetrically to peripheral regions of the brain and may lead to local toxicity due to unnecessary metabolism. Furthermore the infusate flow fields are much pronounced in the tracts of the internal capsule. Figure 7 quantifies the treatment volume 5""as a function of catheter location in the local •rZ""* \" substructures that include striatum, caudate ^ 'U ' nucleus, gray and white matter regions. The total achievable treatment volume using high s 1 °°^ • flow infusion (12 )il/min) was found to be -0.107 cc for thalamic injection. Thus, the thalamic injection is the best injection point Figure 7. Quantification of Drug Targeting amongst other plausible injection sites. The as a function of Injection Site. simulations reveal that higher flow rates of the drug lead to larger treatment volumes (Morrison et al, 1994). However, higher flow rates also increase the necessary pressure >

0.08 -

1

0.06 -

» 0.04 J

^LJI^ Caudate

Striatum

Gray

Injectkxi Site

1

White

Total

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drop causing tissue stresses and strains. This may cause local injury of the brain tissue. Use of higher flow-rates of the drug may also produce backflow along the catheter shaft, thus distributing the drug uselessly into the peripheral regions as opposed to the areas of interest close to the catheter tip (Morrison et al, 1999). 5.

Conclusions A systematic approach for the design of drug delivery policies has been introduced. It will provide physicians with valuable insight in the selection of proper invasive drug therapies. The novel method consisted of three steps (i) accurate reconstruction of the brain geometry, (ii) extraction of unknown transport and kinetic properties from experimental data and (iii) prediction of treatment volumes based on site specific drug delivery. Injection near the thalamic nucleus was found to the best inftision site for administering the drug effectively into the caudate nucleus. Treatment volume was found to be highly sensitive to regional and structural heterogeneity and anisotropy. Accurate catheter placement based on systematic drug delivery design could provide decision support to neurosurgeons for performing clinical trails and to provide innovative therapies to the patient. Better design of therapies through computational methods can minimize the cost of experimentation and can provide better alternatives to the current approaches. References Bohm G., Galuppo P., Vesnaver A.A., 2000, 3D adaptive tomography using Delaunay triangles and Voronoi polygons. Geophysical Prospecting, 48, 723-744. Conn A.R., Gould N.I.M. and Toint P.L., 2000, Trust Region Methods, SIAM, Philadelphia, PA. Date A.W., 2005, Introduction to Computational Fluid Dyanamics, Cambride University Press: New York. DuUien F.A.L., 1979, Porous Media Fluid Transport and Pore Structure, Academic Press Inc: New York. Fletcher C.A.J., 1988, Computational Fluid Dynamics, vol 1, Springer-Verlag: New York. Hamilton J.F., Morrison P.F., Chen M.Y., Harvey-White J., Pemaute R.S., Phillips H., Oldfield E., Bankiewicz K.S., 2001, Heparin Coinflision during Convection-Enchanced Delivery (CED) Increases the Distribution of the Glial-Derived Neurotrophic Factor (GDNF) Ligand Family in Rat Striatum and Enhances the Pharmocological Activity of Neurturin, Experimental Neurology, 168, 155-161. Linninger A.A., Somayaji R.M.B., Xenos M., Kondapalli S., 2005, Drug delivery into the human brain. Foundations of Systems Biology and Engineering, Corwin Pavillion, Univ. of California Santa Barbara Campus, August 7-10. Materialise, Inc., (Mimics), 2005, http://www.materialise.be/mimics/main ENG.html. Morrison, P.F., et al., 1994, High-flow microinfusion: tissue penetration and pharmacodynamics, American Journal of Physiology, 266, R292-R305. Morrison, P.F., et al, 1999, Focal delivery during direct infusion into the brain: role of flow rate, catheter diameter and tissue mechanics. Am. J. Physiol, 277, R1218- R1229. NIH, NINDS, 2005, www.nih.gov. Patankar S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation: Washington. Reisfeld B., Kalyanasundaram S., Leong K., 1995, A mathematical model of polymeric controlled drug release and transport in the brain. Journal of Controlled Release, 36, 199-207. Warner J.J., 2001, Atlas of Neuroanatomy, Butterworth-Heinmann, Boston. Zhang L. and Linninger A., 2005, Newton, Steepest Descent and Trust Region Methods: Application in Unconstrained optimization and Solution of Nonlinear Equations, UIC-LPPD021805.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Mathematical modelling of three-dimensional cell cultures in perfusion bioreactors. Part II Francesco Coletti ^, Sandro Macchietto ^, Nicola Elvassore ^ ^ Dept. of Chemical Engineering, Imperial College London, London SW7 2BY, UK ^ DIPIC, Universita diPadova, via Marzolo 9, Padova 1-35131, Italy Abstract Some applications of tissue engineering require growing cells within supporting scaffolds to obtain structures with adequate functionality for in vivo implantation. A novel general dynamic mathematical model of cells growth and oxygen consumption in a three-dimensional perfusion bioreactor was recently proposed, which includes convection and diffusion, the two main transport phenomena affecting cells growth, combined with cells growth kinetics and oxygen supply. That model is used and modified here to analyze various geometric and operational configurations, including medium flow inversion, initial cell seeding procedures, and a simple vascularization model, represented by micro channels in the scaffold matrix. The significant impact on cell growth, density and density distribution within the threedimensional scaffold is evidenced by the simulation results presented. Keywords: Tissue engineering, Perftision bioreactor. Mathematical modelling. Cells culture. Bio-process simulation. 1. Introduction Tissue engineering, the production of engineered grafts for tissue replacement, presents several challenges, of which achieving significant cells growth in supporting scaffolds is one. Problems include substrate delivery to inner cell layers and non homogeneous cell distribution within a 3D scaffold. In order to help the understanding of the complex interacting phenomena in a scaffold-bioreactor system, several mathematical models have been proposed (Galban and Locke 1999; Pisu et al. 2004; Radisic et al. 2005; Radisic et al. 2006). Most published studies take into account only steady-state behaviour or just a small subset of the relevant transport and cellular processes. A comprehensive mathematical model, presented in a recent paper (Coletti et al, 2006) describes the dynamical behavior of a three dimensional cells culture in a perfusion bioreactor. Figure 1 shows the reference geometry considered. The model (Table 1) includes the fluid dynamics of the nutrient medium flowing by both convection and diffusion through a scaffold, modelled as a porous medium, with cells growth models which account for cell proliferation and contact inhibition, both of which in turn affect scaffold permeability. Realistic parameters for a culture of immortalized rat cells C2C12 on a collagen scaffold are reported in Coletti et al. (2006), together with assumptions, boundary conditions and initial conditions for two typical operations. In this work the above model is used to test some new scaffold configurations and operating conditions aimed at achieving faster growth, higher density and more uniform cells density distribution within the scaffold. These include inversion of the nutrients medium flow direction (Section 2), non uniform initial cells seeding (Section 3) and an initial study of the effects of vascularization, represented as micro channels inside the scaffold (Section 4).

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Figure 1 Reference geometry and system co-ordinates for a total perfusion bioreactor. In the base case the medium flows upwards from entry section ^ i through scaffold Q.^ ^^^ ^^it section Q.^.

Table 1. Model equations and parameters for the three domains Q.^, Q2 ^^^ ^3 • Domain Q.^ and Q3 bioreactor

W

ot

A

c,.,v-

^ = -(V-c,v°) + V-(Z).Vc,) dt

po

=0

D^ constant

p.p

No body force

p

p.j^

No body force

Pcell

Qm^^m

Domain Q.^ • scaffold ^

= _(V.c,v) + V.(A^Vc,) + /?,

Ti ^0,

M-cell ~

— n -Hcell'

^m ^

-v,

r^cell

1^ ^

rr

'^'A^, ^Ri

O2

r'cell

^

C^

,^

P-cell

^d

D^^=eD.lr

€(z, r, 0 = €(z, r, 0) - F,,„p,,, (z, r, 0

max f^cell » ' ^ c ' ^cell ->

e ,T

No cells death: k•^ =0

Pc^K

A ^(2,r,0)

Performance is assessed in terms of the following metrics: 1) Percentage of the scaffold volume with cells density higher than the reference value for physiological human tissues, 2) Percentage of the scaffold volume with oxygen concentration higher than the reference value for cells viability, 3) Maximum cells density, 4) Minimum cells density and 5) Average cells density within the scaffold. Unless stated otherwise, all conditions are the same as in the total perfusion base case of Coletti et al (2006). All calculations were done in Femlab as detailed by those authors.

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2. Medium flow inversion In perfusion reactors, as evidenced by simulations in the cited work, often cells growth well in the first few millimeters of the scaffold, but oxygen concentration rapidly decreases after a very short distance fi-om the entry face of the scaffold, compromising cells growth and viability. A possible alternative is to periodically invert the direction of the medium flow. The medium, with oxygen concentration c = c^", first enters the bioreactor at z =0 and perfuses through the scaffold (direct flow) until inversion time t.. At this time the flux is inverted by feeding the medium at z= L^ + / / + L^ (inverse flow). At time 2 ^. the direction switches back to direct flow, and so on. Numerically, inversion of the flow is simulated by using the solution at time t^ as initial point for a simulation between t. and 21., while also switching the inlet/outlet sections boundary conditions for the material and momentum balances, given in Table 2. Table 2. Boundary conditions at inlet/outlet sections Coordinate z=0

z = Ij + // + Lj

Balance

Inverse flow

Direct flow

material

^=<

1-

momentum

v-2[l-^] ; v» = o

P°=0

material

l=«

momentum

p'=o

c = c'" v: =

-2(l4];v;=0

Figure 2.a and Figure 2.b show the maximum and the minimum cells density obtained with flow switching every 2, 3, 6 and 7 days, compared to the base case of direct flow only (called no switch). With no switch, the maximum cell density after 21 days is 4.5 times higher than in the other cases (but confined to a small entry slice of the scaffold), while the minimum cell density is roughly 2.5 times lower. Since a goal is to obtain a uniform distribution of cells within the scaffold, the average cells density is also important. Figure 3.a shows the average cells density in the scaffold plotted versus time. Flow switching leads to approximately the same average cells density after 18 days. With t.=2 days the average density is constantly higher than with longer switching periods (except between days 12 and 15 with ^,=3). The ratio of the average cells density to the no switch case (Figure 3.b) indicates that up to ~6 days no advantage is obtained from flow inversion while an average cell density 42% higher is achieved after 21 days. 3. Non uniform cells seeding In order to study the effect of initial seeding on cells distribution, two cases with non uniform initial seeding are simulated. In case a, the scaffold domain ^2 is split in two

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^150 \

I

No switches -1=2(^378 1=3 days t.=6days t.=7days

J 1 ^

1

8,100

^

3\

No switches -1=2 days 1.=3 days t =6 days t.=7days

1 :

*

•'

r-"

/ / /'

/f—-

2

J

9 12 Time [days]

-

' ' : •

10 Time [days]

Figure 2 Maximum and minimum cells density. 1.5 J i

No switches t.=2days 1 =3 days t.=6 days 1.=7 days

1.4 H ^

1.3

1=2 days -1=3 days 1=6 days t=7 days

;' f • i

(0

c 0) •o 1.2 m.
/l^

—/

1.0 10 Time [days]

0.9

10 Time [days] (b)

15

Figure 3 Average cell density with and without flow switching (a) and ratio between average cells density with flow switching and that for the no switch case (b) sub-domains with different initial cells densities (Figure 4.a). The first sub-domain, Q2 has a constant initial cells density, p'^^^ , for a depth 5 from the top scaffold surface. In the second sub-domain, Q2» the 'initial cell concentration is set to p'^^iJl. This approximates the experimental situation in which cells are seeded on the upper face of the scaffold only. In case b) both sides of the scaffold are seeded. Domain ^2 is split in three sub-domains (Figure 4.b), with initial density p'"^^ in ^'2 ^ ^ ^ 2 ' ^ ^ P7eii ^^ i^ Q2 • Total perfusion with direct flow is used in both cases. For case a), simulations show that the cells densities in the two domains ^2 ^^^ ^2 remain within 2% of the initial 1:2 ratio during the first 5 days of culture. After this time, the grown cells impede the transport of oxygen to the inner sections of the scaffold. Figure 5 shows the cells density spatial profiles along the axial direction within the scaffold, over time, for cases a and b. In both cases after 7 days there is no variation of cells density at the interface between Q2 ^^^ ^2 • ^^ ^^^^ ^> ^^^^ ^ ^^ys ihQXQ is no more variation of cells density also at the interface between Q2 ^^^ ^2 • Figure 6 shows the average cells densities achieved with both seeding procedures. While two-sided seeding of the scaffold gives a higher initial average cells density (+1.2%), after the 8*^ day it is lower (up to 6% at day 21) than for case a), as cells in the front zone use up the oxygen before it arrives to the inner sections of the scaffold.

Mathematical Modelling ofThree-Dimensional Cell Cultures

P:/2

1703

P'cdl

Oj

p:.n

a;

Pcdl

Q;

1^

^2

mil

>

iiiii (b)

(a)

Figure 4. Scaffold coordinates for case a (seeding on one side) and b (two-sided scaffold seeding). The seeding depth is <5^=1 mm.

6 days - 7 days 8 days 9 days 10 days 11 days 12 days

w

-

•

•

•

•

•

\

-

^""--^^ f 1

2 3 Axial coordinate [mm]

1

2 3 Axial coordinate [mm]

Figure 5 Effect of not uniform cells initial seeding on spatial distribution of cells density. Case of one-sided initial seeding (a) and two-sided initial seeding (b)

4. Simple vascularization The effect of a simple vascularization geometry, represented by 7 equally spaced, vertical micro channels of width g inside the scaffold (Figure 7a) has been investigated. Figure 7b shows the ratio between the average cells density obtained with micro channels (with g=0.02 mm and g=0.1 mm) and that with full perfusion (no micro channels). Initially there is no difference but after 5 days there are notable effects. A reduction in average cells density occurs with the larger diameter channels, as the flow regime becomes diffusion controlled. With the smaller channels width, the convection flow in the scaffold remains high and the average cells density after 21 days is about 5% higher than in the full perfusion base case. 5. Conclusions The effects of various experimental setups for a perfusion bioreactor on cells growth within a three-dimensional porous scaffold were investigated by simulation. The mathematical model used has proved highly flexible and allowed analysis of all cases. The simulations show that a major increase in the average density of the cells grown is achieved by periodic inversion of the nutrients flow, with an increment of 42% after 21 days respect to direct flow. Another 5 % increment relative to total perfusion is given by the right micro channels in the scaffold. However too larges channels produce a negative effect on the average cells distribution. A uniform cells seeding should be achieved as far as possible to avoid uneven cell distributions. Considerable scope was

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1 1

'

' ^-^ ^^^<^^^'

Onesided -Twosided

yy

10 Time [days]

15

Figure 6 Average cells density versus time for one- and two-sided initial seeding procedures

H

n

Hill (a)

(b)

Figure 7 Micro channels configuration (a) and ratio between the average cells density with micro channels (width g=200 jUm and g=20 jUm ), and with full perfusion (base case). revealed for optimizing the scaffold design and reactor operating policies individually and in combination. This will provide a useful tool to those trying to develop their perfusion culture in a porous scaffold. Experimental confirmation of the results presented is required and will be presented in future publications.

References Coletti, F., S. Macchietto and N. Elvassore (2006). "Mathematical modelling of three-dimensional cell cultures in perfiision bioreactors." Submitted for publication. Galban, C. J. and B. R. Locke (1999). "Analysis of cell growth kinetics and substrate diffusion in a polymer scaffold." Biotechnology and Bioengineering 65(2): 121-132. Pisu, M., N. Lai, A. Cinotti, C. Alessandro, et al. (2004). "Modeling of engineered cartilage growth in rotating bioreactors." Chemical engineering science 59(22-23): 5035-5040. Radisic, M., W. Deen, R. Langer and G. Vunjak-Novakovic (2005). "Mathematical model of oxygen distribution in engineered cardiac tissue with parallel channel array perfused with culture medium containing oxygen carriers." Am J Physiol Heart Circ Physiol 288(3): H1278-1289. Radisic, M., J. Malda, E. Epping, W. Geng, et al. (2006). "Oxygen gradients correlate with cell density and cell viability in engineered cardiac tissue." Biotechnology and Bioengineering 93(2): 332-343.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Metabolic Regulatory Network Optimization Using An Information Guided Genetic Algorithm Approach Ying Zheng," Chi-Da Yang, ^ Jen-Wei Yeh, ^ Shi-Shang Jang ^ ^Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, P.R.China ^Chemical Engineering Department,National Tsing-Hua University, Sec. 2, Kuang-Fu Rd No.l01,Hsin Chu,300, Taiwan Abstract In this work, information entropy guided genetic algorithm is implemented to solve the MINLP problem of metabolic regulatory optimization. Alternate optima network structures are first time discussed using the novel approach derived in this work. The novelties of this work include the information theory, local search to improve the efficiencies of the genetic algorithm and cluster analysis to discover the physically meaningful local optima among the qualified solutions. Keywords: Metabolic regulatory network, MINLP, Information theory. Cluster Analysis, Genetic algorithm. 1. Introduction Complex systems such as biological systems are basically complicated with many integer and real variables and numerous reaction, material and energy balance equations, especially in metabolic network problems. Although many discussions in the literatures focused on the optimal design of the metabolic networks, one of the basic problems, the physical meanings of the alternates of feasible local optima, has not yet been studied. The objective of this study is to develop a systematic approach that allocates the feasible local optima, and discuss the rationality of the network structure. Almost every metabolic reaction network is subjected to a regulatory architecture built around it. In many works, the S-system has been widely implemented in the problem formulation of metabolic engineering. For designing biochemical pathways, it is necessary to develop mathematical tools to analyze and optimize integrated biochemical systems. Some literatures (e.g., Voit, 1992) used linear programming approach to solve a simplified metabolic network, the other implemented MILP to solve an optimal solution fi*om a super-structure fi"om a metabolic network (e.g. Hatzimanikatis et al.l996). However, in case of mixed integer programming problems, it has been well-known that the solutions of such a problem may not be unique and subject to rugged objective fimction contour (e.g. Grossmann and Sargent, 1979). Solution of optimal metabolic regulatory network problems can be formulated into a mixed integer nonlinear programming problem (MINLP). Recently, some stochastic approaches attracted many attentions in this area (Cardoso et al., 1998). On the other hand, Bjork and Nordman (2005) showed that genetic algorithm is very suitable to solve

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large-scale heat exchanger network. In this work, stochastic optimization solver, genetic algorithm, is implemented to solve this MINLP problem. The information entropy, one step further, is implemented to improve these solution approaches for MINLP problem. Furthermore, fuzzy c-mean approach is implemented to cluster the feasible and high scored solutions. Different types of solutions, that are of bio-engineers' interest, are given after the cluster analysis. The results show that different regulatory networks at the cluster centers can be found without further mathematic treatments of these complicated system models. 2. The Information Guided Genetic Algorithm 2.1. Genetic algorithm Consider an optimization problem as the following form: Minimize f(X) (1) Subject X&S where / is the objective function, and X = (Xj,..X.,..,X^ ) is a vector of variables in solution space S such that for each variable Xi, there exist a upper bound Xl^ and a low bound X^^. We may call a vector of variables as a chromosome or an individual, and the element of the vector as a gene in genetic algorithm. A set of individuals are called a population. Population size is the number of individuals at the iteration (or generation). Five basic operators have to be considered in genetic algorithm: codingdecoding, fitness evaluation, selection, crossover, and mutation. In a real-coded problem, individuals can be encoded using real numbers. Integer-coded problem is encoded into zero-one codes. Each individual corresponds to one fitness value(score) and which is determined by a fitness function based on the objective function. The suitable individuals are chosen as survivals to perform crossover according to fitness value. The survive individuals are selected to be parents, and are paired randomly to generate new individuals. Usually under a proper probability, genes of individuals are picked randomly to go on mutation. In this work, we implement a premature detector Dp that is the sum of the distance of all individuals in each population. Let D be a threshold, and in case of Dp
E{x) = -Y^p{x)\np{x)

(2)

xeX

where p(x) is the probability of the event x occurring. Information entropy is a measure of how random a variable is distributed. The information guided mutation approach calculates the total information entropy of each variable in MINLP. The variables with highest information entropy are selected to perform IF mutation such that the total information entropy of that variable can be decreased. For details, the reader is referred to our previous work.

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2.3. Local search Ombuki and Ventresca(2004) mentioned local search during mutation will accelerate convergent speed, but extend the computed time. A trade-off between convergent speed and computed time is necessary. 2.4. Penalty function We chose and revised the penalty function proposed by Deb(2000): I fix) ifgj{x)>Q y/- = l,2,...,m, [/max + Z M {SJ (X))

^ , f is the worst [O gi(x)<0 constraint is satisfied ^ " ' ^ reasonable solution of the last generation, g is the inequality constraint. where =<^

{RXX)

Otherwise.

Ri(x)>0

constraint is violated

3. Example In this section the example studied by Hatzimanikatis et al (1996) is solved using the approach described by this work. It is yield optimization in Xanthine monophosphate (XMP) and Guanosine monophosphate (GMP) production. As shown in Fig.l, XJ represents the concentration of the metabolite j (j=1...4); P/ represents the amount of manipulated variable 1 (1=1,...,4); and dashed lines denote inhibition, dashed-dotted lines denote activation. Let Yi = InXt, the optimization problem becomes Maximize {y^) Mass Balances - 0 . 5 J i + 0 . 1 ^ 2 - 0 - 3 ^ 3 - 0 - 3 ^ 4 - ^13^13^3 - ^14^14^4 + 0.6^21^21^1 + 0.6^22^22^2 + 0.6Z23^23>'3 + 0.6Z24^24>'4 + ^'^Z,,£,,y,

+ W,q, " 0 . 6 H ; 2 ^ 2 " 0.41^3^3 = " 4 . 4 9 9 8

0.308Ji -0.482^2 +0.177^3 +0.4^4 -0.37z2i<^2i>^i -0.37z22<^22>'2 -^-'^^z^.s^^y^ -0.37Z24^24>'4 ~ 0.245^346*34^4 - 0.385^436*43^3 + 0.4^53^53^3 + 0.6^64^54^4 +

+0.245^3^3+0.3851^4^4-0.4^5^5

Q31w^q^

-0.6w^q^=\Ji6?>

-0.14ji + 0.409^2 - 0.817373 -0.014^4 -0.455^53^53^3 +0.287z34^34j;4 -0.28x^3^3 +0.455H^5^5 =-4.3212

0.041^2 +0.026^3 -0.799^4 -0.405^^4^64^4 +^^26z,,£,,y, -().26ws,+^A{)5w,q, Bounds on Xj

y = 1,2,3

ln(4.9) > J, > ln(6.0) Bounds o n P ,

=-4.5122

/ = 1,....,6

ln(192) > y2> ln(234)

ln(2176) >y^> ln(2660)

ln(P,'^)>P/ >ln(P/'')

where ^i\l — U 56) and 1^13'^14'^21'^22'^23*^24'^34'^43'^53'^54 j are binary variables, Jy (/ = l v i 4 ) and ^/ (/ = 1,...,6) are continuous variables; L represents lower bound, L'^ represents upper bound.

Y. Zheng et al

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original GA .

+ IF mutation

n Fig. 1. XMP and GMP synthesis pathway

Fig.2. Comparisons of the performance four different GA approaches

Fig.2 compares the performance of four different types GA (the original GA, GA+local search, GA+IF mutation, and GA+local search+ IF mutation) for this particular problem. In these four cases, maximum generation number is set to be 2600, local search is performed in every 400 generations, and the population size is set to be 14. The premature detector parameter, D is set to 1. As shown in Fig.2, it is obvious that by including IF entropy and/or local search, the performance of GA can be drastically improved. After 2600 generations 463 feasible solutions are found. Among them we select 84 high score solutions, which satisfy the following conditions: 1) ^5 "^ ^ ; 2) i.e. x^ > 49021 According to the 16 binary ^ 6 > 0 ; 3) J 4 > 1 0 - 8 parameters >V/(' = U 96) and l^n'^14'^21'^22^^23^^24^^34^^43'^53'^64] . We compute Euclidean distance between these solutions and create hierarchical cluster tree. The dendrogram graph is shown in Fig.3. We can find five solutions of different type according to Fig.3. PR

•XI P2+

l^a

3 29 17 10 21 14 28 30 24 25 2 22 6 13 23 15 27 26 9 5 1 4 8 11 12 16 18 20 19 7

Fig.3. The dendrogram graph

A

/\

Fig.4. The solution when ^ =21

P4+

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Let t represent the leaf node in Fig.3. Fig.4-Fig.8 represent five typical solutions based on different t's. Fig.4 shows the solution when /=21. The values of variables H'/, {^13 5 ^14 9 ^219 ^22 9 ^23 ^^24 9 ^34 9 ^43 9 ^53'^641

J y (/ = U«»»94) a u d ^ / ( / = 1 , . . . , 6 ) a r e 1 1

0 1 0 0, 0 1 0 1 1 1 0 0 0 1, 1.7411 5.4461' 7.6865 10.87, and 1.4722 1.2906 -1.3646 0.42012 0.17219 -0.29841. The maximum concentration of X4 is 52575. Fig.5 shows the solution when /=13. The values of variables are 1 1 1 0 1 0 , 0 1 0 1 1 1 1 0 0 1 , 1.7912 5.3068 7.6894 11.084,1.2309 1.6094-1.6094 0.60501 -0.23515 -0.17344. The maximum concentration of X4 is 65121. Fig.6 shows the solution when t=5. The values of variables are 1 1 0 0 0 1, 0 1 1 1 1 1 0 1 0 1, 1.783 5.4288 7.7101 10.944, and 1.6094 1.6094 1.0534 1.5246 1.6027 0.29228. The maximum concentration of X4 is 56613. Fig.7 shows the solution when /=12. The values of variables are 1 1 1 0 1 1, 0 10 1 1 1 0 10 1, 1.786 5.4192 7.7012 11.151, and 1.534 1.6072 -0.45455 0.77799 0.20221 0.65065. The maximum concentration of X4 is 69633. Fig.8 shows the solution when t=l. The values of variables are 1 1 0 1 0 0, 0 10 0 1 1 0 0 0 1, 1.7408 5.2757 7.688 10.816, and 1.126 1.4727 -0.1548 0.4643 0.036097 -1.2034. The maximum concentration of X4 is 49811. PI+

/\ Fig.5. The solution when ^ =13

7\ Fig.6. The solution when t =5

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Y. Zheng etal Fig.7. The solution when t =12

Fig.8. The solution when t =1

According to the above analysis, it is clear that the local optima at t=12 and 13 give the higher yields (x4= 69633 and X4=65121 respectively) compared with other cases. Notably, the original yield by Hatzimanikatis et al.,1996 was X4=55015.6. Further, the network structures suggested by these two solutions (see Fig.5 and Fig.7) are much simpler than the original work without many contradictory modifications of the networks. Note that five different structures of the above solutions based on our clustering analysis represent different modification of the original metabolic network (Figure 1). As indicated by the original work (Hatzimanikatis et al.,1996), it is undesirable to perform many modifications of the original pathway. In our solution set, some of our solutions require more modification with higher yield and the other solutions require less modification but result lower yield. This is very valuable to metabolic engineers. 4. Conclusion Solution of optimal metabolic regulatory network problems can be formulated into a mixed integer nonlinear programming problem (MINLP). In this work, genetic algorithm is implemented to solve this MINLP problem. The information entropy and local search method are implemented to improve these solution approaches for MINLP problem. Furthermore, clustering analysis is implemented to allocate physically meaningful local optima. Different types of solutions are given after cluster analysis. The results show that the approach is valid and efficient. References Bjork, K. M., Nordman, R. (2005). Solving large-scale retrofit heat exchanger network synthesis problems with mathematical optimization methods. Chemical Engineering and Processing, Vol. 44, No. 8, pp.869-876. Cardoso M.F., Salcedo R.L. and Barbosa D. (1997).A Simulated Annealing Approach to the Solution of MINLP Problems.Computers Chem. Eng. ,Vol. 21,No. 12,PP1349-1364. Deb, K. (2000). An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Engrg., Vol. 186, pp.311-338. Grossmann, I.E., Sargent, W.H. (1979) Optimum Design of Multipurpose Chemical Plants. Ind. Eng. Chem. Process Des. Dev.,Vol. 18,No.2, pp.343-348 Hatzimanikatis, V., Floudas, C.A. and Bailey, I.E. (1996). Optimization of Regulatory Architectures in Metabolic Reaction Networks. Biotechnol. Bioeng., Vol.52, pp.485-500. Ombuki, B.M. and Ventresca, M.(2004).Local Search Genetic Algorithms for Job Shop Scheduling Problem. Applied Intelligence, Vol. 21, pp. 99-109. Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. Journal, Vol 27, pp. 379-423. Voit, E.O.(1992). Optimization in Integrated Biochemical Systems. Biotechnol. Bioeng., Vol 40, pp.572-582. Yeh, C.W. and Jang, S.S. (2005). The Development of Information Guided Genetic Algorithm for Global Optimization", Journal of Global Optimization, In Press.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes Hong Yang, Martin L. Yarmush, Charles M. Roth and Marianthi lerapetritou^ Rutgers, the State University of New Jersey, Piscataway, NJ, 08854, USA. Abstract Extracorporeal bioartificial liver (BAL) devices are the most promising technology for treatment of the liver failure. However, when exposed to plasma from the patient, hepatocytes are prone to accumulate intracellular lipids and exhibit poor liver-specific functions. Our work focuses on understanding the metabolism of cultured hepatocytes used for BAL. In this paper, a logic based programming is used to determine the important reactions in cultured hepatocytes by systemically analyzing the hepatic metabolic network, and investigating whether insulin, amino acid and hormone supplementations upregulate or downregulate certain pathways that control important liver specific functions, such as urea and albumin production. Using elementary mode analysis we were able to obtain 32 independent pathways, which are then used to analyze the results of the logic-based programming approach. Keywords: Hepatocyte metabolism, integer programming, elementary mode analysis. 1. Introduction Liver transplantation remains the best long-term option for the approximately 30,000 patients per year in the United States alone, while roughly 20% die of acute liver failure due to the shortage of organ donors. Extracorporeal bioartificial liver (BAL) devices employ primary hepatocytes or hepatoma cell lines to provide a whole complement of liver specific functions for the treatment of liver failure, but significant technical challenges remain to develop systems with sufficient function capacities. One key limitation is that, during BAL operation, hepatocytes are exposed to plasma from the patient and are prone to accumulate intracellular lipids and to depress liver-specific functions. It has been shown that pre-conditioning hepatocytes in low insulin plasma with supplemented amino acid dramatically improves the hepatic metabolism and increases liver-specific functions (Chan et al., 2003). Further manipulation of the culture environment is likely needed to allow hepatocytes to function at the levels necessary for productive BAL operation. Metabolic pathway analysis has become an important tool of bioinformatics and biotechnology. Without the knowledge of kinetic parameters, it is used to determine the maximal molar yield of biotransformation and as a guideline for reconstruction of metabolic networks based on special cell functions. One of the methods that have been proposed to analyze metabolic networks is elementary mode analysis (Schuster et al., 2000). The elementary modes correspond to the pathways connecting inputs to outputs and comprise a minimal set of enzymes allowing the mass balance for each intermediate metabolite at steady state. Any steady state flux pattern can then be expressed as a nonnegative linear combination of these modes to form a particular pathway. The first aim of this work is to identify the minimal reaction set by logic-based programming for six different cultured conditions after exposure to plasma and to investigate the effects of insulin, amino acid supplementation and hormone in metabolic

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network. We then use the elementary mode analysis framework to analyze the hepatic metabolic network and to investigate the relations between elementary modes and the results of logic-based programming. 2. Modeling and Computational Protocol 2.1. Logic-Based Programming For a metabolic network comprising M metabolites and N reactions, the material balances result in the following set of equations: dX ^ - Z ^ = ZSijVj,

i = lv..,M

(1)

where Xj is the concentration of metabolite i\ Sjj is the stoichiometric coefficient for metabolite i in reaction], and Vj is the flux of reaction j . The sum of the fluxes entering and exiting of the metabolic network can be assumed to be zero based on pseudo steady-state assumption (Schilling, 2000):

Xs,.v.=0,i = l,...,M

(2)

j=i

The metabolic network considered in this work is based on the model developed for cultured hepatocytes (Chan et al., 2003). It consists of 43 unknown fluxes and 34 measured fluxes (tryptophan uptake is also added), and 45 linearly independent mass balance equations. The unknown fluxes are calculated by using the least-square method of minimizing the square of errors of the measured fluxes. Based on this model we used mathematical programming ideas to develop the following optimization model in order to determine the minimal number of reactions required to maintain the mass balances of cultured hepatocytes. Expressing the presence/absence of reactions by logic 0-1 variables, problem (3) is obtained: min^ Subject to: ^SyV. = 0, i = 1,...,M, vpA^ < v^ < v^A^, j = 1,...,N

(3)

j=i

where X.^ is the logic variable that correspond to the value of 1 if the reaction is active and 0 otherwise; Vj is considered as variable between a lower and an upper bound v""", Vj"**"" , respectively; which are determined based on the available experimental conditions (Chan et al., 2003). The problem is modeled in GAMS and solved using GAMS/CPLEX as the optimization solver since the problem corresponds to Mixed Integer Linear Program (MILP/ 2.2. Elementary Mode Analysis One approach for finding qualitatively distinct pathways is to calculate the elementary modes (Schuster et al., 1994). The determination of the elementary modes in a given network depends on the classification of metabolites as internal or external and the reactions as irreversible and reversible.

Minimal Reaction Sets and Metabolic Pathways for Cultured Hepatocytes

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In this paper, the internal and external metabolites are chosen in a similar fashion as for the logic-based programming approach in the previous section. We thus consider 45 independent internal metabolites, which have to be balanced in any pathway, and 29 external metabolites, which are assumed to be unaffected by the reactions in the network, such as energy cofactors ATP, ADP, AMP and CO2, lactate, glucose, urea etc. We treat all of the reactions reversible except reactions 15 producing urea and reaction 69 producing albumin are irreversible. The dimension of the null space depends on the number of free variables in the original stoichiometric matrix (Schilling et al., 2000): dim(Null(S)) = N - rank(S)

(4)

For a frill rank matrix, the dimension of the null space is equal to the difference between the number of reactions and internal metaboHtes. In the hepatocyte network we consider, the dimension is equal to 32 (d = N - M) . Through the application of convex analysis, the solution of the steady-state eqn. (2) must lie in the nonnegtive orthant of the space spanned by all the reactions. Due to the nonnegtive constraints on the fluxes, the unique set of elementary modes is found. If the number of elementary modes is equal to the dimension of the null space, those modes are systematically independent. Because the elementary mode vectors, e^^^, are uniquely determined, any real flux distribution can be expressed as a linear combination of these vectors with coefficients as follows: where \ = {y^Y^^...) is the vector of fluxes of reactions. If all of the reactions participating in one mode are reversible, this mode is reversible and the corresponding weight parameter X can be of any sign whereas if one or more irreversible reactions are included in one mode, this mode is irreversible and in this case the weight X should be positive. As will be shown in the next section, elementary modes can be used to interpret metabolic frmctioning and identify the importance of pathways and reactions in the production of specific metabolites. 3. Results and Discussion The logic-based programming approach described in the previous section is used to estimate the minimum reactions required for maintaining the hepatic frinctions in different cultured conditions. In particular we investigate the following conditions for which experimental data are available (Chan et al., 2003): (a) high/low insulin preconditioned unsupplemented plasma cultures (HIP/LIP), (b) high/low insulin preconditioned with amino acid supplemented plasma (HIPAA/LIPAA), and (c) high insulin preconditioned with hormone supplemented plasma / low insulin with amino acid and hormone supplemented plasma ( H I P H / L I P A A H ) . The results shown in Table 1 illustrate the dependence of the minimal reaction set on the different cultured conditions. Table 1: Number of Minimal Reaction Set (MRS) under Various Cultured Conditions Number of MRS

HIP 60

LIP 51

HIP_AA 64

LIP_AA 64

HIP_H 56

LIP_AA_H 63

Combining logic-based programming with stoichiometric balancing can clarify the effects of insulin on hepatic metabolism. Figure 1 (a) shows that insulin inhibits gluconeogenesis (Flux no.2 to 6) predominantly by suppressing the expression of

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PEPCK (reaction 6) and G-6-Pase (reaction 1) enzymes. Specifically in HIP and HIPAA, the flux of reaction 6 is decreased to 0.372 and 2.05 compared with 0.865 and 2.17 in the LIP and LIPAA, repectively, and also no glucose is released out of the system in high insulin preconditioned plasma. Figure 1(a) also shows that amino acid supplementation significantly increases gluconeogenesis in HIPAA and LIPAA. The difference is that almost all of G-6-P is consumed to produce glycogen and no glucose is released under high insulin preconditionsing, compared to about 28 percent used to produce glucose in low insulin plasma. We also found that hormone addition plays an important role in different culture conditions. In high insulin preconditioned plasma, hormones act similarly to insulin, inhibiting gluconeogenesis and changing the direction of reaction 1 to glycolysis. However, in the low insulin preconditioned plasma supplemented with amino acid, hormones increase the gluconeogenic pathway and almost all of G-6-P is used to produce Glycogen. In summary it is found that low insulin preconditioning with amino acid supplementation is the best culture conditon for glucose production. Furthermore we determined that insulin and hormones have no obvious effect on the TCA cycle (flux no. 9 to 14) of the hepatic network, whereas amino acid supplementation increases significantly the TCA cycle (Figure lb). These results from logic-based programming are in agreement with the experimental data (Chan et al., 2003) in different cultured conditions. As Table 2 shows, fluxes throughout the urea cycle were significantly upregulated by amino acid supplementation, but the albumin synthesis rate decreased to the minimum value. These conditions can be further improved using multiobjective optimization that results in Pareto optimal solutions for urea and albumin optimization (Sharma et al., 2005). (a) (b) TCA Cycle

Gluconeogenic or Glycolytic pathway

•g

""^"

o Q.

A--A

•--•---HIP

A"--^

A

11 E = 1-5 •

& s •il

HIP_M

--*--HIP_H

1

A

E .2 • ^ S 11 -

UP_AA_H

1I 2 « ^

--~*~-UP_M_H

1

/

2

3

4

1

Q.

9-

i

0 -0.5

HIP_H

- - - • - - - LIP

5

2 c

i

9

10

11

12

13

14

7

Reaction number

Reaction number

-1

Fig.l Effect of insulin, amino acid and hormone supplementation on (a) Gluconeogenic/glycolysis pathway, (b) TCA cycle. Table 2: Urea cycle and Albumin Synthesis in the Minimal Reaction Set 15 16 17 69

Arginase Ureal Urea2 Albumin Syn

LIP 0.007 0 0 MAX

LIP AA 1.95 1.822 1.824 MIN

LIP AA H 2.201 2.021 2.033 MIN

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To provide a comprehensive pathway-oriented view of hepatic metabolism, the elementary modes were obtained using FluxAnalyzer (Klamt et al., 2003), which implements the iterative algorithm described by Schuster et al. (2000). Analysis of the hepatic metabolism results in 32 elementary modes. The shortest mode is mode 1, triglyceride storage, the central reaction of which does not include any internal metabolites. Mode 2 consumes glucose to produce glycogen. Modes 4, 14, 15, 16, 17, 18, 24, 25, 26 and 29 include the TCA cycle, serine-pyruvate-cysteine cycle and valinepropionylCOA-methionine cycle. One difference is that these modes include ketone bodies except mode 4. Another difference is that modes 4, 14, 15, 16, 17, 24, 25, 26, 29 include isoleucine uptake, palmitate uptake, cholesterol esterase, glycerol update, lipid, threonine uptake, lysine uptake, tryptophan uptake, and leucine uptake, respectively. Modes 6, 7, 8, 11, 22 and 31 include parts of the TCA cycle and valine-propionylCoAmethionine cycle. Modes 19 and 30 involve the complete electron transport system and gluconeogenesis, respectively. By considering the reactions to produce urea and albumin as irreversible, modes 3, 27, 28 involving the urea production (Figure 2) and mode 32 involving albumin productions become irreversible modes. Since any self-consistent flux distribution can be expressed as a non-negative linear combination of elementary modes, the minimal reaction network obtained by logicbased programming can be reconstructed using the elementary modes. The coefficients of the linear combinations for the different conditions are calculated minimizing the least square error of Eqn. (5). The weights of elementary modes 7, 21 and 22 are negative in all different conditions, which mean that these three pathways will be reversed in these specific conditions investigated in cultured hepatocytes. From the values of coefficients in different culture conditions (data are not shown), we also determined the relative importance of each pathway across the various experimental conditions. For example, modes 3, 27, and 28 involve urea production, so we can determine the importance of urea production by calculating the corresponding coefficients. The sum of these coefficients under the LIP condition is 0.0097; compared to a sum of 0.1853 for L I P A A which means that cultured hepatocytes with amino acid supplementation exhibit significantly greater urea production compared to those cultured in unsupplemented plasma. By comparing the different weight values of modes involving urea production in L I P A A conditions, we found modes 3 and 27 are more important to produce urea than mode 28. In addition, the most important reactions in urea production can be determined to be the inhibition of enzymes carbamoyl-P synthetase I, ornithine transcarbamylase, (reaction 16) argininosuccinate synthetase, and argininosuccinase (reaction 17). Using the results of this analysis, we can investigate the effects of enzyme deletion. For example, blocking fumarase or malate dehydrogenase enzymes will lead to disruption of the TCA cycle and elimination of 19 elementary modes, including all of the modes of 9 amino acid exchange fluxes and albumin production. On the other hand, inhibiting lactate dehydrogenase is likely less important, since it results in deletion of a single elementary mode that may not be essential since its metabolites (e.g. pyruvate) can be produced from alternative pathways. 4. Summary and Future Work In this paper we present an analysis of hepatocyte metabolism based on logic programming and elementary mode analysis. The proposed approach determines the minimal set of reactions required in various conditions, and the importance of individual enzymes. Future work involves the experimental verification of the results obtained as well as the implementation of metabolic control analysis in the hepatocytes network to

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YangetaL

determine the main branch points and compare with the results of the logic programming model. Arginlne

18 [

(

Ornithine

Argjnine

V

y ,

\

- t » /

n

Qucose-e-P

_54^''^

^^^

^

^

Ornithine

•

•

^

^

^

15

\[

CI~^^I>""

Urea

(a) mode 3

(b) mode27 ACAC 49 Beta-OHixityrate

1 |

Arginlne 18 1

1

Glucose

L<1—i

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^ ^ ^

ppp

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/ ^ " ^ ^

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(c) mode 28 Figure 2, Elementary modes producing urea; the red itaUc numbers are expressed as fluxes of reactions (value 1 is abbreviated)

References C. Chan, D. Hwang, G.N. Stephanopoulos et al., 2003, Application of multivariate analysis to optimize function of cultured hepatocytes, Biotenol. Porg. 19, 580-598 C. Schilling, D. Letscher et al., 2000, Theory for the synthemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective, J. theor. Biol. 203, 229-248 N.S. Wang, G. Stephanopoulos, 1983, application of macroscopic balances to the identification of gross measurement errors, Biotenol. and Bioeng., XXV, 2177-2208 N. Sharma, M.G. lerapetritou, M,.L. Yarmush, 2005, Novel quantitative tools for engineering analysis of hepatocyte cultures in bioarticial liver systems, Biotechnol. and Bioeng, 92(3), 321,2005 S. Klamt, J. Stelling et al, 2003, FluxAnalyzer: exploring structure, pathways, and flux distributions in metabolic networks on interactive flux maps, Bioimformatics, 9 (2), 261-269 S. Schuster, C. Hilgetag, 1994, On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst., 2, 165-182 S. Schuster, D. A. Fell and T. Dandekar, 2000, A general definition of metabolic pathways useful for syntematic organization and analysis of complex metabolic networks. Nature Biotechnol., 18,326-332

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Hybrid modular mechanistic/ANN modelling of a wastewater phosphorus removal process J. Peres^, F. Freitas^, M A M Reis^, S. Feyo de Azevedo^ and R. Oliveira^ ^Department of Chemical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal ^REQUIMTE, Departamento de Quimica, Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, P-2829-516 Caparica, Portugal The hybrid modular mechanistic/ANN technique was applied to a wastewater phosphorus removal process with simulation data. The hybrid model was able to describe accurately the process dynamics. With the expectation-maximisation identification technique, the mixture of experts (ME) network converged to a configuration where the two experts developed independent expertise in describing either the phosphorus release state or the phosphorus storage state, which is precisely the main metabolic switch occurring in this process. The final ME network provided a more structured representation of the underlying cellular kinetics. This resulted in higher accuracy and generalisation capacity of the final hybrid mechanistic/ME model. 1. I N T R O D U C T I O N The development of optimal control strategies for bioprocesses is frequently constrained by the availability of sufficiently accurate mathematical models for supporting such developments. Bioprocesses may be generically characterised as complex systems exhibiting non-linear and time-varying dynamics. The main source of complexity arises from the intra-cellular phenomena. Model-based bioreactor performance optimisation studies rarely incorporate detailed descriptions of the intra-cellular phase. The method is simply too "expensive" for routine application in the biochemical industries. It has been pointed out by several authors that hybrid parametric/nonparametric modelling techniques may represent a cost-effective alternative for the analysis of bioprocesses [1,2]. Hybrid parametric/nonparametric systems combine First Principles modelling with chemometric techniques for extracting knowledge hidden in process data. The most widely adopted hybrid structure for bioreactor systems combines macroscopic material and/or energy balance equations with artificial neural networks (ANN) such as the Multiple Layer Perceptron (MLP) or the Radial Basis Function (RBF) networks [3-6]. The job for the neural networks in these structures is the nonparametric modelling of unknown reaction kinetics, which is normally the most challenging part of the process to be modelled in a mechanistic sense. One important feature of living cells is the fact that they process different substrates through different metabolic pathways. Diauxic growth on two carbon sources

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is one of such examples. Another is aerobic/anaerobic growth, depending on the presence or absence of dissolved oxygen in the medium. For example, the S. cerevisiae yeast grows through three different metabolic pathways for exploiting energy and basic material sources and is able to switch between a oxidative metabolic state and a oxido-reductive metabolic state [7]. In the more complex example of mixed cultures, several different metabolic mechanisms may occur simultaneously. In Activated Sludge processes, 3 main families of bacteria are involved, each of them switching between metabolic mechanisms [8] yielding complex kinetic behaviour, viz. - nitrification/denitrification, aerobic/anaerobic, phosphorus accumulation/release states. The biological systems exemplified have inherent non-linear discontinuous reaction kinetics due to switching between metabolic mechanisms. The popular MLPs and RBFs networks have some limitations for approximating discontinuous input-output systems. [9] showed that modular neural networks, such as the ME [10], are well suited for the identification of processes that switch between different operating conditions. Such modular network architectures might be an interesting alternative for bioprocess hybrid modelling. A modular network architecture consists of two or more (small) network modules mediated by an integration unit, which decides how to combine their outputs to form the final system output and which modules should learn which training patterns [11]. This type of architecture performs task decomposition in the sense that it learns to partition a task into two or more functionally independent tasks and allocates distinct networks to learn each task [10]. The main objective of this work is to develop a hybrid modelling method for bioprocesses accounting for the intra-cellular modular structure of the cells subsystem. The combination of First Principles modelling with modular network architectures is thus explored. The remaining of this paper is organised in 4 more sections. In Section 2 a mechanistic/ME hybrid structure for bioreactor systems is described and the ME network architecture is briefly reviewed. In Section 3 the hybrid model employing ME networks is illustrated on a case study. Finally, in Section 4, the main conclusions are presented.

2. M E C H A N I S T I C / M I X T U R E OF E X P E R T S H Y B R I D M O D E L F O R BIOREACTOR SYSTEMS Hybrid model structures have been classified as parallel and/or serial [4,3]. In parallel structures a full mathematical model is available that however is not sufficiently accurate for model-based applications. A nonparametric modelling technique is then combined in parallel with the mathematical model, having access to the same input variables and correcting the mathematical model outputs. In the serial case, there is knowledge concerning the general structure of the system, but parts thereof are not known in a mechanistic sense. Such unknown sub-systems are modelled with nonparametric techniques, which feed with information to mechanistic parts. The hybrid model structures that naturally arise in bioreactor modelling problems tend to be simultaneously serial and parallel [12,13]. The bioreactor system was described by material balance equations while a parallel neural network/mechanistic structure represented the cell population system. In this work a similar structure is adopted that however uses a ME architecture for modelling the unknown reaction kinetics term instead of the usual MLPs or RBFs networks (see Fig. 1)

Hybrid Modular Mechanistic/ANN Modelling cell system

2^

fME 1 • r = KrmecV:>M£; (c, w) {- — •

1719 bioreactor system

= r (c) - Dc + u at

Figure 1. Proposed Mechanistic/Mixture of Experts hybrid structure

This model may be mathematically stated by the following two equations: dc r (c, w) — Dc + u dt c)<^M£;(c,w)^ r = Kr

(1) (2)

with c a vector of n concentrations, X a n x m yield coefficients matrix, r^ec a m x r matrix of mechanistically known kinetic expressions, (PME i^^^) ^ vector of r unknown kinetic functions modelled with ME networks, w a vector of parameters that must be estimated from data, D is the dilution rate, u is a vector of net volumetric input rates (possibly control inputs). 2.1. The mixture of experts architecture The ME architecture has strong statistical foundations since it was inspired by the concept of mixture models from Statistics field [14]. The ME architecture [10,11] consists of a set of K expert networks and a gating network (see Fig. 1). Basically, the task of each expert j is to approximate a function fj-.c-^ cpj over a region of the input space. The task of the gating system is to assign an expert network to each input vector c. The final system output (p^E i^ ^ combination of the expert network outputs ^ME — ^7=19j (^) ^j (^) " where QJ (C) are the gating outputs. The expert modules are simple linear functions for non-linear regression problems or linear functions with a single output non-linearity for classification problems. In some non-linear regression problems it may be necessary to use more complex non-linear experts. MLP networks with the tangent hyperbolic function in the hidden layers and linear functions in the output layer [15] was the expert network structure adopted in this work. Also, different forms for the gating system have been reported. The softmax function suggested initially by [10], is a normalised exponential function of the inputs c and provides a "soft" hyperplane division [16]. A localised gating system based on Gaussian functions provides more flexible "soft" hyper-ellipsoids input space partitions [16] and was adopted in this work.

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It can be formulated as follows - gj{c,a.) = a j P ( c , m^, X l ^ ) / ^ ^ ^ a i P ( c , m^, E^), with P(c,m^-,S^) - ( 2 7 r ) ~ ' ' / ^ | E ^ r ^ / ^ e a : p { - | ( c - m / ) ^ 5 ] - ^ ( c - m ^ ) } . P(c,m^-,E^) is a Gaussian distribution with centre m^ and covariance matrix Jlj (usually only the diagonal is considered). The expression ^'^(c, a) establishes a normalised gating output scaled by the scalar parameters aj. The variable a is a vectored representation of all gating system parameters a = {a^, m^, E^}. The parameters present in each expert and in the gating system were identified using the well known EM algorithm. The Reader is referred to the work of [17] for details of this technique. 3. R E S U L T S A N D D I S C U S S I O N A wastewater phosphorus removal process by activated sludge is addressed. The study was supported by the activated sludge model number 2d (ASM2d) described in [8]. The ASM2d model is a complex morphologically and intracellularly structured model accounting for the existence of 3 types of organisms. In this work, the model was simplified for aerobic phosphorus accumulating organisms (PAO) only. PAOs are facultative aerobic organisms with the capacity of adapting the metabolism for aerobic or anaerobic conditions. The wastewater treatment process is conducted in a sequencing batch reactor (SBR) operated through repeated batch cycles. Each SBR cycle has two main reaction phases (the other phases such as settling, medium withdrawing and replenishing with fresh medium are not relevant for this study). The first phase is the anaerobiose taking a total time of 40 minutes per cycle. This phase is immediately followed by the aerobiose that takes more 20 minutes per cycle. The transition between the anaerobiose and aerobiose is controlled by the aeration rate. The process state is defined by 6 extracellular state variables (concentrations of biomass {XpAo), dissolved oxygen {S02), readily biodegradable substrate (SF), acetate (SA), phosphate (SPOA) and slowly biodegradable substrate (Xs)) and the concentrations of 3 compounds stored intracellularly (stored poly-phosphate (Xpp), stored organic compounds (XPHA) and stored glycogen (XGLY))The main challenge in this case study is to develop an hybrid model capable of describing accurately the process state, but also to bear out if he proposed hybrid structure is able to learn to distinguish between the aerobic and anaerobic phases. A sequence of thirteen cycles was simulated varying the starting concentrations of state variables at the beginning of each cycle. The data was collected at sampling intervals of 1.5 min. The simulation data with noise added was divided in a training partition, comprehending 410 data records, and a validation partition with 123 records. The hybrid model employed has two experts (because the process has two metabolic phases) and a Gaussian gating system. The material balance equations are the following:

I =q(c) Xp^o

(3)

with c = [XpAo, S02, SF.SA, Spo4, Xs, Xpp, XPHA, XGLYV ^^^ Q ^^e respective specific reaction kinetics, q = [lJ^,qo2,qF,qA,qpo4,qs,qpp,qpHA,qGLYVThe only assumption regarding the cells system implicit in Eq. (3) is that the reactions kinetics are specific in respect to active biomass concentration (XpAo)- The job for the ME is thus to model the complex relationship between the kinetic rates (q) and the medium composition (c). The

Hybrid Modular Mechanistic/ANN

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Modelling

)HMiJ:

(a) > 60 O D)

^^ o w

\m 8 P

40 20

50 100 sample point

0

50 sample point

100

5000

4000

3000

1 P' < /

A

f

^ 50 100 sample point

50 100 sample point

50 100 sample point

Figure 2. Results for the validation partition: (a) Gaussian gating network outputs: gi {'' ')^ 92 (~) versus concentrations of 5'o25(o)-(b)-(i) concentrations estimates with a ME with 2 experts (198 parameters): measured values (o), estimated values (-).

experts were MLP networks with 8 inputs (all state variables except XpAo), 5 hidden nodes and 9 outputs corresponding to the specific consumption rates in vector q. The ME equations are the following: qi(c) = W2,i tank (wi,i c + bi,i) + b2,i

(4)

q2(c) ^ W2,2 tank (wi,2 c + bi,2) + b2,2

(5)

q = 9i{c) qi(c) + ^2(c) q2(c)

(6)

with q i the expert 1 outputs, q2 the expert 2 outputs, Wij and hij are MLP parameters, gi and ^2 are the gating system outputs, which are scalar quantities defining the relative contribution of expert 1 and 2 for the evaluation of the specific reaction kinetics q. The training method employed was the EM algorithm and the parameters were identified by cross validation. The modelling results for the validation partition are presented in Figs. 2(b-i). Models and measurements show a good agreement except in what concerns the concentrations of SA and Spo4- An important and most interesting feature of the ME was observed: the ME network was able to detect the switch between the anaerobiose

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and aerobiose as illustrated in Fig. 2(a), and the experts developed expertise in modelling the kinetics of the one or the other metabolic state. The switch between the experts occurs precisely at 0.66 hours in the transition between the anaerobiose and aerobiose. It has succeeded in all batches, either in training or validation batches. This feature was repeatedly observed in several other tests with other processes (not treated in this work). 4.

CONCLUSIONS

The main objective of this work was to explore the possibility of using complex modular network architectures for modelling cells reaction kinetics in a wastewater phosphorus removal process. This idea was motivated by the fact that the metabolism of cells consists of a highly complex modular network of metabolic pathways. As main conclusions it can be stated that hybrid modular mechanistic/MEs trained with the Expectation Maximisation algorithm are able to detect metabolic shifts with the individual experts developing expertise in describing the individual pathways. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

14. 15. 16. 17.

J. Schubert, R. Simutis, M. Dors, I. Havlik and A. Lubbert, Chem. Eng. Technol., 17(1) (1994) 10. R. Simutis, R. Oliveira, M. Manikowski, S. F. de Azevedo and A. Lubbert, J. Biotechnol., 59(1-2) (1997) 73. D. C. Psichogios and L. H. Ungar, AIChE J., 38(10) (1992) 1499. M. L. Thompson and M. A. Kramer, AIChE J., 40(8) (1994) 1328. G. Montague and J. Morris, Trends BiotechnoL, 12(8) (1994) 312. L. Chen, O. Bernard, G. Bastin and R Angelov, Control Eng. Pract., 8(7) (2000) 821. B. Sonnleitner and O. KappeU, BiotechnoL and Bioeng., 28(6) (1986) 927. M. Henze, W. Gujer, T. Mino, T. Matsuo, M. C. Wentzel, G. V. R. Marais, and M. C. M. Van Loosdrecht, Water Sci. Technol., 39(1) (1999) 165. B. Eikens and M. N. Karim, Int. J. Control, 72(7-8) (1999) 576. R. A. Jacobs, M. I. Jordan, S. J. Nowlan and G. E. Hinton., Neural Comput., 3 (1991) 79. S. Haykin., Neural Networks: A comprehensive foundation, Macmillan College Publishing Company, Inc., 1994. R. Oliveira, J. Peres and S. Feyo de Azevedo, M. Pons and J. F. M. van Impe (Eds), Computer Applications in Biotechnology 2004, Elsevier (ISBN: 0-08-044251X), (2005) 195. J. Peres,R. Oliveira and S. Feyo de Azevedo, M. Pons and J. F. M. van Impe (Eds), Computer Applications in Biotechnology 2004, Elsevier (ISBN: 0-08-044251X), (2005) 293. D. M. Titterington, A. F. M. Smith and U. E. Makov, Analysis of Finite Mixture Distributions, New York: Wiley, 1985. C. M. Bishop., Neural Networks for Pattern Recognition, Oxford University Press, 1995. V. Ramamurti and J. Ghosh, IEEE T. Neural Networ., 10(1) (1999) 152. M. I. Jordan and R. A. Jacobs, Neural Comput., 6(2) (1994) 181.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Modelling Morphological Change in Endothelial Cells induced by Shear Stress R.J. Allen^ D. Bogle^ and A.J. Ridley" ^ C O M P L E X , University College London, Wolfson House, 4 Stephenson Way, London, N W l 2HE, UK ^Department of Chemical Engineering, University College London, Torrington Place, London, W C I E 7JE, UK "Ludwig Institute for Cancer Research, Royal Free and University College School of Medicine, London W I W TBS, UK It is well known that following the onset of fluid flow over endothelial cells (ECs) they polarize and elongate in the direction of the flow. Here the cell is described using a continuum approximation and viewed as a passive object. This is the first step to a broader model incorporating a mathematical description of the signal transduction network of interacting molecules which govern the morphological change involving internal cellular structures such as the cytoskeleton. 1. Introduction One of the motivations for this research is to contribute to our understanding of the development of atherosclerosis. This leads to cardiovascular disease which in turn is the leading cause of death in the western world, [1]. The development of atherosclerosis in an artery is characterised by the deposition of lipids and accumulation of cholesterol rich macrophages, which eventually forms a complex 'plaque' in the artery wall. The early stages of atherosclerosis is characteristically deposition of cholesterol in the artery wall associated with lipoproteins, particularly low density lipoproteins (LDLs). LDLs accumulate and are retained in the vessel wall, triggering the ECs to initiate an inflammatory response and inducing monocytes in the blood to cross the layer of ECs lining the blood vessel and differentiate into macrophages, contributing to the forming lesion. These macrophages rapidly take up a modified form of the LDL, leading to the formation of 'foam cells'. Once the foam cells apoptose they deposit their lipid rich contents within the vessel wall. Smooth muscle cells also migrate into the lesion and proliferate there, adding to its growth and development. Heart disease and stroke can occur if a lesion ruptures into the artery. 1.1. E n d o t h e l i u m The endothehum consists of a monolayer of ECs that line the entire vasculature. The ECs are attached to each other by intercellular junctions that confer mechanical properties

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upon the endothelium. More importantly (in this context) these junctions are selectively permeable to materials (for example ions) and cells (for example monocytes) crucial to biological function. Certain points of the vasculature are more susceptible to atherosclerosis: areas of branching or curvature where the pattern of flow is complex are more at risk than tubular areas (where the flow is approximately laminar). Furthermore in the branched areas the endothelium has been shown to be more permeable to larger molecules (such as LDL) [1], which means these areas are more likely to develop atheroscelerotic lesions, [2]. It is well documented in in vitro studies that when endothelial cells are exposed to laminar flow they elongate and align with the direction of flow. Conversely, in regions of more complex flow in vivo the cells do not elongate or align with each other and the junctions between cells are more 'leaky', [1,3]. 1.2. Shear Stress Here we discuss an approach to modelling the passive response of the cell to the mechanical force the fluid flow exerts on the cell membrane. The cell also undoubtedly responds to fluid flow through the activation of mechanosensing receptors and subsequent signal transduction. However, the manner in which the signal from the applied shear stress is received by cells is currently unclear, although several different molecules have been implicated as sensors for the shear stress, [4]. For example, shear stress induces potassium channel activation and G protein activation (within the first minute of the application of shear stress), activation of the GTPases, RhoA, Racl and Cdc42, [4]. as well as mitogen activated protein kinase signalling and activation of transcription factor NF/cB (within the first hour, see [4,5]). These shear stress-activated responses contribute to cell shape change and altered gene expression. Internally the cytoskeleton is remodelled to bestow greater mechanical stiffness to the cell, [5]. This is achieved by induction of actin stress fibre growth and alignment in the direction of flow, [6,7]. 2. Modelling t h e Passive R e s p o n s e . In addition to activation of signal transduction pathways, fluid flow will induce cell shape change as a consequence of the mechanical force imposed on the cell. This is the effect of fluid flow that we address here. The general approach taken here is to model the flow of the blood in the artery via the non-dimensionalised Navier-Stokes equations which describe the velocity of the fluid u at a given point in time and space:

where Re is the Reynolds number of the flow and Fr is Froude number of the flow. This expression along with the continuity equation (V.u = 0), define the flow. The NavierStokes equations are valid for a homogeneous, incompressible Newtonian fluid. These assumptions are questionable when applied to blood, however for the region of interest here (i.e. near the arterial walls) they are more suitable since it has been widely shown that there is a thin layer of the flow that can be modelled as homogenous, and for similar reasoning it also approaches a Newtonian fluid at this point [8,9]. It also - in large arteries is well approximated by the incompressible case. The artery is treated as a simple cylinder. In this region Navier-Stokes can be solved (by imposing boundary conditions of no flow at

Modelling Morphological

Change in Endothelial Cells Induced by Shear Stress

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Side View

j

Top View

I I

Figure 1. Cartoons of both profiles of an endothelial cell. In the side profile the cell is sitting on a substratum (for example the arterial wall), in the top view the substratum is in a plane parallel (and below) the plane of the page

the arterial wall) to give (via experimental measurements of parameters such as common flow rates and typical artery diameter in common carotid arteries) an estimation - to an order of magnitude - of the Reynolds number as 10~^, and the Navier-Stokes equations (in the absence of a body force, b) can be reduced to the Stokes equation: V P = V^u

(2)

In order to calculate the Reynolds number an estimation as to the characteristic velocity of the flow has to be made. The cell is modelled as a two dimensional ellipse sitting above the arterial substratum as in the bottom representation in figure 1. This section is taken to be 1/im above the substratum, and the characteristic velocity is calculated as the flow velocity at this radius. A useful way of approaching the problem is the boundary integral representation (BIR) of equation (2). In physical terms the BIR expresses the total velocity as a sum of velocities resulting from a concentrated point force, with the sum constructed in such a fashion that the boundary conditions are satisfied. By modelling the interior of the cell as a fluid with the same velocity as the exterior fluid the BIR becomes particularly simple, [10]: ^,(xo) = 2/^(xo) - ^

^ A/,G,,(x,xo)dl(x)

(3)

here Gij^K, XQ) is the free space Green's function, u°° is the flow far away from the interface (here we non-dimensionalise by this characteristic velocity so that we can take u ^ as a unit vector in the a;-direction), Af = {a^ — (j^).n is the discontinuity in the interfacial surface force (cr^ and cr^ are the stress tensors for each of the fluids and n is the normal to the membrane) and the dl is an element of the boundary defined by c. In order to close the problem assumptions are needed on the nature of the interface between the two fluids. Here we choose some of the simplest assumptions, namely, that it is an incompressible, inextensible material with an isotropic surface tension, r, acting

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parallel to the 1-D boundary. In two dimensions these assumptions lead (see [10] )to the expressions:

->-V

(^)

where t is a vector tangent to the interface, and dl is a differential element of the interface. This expression is from balancing the forces on the differential element, dl. The second expression is:

'•£ = «

,5,

A finite element method is used to describe the two dimensional boundary (i.e. the cell membrane) and equation. The strategy for evolving the boundary is to solve for the velocity on the boundary at every time step. In order to do this expression (4) is substituted into equation (3) and the resulting expression is substituted into (5). Once the problem has been discretised over the boundary elements this leads to a system of linear equations which can be solved for the tension r:

here the square brackets denote averaging (by integrating along the element) over the zth boundary element. Then the tension of each of the elements can be substituted into a discrete version of equation 3 to solve for the velocity of the boundary elements which can be updated at every time step. Here the boundary initially is an ellipse of a non-dimensionalised height relative to the cell's length perpendicular to the fluid flow. The nodes defining the boundary elements are restricted to move perpendicular to the interface, and every node is updated at every time step via: xr^=x|+<5i(ui.ni)

(7)

where n is the normal to the membrane, 6t is a small time step (here 0.005 time units was used as the time step. Recall that this is a non-dimensionalised time.) and x^ is the position of the zth node at the j t h time step. The number of nodes taken was 150, which through experience was found to give sensible solutions. To fix the cell, seven nodes were fixed (the apex nearest the oncoming flow and three either side of this node). 3. Results An example of the results of this model can be seen in figure 2. In an attempt to compare the model with reality it is necessary to re-dimensionalise. But given that the ellipsical cell is an idealisation (both in terms of actual shape and as a 2D representation) it makes little sense to compare with real cell dimensions after the onset of shear stress. At this stage of the model it makes far more sense to compare the time-scales of the processes. Re-dimensionalising gives that one time step of 0.005 units corresponds to 5 seconds. A stable, repolarised state is reached after 450 steps, which is roughly 20 minutes. This is viewed as a positive result, it is certainly on the same timescale that cytoskeleton is reported to be restructured [5] and comparable to the hour time scale reported for polarization by Stothard and Ridley [4].

Modelling Morphological

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s==100

Figure 2. Simulation results. The initial shape of the cell is in the lighter shade.

4. Discussion A model has been constructed to investigate the endothehal cell response to fluid flow. Current lines of research are into the how the cytoskeleton is restructured, the regulators of this change and how the signal is transduced to the regulatory metabolic network. The model indicates that is plausible that passive response to the force the fluid exerts on the cell could also play an important role in governing morphological change. Given the complex nature of the mechanical properties of the cell membrane (for example the rigidity the cytoskeleton confers, the multiple points of adherence to the substratum etc.) the model is clearly a gross oversimplification, aside from this there is also the organised modification of the cytoskeleton that isn't yet considered in the model. But, the constructed model does demonstrate feasibility of the method - and gives a pleasing result regarding the time-scale of the morphological change. One candidate for sensing the shear stress is thought to be the focal adhesions points, the hypothesis being that tension - transduced by the cytoskeleton - is somehow sensed by a 'mechanosensor', and this may well be the case. The results of the model here lead to a tentative suggestion of another possibility: the regulatory network could be activated by cell shape change itself. The longer term objective of the work is to develop a multiscale model linking the morphological change of the cells to a model of the cytoskeleton and to a model of the cellular metabolism. Changes in cell shape are known to affect the metabolism and hence the function of the cells which affects on athersclerosis. REFERENCES R. Ross, 1999, Atherosclerosis -An Inflammatory Disease, Mechanism of Disease, The New England Journal of Medicine, 340, 2, 115-126. A.L. Hazel and T.J. Pedley, 2000, Vascular Endothelial Cells Minimize the Total Force on Their Nuclei, Biophysical Journal, 78, 47-64. S. Weinbaum, G. Tzeghai, P. Ganatos, R. Pfeffer and S. Chien, 1985, Effect of cell

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5.

6.

7.

8. 9. 10.

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turnover and leaky junctions on arterial macromolecular transport. Am. J. Physiol. 248, H945-H960. B. Wojciak-Stothard, A.J. Ridley, 2003, Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases, Journal of Cell Biology, 161, 429-439. P.F Davies, K.A. Barbee, M.V Volin, A. Robotewskyj, J. Chen, L. Joseph, M.L. Griem, M.N. Wernick, E. Jacobs, D.C. Polacek, N. Depaola and A.I Baraket, 1997, Spatial Relationships In Early Signalling Events of Flow-Mediated Endothelial Mechanotransduction, Annu. Rev. Physiol. 1997, 59, 527-49. K.G. Bimkov, A.A Birukova, S.M. Dudek, A.D. Verin, M.T. Crow, X. Zhan, N. DePaola and J.G.N. Garcia, 2002, Shear Stress-Mediated Cytoskeleton Remodeling and Cortactin Translocation in Pulmonary Endothelial Cells, Am. J. Respir. Cell Mol. Biol. 26, 453-464. A.M Malek and S. Izumo, 1996, Mechanism of endothelial cell shape change and cytoskeletal remodeling in response to fluid shear stress, Journal of Cell Science, 109, 713-726. R.D. Kaum, 2002, Cellular Fluid Mechanics, Annu. Rev. Fluid Mech., 34, 211-32. T.J. Pedley, 1980, The Fluid Mechanics of Large Blood Vessels, Cambridge University Press, Cambridge U.K. H. Zhou and C. Pozrikidis, 1995, Deformation of Liquid Capsules with Incompressible Interfaces in Simple Shear Flow, Journal of Fluid Mechanics, 283. 175-200.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Disturbance propagation and rejection models for water allocation network Xiao Feng*, Renjie Shen Department of Chemical Engineering, Xi'anJiaotong University, Xi'an 710049, China Abstract In this paper, a system modeling methodology for characterizing disturbance propagation is proposed to predict the worst scenario to occur in a water allocation network during synthesis. Then a model-based design procedure for suitable mixing for disturbance rejection is introduced. A case study is given to show the efficacy of the methodology. Keywords: Water allocation network; process modeling and design; disturbance; propagation and rejection. 1. Introduction When a water allocation network is highly integrated, the resultant plant will be structurally more interacted among process units. If a water network is improperly designed, its operation may be unstable or even uncontrollable regardless of the advances of control techniques (Yan and Huang, 2002). Up to now, all the work about optimal water allocation network is based on designing the network for nominal operation conditions and no work on operational aspect, for example, flexibility or controllability, of water networks has been reported. On the other hand, in the synthesis of mass exchanger networks, some methodologies for system flexibility or controllability analysis have been proposed (Papalexandri and Pistikopoulos, 1994; Yan and Huang, 2002; Yang et al, 1999; Karafyllis and Kokossis, 2002). A water allocation network, although sometimes at some extent can be treated as a mass exchange network, has some different features from a mass exchange network as follows. (1) The quality of water may not be affected by a target pollutant, but characterized by some other parameters, e.g., PH, hardness, and turbidity (Huang, et al, 1999). (2) Not all water-using processes can be described with the available models for mass exchangers, for examples, filtration, centrifugal separation (Huang, et al, 1999). (3) When designing or analyzing a water network, people are more interested in the parameters about water rather than about process streams. Although the original disturbances come from some process streams, the target parameters to be controlled are usually some inlet and outlet contaminant concentrations of water streams. This paper is mainly focuses on the development of an analytical tool for evaluating structural disturbance propagation and disturbance rejection in a water allocation network.

All correspondence should be addressed to Prof Xiao Feng (phone: 86-29-8266-8980; FAX: 86-29-8323-7910; e-mail: [email protected])

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2. Process structural related disturbance propagation model 2.1. Unit based model In a water system, the original disturbance usually comes from some process streams. Then the disturbances will be propagated to some water streams with which the process streams transfer contaminants. Further, the water discharged from these processes will propagate disturbance to the downstream processes. When designing or analyzing a water network, people are more interested in the parameters about water streams rather than about process streams. Therefore, the disturbance propagation in process streams should be translated to water streams. Any disturbance from a process stream will lead to a variation of the mass load to be transferred to the water stream in the process. For example, the variation of contaminants mass transfer load for a mass transfer water-using unit is as follows:

dm = dM,{Cf -C,"^) + M,{SC',^ -8Cf')

(1)

Therefore, when a disturbance from the process stream is given and the process is specified, the variation of contaminants mass transfer load, 5m, can be calculated. In this way, any disturbance original from a process stream can be expressed by the corresponding variation in the mass transfer load of contaminants in the process, which in turn is the original disturbance of the water allocation network. Then from the contaminant balance, the disturbance propagating to the water outlet concentration from 6m will be the second term in the right side of Equation (2). Then if the water from the disturbed process is reused in the downstream processes, it will affect the water inlet concentration in the process so that the disturbance will be propagated in the whole water network. When there is ^ in the inlet concentration of water of a process, it will influence the operation of the process and cause disturbances of the outlet concentration of the water stream. The variation of the outlet concentration caused only by the variation of the inlet concentration is the same as the variation of inlet concentration of the water stream. Therefore, the outlet concentration disturbance of water in a process caused by the variation of the contaminant transfer load and the variation of the inlet concentration can be expressed as In a water allocation network, addition to water-using processes, there are some water mixers and water splitters (Yang et al, 1999). For a mixer, the fluctuation of flow rate and that of concentration of the output stream after mixing can be calculated as follows. m

/

1

mi

mi

m

'=' (3) For a stream split to n branches, the fluctuation of the outlet concentration of each branch is the same as that of the stream before splitting. 2.2. System disturbance model To identify an optimal water allocation network, a superstructure for the network is to be used, in which the output water stream from any process can be partially or entirely reused by other processes (Savelski and Bagajewicz, 2001). Therefore, at the inlet of every water-using process, there is a mixer and at the outlet of every water-using process, there is a splitter. If there are N water-using units and one freshwater stream.

Disturbance Propagation and Rejection Models for Water Allocation Network

1731

then there are N potential water reuse streams. Then, the mass balance at the inlet of each water-using process unit should be considered as follows.

J=o

>=i

(4)

(5)

The target parameters to be controlled are usually some inlet and outlet concentrations of water streams. Because every inlet water stream is the outlet water stream from a certain mixer, the fluctuations of the target inlet concentrations about all N processes can be predicted by the following model.

J=' ^^

(6)

can be determined by

For new design, such superstructure will be set up, while for existing network, the network structure is specified, but the concentration variation will be determined by using the above equations. 3. Disturbance rejection model with optimal mixing Mixing some water at lower concentration with the disturbed inlet water stream to a strict controlled process can easily reject the disturbance. Then source disturbances {Sm^ can be rejected through manipulating mixings {SMwt) so that the fluctuations of the target inlet concentrations can be within a certain range. Mathematically, their relationship can be described as

^f'=^f-^, ^f^"'=^f^^-A, (M^, + 5M^,). (Cf ^'^Cf') = m, + Sm, SM,=j;^SM.^,

(8) (9) (10) (11)

7=0

The objective fimctions of the design are the minimum freshwater consumption at normal operation conditions and the minimum freshwater addition for control at disturbances. For the control of an existing network, the objective function is minimum freshwater addition for control at disturbances. Therefore, the optimization problem is a NLP programming.

4. Case study Table 1 gives the limiting water data, the mass load of contaminants, introduced source disturbances, and the permissible target concentration ranges.

X. Feng and R. Shen

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Process No.

m (kg/h)

Table 1 Water data f-^OUT dm ^Max (kg/h) (ppm) (ppm) 25 80 0.2 f~fIN ^Max

Qr^OUT

(ppm) 1

(ppm)

1

2.0

2

2.88

25

90

0.29

1

6

3

4.0

25

200

0.4

2

10

4 5 6 7 8 9 10

3.0 30.0 5.0 2.0 1.0 20.0 6.5

50 50 400 200 0 50 150

100 800 800 600 100 300 300

0.3 3 0.1 2 0.65

4 5 40 0 4 10

6 20 4 20 20

25 L(25^

14.5 (13.5)

(31.4)

11.2 (12.0)

•

25.4 (26.4) 10 rio.6)

38.6 (39.3;L

IT

I 13.9 (13.8) 166.3 (170.1)1

5

16.5 (17.3)

26.9 K30.0) J 2

ro.8) 1.9 (1.9)

23.4 (6.8) (25.0)

(10) 1.9

mi

166.3 171.1)

21.7 (23.2)

10 10 (10.6)

9.5 (10.1)

21.7 (21.5)

0.2

I

SMI 20.3 (19.2)

Fig. 1 Solution A Two network design alternatives are available, both of which reach the freshwater target under normal operation conditions. Solution A is synthesized by Savelski and Bagajewicz (2001), as shown in Fig. 1, and Solution B by the authors featured with simplest structure (Fig. 2). In both the figures, the numbers outside the brackets are the water flowrates at normal operation conditions, and the numbers inside the brackets are the water flowrates at the worst disturbance scenarios. The computed target concentration ranges of each process for each solution are summarized in Table 2. It can be found that both of the solution alternatives cannot meet the control requirements. The freshwater consumption target under disturbances can be determined as 170.1, that is, the minimum freshwater increment, AMwo^'^-^-

Disturbance Propagation and Rejection Models for Water Allocation Network

1733

The next step is to control the fluctuation within the permissible range by mixing some water with lower concentrations. No new pipe is allowed. The adjustment schemes of the two alternatives are also shown in Fig. 1 and 2, respectively. It can be found solutions A can reach the freshwater target but Solution B cannot. Therefore, from the freshwater consumption point of view, solutions A is better. 32

mi

1 k-.

17.1 (17.9)

2 U

40

30 (31.8)

25 (25.9)

e

166.3 32 (33)

5.7

10

10 0.3 |rio.6) 4.0 1(1.3) (0.8) 0.7 (2.9) 18.8 r23.2\

166.3

10 26.8 (28.0)

10 (10.6)

23.1 25 (21-5) (25.9)

•

15 (14.1) Fig. 2 Solution B Table 2 Comparison of the target concentration fluctuation

scf

scf

^f

sc^

SC^

<3

<3

<5

<5

<20

<5

<15

0

0

<2

<4

<5

<20

<5

<9

SC

scr

scr

SC^""'

scr

scr

scr

A

<8

<9

<20

<10

<10

<30

<30

<3.8

B

<8

<9

<20

<9

<10

<30

<29

<4.9

sc

SC{^

A

0

B

Q^OUT

AMm

5. Conclusion Disturbance propagation can be explored by the system disturbance propagation model introduced in this paper, so as to predict the controlled concentration fluctuations. Total disturbance rejection at the level of structure can be achieved by optimal mixing. In this paper, the disturbance rejection model with optimal mixing is also introduced. The objective function in the methodology proposed in this paper is freshwater consumption both under normal operation conditions and under disturbances. When considering capital cost and control complex, a model can be established easily based on the model in this paper.

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Acknowledgment This work is supported by National Natural Science Foundation of China (20376066 and 20436040), and the Major State Basic Research Development Program of China (2003CB214500). The authors are also grateful to the U.S. ACS-PRF for the Summer Research Fellowship as part of a grant to Prof. Yinlun Huang at Wayne State University.

References Huang, C.-H.; Chang, C.-T.; Ling, H.-C; Chang, C.-C. 1999, A mathematical programming model for water usage and treatment network design. Ind. Eng. Chem. Res. 38, 2666-2679. Karafyllis, I., Kokossis, A., 2002, On a new measure for the integration of process design and control: the disturbance resiliency index. Chemical Engineering Science, 57, 873-886 Papalexandri, K.P.,and Pistikopoulos, E.N., 1994, A multiperiod MINLP model for the synthesis of flexible heat and mass exchange networks, Comp Chem Eng, 18, 1125-1139 Savelski, M.J., Bagajewicz, M.J., 2001, Algorithmic procedure to design water utilization systems featuring a single contaminant in process plants. Chemical Engineering Science, 56, 18971911 Yan, Q.Z. and Huang, Y.L., 2002, A disturbance rejection model for designing a structurally controllable mass exchanger network with recycles. Trans IChemE, 80, Part A, 513-528 Yang, Y.H., Yan, Q.Z. and Huang, Y.L., 1999, A unified model for the prediction of structural disturbance propagation in mass exchanger networks. Trans IChemE, 77, Part A, 253-266

Nomenclature C = concentrations of contaminant M = massflowrate m = mass load of contaminant. N = number of water-using processes in the system A = control correction vector for outlet concentration S = change e = control correction vector for inlet concentration Superscripts IN = inlet OUT = outlet = after controlled by mixing Subscripts / = process i j -^i = from process j to process i, m = mixer Max = control precision requirement P = process stream W = water 0 = freshwater.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Energy planning in buildings under uncertainty in fuel costs: The case of a hospital in Greece. George Mavrotas*, Kostas Florios, Paraskevas Georgiou Laboratory of Industrial and Energy Economics, School of Chemical Engineering National Technical University of Athens, Zographou Campus, Athens 15780, Greece "^e-mail: mavrotas@chemeng. ntua.gr

Abstract Hospitals are among the largest energy consumers in the tertiary sector where energy planning may greatly facilitate investment decisions for efficiently meeting energy demand. In this case study the energy rehabilitation of a hospital is examined and due to the recent conditions some new energy alternatives are investigated: a CHP unit for providing power and heat, an absorption unit or/and a compression unit for providing cooling load. The basic innovation of the above mentioned energy model is that it is designed to handle uncertainty in the most uncertain parameters of the objective function, namely, the fuel costs and the interest rate used for the investment discounting of new units. These uncertain parameters are modeled as Triangular Fuzzy Numbers (TFN) and the MILP model is appropriately transformed into a multiobjective MILP model. The solution of the multiobjective model provides the pareto set which means the candidate designs under the existent uncertainty. Keywords: Multiobjective, MILP, Fuzzy optimization, Utility systems 1. Introduction In the present case study the energy rehabilitation of a hospital is examined and due to the new conditions some new energy alternatives are investigated: a CHP unit for power and heat and an absorption or/and a compression unit for cooling loads. The superstructure of the energy network is created and a MILP model is used for the optimization as in [1-2]. The hospital's energy demand is divided in heat, electricity and cooling load, and for each one of them we use hourly data for a typical day of every month. The modeling of demand is performed through the linearization of the typical daily profiles for each one of the three loads in every month. In order to reduce the dimensionality of the model, we examine the possibility of grouping the months into seasons and using approximations of the daily profiles dividing the typical day into periods with constant load. These approximations proved to be successful as the obtained solutions are very close to the results of the fiill model (with 24 periods per day and 12 months). The main innovation of the above mentioned energy model consists in uncertainty treatment for the most "volatile" coefficients of the objective function, namely, the fuel costs and the interest rate that is used for the discounting of the investments. In order to describe the uncertainty in the objective function coefficients we use elements from fuzzy set theory. Namely, Triangular Fuzzy Numbers (TFNs) are used, aiming at expressing the possibility distribution (i.e. the shape of the membership function) of the

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uncertain parameters. Using elements from Fuzzy Optimization, the objective function with the frizzy coefficients is transfr)rmed into three objective frinctions with crisp (real number) coefficients (see e.g. [3]). The resultant Multiobjective MILP model is subsequently solved using the e-constraint method appropriately coded in the GAMS environment. The results provide the efficient solutions of the problem which are in fact the candidate solutions for the design of the hospital's energy system. 2. Methodology The incorporation of uncertainty in the modeling procedure has been among the most developed research areas in Mathematical Programming (MP). Among other techniques of handling uncertainty, frizzy sets are recently concentrating significant research efforts [4-7]. In particular, the term Fuzzy Programming (FP) is used for those problems where imprecise parameters can be represented by frizzy numbers. In many cases the uncertain parameters exist in the objective fiinction and they are modeled as TFNs n

max^c.x. =z

(1)

where ?• are TFNs, defined as C- =(c/7, Ci2, Cis). In this case, according to the algebra of frizzy numbers, the value of the objective frinction is also a TFN, z = (zj, Z2, Z3). In order to determine the (frizzy) optimal solution, the TFNs associated with different feasible solutions, must be compared to each other. In related bibliography, there are various methods in order to compare frizzy numbers [5-8]. In the present study the comparisons rely upon the Ramik-Rimanek's definition of inequality between triangular frizzy numbers [8]: For two TFNs: 7V=(ni,n2,n3) and O = (01,02,03) it holds that: N < 0 if and only if Ui < Oi and n2 < 02 and Us < 03. A solution is considered as frizzy optimal solution to problem (1) if there is no other feasible solution with higher objective frinction's value. In most cases there are more than one frizzy optimal solutions. Therefore, in order to determine the optimal solutions the following multiobjective problem with crisp parameters is derived [3,5]: n

max (zj = X^/i^/' /=i

n

n

^2 = Z^/2^/. ^3 = Yj^n^i) /=i

(2)

/=i

From the definition of the efficient solution in multiobjective programming (see e.g. [9, 10]) and the Ramik-Rimanek's definition of frizzy inequality, it is implied that the set of frizzy optimal solutions of the FP problem (1) is the same with the set of efficient solutions of the multiobjective problem (2). In order to determine these frizzy optimal solutions of an FLP problem with frizzy objective frinction's coefficients, it is sufficient to generate the efficient solutions of the equivalent multiobjective problem as described above. In this case study, the efficient solutions (pareto set) are produced using the econstraint method [9,10]. Briefiy, the e-constraint method isolates one objective frinction and uses the others as constraints varying parametrically and systematically their right hand side (RHS) parameters.

Energy Planning in Buildings Under Uncertainty in Fuel Costs

1131

3. The case study The superstructure of the hospital's energy system is shown in Figure 1.

HTCHPDM

Heat Demand

HTBLAB

ABS

NG

Cooling Demand Electricity sales

PPC

o

Electricity Demand

Figure 1: The superstructure of the hospital's energy system. The new energy alternatives comprise a Compined Heat and Power (CHP) unit, a new compression (CMP) unit and an absorption (ABS) unit. The names in the arcs have three components that indicate the energy carrier the leaving process and the entering process (e.g. 'HTCHPAB' indicates the flow of heat 'HT' from the 'CHP' unit to the absorption unit 'AB'). The cost of natural gas and electricity from the grid (PPC) are expressed as TFNs with values (25, 27, 35) €/MWh and (90,116,135) €/MWh respectively. The price of the electricity from CHP that can be sold to the grid is considered also as a TFN with value (50,56,60) €/MWh. Although the costs and the prices can be considered as correlated parameters, in the specific example we assume that, for the specific ranges, the relative TFNs are uncorrected. Methods for handling correlated fuzzy numbers can be found in [11]. The interest rate used as discounted factor for the investments is expressed as a TFN with value (5%, 8%, 12%). The power demand is given in the form of hourly data (heating load, electricity, cooling load in kW) for one typical day of each month. So, in total there are 12x24 = 288 data for each one of the three loads. In order to reduce the size of the resulted MILP model, we form representative seasons of year and periods of day for which the power demand was assumed constant (pseudosteady state). Concurrently, in order to explore the effect of the model compression to the final results, besides the full 12x24 model, we solve a representative 6x12 (=6 seasons and 12 intra-day intervals) and a simplified 3x4 model. The data compression is carried out in two phases: firstly, a grouping of the months in 3 or 6 seasons is performed and then the typical day of each season is subdivided into

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periods of constant load. The approximations were performed having in mind to adequately represent the maximum demand and required energy implied by the daily profiles. 3.1. Model Building After the creation of the energy network superstructure, a MILP model is formulated for the optimization. In order to maintain the linear characteristics of the model, we use piecewise linear approximations for the characteristics of the equipment as a fiinction of its capacity (e.g. CHP unit is considered to have different power to heat ratio above a specific capacity). The model is formulated as a multiperiod synthesis and operational problem according to the guidelines proposed in [2]. The objective ftinction of the energy model is to minimize the annual cost that incorporates the annualized investment cost and the annual operational cost. Due to the ftjzzy parameters, three objective fimctions are formed according to (2). The decision variables of the model are continuous (that indicate the energy flows and the capacities of the equipment) and binary (that indicate the existence or not of a unit or the operation of a unit during a time period). The main parameters are cost data, demand data, and operational data (efficiencies, lower operational points etc). The model involves the following constraints: • Energy Balances: The energy balances for the CHP, absorption, compression and boiler are taken into account for every period. • Equipment Capacity: The output of each unit in every period is upper bounded by its installed capacity, whether the latter is a parameter (i.e. for existing units) or a variable (i.e. for new units). • Technical Minimum: The output of each machine in every period is lower bounded using a fraction of the nominal capacity of the equipment (usually 30%). • Demand satisfaction: The quantities of heat, electricity and cooling load produced and directed to the demand side in the network must frillfil the corresponding demands in every period of operation. • Reserve margin for cooling load: The sum of capacities of the compression and the absorption unit has to be 20% greater than the annual hourly peak in cooling load. • CHP modeling: The CHP is modeled using two size domains: from 50 to 150 kW and from 151 to 400 kW with different technical characteristics (i.e. efficiency and power to heat ratio). It must also be noted here, that the case of different costs and prices of electricity for specific day periods (peak, off peak) can be also incorporated in the model by inserting new variables and rearranging some constraints. Nevertheless, this was not the case here where a uniform value was assigned to these parameters. The 3x4 model has 234 constraints, 241 continuous and 40 binary variables, while the respective figures are 1254, 1321 and 220 for the 6x12 model and 4926, 5209 and 868 for the fiill model (12x24). 3.2. Results The problem is solved using the GAMS modeling language and the XPRESS solver for MILP. The e-constraint method for generating the efficient points was implemented using GAMS code. The range of each one of the remaining two objective functions as obtained from the payoff table was divided using five equidistant points (25 runs in total

Energy Planning in Buildings Under Uncertainty in Fuel Costs

1739

for each model). The objective function values and the values of the key decision variables from representative efficient solutions obtained from the three models are presented in Table 1. The representation of the pareto set was generated in 30" for the 3x4 model, in 4'20" for the 6x12 model and in Ih 51' 32" for the 12x24 model. From Table 1, it can be seen that in all configurations both the compression and the absorption unit should be installed. The range of the nominal capacities of the devices is also derived from the efficient solutions. Comparing the models, we can observe that they have small differences in the objective function values. However, regarding the design variables, the differences are more significant especially between the simplified model (3x4) and the rest. Specifically the simplified model underestimates the sizes of the CHP and the absorption unit mainly due to the inevitable smoothing of the peak values in the electricity and the cooling demand. Table 1: Representative efficient solutions derivedfromthe three models. zl (€)

z2 (€)

z3 CHP ABS COMPR Electricity ( M W h ) NG(MWh) (kW) supply supply sold (€) (kW) (kW)

239,295 237,165 238,029 245,409

265,701 266,351 265,897 270,045

342,988 350,242 346,849 340,063

305 207 278 245 170

157 157 157 157 157

222 222 222 222 222

0 6.2 0 0 55

933 426 791 627 0

8094 7538 7927 7761 6998

6x12 240,382 242,765 240,578 241,140 248,765

273,100 269,455 271,623 270,376 273,309

365,271 347,765 360,088 354,902 344,531

373 211 325 273 178

283 214 247 247 163

178 203 191 191 222

0 6 0 0 36

1161

8460 7653 8252 8039 7149

12x24 239,347 247,439 241,044 240,136 239,395

271,875 272,054 268,227 269,354 271,439

363,263 343,714 347,782 353,540 361,781

375 186 226 278 360

293 195 241 293 282

197 233 216 197 201

0

1138

27.7

35 512 759

3x4 236,685 267,040 353,635

0.5 0 0

429 971 761 0

1084

8363 7153 7702 7972 8307

4. Conclusions The uncertainty in the objective function coefficients can be effectively tackled with Fuzzy Programming. Transforming the model with fuzzy parameters to a multiobjective problem provides the analytical tools to solve it. The generation of an adequate representation of the Pareto set can be effectively performed using the e-constraint technique which is appropriate for multiobjective MILP models. Regarding the specific case study, we observed that the simplification of the model using data compression is really possible. However the comparative analysis between the full model and the simplified versions reveal that differences may occur in the values of the design variables.

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Acknowledgements The second author is partially supported by the grant 4160 of the Greek State Scholarships Foundation (IKY). References 1. S. Papoulias and I.E. Grossmann, Computers & Chemical Engineering 7 (1983) 695 2. R.R. Iyer and I.E. Grossmann, Computers & Chemical Engineering 22 (1998) 979 3. G. Mavrotas, H. Demertzis, A. Meintani, D. Diakoulaki, Energy Conversion & Management 44 (2003) 1303. 4. M.L. Liu and N.V. Sahinidis, European Journal of Operational Research 100 (1997) 142 5. H. Rommelfanger, European Journal of Operational Research 92 (1996) 512 6. M. Delgado, JL.Verdegay and M.A.Vila, Fuzzy Sets and Systems 37 (1990) 33 7. Y.J. Lai and C.L. Hwang, Fuzzy Sets and Systems 49 (1992) 121 8. J. Ramik J and J. Rimanek, Fuzzy Sets and Systems 16 (1985) 123 9. R.E. Steuer Multiple Criteria Optimization Theory, Computation and Application. 2nd edition. Malabar FL: Krieger, 1989. 10. M.Ehrgott and M. Wiecek, Multiobjective Programming in: J. Figueira, S. Greco, M. Ehrgott (Eds) Multiple Criteria Decision Analysis, p. 667-722, Springer, 2005. 11. M. Inuiguchi, J. Ramik and T. Tanino, Fuzzy Sets and Systems 135 (2003) 123.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubHshed by Elsevier B.V.

Modelling an Electricity Infrastructure as a MultiAgent System — Lessons Learnt from Manufacturing Control Koen H. van Dam, Michiel Houwing, Zofia Lukszo and Ivo Bouwmans Section Energy and Industry, faculty of Technology, Policy and Management Delft University of Technology, Jaffalaan 5, 2628 BX. Delft, the Netherlands Abstract To model the control of an electricity infrastructure incorporating domestic level combined heat and power units (micro-CHP) a Multi-Agent System (MAS) approach is considered. This approach has already successfully been used to control manufacturing systems in the process industry. Because similarities between manufacturing and electricity generation exist it is interesting to investigate how a MAS methodology designed for manufacturing systems is applied to an electricity infrastructure. The interaction between energy companies and households is viewed here in a novel way, namely as a production process. By using an existing methodology for manufacturing control to design an agent-based controller for an electricity infrastructure, issues can be identified that have to be addressed in a control methodology specific for infrastructures. Keywords: multi-agent system, production control, electricity infrastructure, distributed generation, virtual utilities 1. Introduction Combined Heat and Power technology at the domestic user level (micro-CHP) is expected to pervade the electricity infrastructure on a large-scale and in that way will alter the method of electricity generation and supply. Su (2005), Newsborough (2004) and Microgen (2005), among others, describe the market diffusion potential and the technological characteristics of micro-CHP. The introduction of such a distributed electricity generation technology (DG) changes the way electricity networks have to be controlled and the way control is researched (Chambers et al, 2001; Jenkins et ai, 2000). Decision making in a distributed system can be seen as local optimisation within a feasible region determined by other decision makers. This can be done in a hierarchical, coordinated or cooperative way (van Dam et al., in print). The decisions made have an effect on the total behaviour of the system but it can be hard to predict this effect. Control models can help to identify these interactions and the overall effect of local decisions and they can be used to select the best control structure for a certain application. Studying and analyzing the dependencies between different levels of control is essential for improving control methods. To model the control of infrastructures a Multi-Agent System (MAS) approach (Wooldridge and Jennings, 1995; Wooldridge, 2002) is considered. MAS closely resemble the way those systems are organised; agents are autonomous, reactive to changes in their environment, they pro-actively pursuit their own goals and their social ability makes it possible for agents to adapt themselves to different organisational

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structures. Next to that, MAS are flexible, modular and they allow easy reuse of parts of a model. The MAS approach has already successfully been used to control manufacturing systems (aimed at the integration of planning, scheduling and processing) in the process industry. There are many similarities between manufacturing control and the control of infrastructures. Production control is the process of choosing, initiating and monitoring actions in production systems in order to optimise the system (Dean and Wellman, 1991). This can also be said about the control of electricity infrastructures. For both the manufacturing and the electricity domains the classical control approach is hierarchical and schedule driven while the new approach is goal-driven and distributed, promising to be more robust, flexible, and reconfigurable, resulting in agile performance. This makes it interesting to investigate if a methodology developed for the control of production systems can be used for an electricity infrastructure based on the large-scale incorporation of micro-CHP technology. The DACS (Designing Agent-based production Control Systems) methodology, developed by Bussmann, Jennings and Wooldridge (2004) is a new methodology designed specifically for the control of manufacturing systems, but it can also be applied to other domain in which a physical process is controlled by discrete decisions. We consider this approach to be especially interesting because Jennings and Wooldridge are renowned experts of agent-based modelling (See Wooldridge and Jennings, 1995). In this paper we apply this methodology to the case of controlling distributed electricity generation via micro-CHPs in the Dutch electricity sector. This allows us to investigate to what extent the methodology is applicable and what changes in the methodology might be needed to be able to model (electricity) infrastructures. The rest of this paper is structured as follows. Section 2 contains a summary of the DACS approach described in Bussmann et al. (2004). In Section 3 this approach is applied to a different domain, namely the distributed generation of electricity. Section 4 summarises the above and discusses the apphcability of the DACS approach to energy infrastructures in general. Problems with the application on an electricity infrastructure are identified which leads to recommendations for further research. 2. Summary of the DACS methodology Bussmann et al. demonstrate in their book (2004) that both data-oriented (focus on input/output) and structured (focus on functions) methodologies do not work well for the new coordinated control. Object-oriented methodologies are not applicable to the design of MAS either and cannot be used to model the new cooperative production control method because objects are passive and they lack support for structuring organisations. Existing manufacturing control design methodologies (based on discrete event systems or petri-nets, for example) as well as traditional methodologies for agent design (e.g., CommonKADS) also are not sufficient, mostly because they lack focus. DACS is new because it incorporates the appropriate and descriptive requirements for the production control domain and it allows for re-use of interaction protocols. The input for the DACS methodology is a specification of the production control problem (Subsection 1). The major steps of the methodology are analysis of control decisions (Subsection 2), followed by identifications of agents (Subsection 3) and finally the selection of the interaction protocols (Subsection 4). The output after following these steps is an agent-based design of a controller that can then be implemented and executed.

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as a Multi-Agent System

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1. Specification of the production control problem The input to this design methodology, and any suitable methodology for that matter, is a specification of the production control problem to be solved. The specification of the control problem must contain a description of the physical production process to be controlled and the available interfaces with mechanical components as well as a specification of the operation conditions (input, output, possible changes and disturbances). A set of goals and requirements for the production system, such as high productivity or high throughput, determines the performance criteria. 2. Analysis of control In this step the control problem is analysed and decomposed into a decision model. First decision tasks that have to be taken to run the process under certain constraints and that have an effect on the physical aspects of the system are identified. For each of these decision tasks the parameters, control interface, triggers, decision space and local decision rules are described. Decision tasks are classified as being dependent on one another when one task sets constraints for another task or their effects are linked. 3. Identification of agents The control tasks are executed by agents, but often it is beneficial to give multiple tasks to one agent instead of creating a separate agent for each control task. When dependencies between tasks are ignored this might result in sub-optimal performance. Decision tasks can be executed by one agent if tasks can be clustered based on interface cohesion (if the decision has an effect on the same physical device through a control interface), state cohesion (if the decision has an effect on the state of a device) or interactive coupling (if the control task can only be solved by also solving another task). When it is difficult to distribute a task among agents, an attempt can be made to split up the task in smaller subtasks. The result of this step is a list of all control agents and their responsibilities. 4. Interaction protocols Agents that are made responsible for tasks that are dependent will have to communicate with one another. For this interaction a library of protocols is available or a protocol can be customised. 3.

Application of DACS to a distributed electricity generation system

In this section the DACS methodology is used to design an agent-based controller for a distributed electricity generation system based on micro-CHP technology. The fixture interaction between an energy company and households operating micro-CHPs is explained in detail in Houwing et al (2006) using a multi-level optimisation approach. Here we view this interaction in a novel way, namely as a production process. We extend the energy hub concept of Geidl and Andersson (2005) and define a household as an energy hub. We consider multiple households and their energy supply company and regard the total system as one production process. The process described here is not unrealistic; fixture electricity generation could very well develop towards this scenario. In the Netherlands the network management activities are separated from the commercial generation, trade and supply of power. Energy companies dealing with generation, trade and supply obtain an additional option of power supply to customers when control of household DG technologies is envisaged. This is like the virtual utility concept, where an energy company runs DG units and can inject power into the grid in order to optimise overall economic performance. Here, however, we assume households themselves operate units themselves and respond to signals from their supplier.

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K.H. van Dam et al. Changes I

Di$turbartC0S I

^t_ _ _ Bnerg^ (gas •«. efactficity) suppiier

Output

Input Micro-CHP

jHs^/e e-h conv --M tmBi

Household system boundary

Figure 1. Households and their energy suppUer represented as a production process (e = electricity, g = gas, h = heat)

1. Specification of the production control problem Figure 1 shows a number of households with a micro-CHP unit and an electric heater ("e-h conv.") that are used to meet their electricity and heat needs (e-sink and h-sink). The connection of each household to the gas and electricity grid is facilitated by an energy supplier. The supplier sells gas for fuelling the micro-CHPs and buys or sells additional electricity from/to households. Micro-CHPs produce electricity and heat with a fixed proportion. In the Dutch electricity system there is a market where suppliers can buy or sell the difference between the predicted demand (bought on the spot market or via bilateral contracts) and actual electricity demand. These balancing costs represent a large share of a supplier's operational costs. In the situation presented here the supplier has the additional option of influencing household behaviour regarding energy generation to minimise his balancing costs. The physical components that have to be controlled in this system are the micro-CHP units, electric heaters, heat outlets (for heat not needed by a household) and an intelligent meter (a small computer in a household that monitors electricity consumption and prices) that all have their own control interface. So, analogous to Bussmann et al. (2004) we define inputs (electricity and gas), outputs (electricity and heat), changes (predicted electricity and heat demand patterns) and disturbances (e.g., weather) of our production process. Besides physical processes there are also non-physical processes to be controlled in this system, as energy supply is mainly an administrative business. The overall goal of the production system is efficient generation of electricity and heat. 2. Analysis of control In Table 1 the operational decision tasks that can be distinguished in the production process are presented together with parameters influencing the decisions, control interfaces, decision space and local decision rules. All listed decision tasks are triggered every 15 minutes as the trade periods on the Dutch electricity market are defined this way (Tennet, 2005). The energy flows in a household (gas, heat and electricity) are connected via mathematical energy balances. Therefore decisions tasks 1, 2 and 3 are fully dependent. Houwing et al. (2006) show that all three decisions have to be taken into account at the same time to reach efficient electricity and heat generation. Task 4, 5 and 6 are also coupled because they share exactly the same parameters.

Modelling an Electricity Infrastructure

as a Multi-Agent

MAS

System

Table 1. Operational decision tasks Task Name

Parameters

1) Set micro-CHP power level

Gas price, electricity price buy/sell, e-sink, h-sink, heat flow from electric heater

2) Set amount of heat discharged

Heat flows from microCHP and electric heater, hsink Electricity and heat flow from micro-CHP, gas price, electricity price buy/sell, esink, h-sink Balancing market prices, electricity trade cost, gas price Balancing market prices, electricity trade cost, gas price Balancing market prices, electricity trade cost, gas price

3) Set electric heater power level 4) Electricity price setting 5) Payback price setting 6) Buying electricity from balancing market

Control Interface Micro-CHP unit

Decision Space Power level between min. and max.

Heat outlet valve

Valve between 0% and 100% Power level between min. and max.

Electric heater Intelligent meter Intelligent meter System operator

Electricity price [euro/kWh] Electricity price [euro/kWh] Amount of electricity [kWh]

Local Decision Choose level to meet e-sink and h-sink for lowest price Set the valve to meet h-sink Set power level in order to meet h-sink for lowest price Profit optimisation Profit optimisation Profit optimisation

3. Identification of agents For each cluster of highly coupled decision tasks an agent is created that is responsible for executing these decision tasks. Following the clustering guidelines and the dependencies addressed above, we arrive at a MAS with one agent responsible for decision tasks 1, 2 and 3 (Household agent) and one agent responsible for tasks 4, 5 and 6 (Supplier agent). It is important to keep in mind that there is a large number of households in the system so there are multiple agents of the first type, each with their own goals and values for decision variables (different comfort levels), but based on the same decision tasks. 4. Interaction protocols Agents that have dependent control decision tasks have to communicate in order to exchange information about these tasks. In this case the Household agent and the Supplier agent have to communicate about prices and flows. DACS provides a set of standard interaction protocols but we choose to customise a communication language based on an ontology. This is a formalised specification of concepts (Gruber, 1993) used by the agents to communicate at a semantic level. Van Dam et al (2006) and Nikolic and Dijkema (2005) describe a process decomposition method that results in such a formal domain description. This makes it possible to extend the system and reuse agents in other models. 4.

Conclusions and Recommendations

The distributed electricity generation by micro-CHP and the facilitating energy suppliers can be regarded as an electricity production system that can be an efficient alternative for central generation. The result of applying the DACS methodology to this system is the conceptual design of an agent-based controller for an electricity infrastructure. It was shown that a design methodology for manufacturing control can be applied to a socio-technical infrastructure if we include the control of non-physical

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processes. This is a new application that goes beyond the type of systems intended by the developers of DACS. The aim of the designed agent-based controller is to support operational decision making. This design will be implemented so that experiments with different control strategies can be executed. An agent-based model is an excellent way to experiment with different control strategies because it is developed in a bottom-up approach in which the structure is not fixed in the design. In our research we focus on the design of models that can be used to compare control concepts rather than designing novel control functions and algorithms. Because we successfully applied a design methodology intended for manufacturing control to an electricity infrastructure we plan to extend this to other (energy) infrastructures as well, which allows for a comparison of control strategies between domains. Acknowledgments This project is supported by the Next Generation Infrastructure Foundation (www.nginfra.nl).

References Bussmann, S., Jennings, N. R., Wooldridge, M., (2004), Multi-agent systems for manufacturing control: a design methodology, Berlin, Springer Chambers, A., S. Hamilton, et al. (2001). Distributed Generation: A Non-technical Guide. Tulsa, Oklahoma, US, Penn Well Corporation. Dam, Koen H. van, Igor Nikolic, Zofia Lukszo and Gerard P.J. Dijkema. (2006) Towards a generic approach for analysing the efficiency of complex networks. Accepted for the 2006 IEEE International Conference on Networking, Sensing and Control, Ft. Lauderdale, USA. Dam, Koen H. van , Verwater-Lukszo, Z., Ottjes, J.A. and Lodewijks, G. (in print) 'Distributed intelligence in autonomous multi-vehicle systems'. Int. J. Critical Infrastructures. Geidl, M. and G. Andersson (2005). A Modeling and Optimization Approach for Multiple Energy Carrier Power Flow. IEEE, PES, PowerTech' (2005) Conference, St. Petersburg, Russia Gruber, Thomas R. (1993). A Translation Approach to Portable Ontology Specifications. Knowledge Acquisition, 5(2), 1993, pp. 199-220. Houwing, M., Petra W. Heijnen and Ivo Bouwmans. (2006) Deciding on Micro-CHP; A MultiLevel Decision-Making Approach. Accepted for the 2006 IEEE International Conference on Networking, Sensing and Control, Ft. Lauderdale, FL, April 23-25, 2006 Jenkins, N., R. Allan, et al. (2000). Embedded Generation. London, UK, The Institution of Electrical Engineers. Microgen (2005) website http://www.microgen.com/ Newborough, M. (2004). "Assessing the benefits of implementing micro-CHP systems in the UK." Proceedings of the I MECH E Part A Journal of Power and Energy 218(4): pp. 203-218. Nikolic, I. and G.P.J. Dijkema. (2005) Intelligent Infrastructures in a Changing Environment: Innovation and Evolution of Industry-Infrastructure Networks, Proceedings of the 2005 IEEE International Conference On Networking, Sensing and Control Tucson, Arizona, U.S.A. Su, A. (2005). Exploring the Unknown Market; The Anticipated Diffusion of Domestic MicroCombined Heat and Power (CHP) in the Netherlands. Faculty of Technology, Policy and Management. Delft, Delft University of Technology. Tennet (2005) website http://www.tennet.nl/ Wooldridge, M. J., and Jennings, N. R. (1995). Intelligent agents: Theory and practice. Knowledge Engineering Review 10(2) Wooldrige (2002) An Introduction to Multi-agent Systems, John Wiley & Sons (Chichester, England). ISBN 0 47149691X.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B,V.

Global Optimization of Multiscenario Mixed Integer Nonlinear Programming Models Arising in the Synthesis of Integrated Water Networks under Uncertainty Ramkumar Karuppiah and Ignacio E. Grossmann* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA. The problem of optimal synthesis of an integrated water system is addressed in this work, where water using processes and water treatment operations are combined into a single network such that the total cost of building the network and operating it optimally is globally minimized. The network design has to be feasible and optimal over a given set of scenarios in which different operational conditions hold. We propose a superstructure whose optimization is formulated as a multiscenario non-convex Mixed Integer NonLinear Programming (MrNLP) problem. A spatial branch and cut algorithm is proposed that uses Lagrangean decomposition for global optimization of the large multiscenario model. An example is presented for the global optimization of an integrated network operating under uncertainty using the proposed algorithm. Keywords'. Global optimization; Integrated water networks; Uncertainty 1. INTRODUCTION Process synthesis under uncertainty is in general a very challenging problem. A number of parameters usually change during the operation of a process network and for which the data is not known exactly. The main objective when synthesizing a network operating under uncertainty is that the design should be optimal and feasible over a range of values of the uncertain parameters. The problem of ensuring feasibility of design has been addressed by Grossmann et al. [1] where the control variables in the system can be adjusted for the parameter changes. In a stochastic programming based approach, the emphasis is on achieving optimality accounting for the fact that the recourse variables can be adjusted for each parameter realization (see Acevedo and Pistikopoulos [2] and Liu and Sahinidis [3]). A recent review of the major techniques for optimization under uncertainty is given in Sahinidis [4]. This paper addresses the design optimization of integrated water networks operating under uncertain operational conditions. We formulate the corresponding two stage stochastic programming problem as a deterministic multiscenario Mixed Integer Non-Linear Programming (MINLP) problem since the uncertain parameters can take on a finite number of realizations. An *Corresponding author. Tel: +1-412-268-2230; fax: +1-412-268-7139. Email address: [email protected] (I.E. Grossmann)

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algorithm is proposed to solve the problem to global optimality. We present an example to illustrate that the algorithm solves the problem in significantly less time than the MINLP solver BARON (Sahinidis [5]). 2. PROBLEM STATEMENT In this work, we consider the optimal synthesis of an integrated water network consisting of water using process units, water treating units and mixers and splitters, operating under uncertain operational conditions. The amounts of contaminants generated in the process units and the contaminant removal ratios in the treatment units are the uncertain parameters which take on different values in each scenario. The objective is to synthesize a network such that the costs of constructing the network and the expected cost of operating the network optimally over all scenarios is minimized. The first stage capital costs include the investment cost for piping which depends on the maximum flowrate allowable in a pipe, and the capital cost of each treatment unit, which is dependent on the maximum flow of wastewater to be handled by that treatment unit. The operating costs of the network appear in the second stage, which include the cost of obtaining freshwater for use in the process units, the cost of pumping a certain flow of water through the pipes (this flow should be less than the maximum flow allowable in the pipes) and the operating costs of treating wastewater in the treatment units. The synthesis problem is formulated as a multiscenario non-convex MINLP which is solved to global optimality. 3. MODEL We extend the non-convex NLP formulation for the synthesis of integrated water networks given in Karuppiah and Grossmann [6] to build the multiscenario MINLP model. A detailed nomenclature of the terms used here is given in [6]. Objective function: (z)=^i^^(^c;y+/p'(F'f

+^Z^«

Z^^^"

^"YPn^FwFW^ +AR^Icirf +H^„ YpCFi,

(1) Here, pn is the probability of occurrence of scenario n, Cj^is the cost coefficient corresponding to existence of a pipe /, IP (F')

is the investment

cost of pipe /, while PM F^ is the cost of pumping water inside a pipe / in scenario n. The design variable y is binary and pertains to the existence of a stream/pipe /. The other first stage design variable F ' pertains to the maximum flow allowable in a pipe / while the vector FJ is the set of second stage state variables which corresponds to the flows in the pipes in each scenario n. Mixer Units: F/

= ^

Fj

^me

MU , ke m^„, , V«

G N

(2)

Global Optimization of Multiscenario Mixed Integer Nonlinear Programming Models 1749 F'C%

= Y^ F!,C)„

\/j, ^m^

MU , ks m„„, ,V« 6 N

(3)

Here C)^ is the concentration of contaminanty (ppm) in stream / in scenario n. jn Splitter Units: F / = X ^«

yseSU,keSj„,\/neN

(4)

ie Sg^f

C)n =C%

'^j,ysESUyies,,,,kes,„,\/neN

(5)

Process Units: F /

=

F;

=

P

^

\/ps

PU , ie p,„, ke

P^C;.„ +L5„xlO^ =P^C}„

p,,, ,\/n e N

(6)

Vy,VpG Pf/, /G p,„,^G p,„,,V«G N

(7)

The load of contaminanty inside a process unit/? for each scenario n is different and is given by L^„. Treatment Units: F/ = F;

\/te

C;„ = y ^ ; c ) „

TU , ie t,,, , ke

t,„,yne

N

(8)

V7,V^G rC/, ie t,,, , k e t,„,yne

N

(9)

The contaminant removal ratios in the treatment units are different in each scenario n and so J3j which is defined as y^j = 1 - {(Removal ratio for contaminant 7 in unit t (in %)) / 100} takes on different values (pj^j in each scenario n. Bound strengthening cuts: X i5„ X10 ' = X (l - n pe PU

K*C;„ +Fr C;:

yjyne

N

(10)

te TU

Design Constraints: F'^^y' F; V/,V« G N (12) The multiscenario MESfLP model (P) comprises equations (1) - (12). 4. SOLUTION STRATEGY The multiscenario models such as (?) grow quickly in size with the number of scenarios and are very difficult to solve to global optimality without the help of specialized techniques. We use a spatial branch and cut algorithm to solve the multiscenario MESfLP to global optimality. Lower bounds are obtained at every node of the tree by solving a convex relaxation of model (P) with certain cuts added to it. In order to generate cuts, a decomposition scheme is proposed, where we use Lagrangean relaxation to decompose the non-convex model (P) into single scenario sub-problems at every node of the search tree. These sub-problems are then solved to global optimality and their solutions are used to generate the bound strengthening cuts. A heuristic is used for the

1750

R. Karuppiah and I.E. Grossmann

generation of good upper bounds at each node. The lower and upper bounds are then converged to within a specified tolerance in the branch and cut algorithm. 4.1. Generation of tight lower bounds: In order to construct a Lagrangean relaxation of the original MINLP problem, we dualize the linking constraints (eq (12)) between the different scenarios. To do this, we create copies of the design variablesF'andj^'for each scenario, which are given by FJand

y\

respectively, and replace F ' and y^ by these newly created variables in model (P). Hence, eqs (11) and (12) get modified to yield eqs (13) and (14) respectively. F^'yl
> Fj

\/ i,y

n e N

(14)

The objective function is also altered as shown in eq (15), where F ' i n the original objective function is replaced by F / . mm = Al\^(cly[+IF{F;J]\+H'^p^^^ \_ i ^

^J

77

/

n

teTU

n

teTU

'" (15) Finally, we add eqs (16) and (17) to (?) to get a reformulated model (RP). F;

- F;^^ = 0

fn-yLx=^

y i,\f n^ N ,n <\N\

(16)

ytyrie

(17)

N,n<\N\

Further, we multiply the eqs (16) and (17) with Af^i^ne N,n<\N\)

and

/l^„(yneN,n<\N\) respectively and transfer these constraints to the objective function to get a Lagrangean relaxation of the original problem (P), which is denoted by (LRP) and is decomposable into smaller sub-problems that are easier to solve. The parameters A^ and A]^ are known as Lagrange multipliers. The model (LRP) is then decomposed into [A'^I smaller models that contain variables pertaining to only one scenario. It is to be noted that the bounds of the second stage variables in all the sub-problems are the same as in the original problem, while the bounds of the newly created variables, F^ and y'^ are the same as the corresponding design variables F'andy^ problems is as follows: minz„ = r„ +HYPnPM'K +Hp„C,^FW„ +H^P„OC'F;, %T, s.t. eqs {2)-(lO),(l3),{U)

respectively. A set of decomposed + ^ W { -A(„_,)F^ + ^ ^ ^ ' '

where ^ = 0 , A^ =0 ,;i;|^| = 0 , ^^^ =0 and AR\ X f c > ; +/P'(Fjf ] +AR^IC'{F^f [0

n=1 n^2,...,\N\

-^-M] n = \,...,\N\ J

jlobal Optimization of Multiscenario Mixed Integer Nonlinear Programming Models 1751 Each of these sub-problems is globally minimized to obtain a solution z*. The sum V z* yields a valid lower bound to the solution of (P) at a node in the branch and bound tree. Instead of using such a lower bound we generate valid cuts in the space of the original design and state variables based on the solutions z„ , which are given in eq (18).

(18) AR\

where,

YicWMF-1

+ AR

xiciF^r

n=\

Xn=\ n = 2,...,\N\

These cuts are then added to the model (P). Futhermore, the Lagrange multipliers can be updated using sub-gradient methods to derive additional cuts, in the same way as before, to add to the original problem (P) and this procedure of updating the multipliers and adding cuts can be performed any number of times. The initial values of the Lagrange multipliers are chosen arbitrarily. The problem (P) with these cuts added is convexified by constructing convex envelopes for the non-convex nonlinear terms and the resulting MILP (model (R)) is solved to predict a valid lower bound to the solution of (P) over the subregion corresponding to a particular node of the search tree. 4.2. Upper bound generation: A heuristic procedure is used to generate upper bounds at every node of the branch and bound tree. We solve the single scenario model (obtained from (P) by taking a single element in the set N) for all the given scenarios « G N to global optimality, and superimpose the resulting structures. That is, we fix the design variable y' to 1 if there exists a non-zero flow FJ in the solution of at least one of the|A^| single scenario sub-problems. The problem (P) is transformed from a non-convex MINLP to a non-convex NLP which is solved to get an upper bound. 5. NUMERICAL EXAMPLE We consider a network consisting of two water processing units and two water treatment units whose superstructure is shown in Fig. 1.

Freshwater

Fig. 1 Superstructure of a 2 Process unit - 2 Treatment unit integrated network It is a system involving two contaminants A and B, which are generated in the process units and removed using the treatment units. The concentration of

R. Karuppiah and I.E. Grossmann

1752

these pollutants has to be reduced to less than 10 ppm in the effluent stream discharged into the environment. This system operates over a set of 10 scenarios in one year, where the uncertainties correspond to the contaminant loads in the process units and the contaminant removal ratios in the treatment units. The data used for optimizing this integrated water network can be obtained from the authors. This multiscenario MINLP corresponding to this example involves 24 binary variables, 764 continuous variables, 928 constraints and 406 non-convex terms and was initially attempted to be solved using GAMS/BARON 7.2.5 on an Intel 3.2 GHz machine with 1 GB memory. The termination criterion used was that the gap between the upper and lower bounds should be less than the specified tolerance of 1 %. On directly using BARON to solve the problem, it was found that the solver could not verify global optimality even after 10 hours, and yielded a relaxation gap of 5.5 %. The application of the proposed algorithm yielded an expected cost of $651,653.06, which is the global solution to the problem. It was also found that the lower and upper bounds converge to within the specified tolerance at the root node of the branch and bound tree. The proposed algorithm took a total of 62.8 CPUsecs to solve which is drastically less than the time taken by BARON to optimize the original model. The optimal network topology is shown in Fig. 2 where, alongside the pipe connections, the maximum flowrates that can be handled by the pipes are shown.

(m^ ^ 4 4 . 7 9 *

TU1

_ 4 4 . 7 9 - ^(

S4

^~~~~~-~-^447S 0.89

(M2)

^ 4 )

^50.89>

TU2

1 S5

—50.89-»i

Fig. 2 Optimal solution for a 2 Process unit - 2 Treatment unit system operating under uncertainty Acknowledgement. The authors gratefiilly acknowledge financial support from the National Science Foundation under Grant CTS-0521769.

REFERENCES 1. I.E. Grossmann, K.P. Halemane and R.E. Swaney, Computers and Chemical Engineering, 7(1983) 439. 2. J. Acevedo and E.N. Pistikopoulos, Computers and Chemical Engineering, 22(1998) 647. 3. M.L. Liu and N.V. Sahinidis, Industrial and Engineering Chemistry Research, 35(1996)4154. 4. N.V. Sahinidis, Computers and Chemical Engineering, 28(2004) 971. 5. N. Sahinidis, Joumal of Global Optimization, 8(1996) 201. 6. R. Karuppiah and I.E. Grossmann, Global Optimization for the Synthesis of Integrated Water Systems in Chemical Processes, to appear in Computers and Chemical Engineering (2006).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Hierarchical Markov Reliability / Availability Models for Energy & Industrial Infrastructure Systems Conceptual Design A.N. Ajah,"'^ P. M. Herder," J. Grievink,^ and M. P.C. Weijnen," "Energy and Industry Group, Faculty of Technology, Policy and Management, Process Systems Engineering Group, Faculty of Applied Science, Delft University of Technology, 2600 GA, Delft, Netherlands. Abstract Modelling infrastructure system's deterioration and repair behaviours through a markov model is not only essential for accurately predicting the system's future reliability condition but also act as key inputs for effective infrastructure systems maintenance. During the conceptual (re)design of these systems, myriads of components are usually involved. A multi-state markov modelling of this component is emphasised in this work. However, the exponential explosion in the size of the markov model as the number of such components increase may pose great limitation in its application at this stage of design. We also present a hierarchical modelling approach that could aid the designer in not only overcoming this limitation but also in the detailed screening and analysis of the reliability and availability of such infrastructure systems' components. The application effectiveness and utility of the proposed approach is tested by means of a case study, the reliability modelling of a proton exchange membrane (PEM) fuel cell power plant. Keywords: Reliability & Availability, Markov model. Conceptual Design, Infrastructure systems. 1. Introduction Infrastructure system's (energy, gas, water) reliability, just like the reliability of any engineered system, is increasingly becoming an important performance indicator. This increasing importance and the associated pressure on the system designers calls for a drastic change in the ways the infrastructure systems are currently conceptually designed. Most of the common practices of estimating infrastructure system's reliability and overall availability at the conceptual design stage are either done on an ad hoc basis or basically rely on some predefined and assumed component availability (usually 80-95%). Two major reliability analysis can be distinguished; measurement and model-based (Sathaye et al., 2000). During systems design, measurement based analysis may be infeasible and expensive, hence model based approaches are often relied upon. However, most of such reliability models utilize the non-state-space model which assumes that the components are independent of one another in their failure and repair behaviours. The markov model could well predict the dependencies and future conditions of these infrastructure components, systems and networks through the characterization of such deterioration and repairs in a probabilistic continuous or discrete-state manner. Capturing such infrastructure system's deterioration and repair behaviours and their effects on the overall system, through markov model is not only essential for accurately predicting the system's future reliability condition but act as key inputs for effective system maintenance (corrective and preventive). However, the explosion in the size of the markov model

1753

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A.N. Ajah et al.

as the number of such systems, components or networks increases is being perceived as a major hmitation in its apphcation during the early phase of conceptual design where time is often a constraining factor. Due to this exponential growth, it may not be easy to markov model complex infrastructure systems reliability as a once-through entity. What is needed is an approach that can decompose the reliability problem into manageable levels of abstraction, address the reliability issues separately at the decomposed levels and then aggregate the results at each level into final system reliability. This work builds on this concept as it dwells on a hierarchical markov modeling approach to circumvent this limitation and also aid the designer in the proper screening and analysis of the reliability and availability of infrastructure systems at the early phase of the conceptual design process. 2. Hierarchical markov modeling approach In the hierarchical markov modeling being proposed, the reliability problem is decomposed into three manageable markov levels (components, units and system), based on the structural and behavioral complexity of the system. This decomposition reduces the number of state space problems to a size that could be easily handled. At the component level, each component is markov-modeled separately. This gives the designer, extra degree of insight into the dynamic performance of the components and hence in the selection of components to feature in the design. At the unit (sub-system) level, for a given flow diagram of an infrastructure system, an aggregation of the equipment based on functional and structural similarities is carried out. This aggregation reduces the number of equipment to be markov-modeled. Lastly, at the higher (system) level, the evaluated subsystems markov reliability and availability are combined into the total system reliability and availability. In this way, the size of the problem as well as the computational efforts is significantly reduced. The attributes of decomposition levels is as shown in table 1. Table 1: attributes of decomposed levels Attributes Analysis type States

Level of decomposition Component

Unit

System

Markov

Markov

RBD + Markov

Multi-states

Reduced

Reduced

Structural model

No connectivity

State space diagram connectivity

Series-parallel

Behavioral model

Independent

Dependence of units

Dependence of units

Reliability and availability modeling at the component level: At the component level, the essence of reliability and availability analysis being proposed is to assess the reliability characteristics and states of each of the components with a view to selecting the most reliable and promising ones. However, the conventional attitude is to assume that a component has binary basic states-operable and failed states normally designated as 1 and 0 respectively. In most infrastructure systems, there are some groups of system failures that may not be immediately observed upon occurrence. Such system disorder do manifest as small errors or defects such as pump fouling, pipeline partial blockage etc that do not cause immediate total or catastrophic failure. Nonetheless, if left undetected and unrepaired, such failures still grow to cause a larger failure that result in unscheduled downtime. Before the catastrophic or total system downtime, occurrence of such menial disorder may force the system into a state of reduced functionality vis-a-vis its incipient operational state. These sorts of failures that do not result into immediate catastrophic failure of the system but can lead to diminishing functionality have been modeled as transient failures and the states at which they occur, as transient states, states r and d in

Hierarchical Markov Reliability/Availability

Models

1755

figure7. Considering and modeling these transient states using the markov approach will give a more realistic steady state and dynamic reliability characteristics of the components and will help in the proper screening and selection of components to be featured in the system being designed. The transition diagram as depicted below covers possible states and transitions that a component can realistically experience. Transition from state u (fully operational) to transient states r or d may be caused by environmental and human induced problems respectively while the transition from state r or d back to state u is caused by the removal of such a failures (repairs). If at states r or d, no action is taken, the component might finally transit to the state / (permanent failure) which may be restored to either states u (full restoration) or to states r and d (partial restoration), depending on the nature of the maintenance action taken. The direct transition from states u to state / depicts mechanical and other unknown problems which lead to instantaneous permanent failures.

,<S^

<S>--

Fully operational state

Reduced Functionality states

Fully unoperatJonaf state

Figure 1: Transition diagram for a multi-state markov modeled component It has been reported that human errors and environmental factors contribute about 40% of total equipment downtimes (w^ww.plantweb.emersonprocess.com); hence, we have differentiated between states r and d to highlight the effects of operators and environmental factors on the availability of components of especially energy infrastructures. Assuming there is enough data, we think that considering them early in the design process; will aid the designers in the critical assessment of these factors and thus the screening, differentiation and better choice of the more reliable components. The probability of components (Pi) to be in any of the M states could be obtained from the solutions of sets of equation 1. X and ji are components failure and repair rates.

dP^(t)_ dt

^

^

^

^

Y,^i^j \PXt) + \ Z / ^ ; - h W

V J

i = 0,...,M;M>2

(1)

V J*i

Reliability and availability modeling at the unit level: Having identified the most reliable components to feature in the design based on the detailed component level markov modeling, at the unit (sub-system) level, the combinatorial dependencies and redundancies associated with these components is also markov-modeled. It is envisaged that for complex infrastructure systems, involving units with components in the order of tens and hundreds, state space explosion problem will be imminent. To circumvent this, we propose unit's component aggregation. The basic idea behind this is to reduce the number of equipment in the units of such complex large-scale systems by substituting some sets of equipment whose individual unavailability does not clearly affect the system performance, or whose failure and repair rates are similar, with a single representative component. For a unit with N number of components, the state space size is estimated by 2^, but with the approximate state space aggregation technique (Lefebvre, 2002) the size reduces to:

ri(A^,+i)

(2)

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where Nt is the number of components of the ith family, k is the total number of different aggregate families. Assuming there are a total of 6 components (i.e. N=6 with 2 components aggregated into each family (Ni=2) then k becomes 3); the total number of unreduced state space is 64 while the approximate reduced state space is 27. However, such aggregation is valid for set of equipment which have single (or multiple) input(s) and single output to the remaining components of the system, and for certain types of multiple input-output subsystems (Van Rossen, 1994). Since aggregation policies and rules are often based on dedicated structural and behavioral heuristics, caution, experience and sound engineering judgment are needed of the designer in the application of these aggregation techniques. Reliability and availability modeling at the system level: At the system level, using a reliability block diagram depicting the connections(series, parallel, series-parallel etc) and other reliability characteristics the aggregated individual units of the system is drawn and the overall system structure and reliability, analyzed using the network reduction (Sahner and Trivedi, 1986; Knegtering and Brombacher,2000) approach. From the network reduction technique, given the combined markov models and RBD constructs, if a system has a series-parallel structure, its overall reliability can be obtained using :

(3)

Rs,s(t)=^A^-Kr.uM,aralM{^-K,uM,ar.lM-\^-Kr,uM,araneH^^

Where Rgys (t) is the system overall reliability at time t, Ruint i (t) is the reliability of unit i at time t as obtained from the unit markov reliability modeling. If all the units in the system have parallel connection structure, the last factor of equation 3 can be neglected. 3. Illustrative Case study As an illustration, the proposed decomposed markov reliability model is applied to the analysis of the reliability characteristics of a proton exchange membrane (PEM) ftiel cell power plant as shown in a condensed block diagram of figure 2. Air (oxygen) Hydrogen

DC Power FCP Stack,

REFORMSRt

Natural Gas

Fuel Converter (REFORMER)

Fuel cell i^wer Stack

DC/AC Power Inverter

Transfrnmer

REFOWeR j

(a)

FCPStackj

(b)

Figure 2: a) Condensed block diagram of a PEMFC power plant, b :) Aggregated units of the PEMFC Fuel cell power plant is a heterogeneous system featuring the interactions between chemical, mechanical, thermal and electrical components and subsystems for converting fuels such as gasoline and natural gas to alternating current. Natural gas flows into the reformer where it is converted into hydrogen. The hydrogen produced together with oxygen from air are then routed into the (PEM) cell stack assembly where the O2 and H2 streams are electrochemically converted into electrical power, steam and water. The steam is recycled to the reformer for the reforming process. The electrical power generated, in the form of DC power flows to the power inverter for inversion into an AC power before being routed to grid.

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Component level: The case study described above has more than ten components such as compressors, pumps, humidifiers, reactors, transformers, membrane, control equipments etc. For the compressor, table 2 shows the state probabilities. Steady state is assumed at 250 hrs. Table 2: compressor steady-state probabilities Time(hrs)

State u

State /

State;

State 1

0

Compressor 1&2

1

0

0

0

250

Compressor 1 Compressor 2

0.9679 0.9736

0.0093 0.0085

0.0152 0.0113

0.0076 0.0066

From the foregoing, it could be deduced that the compressor 2, with higher chances of being operational and lower chances of environmental and human failures (states / 8c j) outperforms compressor 1 and thus would win the selection process. Unit (subsystem) level: Using the concept of components aggregations as discussed in section 2, and assuming redundancy, the components of the PEMFC power plant have been lumped to form the basic unit as shown in figure 2b. In all, five units with 32 state and 80 transition paths as depicted in figure 4a is identified. If a second degree aggregation is carried out, the number of states reduces to about 18, with 32 transition paths (figure 4b). From figure 4a, sets of markov differential equations (equation 5) results, assuming different failure and repair rates for the various transitions to and fro states. The solution of these differential equations, gives the probability of the system to be in each depicted state. Such solution could be obtained using any of the numerical methods of Euler, Runge Kutta and LU decomposition or with software such as MATLAB. Table 3 shows the results of the state probabilities for the second degree aggregated units of 18 state spaces.

r - ^ (/^sto~"^ - - * • - / V-*---. >-—;—-ji " > - " ^ ^ (,pnio))^X'ai'2i ([.i?ioi) ((11^001 ; Ct5U0iiJ (ifllnj

X---^-/ (tofilij)

V" Ctio

(b) Figure 4: a) 1st degree reduced state space [Si denotes State i, jUij and /lij denote repair and failure transition ratesfromstate / to statey]; b) 2"** degree reduced state space of the illustrative case study.

A.N. Ajah et al.

1758

-z\<

r^,(oi d dt

A(0

-"«,!

- E(^2,,;/'2.,)

•••

fnit).

>«„,:

(•5;

• £ -"«,. Table 3: State probabilities of the 2 order reduced units

Ohrs

SI 1

S2 0

S3 0

S4 0

250hrs

0.9725

0.0491

0.01452

1.91E-6

Time

S5

S6

S7

S8

0 3.4E-6

0 0.0081

0 2.7E-6

0 3.8E-6

S9 0 2.7E-6

SIO 0

Sll 0

S12 0

S13 0

S14 0

S15

S16

S17

0

0

0

S18 0

4.2E-5

1.12E-6

1.812E-6

4.13E-7

7.02E-5

9.73E-9

5.63E-7

1.705E-8

1.52E-10

System level: At this level, the components and subsystems, initially abstracted are integrated using the network reduction techniques, to obtain the system reliability. Using equation 3, and the second order reduced units of figure 2b (with Rrefomier 1 =Rrefonner 2 = 0.9725; RFCP stacki = RFCP stack 2 = 0.95 and Rtransformer = 0 . 8 ) thc ovcrall systcm reliability at steady state have been estimated at 0.79998. However, if another redundant transformer (same reliability of 0.8) is introduced into the network, the system reliability goes up to 0.99999, thus reflecting the effect of redundancy on reliability. 4. Conclusions: Most reliability models assume a binary component or system reliability (up and down states). However, many real-world components of energy and water infrastructure systems are often of multi-state nature, with different performance levels and several failure modes that induce degradation effects on the entire system performance. We present a multi-state component reliability model and the hierarchical decomposition of infrastructure system markov-reliability modeling into three levels. This decomposition approach, apart from its utilization in the management of the state space explosion also helps the designer in the proper use of reliability as performance criteria for components selection. The utility of the proposed approach is tested by means of a case study, proton exchange membrane (PEM) ftiel cell power infrastructure system. References. Knegtering B and A.C. Brombacher (2000), A method to prevent excessive numbers of markov states in markov models for quantitative safety and reliability assessment, ISA Trans, 39, pp 363-369 Sahner S.A. and K.S.Trivedi, (1986), A hierarchical, combinatorial-markov method of solving complex reliability models, in proceeding of joint computation conference pp 817-825, Sathaye,A., Ramani S., K.S. Trivedi (2000) Availability models in practice, Conf proceeding intl, workshop on fault tolerant control and computing. Van Rossen J.C.P. (1994 ) Criticality Rating and Safety Analysis in FRAMS. Center for process Systems Engineering, London. Yannick lefebvre (2002), Approximate aggregation and application to reliability, 3rd Intemational Conference on Mathematical Methods in Reliability

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 Published by Elsevier B.V.

Agent-enabled dynamic management system for process plants Antonis Kokossis^, Zhigang Shang^, Elaine Gao^ ^ Department of Chemical & Process Engineering, University of Surrey, Guildford, Surrey GU2 7XH, UK ^ Department of Process & Systems Engineering, Cranfield University, Cranfield, MK43 OAL, UK Abstract The paper presents an agent-enabled environment to support decisions and the dynamic trading of utility services. The developments are set up to emulate different departments of a total site, individual production processes, the utility system, and trading departments. The proposed approach reviews ways to assess investment schemes, as well as design and operational scenarios. The agents make use of knowledge models that communicate with individual processes and assess scenarios for energy efficiency. Optimization models take into account objectives for improvements, whereas agents take into account the dynamics of the communication. Knowledge and optimization models are linked with databases that contain planning and operational data useful to manage and support decisions. The approach is illustrated with two case studies. Keywords: dynamic management, agent, planning, optimisation, utility system. 1. Introduction Most process plants operate in the context of a Total Site where chemical production processes consume heat and power supplied by a central utility system. The processes consume and/or generate steam at various levels. Generated steam can be supplied to the steam mains and consumed by other processes. The potential for indirect interaction between processes may lead to significant savings. The problem to define solutions becomes quite complex though. Conventional problem descriptions would assume a set of tasks, steam and power demands. The site utility system would then be designed to determine optimal pressure levels and optimal allocations of units. In a conventional description of the problem, the utility system addresses the strict needs of the site. In more competitive environments, service units operate for economic viability, they become increasingly independent, and are expected to scope for additional profits by trading services outside the promises of their corporate environment. To accomplish such an objective, they need to monitor opportunities and integrate online trading with decisions on scheduling. The dynamic management requires technology able to support decisions and the dynamic trading of services. Decisions account for background cases for negotiations, the development of profitable scenarios, schemes for discounted services to attract customers, and application of yield management practices. Dynamics should translate, communicate and adjust scenarios according to acceptable terms between trading partners, ensure smooth dynamics in the network operation and the application of optimization principles in an automated and coordinated manner. The paper presents an agent-enabled environment to this purpose. The developments are set up to emulate different departments of the total site, individual production processes, the utility system, and trading departments. Agents are assigned

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the review results from optimization runs, prices of services sold to local production processes, prices of services sold to external customers and the possible trade with the main grid. Agents make use of knowledge models that model the network and its links, mathematical models that optimize for energy efficiency, and synthesis models that are used for design and operational scenarios. Knowledge and optimization models are linked with databases that support scheduling and planning decisions. 2. Design and implementation Agents share information, knowledge, and tasks ([1], [6], [7], [10]). Agents communicate over a specific domain, agree on a terminology, and share a common ontology ([2], [4]). Ontologies are conceptualisations of shared domains ([3]). Once committed to a common ontology, interacting agents are able to interpret communication interactions and accomplish a mutual understanding. Equipped with common communication language and a shared ontology, the agents are able to communicate with each other in the same manner, in the same syntax, and with the same understanding of the domain. The section presents the agent-enabled system that is employed in this work. It highlights the system architecture that comprises a knowledge base, software agents, and a user interface. The knowledge base comprises process models, heuristic rules, process-related data and contextual information. The models include mathematical models for utility Site and ^^•••^^ Optimization, scheduling broker agents ^ • g ^ ^ ^ ^ ^ ^ H ^ l ^ ^ H jnodels and models for design and retrofitting decisions. Such models are MILP formulations developed in previous Task Agents research ([8], [9]). Data include historical and real-time operation data. Heuristic rules resolve trading decisions Knowledge base customizable to user preferences. The system processes information on steam and power demands and initiates dynamic trading. The agents are organized in a layered system that comprises a site agent, a broker agent and a group of task agents. Site agents receive tasks from the users and return results. The broker agent distributes the tasks, coordinates the performance of the task agents, assists human decision-making and notifies the result to the site agent. Task agents are designed to perform decision support tasks that acounts for optimization, scheduling, data analysis and dynamic trading. Scheduling agents optimize the operation and allocate units in time with the application of deterministic models. Data analysis agents collect, aggregate, and analyze transaction data. Heuristic rules are applied to assist negotiation around selected trading scenarios. Integer cuts provide a set of backup scenarios that support negotiation tasks. Decisions supported by the agents relate to the type of optimization required to apply, the management of time during the negotiations, and the management of the different scenarios generated as acceptable deviations from the targets. Agents negotiate

Agent-Enabled Dynamic Management System for Process Plants

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according to available trade-offs between internal prices of steam (or electricity) and external demand for utitlities. The higher the external demand (where profit margins are always higher) the lower the potential to discount the internal prices (so that to exploit the potential of the external market). Each trade-off represents a Pareto point calculated with the use of deterministic models. According to the application, agents may activate models for operation (scheduling) or analysis (synthesis). Results determine the scope for negotiation and is stored to avoid repetitions, to support the development of scenarios, and to enable agents to negotiate. Time controls are based on default parameters that manage time and deadlines. The generation of options is based on a heuristic scheme that creates incremental discounts in prices. Prices are negotiated with internal and external users and could make use of limits to accept discounts or satify contractual obligations to the internal users. The heuristics are simple and do not generate conflicts in the current version of the work. The agent-based system has been developed using JADE. The figure on the left illustrates the user interface. There are three communication categories in the multi-agent system: agent-internal resources (such as models, databases), agent-agent and agent-external data. The communication protocol depends on the type of agents and the information exchanged. Agents to database communication is based on ODBC/JDBC using SQL requests and commands. The cooperating agents communicate through FIPA ACL. ^^^e' Agents could run on the same or different machines. 3. Case Studies The utility system (figure on the right) services 4 plants (5 steam levels, 5 boilers, 6 steam turbines, 1 deaerator). The application is illustrated with two case studies. The first considers the debottlenecking of a utility operation. The second considers the reconciliation of customer demands for maximum profit. The

^

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utility system is interconnected with the public grid. Occasional shortage of power is addressed by importing power. Cost data for the utilities are given in Table 1. The current operating cost is 82.2M$/year. The current site power demand is 212.5 MW. The demands of VLP, LP, MP and HP are 40 MW, 80MW, lOOMW and 150 MW respectively. The maximum power to be allowed to export is 65 MW. The maximum capacity of the gas turbines used is 40 MW. Table 1. Utility cost data (US$/kWh) Fuel(Bl)

Fuel(Bl)

Fuel(Bl)

Electricity

HP

MP

LP

VLP

0.0095

0.0097

0.0125

0.1

0.038

0.036

0.034

0.032

Case 1 The objective of this case study is to identify the best investment scheme for the site utility system by negotiating the electricity demand and price with each internal consumer and external consumer. The site utility system is optimised for different negotiating scenarios. The scenarios reviewed in the case are shown in the figure below. Scenario A (no new unit) For this scenario, no units are required. By negotiating with both internal and external customers, a new agreement can been achieved where the utility supplies 80% of the internal electricity demand; internal users pay 90% of the standard price (0.1US$/kWh). The utility system exports 42.5 MW electricity to external users at 0.13 US$/kWh. Table 2 gives the economics of the scenario

Existing system

Negotiate O.lS/kWh

Scenario A

Cost(M$/year)

Unit price

82.2

90

Scenario B

92.5

95

Scenario C

102.96

100

Table 2. Economics of Scenario A Cost

Revenue of selling electricity (external)

Revenue of selling electricity (internal)

Revenue of selling steam

Profit

82.2 M$/year

48.4 M$/year

134M$/year

116M$/year

217M$/year

• Scenario B (one new gas turbine + one steam turbine) One new gas turbine and one steam turbine are required. By negotiating with both internal and external customers, an agreement has been achieved where the utility system supplies 90% of the internal electricity demand; in return, internal users only pay 95% of the standard price. The utility system could export 65 MW electricity to the external users at a price of 0.13 US$/kWTi. Table 3 gives the economics of the scenario. Table 3. Economics of Scenario B Cost Revenue of Revenue of Revenue of Profit selling electricity selling electricity selling steam (internal) (external) 92.5 M$/year 74M$/year 170 M$/year 116 M$/year 268 M$/year

Agent-Enabled Dynamic Management System for Process Plants

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Scenario C (one new gas turbine + three steam turbines)OnQ new gas turbine and 3 new steam turbines are required. This is the optimal design obtained by optimising the utility system satisfying both internal and external demands without involving negotiations. The utility system can supply all the internal electricity demand the standard price and export 65 MW electricity to the external users at a price of 0.13 US$/kWh. Table 4 gives the economics of the scenario.

Cost

103 M$/year

Table 4. Revenue of selling electricity (external) 74 M$/year

Economics of Scenario C Revenue of Revenue of selling electricity selling steam (internal) 186 M$/year 116 M$/year

Profit

274 M$/year

Case 2 The objective of this case study is to identify the optimal operation of the site utility system by negotiating the electricity demand and price with each internal consumer and external consumer. The site utility system is optimised for Demand A different negotiating scenarios. The Negotiate scenarios reviewed in ,095$/kwh Demand B the case are shown in the figure on the left. r " ^ , Neg. jgotiate

<jy

Schedule operation

0.087$/kwh

• Scenario A (Base Case): For the base 212.5 0 Base case case, the utility 170.0 Scenario B 5 system only 147.2 Scenario C 13 supplies the internal electricity demand at the standard price (0.1US$/kWh). Thus the utility system does not have any additional power for export. Table 5 gives the economics of the scenario. Site Power Demand (MW)

Cost

82.2 M$/year

Discount (%)

>/

Table 5. Economics of Scenario A Revenue of Revenue of Revenue of selling electricity selling electricity selling steam (external) (internal) 0 186 M$/year 116 M$/year

Profit

220 M$/year

Scenario B: By negotiating with both internal and external customers, the new agreement for Scenario B achieved is as follows: the utility system only supplies 90% of the internal electricity demand, in return, the internal users only pay 95% of the standard price (0.1US$/kWh). Thus the utility system could export 21.25 MW electricity to the external users at a price of 0.13 US$/kWh. Table 6 gives the economics of the scenario.

1764

Cost

82.2 M$/year

A. Kokossis et al Table 6. Economics of Scenario B Revenue of Revenue of Revenue of selling electricity selling electricity selling steam (external) (internal) 24.2 M$/year 159 M$/year 116 M$/year

Profit

218 M$/year

Scenario C By negotiating w^ith both internal and external customers, the new agreement for Scenario B achieved is as follows: the utility system only supplies 70% of the internal electricity demand, in return, the internal users only pay 88% of the standard price (O.lUSS/kWh). Thus the utility system could export 63.75 MW electricity to the external users at a price of 0.13 US$/kWh. Table 7 gives the economics of the scenario.

Cost

82.2 M$/year

Table 7. Revenue of selling electricity (external) 73 M$/year

Economics of Scenario C Revenue of Revenue of selling electricity selling steam (internal) 113M$/year 116M$/year

Profit

222 M$/year

4. Conclusions Utility networks are challenged to participate in open markets and competitive environments. Conventional formulations typically assume constant demands and search for the most economical ways to address the needs of internal customers. The paper presents technology to enable these networks to uphold more aggressive policies, negotiate and settle prices, manage and coordinate demands from different customers and settle for the most profitable prices at each time. The work combines optimization capabilities with the ones in knowledge management, proposing an agent-enabled envirormient equipped with an authoring system to negotiate and trade. Indicative results are provided on the basis of an actual network but with rules developed to demonstrate the underlying principles. References 1. Bradshaw, J.M., Dutfield, S., Benoit, P., Woolley, J.D., 1997, KAoS: Tov^ard An IndustrialStrength Open Agent Architecture, Software Agent, MIT Press, pp: 375 - 418 2. Finin,T., Labrou, Y., Mayfield, J., 1997, KQML as an agent communication language, Software agents, MIT Press, pp. 291 - 316 3. FIPA00023, 2000, FIPA Agent Management Specification. Foundation for Intelligent Physical Agents, http://vv^w^.fipa.org/specs/fipa00023/ 4. Genesereth, M.R., Ketchpel, S.P., 1994, Software agents. Communications of the ACM, 37,7, pp. 48-53 5. Gruber, T. R., 1993, A translation approach to portable ontology specifications. Knowledge Acquisition, 5, pp. 199-220 6. Nwana, H., 1996, Software Agents: An Overview, The Knowledge Engineering Review 11, 3, pp.79-92 7. Wooldridge, M., 2002, An Introduction to Multi-Agent Systems, John Wiley and Sons Limited. 8. Shang Z, AC Kokossis, 2004, Computers & Chemical Engineering, Volume 28, Issue 9, ppl673-1688 9. Shang Z, AC Kokossis, 2005, Chemical Engineering Science, 60 (16), pp 4431-4451 10. Sycara, K.P., 1998, Multiagent Systems, Artificial Intelligence Magazine, 19, 2.79-92

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1765

Methodology for the design of industrial hydrogen networks and the optimal placement of purification units using multi-objective optimisation techniques Luc Girardin^, Franfois Marechaf, Pascal Tromeur^ ^Laboratory for industrial energy system, Ecole polytechnique federate de Lausanne, CH-1015 Lausanne, Switzerland. ^Air Liquide, Centre de Recherche Claude-Delorme, Jouy-en-Josas, France. Abstract The optimal design of hydrogen networks aims at minimising the consumption of fresh hydrogen by improving recycling and reuse of process hydrogen. To solve this problem, a new graphical representation has been developed to characterize the minimum hydrogen requirement, and make a preliminary selection of the compatible purification units. From this preliminary analysis, a multi-objective optimisation method is applied in order to define the best hydrogen network and the proper integration of purification units. The proposed method decomposes the problem into two sub-problems : a mixed integer linear programming for network design at the lower level and an evolutionary algorithm strategy to solve the optimal design of the purification units at the upper level. Keywords: Hydrogen network design, Hydrogen Pinch analysis. Multi-objective optimisation. Purification unit placement. 1. Introduction LLProblem statement In the refining industry, growing amount of hydrogen is needed for petroleum conversion and clean fuel production. The design of hydrogen networks in industrial production sites aims at minimizing the consumption of fresh hydrogen by optimizing the operating parameters, more and more by trying to recycle the degraded hydrogen produced in the process units, and by integrating hydrogen purification units. In this study, process units of the refinery are defined as hydrogen sources and sinks, each being defined by a hydrogen flow, purity, contaminant and pressure. Considering the analogy heat load/flow rate (quantity) and temperature/purity (quality), the minimum hydrogen requirement may be defined by applying pinch based techniques A first representation has been proposed by [5], it was improved by [3] in the form of total flow rate — purity diagrams and surplus diagrams. The graphical methods suffer from the fact that these are not able to consider simultaneously purity and other contaminant constraints or the pressure constraints. Furthermore, these are not able to identify hydrogen consumption reduction obtained by reusing low grade hydrogen by mixing, although this operation is of common practice in the refining process. The use mathematical programming techniques appears to be more appropriate in order to consider simultaneously pressure constraints, contaminants restrictions and network topologies. A MINLP (Mixed Integer Non Linear Programming) hydrogen network model, which integrates compressor units, was proposed by [4] and solved after linearization as a MILP (Mixed Integer Linear Programming) problem. A method for integrating purification units in the hydrogen network was proposed by [1]. It uses

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relaxation techniques to allow algorithm convergence. This paper describes a multiobjective optimization methodology based on pinch concepts and mixed integer linear and non-linear programming to solve the minimum hydrogen requirement and the hydrogen network design problem. The major advantage of using the multi-objective optimization strategy is its ability to generate a list of Pareto optimal hydrogen system configurations that may be compared to synthesize the best solution. 2.Hydrogen composite curves When using composite curves of suppliers and consumers using hydrogen flow rate (hi)—purity(xi) diagrams [3], it is easy to understand that the computed minimum hydrogen requirement will not be appropriately calculated by this method because this representation will not allow the identification of hydrogen reuse by mixing. Recycling by mixing is similar to a heat pump effect : a consumer takes low purity hydrogen below the pinch and upgrades its quality by mixing with high purity hydrogen to produce an hydrogen requirement of medium quality above the pinch. Therefore, in order to represent the mixing effect, we considered that a mixer is a device in which hydrogen sources decrease their hydrogen purity up to a depleted level (XQ = 0) while hydrogen sinks increase their purity from the depleted level to the required purity. The analogy with the hot and cold composite is therefore trivial and allows computing the suppliers and consumers hydrogen composite curves and the corresponding minimum hydrogen requirement. Combining the two curves defines the hydrogen Grand composite curve, a limiting curve that is similar to the one of the water pinch representation [2] and against which the fresh hydrogen curve (the utility stream) can be plotted to define the minimum fresh hydrogen flow as a function of its purity (figure 1). In this representation, the total hydrogen flow is the slope of the utility curve whose fixed points are the fresh hydrogen purity and the depleted level that is common to all the hydrogen streams. According to the activation or not of the pinch points, the hydrogen flowrate may be higher then the minimum required and an excess of hydrogen will be read on the left of the Y axis. The use of purification units will allow to reduce the fresh hydrogen flow, by changing the pinch point location. Hydrogen of purity below the pinch point will feed a purification unit (figure 2) to produce high quality hydrogen above the pinch point and low purity hydrogen. The high quality hydrogen allows again an additional hydrogen reuse by mixing. This effect is illustrated on the integrated composite curves [7] of the purification unit given on figure 2. The modification of the hydrogen pinch point resulting from the integration of a purification unit allows the reduction of the fresh hydrogen flowrate.

Hydrogen Flow Rate [Nm''H2^]

Figure 1 : integrated composite curve of the fresh hydrogen integration

Hydrogen Flow Rate [Nm'^H Th]

Figure 2 : integrated composite curve of the purification unit integration

Design of Industrial Hydrogen Networks and Optimal Placement of Purification Units 1767 1. Multi-objective optimization strategy Although the graphical representation allows understanding the hydrogen network integration, to compute the hydrogen requirement and to identify the possible integration of purification unit, it does not allow to consider all the important constraints of the hydrogen network design like contaminant and/or pressure and compression constraints, piping and repiping, as well as depleted hydrogen valorization. For solving such problem, mathematical programming formulations have to be used. The optimal hydrogen network design problem is stated as : • Minimize: Cyear(2£)~CH2+ICpipe+ICcomp + ICpur+ OCcomp + OCpur " S H 2 • Under constraints : g(x,y)=o, where Cyear(x) is the sum of annual operating and investment costs. Cm fresh hydrogen cost, it accounts for the investment and the production costs of the hydrogen sources. ICpipe, ICcomp? ICpur respcctivcly annualised investment cost of piping, compressors and purification units, OCcomp, OCpur respectively operating cost of compression and purification untis, SH2 selling of the depleted hydrogen. It is interesting to note that as the major contaminant is methane, the lower heating value and therefore the selling price of depleted hydrogen is increasing when the purity is decreasing. Recycling of hydrogen with the highest possible purity is then favored by this formulation. X are the decision variables: i.e. the existence and the flowrates in the pipes of the network, the selection and the flowrate of the fresh hydrogen sources as well as the existence, the operating conditions and the sizes of the purification units. gfe^y) are the linear and non linear constraints equations. Purification units models

IMILP Network Design

Nonlinear cost and perfonmances

Si2S, o|K!mtiiig conditiojis

Evolutiofiaiy algorithm Multiple configurations for evaluation

• Clustering • Mufti-obJective Pareto Frontier • Fixed number of Iterations

Figure 3 : the solving procedure algorithm. As described on figure 3, this MESfLP problem is solved by decomposing it (variables and constraints) into two sub-problems. The upper level uses an evolutionary algorithm to compute the characteristics and optimal operating conditions of the purification units while the network is designed by solving a Mixed Integer Linear Programming problem at the lower level. 1.1. The hydrogen network superstructure. The hydrogen network is modeled as a superstructure that contains all the feasible connections between hydrogen sources and sinks of the process. The decision variables for the MILP problem are : the hydrogen flow rate between sink and sources (A^ ^ [IS[m^H2/h]), the connection existence (j^^^ {0,1}), the units utilisation level (^^ [-]). In this model, the purity, contaminants, pressures and hydrogen flowrate are assumed to be constant and it is assumed that the operating conditions of the process units will not

L. Girardin et al

1768

change if the hydrogen purity is higher than the present consumption. When defining the requirement, we consider the hydrogen quaUty that is really required at the inlet of the process units. When a recycling system is already in use, it only defines an existing connection that will be favored but it is not imposed a priori in order to allow new connections by repiping. A utilization level (qu) is associated with each process units and will multiply all the flowrates (sources and sinks) associated with the unit. A purification unit is defined in the same way, introducing one hydrogen sink (medium purity) and two hydrogen sources (one of high purity and one of low purity (lean stream). For these units, the purity and the nominal flows are considered as constant and will define the unit size and the corresponding investment. For such units, the level of utilization will only affect the operating cost. Each unit is geo-referenced and a piping path is estimated in order to account for pressure drops, existing pipes and new piping costs. The linear programming model is stated as follows. The investment costs are linearised functions. M i n i m i z e Cyear(x)=CH2+ICpipe+ICcomp + ICpur +OCcomp+ OCpur - SH2

Hydrogen production constraint

ndp) ,t,yp = l„..,np

/hpc=mcqu{c)yc

Hydrogen demand requirement

= \,...,nc

p=\

Impurity demand restrictions

^p,[spec]

^ f^c^ui^) . ^max

Specific impurity demand restrictions

y^ _ J

c,[spec]

p=i

Flow rate constraint

yp,c-hmin ^ hp^c ^3^^,^ • l . l m i n ( m ^ ^ , m ^ ^ ) 1 yp,c"PJC

1 ^c

•k-R'Tr,

r-1

[Pcl[pp-^p,c)]y

if Pp
Compression power

0 if Pp>Pc

Unfeasible connections A parametrized heuristic rule is added to limit the number of feasible connections by considering a maximum expected pay back time of each pipe. The limiting pay back time is considered as a decision variable of the upper level. ;^;7,c=0,pe{l,...,«^},ce{l,...,«^}

1.2. The upper level model The definition of the hydrogen sources and sinks of the purification units are defined in the upper level using an evolutionary algorithm. In the non linear problem, the decision variables for each of the proposed purification units are the hydrogen nominal flowrate in the purification units (hf [ Nm^H2/h]), the feed purity (Xf [kmolH2/kmol]), the feed total pressure (Pf

[bar]), the product total pressure (Pp [bar]), the product hydrogen

purity (x^ [kmolH2/kmol]) and the geo-referenced unit position {px,py[m\).

Using

thermo-economic data from [8] for purification units, the performances of the units are

Design of Industrial Hydrogen Networks and Optimal Placement of Purification Units 1769 modeled and define the data for the network design model. Cost are estimated considering the nominal size defined in the upper level. 1.3. Solving the multi-objective optimization problem The use of an evolutionary algorithm to solve the upper level problem allows to use a multi-objective optimization strategy. The evolutionary algorithm [6] generates Pareto optimum frontier. It does not require the coding of continuous variables and works with clustering techniques that allow isolating different systems configurations. The objective functions are the operating cost (that relates to the hydrogen recovery) and the investment cost. From the results of the lower level problem, non linear costing of the piping and the unit investment are performed. The evolutive solver tries to improve solutions by proposing new admissible operating points. The algorithm proceeds like this until it reaches a fixed number of iterations. This resolution scheme has several advantages over schemes that solve simultaneously a set of MfNLP equations: • its modularly aspect is suitable for adding new models equations. • it can track several design objective simultaneously, • it generate multiple clusters of solutions. • It allows to handle problem discontinuities (e.g. unit design conditions and purity optimisation)

2. Case study The problem to be solved is the retrofit of a hydrogen recovery network. The process sources and sinks are given in table 1. The fresh hydrogen actual consumption is of 15691 Nm^H2/h (figure 1). The results are presented in the form of the Pareto curve of figure 4. The different tics shapes are used to identify the different solution clusters found by the algorithm. Each point corresponds to a different hydrogen recovery configuration. For the analysis, each configuration is represented automatically. Figure 5 gives an example of a network configuration, the dashed lines represent new connections, the size of connections are proportional to the hydrogen flow. The results show a hydrogen saving ranging from 24% to 29% with pay back time from 1 to 5 years. Table 1 : producers and consumers for the problem Flow rate Purity Pressure Flow rate Purity Pressure [%vol] [bar] Producers [Nm^/h] [%vol] [bar] Consiuners [Nm^/h] G_IN 35.4 FRESH 18246 75.6 23.6 75.6 29.6 A BP 20 A_IN 11517 558.6 17 75.6 29.6 BIN A_HP 10323 79.3 17 5170.8 79.1 31 160.2 25 C_IN 7232.4 BBP 26 79.2 46.4 BHP E_IN 852.6 78.6 26 35.4 75.6 29.6 C_BP 2899.8 54.2 F_IN 33.9 2494.8 75.6 32.6 C HP 2611.8 79.1 33.9 DIN 8406.6 79.3 17 D OUT 8938.8 79.3 17

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o u iP

a»

<^ <

V»o^

E j/j

100

>

^i. 80

-1300 -1200 -1100 -1000

-900

-800

-700

J?^AL„

-600

Operating Costs [kEuro/year] Figure 4 : Pareto optimum Frontier for the problem

-500

-400

Figure 5 : example of network representation

3. Conclusion Combining evolutionary algorithm and MILP network design models, a multi-objective optimization methodology has been developed to solve the hydrogen recovery network design problem considering the integration of purification units. The major advantage of using the multi-objective optimisation strategy is the generation of a list of Pareto optimal process configurations. Each of the configurations can then be analysed from practical point of view in order to define the best configuration. The developed methodology has been applied to solve real scale problems. The best configurations show hydrogen savings as high as 29% with reasonable pay back.

References [1] Liu, F. and Zhang, N., Strategy of purifier selection amd integration in hydrogen networks. Chemical. Engineering Research and Design, Vol 82, pp. 1315-1330, (2004) [2] Y. Wang and R. Smith. Wastewater minimisation. Chemical Engineering Science, 49(7):1981-1006,(1994). [3] Alves, J. and Towler, P., 2001. Analysis of Refinery Hydrogen Distribution Systems Ind. Eng. Chem. Res., Vol 41, pp. 5759-5769, (2002). [4] Hallale,N. and Liu, F., Refinery hydrogen management for clean fuels production. Advances in Environmental Research, Vol 6, pp. 81-98, (2001) [5] Towler, G.P., Mann, R., Serrierre, A. J-L. and Gabaude, C , Refinery Hydrogen Managment: Cost Analysis of Chemically-Integrated Facilities, Ind. Eng. Chem. Res., Vol 35, pp. 23782388,(1996). [6] Molyneaux, A., Favrat, D., and Ley land, G.. A New Clustering Evolutionary Multi-Objective Optimisation Technique. In Third International Symposium onAdaptative Systems, Institute of Cybernetics, Mathematics and Physics, pages 41-47, 2001. [7] Marechal, F. and Kalitventzeff, B. Targeting the minimum cost of energy requirements : a new graphical technique for evaluating the integration of utility systems. Computers chem. Engng, 20(Suppl.):S225-S230, 1996. [8] Bhide, B.D, Stem, S.A, Membrane processes for the removal of acid gases from natural gas, Joumal of Membrane Science, Vol 81, pp. 209-237, (1993).

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Modelling and Simulation of a Tyre Gasification Plant for Synthesis Gas Production Narendar R. Mitta, Sergio Ferrer-Nadal, Aleksandar M. Lazovic, Jose F. Perales, Enric Velo, Luis Puigjaner Chemical Engineering Department-CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain Abstract Gasification is becoming one of the best alternatives for waste solids reuse, especially for those, as tyres, which can cause a significant environmental impact. The proposed gasification model improves the understanding of the process and can be used as a predictive tool at the optimization stage. Validation of this model is carried out using the gasification pilot plant located at the Chemical Engineering Department of Universitat Politecnica de Catalunya (UPC). Keywords: Waste tyres, gasification technology, process modelling. 1. Introduction The rapid increase in vehicle usage since the past two decades results in the generation of waste tyres to an alarming rate. Around 2, 6000,000 metric tons of used tyres were produced in the European Union in the year 2000 (Mastral, 2002) while only a small percentage of these waste tyres goes to reuse. At the moment, the most frequent option for this waste solid removal is land filling which causes environmental and hygiene problems. Furthermore, land filling is a potential danger because of the possibility of accidental fires with high emissions of hazardous gases. A better alternative is their employment as substituting fossil fuels in some industries like cement industry. But the complex nature of the tyres and the stringent environmental regulation makes it difficult to recycle through incineration. Nowadays, gasification is a commonly used technology for extracting the energy from solid materials like coal, coke, biomass, scrap tyres, etc. (Pan, 2000). This technology can make use of the high energy content of the tyres. These solid materials are gasified to produce a gas containing mainly carbon monoxide and hydrogen. The gases are utilized in gas turbines of Integrated Gasification Combined Cycle (IGCC) systems. The use of hydrogen in fuel cells is another very attractive alternate for power production. In general, a typical gasification system essentially consists of a gasifier unit, a purification system and an energy recovery system. Gasifier reactors are basically classified as fixed beds, fluidized beds and entrained beds. Fluidized bed reactors have an excellent gas-solid contacting leading to very good heat transfer together with the ease of solids handling. In this work a rigorous model of a fluidized bed gasifier unit has been developed which could be used for enhanced hydrogen production. Numerical simulations are necessary to help in finding out feasible operating conditions to achieve better process performance. The introduction of solids in a process changes the heat and mass balances, even if the solid essentially passes through the process as an inert component. Aspen Plus© is chosen as a simulation tool because of its capability on the solids handling. Aspen Plus includes particular physical property model and

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accurately represents the solid particle. In addition, FORTRAN and MATLAB calculation routines are introduced in the model. The development of the model is described in the following sections. 2. Gasification model This gasification model has been divided into three different stages: drying, devolatilization-pyrolysis and gasification-combustion. The Aspen Plus flowsheet of the model is shown in the Figure 1. Qog) - j WATER \-

Figure 1: Aspen Plus flowsheet of the Gasification model When the raw material is fed, the first step is the heating and drying of the particles. A RSTOIC module has been used to model this instantaneous drying. Due to the high content of volatiles in the tyre it is important to consider the devolatilization step of its conversion. This devolatilization process, namely fast pyrolysis mechanism, produces volatile gases, tars and char. There is no general model for the prediction of the volatiles composition, being necessary to complement a good model with experimental results. RYield block is used to model the pyrolysis/devolatilization part of the model. For modelling purposes it is essential to know the mass fraction of the initial fuel, which is pyrolysed. It is assumed that the total yield of volatiles equals the volatile content of the parent fuel determined by the proximate analysis. The RYIELD module can convert its feed into a stream made up by the equivalent elemental components of the feed at the same enthalpy level. From the pyrolysis and kinetic experiments with the tyre it is observed that the temperature range for pyrolysis is between 300 and 500 °C (DTG maximum is observed at 397 °C). From these studies, the temperature for RYIELD is fixed to 500 °C. RGIBBS reactor module is used to model the gasification and combustion reaction. The stream from the RYIELD block as well as the preheated oxygen and steam are directed into the RGIBBS module, which can predict the equilibrium composition of the produced gas from RYIELD at specified temperature and pressure. The ash of the gasification process will be removed fi'om the RGIBBS module. In the model, an overall equilibrium approach was employed by neglecting the hydrodynamic complexity of the gasifier. Although higher hydrocarbons, tars and oils, are produced in the gasifier they have been considered as non-equilibrium products to decrease the complexity of the model. Therefore, CH4 is the only hydrocarbon taken into

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consideration in the calculation. All the results from the model were normalized to make them free from tars. The sulphur in the tyre is assumed to be converted mainly into H2S. Steady state conditions are assumed in the model. 3. Experimental 3.1. UPC Pilot scale plant The pilot plant is depicted in Figure 2. It consists of a fluidized bed reactor able to operate under a wide range of different feeds and conditions. The reaction chamber is a cylindrical vessel (stainless steel, ANSI-904-L) provided with electronically controlled rotary screw-feeders located over the gas distributor grid. Heating system consists of three individually controlled electric heaters which supply heat not only for the start-up but also maintain the steady temperature during operation. Rotary blades facilitate the discharge of materials from the bottom. The gasifying agent is pressurized air from a compressor and is injected to the reactor together with an adjustable amount of steam. Two cyclones, a filter and a condenser-cooler comprise the gas clean-up section. After measuring the composition of the produced gas a burner is provided to bum-off the produced gas before the vent. The plant is continuously monitored to make available measurements of operating conditions and outlet gas stream composition. Further details can be found in Pan et al. (2002).

Figure 2: Overview of the Pilot scale plant

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3.2. Procedure The mean particle diameter of the raw material is around 1mm. Hydrodynamic experiments were performed to calculate the required amounts of the fuel, air and water for the gasification experiment. The operating temperature of the gasification experiments was fixed to around 950 °C. Distilled water and the air from the compressor are pre-heated until 500 °C before feeding into the gasifier as gasifying agent. N2 is used as a purge gas during the start-up of the system. Around 3 to 4 hours time is required to reach the required experimental conditions. This gasifying agent, pre-heated air and steam, is introduced below the distributor. As this feeding material (tyre) is new to the system and to avoid further complexity, constant feed rate was chosen to perform the experiments. The gas composition is determined online by a continuous analyser and microchromatography of gases. CO2 and O2 are measured using IR and electrochemical sensors whereas measurements of concentrations of H2, O2, N2, CH4 and CO are available every 90 seconds using a thermal conductivity sensor. In the tests, once the steady state is reached, an outlet gas is obtained with a very uniform composition with time. 4. Results and discussions 4.1. Sensitivity Analysis Sensitivity analysis is performed to monitor the dependence of different parameters on the composition of the produced gas from the reactor. In this work, the temperature and feed composition are analysed in order to evaluate their effects on the composition of the produced gas. 4.1.1. Effect of Temperature The sensitivity analysis for the reactor temperature effect on the final gas composition between 750 and 1100 °C is shown in Figure 3. For this sensitivity analysis, the feed conditions of the experiment are used in the model. From the Figure 3, the increase in CO and H2 and the decrease in CO2 and CH4 is may be because of the exothermic steam methane reforming and CO2 reforming reactions. Temperature

-co -C02 CH4

a

-H2

750

800

850

900

950

1000

1050 1100

Temperature °C

Figure 3: Effect of temperature on final gas composition. Table 1 presents the composition values of the final gas obtained from the model as well as from the experiments at the same temperature and pressure. The results from the model are deviated in a small percentage because of the several simplications in the model.

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Table 1: Composition of gas producedfrommodel and gasification experiment at 950*" C and latm. Components

Model (% Vol.)

Experimental (% Vol.)

H2

15.5 8.1 17.8 11.6

16 7.3 15

CO C02 CH4

4.1.2. Effect of Fuel/Air ratio Figure 4 shows the effect of fuel to air ratio on the CO and H2 composition of the product gas. Here the flow rate of water is considered as constant. Sensitivity analysis is made by varying the fuel to air ratio from 0.2 to 0.8. A clear increase in the H2 and CO is observed with the increase in the ratio. Fuel/Air 25 n 20-•—CO ^

--^h^H2

105 0^ 0.2

0.4

0.6

0.8

Ratio

Figure 4: Effect of Fuel/Air ratio on compostion at operating conditions 950 ^'C and 1 bar and constat water flow 4.1.3. Effect ofFuel/H20 ratio Figure 5 shows the effect of fuel water ratio on CO and H2 composition of the product gas. Sensitivity analysis is made by varying the fuel to water ratio from 0.2 to 0.8. Here decrease of both concentrations is observed.

Figure 5: Effect of Fuel/Water ratio on composition at operating conditions 950 ^'C and 1 bar and constat flow of air

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5. Conclusions In this work, the gasification process is modeled using ASPEN Plus© process simulator. The developed model is able to predict the composition of the produced gas under various working conditions, including the flow rate, composition and temperature of the feed materials, as well as the operating pressure and temperature. Within a range of smaller deviations the model showed similar results from the experiment. The differences may be caused by the assumptions of complete conversion and several simplifications. The sensitivity analysis showed the effect of operating temperature of reactor on the composition of final gas. The model can be used to gain the primary understanding about the gasification process and for optimization purposes which are underway. According to the above discussion this model can be used to find the final gas composition from the gasification process of other soHd waste fuels. The future modeHng efforts will focus on including the chemical kinetics and possible gasification reactions in the model. The possibility of including the hydrodynamics of the fluidized bed reactor in the model is also a future interest. Acknowledgement Financial support received from the European Community projects (MRTN-CT-2004512233, RFC-CR-04006, INCO-CT-2005-013359) and the Generalitat de Catalunya (FI grant with the European Social Fund) is fully appreciated. References Mastral, A. M., Murillo, R., Garcia, T., Callen, M. S. & Lopez, J. M. (200.2) Study of the viability of the process for hydrogen recovery from old tyre oils.Fuel Processing technology 75, 185-199. Pan, Y.G., Velo, E., Roca, X., Manya, J.J. & Puigjaner, L. (2000). Fluidized-bed co-gasification of residual biomass/poor coal blends for fuel gas production. Fuel 79, 1317-1326. Sofer, S.S. & Zaborsky, O.R (1981). Biomass Conversion Processes for Energy and Fuels. Gomez, C.J., Manya, J. J., Velo, E. & Puigjaner, L. (2004). Further applications of a revisited summative model for kinetics of biomass pyrolysis. Ind. Eng. Chem. Res. 43, 901-906. Nougues, J.M., Pan, Y.G., Velo, E. & Puigjaner, L. (2000). Identification of a pilot scale fluidised-bed coal gasification unit by using neural networks. Applied Thermal Engineering 20, 1561-1575

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Library for modeling and simulating the thermal dynamics of buildings Juan I. Videla ^ and Bernt Lie^ * ^Telemark University College, P.O. Box 203, N-3901, Porsgrunn, Norway. Abstract Today's buildings consume more energy than any other sector of the U.S. economy, including transportation and industry; a similar importance can be expected in most European countries. Due to the increased interest in saving energy in buildings, new dynamic thermal models that describe transient response in more flexible modeling languages become necessary. Traditional building simulation software (e.g. TRNSYS or EnergyPlus) are based on almost intractable simulation codes, difiicult to maintain and modify, that predict system quantities at fixed time intervals. More clear code, properly separated from the simulation envirormient, with variable time step solvers would be necessary for the assessment of HVAC system performance with quicker dynamics. Following some ideas from a previous building thermal behavior library, a new enhanced Modelica library for modeling buildings is presented. The library basically consists of a combination of lumped parameter models and one-dimensional distributed parameter models that interconnects with each other through a set of common interfaces. Object-oriented features like class parameters and multiple-inheritance are used to improve the library structure making it easy to read and use. Complex building topologies can be built-up from component blocks that result in physically correct compound models that can be efiiciently simulated and studied in any Modelica simulation environment. Keywords: Building modeling, Object-Oriented Modeling, Modelica. 1. INTRODUCTION Buildings are complex nonlinear dynamic systems that involve many physical aspects — heat conduction, convectiveflow,radiation, mass flows, etc.— that must be properly addressed. Due to the complexity and highly coupled nature of these phenomena, simulation seems to be the most cost effective way to study how to improve the energy efiiciency in buildings (Clarke 2001). Although there are several building simulation tools available (Underwood & Yik 2004), typically composed of component libraries or template like forms to define model parameters, the great majority of the underlying models remain as blackbox modules for the user or are very difiicult to modify. Modelica is an equation-based object oriented language for modeling complex and heterogeneous systems, that perfectly fits with building models' requirements. Today, there are at least three simulation tools based on the Modelica language; in this work, it has been used, Modelica the building library language, and Dymola as the simulation environment (Dymosim 2006). * [email protected]

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Buildings have a strong multi-engineering hierarchical structure where several levels of detail can be properly structured in a library and well defined using modem programming techniques like object-oriented modeling, allowing extensive reuse of components through inheritance, acausality, and aggregation. Thus, Modelica components can be easily interconnected increasing complexity while keeping the library clear for the end-user. Equation-based code is open for model developers in order to enhance or modify the library to their own needs. The models in the building library are designed for system level simulation. A multi-zone modeling approach to buildings has been used, where models are formulated using mass and energy balances for the lumped control volumes, and a steady-state hydraulic fluid assumption for the bi-directional mass flow components. The nodal network method used, treats a multi-zone building as a set of pressure nodes represented by rooms where mass flow connections are defined using components for cracks, doors, ventilation, etc. This approach allows modeling inter-zone convective energy flow. Several thermal dynamic heat transfer modeling methods for buildings exist (Kallblad 1983). Among the heat balance methods, the treatment of conductive heat transfer through the elements that form the building envelope has generally been used as the main classifier of thermal model for a building. Thermal capacitance has been considered in conductive elements where the method of lines has been used to discretized the PDE governing the one-dimensional conduction through the several layers of material, radiation inter-exchange has been treated through a radiation network among gray surfaces, and an empirical heat transfer coefficient has been used to model convective heat transfer components within a lumped air volume. The building library is constructed from physical principles, and verified by comparing its performance with a previous building library developed at Universitat Kaiserslautem (Feigner et al. 2002). 2. LIBRARY STRUCTURE The library is organized in packages that contain the building components. Basic Elements describe single physical phenomena like conduction, convection, or radiation. Most of the constructive elements are formed through inheritance and aggregationfi"omthese basic elements representing combined complex phenomena like heat transfer in a wall by the combined effects of radiation, convection, and conduction. Complex elements are template-like components built typically from aggregation, and meant to be used after a simple customization. Interfaces defines all the connector classes required by the components while in Boundary conditions, several sources and sinks are defined. • Complex elements - several room and story templates. • Constructive elements - Walls, ceiling, windows, doors, air volume, etc. • Basic elements - conduction, convection, radiation, heat/mass storage, and orifice models. • Interfaces - all connector and components used in interconnections. • Boundary Conditions - Variable and fixed sources and sinks. 2.1. Connectors A correct design of components requires explicit interfaces (Fritzson 2004). Explicit interfaces are implemented through connectors that reflect interaction with the surroundings. This library mainly uses two types of connectors: the Mass/Enthalpy transport connector that describes mass and enthalpies flows between components, and the Heat transfer connector used for thermal energy transfer without mass flows, typically used in purely conductive processes (see Figs. 1 and 2). Within connectors, the required intensive and extensive variables to

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interconnect components are defined. pastial coniMCtor hpHM_Por"Mass/enthalpy <

p a r t i a l coniMctoi; Heat_ "Heat t r a n s i

SI.Pressure p ; SI.SpecificEnthalpy h; flow SI.MassFlowRate M_flow; flow SI.EnthalpyFlowRate H_flow;

satFlowRate Q_flow;

Kid hpHM_Port;

Fig. 1 Mass-Enthalpy flow Modelica connector.

Fig. 2 Heat transfer Modelica connector.

Fig. 3 Wall UML graph.

2.2. Control volume model The air volume inside a room can be modeled using the mass and internal energy balances within a lumped control volume (see eqs. 1 and Fig. 5 for Modelica code): H{t) U(t) H{t)

=

= =

m{t)h{t) m(t)u{t)

(1)

U{t)+p{t)V{t)

where m is the mass within the control volume, rhj the mass in/outflows, Hj the enthalpy in/outflows, Qk the heat in/outflows, U the internal energy, H the enthalpy, p the pressure, V the volume, and h and u, the specific enthalpy and specific internal energy, respectively. The ideal gas law is used since nominal temperatures, pressures, and densities for buildings are well within the range of validity of this thermodynamic model. The ideal gas law is implemented in a partial model class, and used when necessary in other components through inheritance (see eqs. 2 and Fig. 4 for Modelica code). The preferred states and their initial conditions are then selected, and model reformulation is automatically done by Dymola using its built-in symbolic math capabilities for dealing with DAE (Differential Algebraic Equations): p{t)v(t) u{t)

= RT[t) = uo + c^{T{t)-To)

h{t)

=

u{t)-\-p{t)v{t)

(2)

where, v is the specific volume, T the temperature, UQ the specific internal energy at To, and R the gas constant. 2.3. Conduction over a homogeneous slab/wall Two phenomena are of interest when modeling conduction through a slab/wall layer: their heat capacity and their resistance to heat transfer. The heat conduction through a homogeneous isotropic solid (e.g. layer of a wall), with properties of the material independent of temperature, can be modeled as a ID-PDE of the form: dT{x,t) dt

k dqix,t) pc dx

Acond

~

udT{x,t) ~'^ dx

(3)

where T (x^t) is the temperature, Qcond {x, t) is the conductive heat flow rate, Acond, k, p, and c are conductive area, thermal conductivity, density and specific heat capacity, respectively. This equation can be solved with several analytical methods, like temporal analysis through the Convolution theorem, the Laplace domain analysis, the Response Factor Method, the Transfer Function Method, and the Admittance Method. The most practical way to implement it in Modelica, which only has time as independent variable, is using the method of lines. Several models representing these different mathematical solution can be interchanged in the final model implementation through class parameters.

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2.4. Other Basic components Radiation heat exchanges among the surfaces of a room are modeled using the radiation heat transfer network analysis method, based on the assxmiption that all the surfaces are plain grey surfaces. The radiation heat flow rate gain of each of the grey surfaces that are exchanging radiation with each other, can be described by: (4) (5)

where 6^^ is the emissivity of wall i, Qr^is the incoming heat flow rate of wall i due to longwave radiation, A^i is the area of wall i, E^^, is the emissive power of a black body at temperature T^y., J^y. is the radiosity of the internal surface of the wall i, and a is the Stefan-Boltzmann constant. A heat balance consideration for each surface accounting for the radiosity exchange among the other surfaces forming the enclosure, can be established with eq. 5, where matrix F corresponds to the view-factor values calculated fi'om the room geometry. Convection takes place in the boundary region between any solid surface (e.g. walls, ceiling, floor) of a room and the air volume inside. The heat transfer coefificient is considered constant (average value), and no heat capacitance is considered, that is, Qc^ + Qc^^^ = 0 ,(0

h (T^, {t) - Tair {t))

(6)

where Qc^ is the heat flow rate leaving the boundary due to convection, h is the average heat transfer coefiicient, A^^ is the wall area of wall i, Ty^^ is the internal wall surface temperature, and Tair is the room temperature. The orifice equation assuming isentropic flow and no mass storage in the flow component can be used to describe mass flows through openings (see the Modelica code in Fig 6) like ventilation, open windows or doors:

rr^it) -

C,A.^^pS)^|^

H{t)

=

m{t)hi{t)

(J)

where Cd is the discharge coefficient, Ao the opening area, Ap the pressure difference across the opening, p^ the density of the incoming air, H is the enthalpy flow rate, and hi is the upstream specific enthalpy. {Mirtial modal IdealGasLa»

! Heat Store

modttl OpenL I IdealGasLaw; terfaces.hpHM_TwoPin

SI.Pressure P; SI.SpecificVolume v; SI.Temperature T; SI.SpecificInternalEnergy i SI.SpecificEnthalpy h; der(in) = M_flow; der(0) = H_flow + Q_£low; H = U + P*V;

1 IdealGasLaw;

I,inearRoot(2*(Pl - P2) *v, if ((Pi - P2) >» 0) th«n P = PI; h - hpHM_Port_al.h; HI = hpHM_Port_al.h*Ml; •Is* P = P2; h = hpHM_Port_bl.h; HI = hpHM_Port_bl.h*Ml; •nd if;

•ad Mass Heat Store; •nd OpenL;

Fig. 4 Ideal gas law.

Fig. 5 Control volume.

Fig. 6 Bidirec. orifice eq.

Numerical difficulties arise when the orifice equation is used at flows speeds close to zero due to the singular derivative of the root fiinction. This can be avoided by using a linear interpolation instead of the root fiinction in a user-definable region around zero flow.

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3. CONSTRUCTIVE AND COMPLEX ELEMENTS 3.1. Building more complex components More complex components like a wall addressing radiation, multi-layer conduction and convection can be easily constructed through aggregation and inheritance. Ventilation, doors, windows, ceiling, floor, junctions and cracks can be defined firom the basic components already defined. As an example a wall is constructed following the UML diagram shown in Fig. 3. In general, each constructive component can be open and its internal components can be further customized. 3.2. Room template A simple room component is shown in Fig. 7, where constructive components have been used to compose it as a general template-like component. Internal component parameters are directly defined through the room parameters. Although the room component has a fixed geometry, it can be easily modified by adding the necessary additional constructive components.

D

g = --D

CK D-

-D

L>i

Fig. 7 Room internal elements.

Fig. 8 Building with boundary conditions.

SIMULATION A simple ten room building is used to illustrate the building library.(see Fig. 8). Boundary conditions accounting for daily temperature and pressure variation are used. The building is left passive without any HVAC system acting and without any agent model. As seen in Fig. 9, the pressures rapidly converge to a common value producing an enthalpy flow (inter-zone convection). After this initial transient response, the slower thermal dynamic of the building reaches the stationary response after a few days, as seen in Fig. 10 for two neighbor rooms. The thermal response has been compared with the ATPlus library (Feigner et al. 2002) and both libraries show similar transient behavior but different stationary response; one reason can be the mathematical solution used to solve the ID-PDEs (e.g. level of discretization of each layer) representing conduction, or the view factors used in the long-wave radiative exchange within the enclosure. 5. CONCLUSIONS While simulation tools gain generality and more computational power is available, more detailed dynamic models, advanced controllers, and energy management strategies can be studied. Using a library of models is a widespread way to structure building simulation and modeling

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applications, but since a few of these tools expose the underlying models, typically the user has to rely on the quality and completeness of the provided documentation (Tummescheit & Ebom 1998). The primary goal of this paper is to present a base library with common model parts which are user-extensible with a consistent logical structure, that is, neither too abstract and difficult to use, nor obvious and over-crowded of components. The presented library can be easily modified and customized to special needs. Room 1.T (HiTLib)

Room 1 .T (ATPIus) 290 ••<

\ \

e 285 £280 275

V-^ \/p^k^^

270

Fig. 9 Pressure response of the building. Fig. 10 Thermal response for two libraries. The Modelica/Dymola simulation tools allow the implementation of advanced controllers in a general simulation environment, and even agent interaction/perturbation (opening a window, turning on the oven) can be modeled. Modelica/Dymola can also be interfaced with Matlab/Simulink where powerful control toolboxes already exist. The building library correctly models the inter-zone convective energy flows and building thermal response. The radiation network treatment has proved to be rigorous enough for the system level simulation, and the DASSL integrated DAE solver has proved to be able to handle this kind of stiff problems (quick pressure variations with slow thermal response). Future work includes developing HVAC system components and meteorological models. A validation test of the library will be implemented as well (Tiller 2001). REFERENCES Clarke, J. A. (2001), Energy Simulation in Building Design, Butterworth-Heinemann. Dymosim (2006), Dymola. www.dynasim.se. Feigner, F., Agustina, S., Bohigas, R. C, Merz, R. & Litz, L. (2002), Simulation of thermal building behavior in modelica, in 'Proceedings of the 2nd International Modelica Conference', Oberpfaffenhofen, Germany, pp. 147-154. Fritzson, P. (2004), Principles of Object-Oriented Modeling and Simulation with Modelica 2.1, IEEE press. Kallblad, K. (1983), Calculation methods to predict energy savings in residential buildings. Vol. Anex III,D4, Swedish Council for Building Research, International Energy Agency, Stockholm. Tiller, M. M. (2001), Introduction to Physical Modeling with Modelica, Kluwer Academic Publishers. Tummescheit, H. & Ebom, J. (1998), Design of a thermo-hydraulic model library in modelica(tm), in 'ESM'98: Proceedings of the European Simulation Multiconference', Manchester, UK. Underwood, C. P. & Yik, F. W. H. (2004), Modelling Methods for Energy in Buildings, Blackwell Publishing.

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A New Method for Designing Water Network Based on Variable Removal Ratio of Treatment Process SONG Lili, DU Jian, CAI Shaobing, YAO Pingjing Dalian University of Technology, Dalian,! 16012, Liaoning, China Abstract This paper presents a method for water network design considering removal ratio of treatment process as a parameter based on biochemical treatment model. The removal ratio is calculated by model of plug-flow reactor of Monod and Andrews. Compared with traditional method, the approach can assure that the calculated values of the concentration at the outlet of treatment process are approximately same with the practical values; hence the more feasible water network can be obtained. On the basis of the variable removal ratio of treatment process, the combination of pinch analysis and mathematical programming is a more efficient and feasible method for water network design in targeting the minimum total annual cost. The proposed approach is demonstrated with an example and the result is more close to practical values compared with literatures. Key words: water network design; variable removal ratio of treatment process; combinational method 1. Introduction In recent years, there has been considerable development of systematic methods to achieve fresh water and wastewater minimization in industry. This has been driven by water scarcity and stricter environmental legislation on industrial effluents, as well as the rising costs of fresh water and effluent treatment. Takama (1980) addressed an approach for optimal water allocation in a petroleum refinery based on a superstructure of all possible re-use and regeneration opportunities. Then El-Halwagi (1989) defined the composite curve to denote mass exchange operation, which was adapted from the methodology developed for heat exchanger networks by Linnhoff and Hindmarsh. Remarkable work was done by Wang and Smith (1994) by introducing the important concepts of 'water pinch' and 'limiting water profile', which is further improved later by Doyle & Smith (1997), Kuo & Smith (1998) and PoUey & Policy (2000). The graphical method can give the minimum fresh water in a direct way, but when multiple-contaminants are present, graphical methods require assumptions for ease of implementation, some of which may be difficult to justify. Huang, Chang, Ling and Chang (1999) presented a mathematical programming solution by solving nonlinear problem (NLP). Recently, Bagajewicz and Savelski (2000, 2003) showed how to reduce a nonlinear program to a linear program (LP), by inserting the maximum outlet concentration conditions. In recent years, mathematical programming have greatly developed by combining with advanced algorithms and optimizing approaches. The mathematical programming method is effective in optimizing largescale systems, but is difficult to interpret, giving designers fewer insights compared with graphical methods.

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In all of above literatures, the optimal water networks were determined by given process conditions with fixed coefficients, such as the removal ratio of wastewater treatment processes, namely, each treatment process had the same removal ratio to all treated streams in different concentration of contaminant. However, it is difficult to fulfill for practical streams with relatively high or low concentration. This paper presents a new approach to overcome the problems. 2. Model of Wastewater Process Since biological wastewater treatment has been widely used as core part of treatment process, it is essential to understand the characteristics of each biological process to ensure that the proper environment is produced and controlled effectively. 2.1 Models of Monod and Andrews Environment conditions have an important effect on the survival and growth of bacteria and effective environmental control in biological wastewater treatment is based on an understanding of the basic principles governing the growth of bacteria. The included conditions are temperature, PH, concentration of substrate or nutrient, the mean hydraulic retention time, etc. Here, we only discuss the effect of substrate. Assuming other conditions have been settled properly, it has been found experimentally that the effect of a limiting substrate or nutrient can often be defined adequately using the following expression proposed by Monod: // = //

*

(1)

The Monod equation is applicable to no toxic system. But if there are some toxic compounds and the substrate concentration reach a certain value, the growth of bacteria will be restrained. Andrews proposed the following amendatory equation: /^ = /^.ax*

—,

(2)

K^+S + S IK^

2.2 Models of Plug Flow Reactor of Monod and Andrews Fluid particles pass through the tank and are discharged in the same sequence in which they enter. The particles retain their identity and remain in the tank for a time equal to the theoretical detention time. This type of flow is approximated in long tanks with a high length-to-width radio in which longitudinal dispersion is minimal or absent. The plug-flow system is shown in Fig 1. •

*

^

w

\\

I I

Fig 1 Definition sketch used for the analysis of flug-flow reactor A kinetic model of the plug-flow system is mathematically difficult, but Lawrence and McCarty have made two simplifying assumptions that lead to a useful kinetic model of the plug-flow reactor: (1) The concentration of micro organisms in the influent to the reactor is approximately the same as that in the effluent from the reactor. The resulting average concentration of micro organisms is symbolized as X .

A New Methodfor Designing Water Network

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(2) The rate of substrate decrease is equal to the rate of substrate utilization as the waste passes through the reactor, given the following expression: dS I dt

V S

X

Ks+S

max

(3)

Eq. (3) can be simplified to yield K dS + SdS = -V XSdt S

max

(4) ^

^

Above equation can now be integrated between the limits of-S'^ to So and 0 to z' to yield K^ \n(S^ /S^) + (S-S^) = v^^Xt (5) The result of Andrews model apply to plug-flow reactor is similar to the model of Monod, receiving the equation as follows 1 (6) {S'-S:) + {S-S^) + K^\n{SIS^) = v_Xt 2K, 3. Wastewater Treatment Network with Variable Removal Ratio of Treatment Process The wastewater treatment network is described as a system, in which wastewater can be treated more than once, including multiple streams and multiple wastewater treatment processes. The flowrate to each wastewater treatment process have been given. The removal ratio of treatment process is a parameter, which can be confirmed according to the models of plug flow reactor of Monod and Andrews. 3.1 Set up mathematical model to solve the wastewater treatment network Entering wastewater treatment network, each stream flows into a splitting node, where stream can flow to all wastewater treatment process. There is a mixing node in front of each treatment process, where all streams coming from other operation units are mixed. There is another splitting node behind each treatment process, in which stream may flow to discharge node or mixing nodes. And all wastewater streams flow to a mixing node at last, namely the discharge node, where the concentration of contaminant must satisfy the environmental regulations. In Fig.2, superstructure model of wastewater treatment network includes three wastewater streams, a discharge pool and two wastewater treatment processes, one or two which are biological treatment, complying with the model of plug flow reactor of Monod or Andrews.

Figure 2 Superstructure model of wastewater treatment networks 3.2Decease uncertain parameter by water pinch analysis

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The water pinch can be solved from the procedure presented by Wang and Smith. Streams whose contamination concentration is higher than that of the pinch are fully treated by the treatment process, those that are equal to the pinch are partially treated and partially bypass the treatment, and those that are lower than the pinch completely bypass the treatment. According to the design rule, the streams partially treated are regarded as parameters, while streams fully treated and completely bypass the treatment are regarded as constants. So the amount of uncertain parameters is deceased largely. 3.3 Set up objective function The capital cost and operating cost functions of treatment process are expressed by wastewater flowrate, and the mathematical model is shown as follows: Objective function: Min^i^xAff

^y^^.D

(7)

Subject to: (1) Mass balance of wastewater treatment process t: (8) (2) Contaminant mass balance of wastewater treatment process t: I^s Hc,,j,^ lc,j,oJHc,j,„

-c,j,^)=v^Xt

o'- K, Hc,jj„ /c,j^o.)Hc,jj„ -c,j,J>+^{c,J

(9)

-c,jJ)=v^'Xt

Where Ks, Kj, Vmax and xt have been known. (3) Mass balance of wastewater stream w: x +x =/

Z

t,w

e,w

(10)

(11) V

J w

/

t

(4) flowrate limiting of wastewater treatment process t:

Z^'..v^^.

(12)

(5)limitmg concentration of wastewater in discharge pool: Z

^efuj^ou, + Z t

W

^.,wCw,y,o«, A Y . L \ \ w

<^e.j

(13)

J

It is often difficult to obtain the global optimum solution for the kind of non-linear programming. Applied the rule of pinch analysis to deceased the amount of uncertain parameters, the superstructure is easy to solve the global optimum solution. 4. Example The case is taken from Kuo and Smith (1997). Three wastewater streams are produced and must be treated before discharge. The flowrates of streams and the concentration of a contaminant (H2S) involved are given in Table 1. The parameters of treatment processes, removal ratios and the cost functions are given in the Table 2. TPl is biological treatment, and complies with the modds of plug flow reactor of Andrews model. And the value of constant Ks, Kj, Vmax and Xt are determined at proper condition with ideal removal ratio, respectively 1500, 9000, 5h"\ and 37171.6. Table 1. Wastewater stream data for the case study

A New Methodfor Designing Water Network

Stream No. 1 2 3

Flowrate (t/h) 13.1 32.7 56.5

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H2S (mg/1) 390 16780 25

Table 2. Removal ratio and cost function for treatment process Treatment process Ideal Removal ratio Operating cost($) Capital cost ($) No. (%) 1.0*F TPl 99.9 16800*F^0.7 0.0067*F TP2 90.0 12600*F^0.7 Note: Annual rate of return 10% Operating hour 8600h/a According to the given parameters and data, set up the superstructure mathematical model. In the guide of pinch analysis to deceased the amount of uncertain parameters, the superstructure is solved and we obtain the following optimal wastewater treatment network as shown in figure 3. The optimal annual cost is 383414.6469$, whereas the result of Kou and Smith was 384489.7694$.

Figure 3. The optimal wastewater treatment network in the paper 5. Discussion Compared with the result of Kou and Smith, the annual cost of the optimal wastewater treatment network in this paper is a little lower in quantity. But the design method considers the removal ratio of biological wastewater treatment as variable, which is identified by model of plug-flow reactor of Monod and Andrews. In the case, the outlet concentration of TPl is 68.32mg/l, and the removal ratio is 99.55%, which is lower than ideal removal ratio. With the inlet concentration increasing unceasingly, the outlet concentration increases sharply and the removal ratio decreases evidently. Furthermore, it can be learned that the two networks are mainly different in the reused stream out of TP2. It is evident that decreasing the flowrate to TPl can reduce the annual total cost because the operating cost of TPl is much higher than that of TP2. At the same time, the inlet concentration of TP2 will increase while the outlet concentration of TP2 increase sharply, which causes the reused stream flowrate and the operating cost increasing. Conceptual design approach adopted by Kou and Smith didn't consider the change of removal ratio of treatment process, and the reused stream out of TP2. Mathematical programming can set up all possibilities of the network automatically and get the more optimal result. 6. Conclusion In this paper, a new network design procedure is introduced to solve wastewater treatment network. Firstly, this paper considers removal ratio of treatment process as a

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parameter based on biochemical treatment model. The removal ratio is calculated by model of plug-flow reactor of Monod and Andrews. Compared with traditional method, the approach can assure that the calculated values of the outlet concentration are approximately same with the practical values; hence the more feasible water network can be obtained. Then, set up superstructure mathematical model. Finally, solve the wastewater network by mathematical programming in the guide of the rule of pinch analysis. The method overcomes the drawback of the conceptual design approach and mathematical programming, and obtains the general optimal solution. The new method is illustrated by a case, and the result is more close to practical values compared with the solution of Kou and Smith. Nomenclature At capital cost coefficient of treatment process t Bt operation cost coefficient of treatment process t Ce the environmental discharge limiting concentration Ctj^in concentration of contamination j at inlet of treatment process t Ctj.out concentration of contamination j at outlet of treatment process t Cwj.out wastewater concentration of contamination j y^ flowrate of wastewatr stream w Ks half-velocity constant Ki restrain constant It imiting flowrate of water treatment process t S concentration of growth-limiting substrate in solution Xe,t discharge flowrate from treatment process to entironment Xe,yv discharge flowrate from wastewater stream to entironment Xt^w flowrate of wastewater stream w to treatment process t Wb reuse flowrate of wastewater u specific growth rate u mca maximum specific growth rate Subscripts in inlet j contamination out outlet / treatment process w wastewater stream p exponential coefficient of capital cost of treatment process t

References S. J.Doyle, 1997, Targeting water reuse with multiple contaminants. Transactions of Intemational Chemical Engineering, Part B, 75(3), 181-189. M. M. El-Halwagi, 1989, Mass exchanger networks, American Institute of chemical Engineering Joumal, 35(8), 1233-1244. C.H.Huang, 1999, A mathematical programming model for water usage and treatment network design. Industrial Engineering Chemical Research, 38(7), 2666-2679. W. C. J.Kuo, 1997, Effluent treatment system design. Chemical Engineering Science, 52(23), 4273-4290. G. T.Polley, 2000, Design better water networks. Chemical Engineering Progress, 96(2), 7-52. M.Savelski, 2000, On the optimality conditions of water utilization systems in process plants with single contaminants. Chemical Engineering Science, 55(22),5035-5048. M.Savelski, 2003, On the necessary conditions of optimality of water utilization systems in process plants with multiple contaminants. Chemical Engineering Science, 58(24), 5349-5362.

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N.Takama, 1980. Optimal water allocation in a petroleum refinery, Computers & Chemical Engineering, 4(4), 251-258. Y. P.Wang, 1994, Wastewater minimization. Chemical Engineering Science 49(7), 981-1006. F.L.Burton, 1991, Wastewater Engineering Treatment, Disposal and Resue. Metcalf & Eddy, Inc.

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Environmental life cycle impact and cost minimization in the steam and power generation plant Pablo Martinez and A.M. Eliceche Departamento de Ingenieria Quimica, UniversidadNacional del Sur, PLAPIQUICONICET,Camino La Carrindanga km 7, 8000 Bahia Blanca, Argentina. Abstract The selection of the operating conditions of the steam and power generation sector of an ethylene plant is carried out minimizing the environmental life cycle impact and operating cost solving a Mixed Integer Nonlinear Programming problem. The battery limits of the utility plant are extended to include the main environmental impacts corresponding to the electricity imported generated by nuclear, hydroelectric and thermoelectric plants. When the environmental life cycle impact and operating cost are minimized the optimal solutions are similar, while if the environmental impact is evaluated inside the utility sector battery limits, a different solution is found mainly associated to a different selection of drivers. Keywords: Environmental, Life Cycle, Impact, Operation, Utility. 1. Introduction The need to incorporate Environmental Impact objectives in process optimization has been recognized in the last decade by authors like, Dantus and High (1999), Cabezas et al. (1999) and Young et al. (2000). Life Cycle Assessment (LCA) has been traditionally used to quantify and assess the environmental performance of a product. Azapagic and Clift (1999) have proposed the use of the environmental life cycle assessment in the selection of alternative technologies for a given product, by using linear models. The utility sector has been chosen as the case study due to its significant contribution to the energy consumption in the process plant. For the ELCI approach the battery limits have been extended to include the life cycle of the energy generation imported by the utility plant. Diwekar et al. (2003) have proposed the minimization of cost and greenhouse gas emissions of three chemicals with a multi objective framework for utilities. Hashim et al. (2005) studied the optimization of the Ontario energy system with linear models considering CO2 emissions. The main objective of this work is to select the optimal operating conditions of the steam and power generation plant to reduce the Environmental Life Cycle Impact and the operating cost. The temperature and pressure of the high, medium and low pressure steam headers and alternative drivers such as steam turbines and electrical motors are selected optimally.

2. Environmental Impact Evaluation 2.1. Environmental Impact Categories The following environmental impact (EI) categories are evaluated as suggested by Heijungs et al. (2002):

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Global warming is caused by emissions of the greenhouses gases, e.g. CO2, N2O, CH4 and other VOC. These molecules can absorb the reradiated energy from the earth increasing the temperature of the atmosphere, this effect is called the greenhouse effect or global warming. Depletion of stratospheric ozone quantifies stratospheric ozone breakdown as a result of human activities. Because of the dilution of the ozone layer, a large fraction of the sun's UV-B radiation reaches the surface of the earth. Ozone depletion indicates the potential of chlorofluorocarbons (CFCs) and chlorinated hydrocarbons for depleting the ozone layer. Acidification is based on the contributions of SO2, NOx, HCl, and NH3 to the potential acid deposition, i.e. on their potential to form H^ ions, and consequently produces acid rain. Photo oxidant formation is the formation of reactive substances, which are injurious to human and ecosystem health (e.g. smog formation). Photo oxidant can be formed in the troposphere via photochemical oxidation of volatile organic compounds or carbon monoxide in the presence of NOx and under the influence of UV light. Nitrification or eutrophication is defined as the potential to cause over-fertilization of water and soil. Emission of NOx, N H / and P are considered to be the main responsible for eutrophication. Human toxicity is related to chemical released to air or water that are toxic to humans either by inhalation or ingestion. They are calculated using acceptable human daily intake of the toxic substances. Eco-toxicological impacts are the effects of toxic substances on aquatic and terrestrial ecosystems. This impact category depend on the maximum tolerable concentrations on water and soil, that represent the concentration considered to protect the 95 % of the species in a certain ecosystem. 2.2. Overall Environmental Impact Assessment The contribution of the emission of component A: to a given environmental impact category 7 is evaluated multiplying the flow rate qk emitted into the environment by the factor hkf published by Heijungs et al. (2002). The Heijungs factor hig represents the effect that chemical k has on the environmental impact category/ ¥hj=qkhkj

(1)

The environmental impact of each category 7, y/p is calculated adding the contribution of all the components ^ within these category as follows: ¥j=cCjY,y^i^

(2)

k

A normalizing factor cCj has been suggested by Cabezas et al. (1999) and can be calculated as the inverse of the average value of the Heijungs factors of the components contributing to categoryy:

k

where «, is the number of chemical compounds k that contributes to the environmental impact category/

Environmental Life Cycle Impact and Cost Minimization

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The overall environmental impact is calculated as the sum of the contribution of each environmental impact (EI) category y/j, with cOj representing the relative weighting factor of EI categoryy. yf = Y^cOjii/j

(4)

j

Considerable uncertainty and lack of information can be found along the extended limits of the environmental life cycle assessment. So the environmental impact quantification relays on the data available outside the battery limits of the plant being analyzed. 3. Optimization Problem Formulation The objective function is the overall environmental impact {^as defined in Eq. (4), including the environmental impacts due the utility sector and the generation of imported electricity. Alternative objective functions like the operating cost have also been used. The following optimization problem is formulated to select the operating conditions of the steam and power plant: Min

Y{x,y)

x,y

h(x)--= 0 g{x) + A{y)'.<0

S,t.\

L

X

(PI)

<x<x^

xe R" y^

mr

Vectors x and y are the continuous and binary variables, respectively. Superscripts U and L, indicates upper and lower bounds on vector x, respectively. The equality constraints h(x) = 0 are the system of non-linear algebraic equations that represent the steady state modeling of the process plant, including mass and energy balances; enthalpy and entropy prediction. The inequality constraints g(x) + A(y) < 0 represent logical constraints, minimum and maximum equipment capacities, operating and design constraints, etc. The A matrix includes linear relations between binary variables such as logical constraints. Pressures and temperatures of high, medium and low-pressure steam headers, deaerator pressure and letdowns are the continuous optimization variables. The vector y, represents integer variables to select between alternative drivers such as steam turbines and electrical motors. The flow sheet of the steam and power generation plant is shown in a previous paper by Martinez et al. (2005). The power and steam demands of the ethylene plant are posed as equality constraints. The main power demands correspond to the cracked gas, ethylene and propylene refrigeration compressors. Other power demands are for condensate pumps, air fans, boiler water pumps and cooling water pumps. Energy is recovered from the Cracking Furnace to generate high steam pressure. The binary variables are used to select the alternative drivers of: water tower pumps, lubricating pumps, condensate pumps, boiler water pumps, cooling water pumps, air fan and boilers selection. The problem was formulated in GAMS (2003).

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4. Contribution of tlie environmental categories. In the utility sector of an ethylene plant the pollution comes mainly from the combustion emissions, the purged water from the boilers and water discharges from the cooling towers. The gas emissions form by the exhaust gases of the boiler stack were evaluated using the emission factors published by USEPA AP-42 (1998). The effluents were calculated using the emission factors presented by Elliot (1998) and the EPRI report (2000). In the life cycle assessment context, the limits of the utility plant are extended to contemplate raw material extraction, transport, production, use and waste disposal. The raw material is the natural gas in the steam and power generation plant. The interconnected network in Argentina has approximately the following distribution: 53 % of thermo electrical, 35 % of hydro electrical and 12 % of nuclear generation. The main environmental impacts of imported electricity generated by thermoelectric, hydroelectric and nuclear plants are evaluated. The combustion process of oil, coal and natural gas releases pollutants, such as NOx, CO, particulate matter, SO2, VOCs, organic hydrocarbons and trace metals into the air are calculated as reported in the emissions factors published by USEPA AP-42 (1998). The emissions associated with exploration, extraction, transport and refining were taken into account for oil, coal and natural gas consumed in the electric power generation. For the hydro electrical power source, the emission factors take into account the dam building, the material transport and submerged biomass decay, stages that emit greenhouse gases. For Nuclear Power Generation during the energy production stage, the emissions are mainly aqueous. The emission factors due to friel fabrication and power plant construction stage are also taken into account, AEA (1998). In the present work the (Oj factors of equation 4 were set equal to one, except for Depletion of Stratospheric Ozone and Human Toxicity Air EI categories that were set equal to zero. Emissions for one hundred chemical compounds were evaluated in this example. The operating conditions are evaluated minimizing the Environmental Life Cycle Impact (ELCI) in problem PI. The number of equations included in GAMS is 10548, with 10555 continuous (seven optimization variables) and 24 binary variables. A Mixed Integer Non Linear Programming problem is formulated and solved in GAMS. The MINLP problem was solved with the code DICOPT, CONOPT++ and CPLEX codes for NLP and MIP sub problems respectively. The solution was found in 26.85 seconds and three major iterations in a Pentium III, 700Mhr workstation. The contribution of each environmental impact category to the global environmental impact is quantified in Table 1. Table 1. Environmental life cycle impact categories. Environmental Impact Category Global Warming (GWPioo) Human Toxicity Air (HTA) Acidification (Acid) Photochemical Ox. (POF) Human Toxicity Water (HTW) Eutrophication (Eutrop.) Ecotoxicity Water (Ecot.) Str. Ozone Depletion (SOD)

Values 22212.956 61.753 53.232 0.63 0.421 0.036 0.017 2.152E-06

kg CO2 eq. kg kg SO2 eq. kg ethylene eq. kg kg P04-^ eq. m^ kgCFC-lleq.

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The environmental burdens (e.g. LCA Inventory) of each impact category includes the steam and power plant and the electricity imported. GWPioo is the major contributing category, accounting for 99 % of the global potential environmental life cycle impact. In Fig. 1, the contribution of the utility sector, thermoelectric, hydroelectric and nuclear generation to each environmental impact category, is shown. The utility plant contributes with more than 98 % of the environmental life cycle impact in GWPioo, Acidification, POP and HTA impact categories and 80 % in the HTW impact category. Thermoelectric power generation is the only contributor to SOD impact category. Nuclear power generation has a contribution of 28 % in Eutrophication. Hydroelectric power generation has a contribution less than 0.01 % in all the environmental impact categories.

80%

40%

0% GWP

Acid.

PDF

HTA

HTW

Ecot.

Eutrop.

SOD

Fig. 1. Contribution of the utility sector Q , thermoelectric B , hydroelectric H and nuclear • plants to each impact category.

5. Minimizing Environmental Life Cycle Impact and Cost The operating conditions have been chosen minimizing three alternative objectives functions: operating cost, ELCI and the Environmental Impact inside the battery limits of the Utility sector. The values of the operating cost, ELCI and Utility EI for the three solutions and the difference with the corresponding minimum value are reported in Table 2. The first two columns minimizing operating cost and ELCI show similar solutions, with a difference of 0.08% in ELCI and 0.42 % in the operating cost ($/hr). In the utility plant analyzed, both objective fiinctions leads to similar solutions. On the other hand, the solution minimizing the Utility EI inside the battery limits, in the third column, the operating cost is 5.1 % more than the minimum cost in the first column. Table 2. Comparison of solutions minimizing operating cost, ELCI and utility EI. Objective Function Cost ($/hr) ELCI Utility EI

Min Cost

Min ELCI

% Var. 1696.268 528.339 522.960

0 0.083 4.289

Min Utility EI

% Var. 1703.44 527.900 519.703

0.421 0 3.689

% Var. 1787.504 527.930 500.529

5.104 0.006 0

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Thus minimizing the EI inside the battery Umits of the utiHty plant leads to a different solution than minimizing operating cost and environmental life cycle impact. The main difference in terms of the operating conditions between operating cost and ELCI solutions (first and second columns) and the Utility EI solution (third column) is that in the last case, most of the drivers correspond to electrical motors instead than steam turbines because there is no impact associated to the electricity imported. 6. Conclusions It has been shown that for this steam and power generation plant minimizing the environmental life cycle impact (ELCI) and operating cost lead to similar solutions. This is not the case, if the environmental impact is evaluated only inside the utility battery limits. Therefore if environmental impact objective functions are used in process optimization it is important to extend the battery limits to consider the main environmental impact in the life cycle context. The contribution of global warming due to greenhouse gas emissions accounts for nearly 99 % of ELCI in the utility sector. Thus the methodology presented is an important tool to reduce Carbon Dioxide emissions contributing to the fulfillment of the Kyoto protocol. The minimization of environmental life cycle impact and cost lead to similar solutions in processes were the energy consumption and its corresponding emissions can be reduced simultaneously by increasing the efficiency, as it is the case of the utility sector. References AEA Technology Environment, 1998. Power generation and the environment - A UK perspective.Technical Report AEAT 3776. Azapagic, A., Clift, R., 1999. The application of life cycle assessment to process optimisation. Comp. Chem. Eng., 23, 1509-1526. Brooke, A., Kendrick, D., Meeraus, A., Raman, R., 2003, GAMS development Corporation. A user guide. Cabezas, H., Bare, J., Mallick, C , 1999. Pollution prevention with chemical process simulator: the generalised waste reduction (WAR) algorithm. Comp. Chem. Eng., 23, 623-634. Dantus M., High, K., 1999. Evaluation of waste minimization alternatives under uncertainty: a multi objective optimisation approach. Comp. Chem. Eng. 23, 1493-1508. Electrical Power Research Institute, 2000. Aqueous discharges from steam electric power plants, data evaluation. Updated Report, http://www.epri.com. Elliott, T., 1989. Standard Handbook of Power Plant Engineering, Mc Graw Hill, New York. Fu, Y., Diwekar, U., 2003. Cost effective environmental control technology for utilities. Advances in Environmental Research, 8, 173-196. Hashim, H., Douglas, P., Elkamel, A., Croiset, E., 2005. Optimization Model for Energy Planning CO2 Emission Considerations. Ind. Eng. Chem. Resources, 44, 879-890. Heijungs, R., Huppes G., Kleijn, R., Koning, A., van Oers, L., Sleeswijk, A., Suh, S., Udo de Haes H., Jeroen Guinee (final editor), 2002. Handbook on Life Cycle Assessment, Kluwer Academic Publishers, Dordrecht. Martinez, P., Corvalan, S., EUceche, A.M., 2005. Environmental Life Cycle Assessment as a tool for Process Optimisation in the utility sector. In: L. Puigjaner (Ed.), Proceedings of the European Symposium on Computer Aided Process Engineering-ESCAPE 15. Barcelona, Spain. Volume 20, Part A, pp 853-858. U.S. Environmental Protection Agency, 1998. AP-42. Compilation of air pollutant emission factors. Young, D., Scharp, R., Cabezas, H., 2000. The waste reduction (WAR) algorithm: environmental impact, energy consumption and engineering economic. Waste Management, 20, 605-615.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Case Study of a Regional Network for the Recovery of Hazardous Materials J. Duque^ A. P. F. D. Barbosa-Povoa*^ and A, Q. Novais^ ^Departamento de Modelagdo e Simulagao,DMS- INETI, LP.,Lisboa, Portugal ^Centro deEstudos de Gestao, CEG-IST, 1049-001, Lisboa, Portugal Abstract This work presents the development of a model for a regional recovery network of industrial residual products. The aim is to optimise simultaneously the design and scheduling of the network, so that the model may be suitable as a managerial tool for the elimination/reduction of environmental impacts while ensuring economic viability. The geographical dispersion and the potentially large number of entities involved impose a large number of transport routes, which usually lead to very complex optimisation problems. Several ways of modeling transports are employed and the corresponding network results compared for a real case, in order to assess the relative merits of these transport approaches in achieving valid results for large problems within reasonable computational resources. Keywords: Recovery Network, Transport modeling. Environmental impacts. 1. Introduction The present work describes the case study of an optimal design network for managing the recovery of residual products originated at industrial plants. It addresses the recovery of the sludge obtained from Aluminum surface finishing plants, which after some physical conditioning can be employed as co-adjuvant for the treatment of other industrial, as well as municipal effluents. The aim is to optimise the design and scheduling of a network of geographically dispersed and collaborative entities - producers, end-users, processing and transport agents - subject to environmental constraints on generated pollutants that may evolve into a fate analysis as part of the Eco-Indicator 99 [1]. The model is to be used as a managerial decision support tool. It is therefore conceived to generate the optimal network design and schedule for a chosen cost related function, subject to environmental constraints for the pollutants, accounted for by the environmental impact indices. With this aim a novel cyclic production multitask model based on the maximal State Task Network representation (mSTN) is developed [2], which incorporates Stephanis' generalization of the minimum environmental impact methodology (MEIM) [3]. The system involves a vast number of agents and different types of transport that ensure an adequate territorial coverage, as well as two types of sludge, which lead, after recovery, to two final products. Two modeling approaches are studied to deal with the transport modes of operation. A first one where each transport is modeled as a general transport task with associated environmental impacts [4] and a second one where transports are treated as simple Author to w h o m all correspondence should be addressed. Tel: +351-218419014. Fax: +351-218417979. E-mail: [email protected]

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connections, with associated transit times and pollutant production. The relative merit of each transport model is analysed and discussed, in terms of the solution obtained and of the model performance. The first modeling approach (App.l) is likely to increase markedly the order of magnitude of the resulting mixed integer linear programming (MILP) problem, compromising the tractability of the problem, a difficulty expected to be overcome while using the second approach (App.2). Two variants within each approach are studied. These approaches, either directly or with some modifications, are applied to moderate size examples based on real data, leading to several recovery network models, whose results are evaluated and compared. 2. Problem Definition and characteristics 2.1. Problem definition A short description is provided. For a full description check [4] Given: A recovery network superstructure (STN/Flowsheet), the cost data, the operational data, the environmental data - Critical air mass (CTAM) limits, critical water mass (CTWM) limits, solid mass disposal (SMD) limits, global warming potential (GWP), photochemical oxidation potential (POCP) and stratospheric ozone depletion potential (SODP) - (see [3 and 5]) The demand data: Product demand Nominal, maximal and minimal product demand. Determine: The optimal network structure (processing operations, storage locations and transfer routes of materials) The optimal operating strategies (scheduling of operations, storage profiles and transfer occurrences). So as to optimise an economic performance index, such as the maximum network profit, while satisfying the imposed environmental legal limits. 2.2. Problem Characteristics The mSTN representation is used to model the general network superstructure, which is coupled with a generalization of the MEIM so as to account for the waste generation at utility production and transportation levels. Since the recovery of hazardous products is being addressed, the system frontier for the environmental impacts is defined at the level of the feeds, including any type of utilities used. The amount of pollutants computed may also be used, with some minor modifications to the model, as input to a fate analysis envisaging the use of the Eco-indicator 99 methodology [1] The model has the particularity of considering all possible concurrent transportations and transformations, as well as all raw material (the sludge, i.e., the actual hazardous material) producers and re-users. The resulting pollutant materials are added up for all the different types of waste. The limits on the total waste production and global environment impacts are introduced in the form of waste and pollution vectors, which normally derive from legal regulations and enter the model as additional constraints. The transports are modeled using two main approaches, Appl where transportation are considered and App2 where transports are modeled as simple connections. For the first approach two cases are considered: (1) transportation tasks with utility consumption and

Case Study of a Regional Network for the Recovery of Hazardous Materials

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polluted waste production; (2) direct transportation tasks, which collect and deliver directly from storage, thus reducing the total number of model variables (connections). For the second approach again two cases are studied: (3) simple connections with a transit time, greatly reducing the number of binary variables, but at the expense of a coarser estimation of utility consumption and waste production; (4) connections with allocation binary variables, and hence with a fme estimation of utility consumption and polluted waste production. For (1) and (2), two additional model variants can be derived, (la) and (2a), by allowing the overlap of transports at successive time intervals, i.e., a given transport task, whose duration spans several time intervals, once initiated, may undergo repeated re-initiations at subsequent time intervals, while the preceding transport instances are still in progress. 3. Recovery route example The proposed model was applied to the optimisation of a recovery route for an Al-rich sludge. Two different types of sludge, SI and S2, are considered that may be transformed by two different sequences of operations, leading to dilute or dry products, respectively S3 and S4, whose recipes are: Dilution: - The feed materials, 0.6 of SI and 0.4 of S2, are diluted under agitation with 5% of water, producing, after a certain mixing time, a stable product S3, at batch proportions of 1 to 0.99. The process lasts for one day and generates waste at a batch proportion of 1 to 0.1. The mixing consumes electric power. Drying: - The feed materials, 0.25 of SI and 0.75 of S2, are dried at 30''C, ground to a powder and mixed, leading to a stable product S4 at batch proportions of 1 to 0.99. The process has duration of two days and produces waste at a batch proportion of 1 to 0.1. The drying and grinding operations consume electric power. All the dust produced during the grinding process is retained by filtration. For the present analysis it is important to retain that both states, S3 and S4, are stable and suitable to use as a coagulant and flocculant for the treatment of industrial and municipal effluents. They are to be prepared over a production time horizon of 30 days for a periodic operation of 5 days duration. Materials are transported in standard containers (of 6, 10 or 20 m^) that are left in situ, at a cost of 40, 40 or 85€ per month, respectively. Each transportation can be done by two alternative types of transport differing both in the average speed and cost. For type I, these are respectively 80km/h and costs are calculated based on transport duration at a rate of 1.8 €/km, while for type II these values are halved; diesel consumption is estimated at 30 lit/100km and variation with the actual truck load is taken into account. The number of entities on the network may be varied, along with their location, which is always considered at a district level. The simplified general network flowsheet is shown in figure 1. All models were solved on a PC PENTIUM IV, 3.0 1GB RAM using GAMS Rev 141 with CPLEX 9.0.2. 3.1. Motivating example To test the different models, a motivating example was generated where the number of entities was limited to 2 producers, 3 transformers and 3 final clients. Raw materials and product storage are assumed unlimited. All final clients have the same demand values for S3 (150, 180, 120 for nominal, maximal and minimal demand, respectively).

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J. Duque et al Al rich sludge Producers

Transformer entities

Transports

Client entities

Transports

Producer t

c ci

Sriidge I \

; Sludge 1 , storage

Tranformer

\

Product 1

instalations '•

Skjdg8 2 \

'l

1

Pr«iudt 1 vtorage

i

Producdt 2 storage

Producer M

*R^; f

4-<

Slijdcje 1 Storage

i S''^ ^ storage

: Tranformer > . ^ , .. instalations

Product 1 storage

Produt.t 1

^lU

K.y [ Sludge 2 i storage

Sludge 2 ^

i

t

PrcKiuct 2

Figure 1 - Network recovery route flowsheet and for S4 (120, 140, 100 tons, respectively, for the equivalent quantities). The transformer design capacities are available in continuous values between 10 and 30 tons. The maximal and minimal occupation factors of the transformation capacity, for any selected transformer capacity, are of 1 and 0.1, respectively. The model statistics and solution resources are shown in Table 1, while Table 2 shows global environmental impact indices of the obtained solution. Table 1 - Model statistics for Motivating Example Transport

Iteration count

model

Number of Variables Single

Resource usage

Discrete

Optimised Values Solution

Best

Relat. Err

(1)

59914

5509

802

27.578

325.4654

328.717

9.989e-3

(2)

5213

3949

682

3.312

328.1531

330.947

8.514e-3

(1a)

59914

5515

802

26.250

325.4654

328.717

9.989e-3

(2a)

5213

3949

682

3.281

328.1531

330.947

8.514e-3

(3)

621

3563

296

0.234

325.9691

326.409

1.349e-3

(4)

100638

3863

596

24.640

325.6678

328.552

8.856e-3

Comparing the optimised values obtained versus the computational resources used (Table 1), it can be seen that model 3 performs considerably better than all other models, at the expense of less than 1% of the optimal value. Since performance can be expected to be a major hindering factor in a real life size problem, model 3 was selected. Most global environmental impact indices values (Table 2) remain insensitive, with the exception of those directly related to the transport solution used.

Case Study of a Regional Network for the Recovery of Hazardous Materials

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Table 2 - Motivating Example: Global Environmental Impact (GEI) indices values

Indices values

GEI Type

(1)

(2)

(la)

CTAM

0.07

0.07

0.07

0.07

0.07

0.07

CTWM

1.29

1.29

1.29

1.29

1.29

1.29

GWI

0.02

0.01

0.02

0.01

0.02

0.02

POI

0.27

0.16

0.27

0.16

0.26

0.27

(2a).

(3)

(4)

3.2. Case Study Using the model 3 tested above, a real case recovery route is to be designed. This covers the entire country, where a transformer and a client are considered in each of the 18 districts, also coupled v^ith 8 possible producers. The demand for S3 and S4 for the Lisbon district are respectively 330, 380, 270 tons (for the nominal, maximal and minimal demand), and 300, 360, 240 tons (for the equivalent quantities). For each of the remaining districts, demands are based on the Lisbon values, v^hich are multiplied by a factor reflecting the district population relative proportion.

Figure 2 - Case Study (a) network superstructure; (b) Optimised network structure

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J. Duque et al

The entities locations are illustrated in figure 2(a) while figure 2(b) shows the optimised network with the transports indicated by overlaid arrows. When comparing both structures it can be seen that the optimised solution differs from the superstructure in two non-installed transformers (Castelo Branco and Evora). This example also shows that local or closer locations are preferred, leading to regional clusters of recovery networks, thus emphasizing the relative importance of transports. The optimised solution produces a total of 1756 tons of S3 and 1660.2 tons of S4 with the reported statistics indicating that the model used 88676 variables, of which 6296 are discrete, and took 35.265 CPU seconds and 5919 iterations, to obtain a solution of 895.421 k€ for the network added value, with a relative solution deviation of 0%. The optimised network solution used 168.29 kWh of electricity, 14.78 m^ of water and 580 1 of gas oil. The computed values for the impact indices were respectively CTAM: 0.22, CTWM: 5.08, SMD: 0, GWI: 0, POI: 0.03 and SODI: 0.01. 4. Conclusions and future work A network design model was developed where different model transport approaches were tested. A first one (Appl) considering a more exact and complex model and a second one (App.2) based on a simpler model. In conclusion, the validity of the latter was established, in the form of model 3 - simple connections with transit time - with a view to tackle complex networks effectively and at a low computational cost. Based on this model, a real recovery route was designed where different entities locations were investigated. An optimised network in terms of costs, operability and environmental impacts was obtained. The operation cost of the entire network was found to be covered by the alternative cost associated to the direct landfill deposition. It should be noted that the landfill deposition, apart from its bulk cost, in practice also involves transports and associated pollutant loads, which were not taken into account in the present study, and which contribute markedly to fiirther discourage the deposition option. As future development to the model it is considered the inclusion of Eco-indicator to give value to the reduction on land occupation with dangerous industrial residues. Also to be implemented in the model is the limited storage capacity and to instead of the demand satisfaction enforces the treatment of the estimated limited production of industrial residual products, at each producer entity and based on its monthly production rate.

References 1. The Eco-indicator 99. A damage oriented method for Life Cycle Impact Assessment, available on-line at www.pre.nl 2. Barbosa-Povoa and Macchietto, 1994, Comp. Chem. Engng., 18, 11/12, 1013-1042 3. Stefanis, S.K., A.G. Livingston and E.N. Pistikopoulos, 1997, Comp. Chem. Engng., 2 , 21, 10, 1073-1094 4. Duque J.; Barbosa-Povoa, A.P.; Novais, A.Q. (2005), "Synthesis and Optimisation of the Recovery Route for Residual Products under Uncertain Product Demand", Special Issue of Computers & Operations Research on Hazardous Materials, 2005 5. Pistikopoulos E.N., S.K. Stefanis and A.G. Livingston, 1994, AIchemE, 90 (303),139-150

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Optimisation of a pertraction process for wastewater treatment and copper recovery A. M. Eliceche," M. F. Orlandi,^ A. M. Urtiaga" and I. Ortiz" ^Departamento de Quimica e Ingenieria Quimica, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Espana. ^Departamento de Ingenieria Quimica, Universidad Nacional del Sur, CONICET, Camino La Carrindanga km 7, 8000 Bahia Blanca, Argentina.

Abstract The optimal operating conditions of a pertraction process for the recovery of Copper used as catalysts in a Wet Peroxide Oxidation (WPO) effluent is presented. Separation targets related to the maximum Copper concentration in the effluent for final disposal and minimum Copper concentration of the product are posed as inequality constraints of the optimization problem. A rigorous modeling of the pertraction process is used represented by a set of algebraic and differential equations which are posed as equality constraints in the optimization problem formulated in gPROMS and solved by gOPT. A reduction of 41 % in the batch time is achieved, indicating the room for improvement available in this cleaner and new technology. Variations of the batch time with respect to changes in the operating variables are reported. Keywords: Pertraction, Copper recovery. Operation, Effluent treatment 1. Introduction Higher demands are put onto new separation technologies due to environmental legislation and needs for cost reduction. To meet these ever increasing demands, there is a tendency to propose new processes that are technically and cost effective. One of those technologies is the emulsion pertraction that combines the efficiency of emulsion liquid membranes with the advantages of using hollow-fiber modules. In the emulsion pertraction process, the water phase is kept apartfi*omthe emulsion phase by a hydrophobic microporous membrane. The emulsion phase consists of an organic solvent with a dissolved extractant as a continuous phase with aqueous droplets of stripped liquid dispersed in it. The contact surface between the aqueous feed phase and the emulsion phase lies in the pores of the membrane. The metal to be removed from the wastewater stream is bounded by the extractant present in the pores of the membrane. In the shell side of the membrane, the extractant is regenerated by the backextraction or stripping solution. The hydrophobic nature of the membrane keeps the wastewater and the emulsion always separated using a slight over pressure on the wastewater side. Initial work on the selection of the operating conditions of on a non-dispersive solvent extraction process for effluent treatment and Chromium recovery was reported by Eliceche et al. (2000). The modeling and optimization of a pertraction process for removal and concentration of hexavalent Chromium was presented by Ortiz et al. (2003). The design of a pertraction plant for Cr(VI) removal and recovery was presented by Eliceche et al. (2005). The modeling of a copper recovery from a WPO effluent was developed by Urtiaga et al. (2005) and will be used in this work.

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A.M. Eliceche et al

The aim of this work is to analyze the behavior and select the optimal operating conditions to improve the performance of a new emulsion pertraction process for copper removal, recovery and reuse as a catalyst in a WPO effluent treatment plant. 2. Emulsion pertraction process for copper recovery In a emulsion pertraction process (EPP) as shown in Fig.l, the aqueous stream runs along the lumen side of the hollow fibber membrane module while the emulsion runs counter currently through the shell. The emulsion consists of an organic phase in which an aqueous stripping phase is dispersed. The extraction from the effluent into the organic phase is assisted with hollow fiber membrane modules while the back extraction from the organic into the stripping phase is carried out in the emulsion. A chemical dissolved in the organic phase extracts the pollutant from the effluent reacting at the membrane interface. A second reaction takes place at the aqueous bubbles interface, releasing the compound of interest to the stripping phase in which the contaminant is concentrated. After leaving the membrane module, the emulsion goes to the emulsion tank and back to the membrane module up to time in which the effluent reaches the maximum allowed metal composition and the stripping phase reaches the minimum desired concentration. At the end of each batch the emulsion is decanted and the aqueous and organic phases are separated. The aqueous stripping phase concentrated in the pollutant is the product. The organic phase is reused in the next batch to form the emulsion.

Membrane module — ^ Effluent tank Emulsion tank

Fig. 1. Discontinuous emulsion pertraction process.

In this work a discontinuous operation for the effluent and emulsion phases is considered. Thus the maximum allowed contaminant composition in the effluent for disposal and the minimum concentration in the concentrated aqueous phase of the emulsion should be achieved in the batch time. The application studied corresponds to the removal of Copper from the effluent of a WPO treatment plant, the copper recovery and concentration in the stripping phase as the product to be recycled to the WPO process as a catalyst. The equations that describe the kinetics of the extraction and back-extraction reactions of copper were reported by Harada et al. (1989) and have been used by other authors in the description of copper mass transfer in hollow fiber contactors, using different extractants containing hydroxyoximes as functional groups. The reaction between Cu^^ and the organic extractant can be represented as follows: Cu 2+ + 2HR^CuR2+2H'^

(1)

where HR represents the organic extractant and CuR2 the organo-metallic complex. A hydrophobic microporous membrane is wetted by the organic phase. Thus the effluent-organic interface is located at the inner side of the membrane wall.

Optimisation of Pertraction Process for Wastewater Treatment and Copper Recovery 1805 Mass transfer of the metal species from the effluent to the stripping phase takes place in four steps: (i) diffusion of copper ions in the effluent phase though the liquid boundary layer to the organic interface; (ii) interfacial chemical reaction of the copper cation with the organic extractant to form the organo-metallic complex species; (iii) diffusion of the complex through the organic phase impregnated in the porous membrane and, (iv) interfacial chemical reaction of the complex with the back-extraction agent at the outer surface of the stripping phase globules of the emulsion. The main objective of this work is to select the optimal operating conditions.

3. Selection of the operating conditions The operating conditions of the EPP process in the discontinuous mode are found solving the following optimisation problem: Min V

s.t.

tfiy) -^

f{x{t),x{t),v)

= 0

g {x{t),x{t),v)

< 0

I {x(0),v)

= 0

.min ^

.

te[ojf] (PI)

^ .max

w'"'"< w(t^)

<w'"^

The mass balances, equilibrium, and interconnection relationships are represented by a set of differential and algebraic equalities constraints / while the restrictions on the effluent output concentration and the recovered metal concentration are represented by inequalities constraints g. The initial conditions for the differential equations are given by /, where x{t) are the time derivatives of the variable x{t). There are lower and upper bounds on the end point constraints w(tf\ time invariant parameters v and time horizon tf. The numerical results reported in this paper correspond to the minimization of the batch time tf although different objective ftinctions have been used. The optimization variables considered are the following time invariant parameters v: aqueous and emulsion flow rates, the initial copper concentration in the organic and in the stripping phases, the initial extractant concentration in the organic phase, the initial aqueous solution of sulphuric acid in the stripping phase and the pH in the effluent tank. 4. Optimal operating conditions for Copper removal and recovery The case of study selected in this work deals with the recovery of copper from WPO residual waters by means of the emulsion pertraction technology. The non steady state model of this process has been formulated by Urtiaga et al. (2005), the numerical values of the parameters related to chemical equilibrium constants and mass transfer coefficients were estimated from experimental data obtained in the laboratory. A system of differential and algebraic equations describe the behavior of the effluent and emulsion phases in the membrane module, effluent and emulsion tanks, chemical equilibrium and connectivity. Thus the modeling equations formulated as equality constraints in problem PI correspond to the differential and algebraic system of equations published by Urtiaga et al. (2005) and also the upper and lower bounds on the

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A.M. Eliceche et ah

operating conditions correspond to the limits in which the experimental work was carried out. The volumes of the effluent and emulsion tanks are equal to 1 liter. The emulsion is formed by 0.8 liters of the organic phase and 0.2 liters of the stripping phase. The emulsion phase flowrate can be expressed as follow: pem ^po^ps

^^^^^

F"" =4-F'

(2)

The emulsion tank is agitated continuously to prevent phase separation. The membrane module is a microporous polypropylene hollow fiber module of 1.4 m^ of area. A pH control of the effluent tank is used. The operating conditions were selected for a nominal copper concentration in the WPO effluent equal to 3.2 molW. The maximum allowed effluent Cu concentration is 0.0295 mol/m^ and the minimum Cu concentration of the stripping phase in the emulsion tank is 14.164 molW. These two separation targets are posed as inequality end point constraints. The initial point and the intervals on the operating conditions correspond to the values reported by Urtiaga et al. (2005) for the case of a concentration of the organic extractant of 18.364 molW (1% v/v of LIX622N in kerosene). Table 1. Optimal operating conditions.

cL,i (mol/m^) CH (mol/m^) CEX,! (mol/m^)

F'

(m^/h)

Ccu,i (molW) Ccu,i (mol/m^) F° (m^/h) cL,f (mol/m^) C'cu,f (mol/m^) Batch time tf (mm) % Reduction tf CPU time (sec)

Initial point

Solution point

Lower bound

Upper bound

2000 1 18.364 0.0266 0 0 0.0219

4000 0.100 22.954 0.033 0.100 0.100 0.024 0.029 14.164 17.6 41.3 14.32

1000 0.100 18.364 0.024 0 0 0.019 0.001 14.164 5

4000 10 183.640 0.033 0.100 0.100 0.024 0.029 100 60

30

Problem PI was formulated and solved with the gOPT code of gPROMS (2004). It can be observed in Table 1, that the batch time has been reduced by 41 % at the solution point of problem PI, indicating that when a rigorous modeling is available a systematic EPP optimization improves significantly the process performance. The batch time has been minimized, although different objective functions have been solved. The optimal values for the operating conditions lie mostly at their lower or upper bounds due to the monotonic behavior of the batch time with respect to these variables. The solution shown in Table 1 requires 14.32 seconds of CPU time on a Pentium III, 700 MHz workstation. The Copper concentration profiles in the effluent tank and stripping phase of the emulsion tank versus time are drawn in Fig. 2 (a) and (b) respectively. The copper composition profiles are shown for the initial (30 minutes) and optimal (17.6 minutes) points reported in Table 1.

Optimisation of Pertraction Process for Wastewater Treatment and Copper Recovery 1807

0

tD ^r"20 Time (min)

30 Time (min)

(a)

(b)

Fig. 2. Copper concentration profiles in the effluent (a) and stripping phase of the emulsion (b) tanks varying with time. Initial point (A) and optimal point (•) of Table 1. The same copper composition in the effluent and concentrated product are achieved in 30 and 17.6 minutes by choosing optimally the operating conditions. 5. Influence of the operating conditions in batch time The influence of the operating conditions in the batch time is reported in Table 2. All the operating variables are fixed at their optimal values as reported in Table 1. The values of each operating variable is changed one at a time, between the upper and lower bounds reported in Table 1. The batch time required at the upper and lower bounds are evaluated and the difference is reported as A/ /:

M)=[tf(vr)-tf(yr)i

V • = J

V°P' J

yj*k

(3)

Table 2. Changes in batch time with the operating conditions. Optimization variables

Increment (UB - LB)

A/ f (min)

ACH,i

3000

-93.55

-530.48

ACH

9.9

84.87

481.27

165.27

49.55

280.98

0.009

-2.47

-14.02

0.1 0.1

-1.21

-6.87

-0.28

-1.61

0.0048

-0.21

-1.19

o

AC Ex,i

ACc>,i ACcu,i AF"

tf change %

k

The percentage of changes in Table 2 are calculated as (A^ / / tf^). Analyzing the numerical results reported, it is clear that the influence of the initial stripping sulfuric acid concentration, the effluent tank pH and the initial extractant composition in the organic phase have a strong influence in the batch time. In the range between the upper and lower bounds for these compositions, the batch time increases up to five (530%) and three (281 %) times bigger than the optimal batch time approximately, indicating that these three compositions should be chosen carefully. These compositions influence the extraction and back extraction reactions as can be deduced from Eq.(l). The sulfuric acid concentration in the stripping phase lies at the upper bound favoring the back extraction reaction. It is important to have a pH control in the effluent tank to keep it as

1808

A.M. Eliceche et al.

high as possible, and the CH as low as possible, to favor the extraction rate of reaction. Thus the optimal value of CH is at the lower bound. The pH will decrease along the membrane module. The kinetics of copper stripping is also favored as the extractant concentration in the organic phase is low.

6. Conclusions It is important at the conceptual design stage to attempt a systematic optimization of new technologies to increase the insight and improve the performance. In the emulsion pertraction process for copper recovery analyzed, a significant reduction of 41 % in the batch time has been achieved selecting the operating conditions with a rigorous simulation of the membrane module. The pH in the effluent tank, the initial sulfuric acid concentration in the stripping phase and the initial extractant composition in the organic phase have the strongest influence in the batch time as shown in Table 2, in the intervals studied. This is a initial step to gain insight and analyze the economical viability of this new and cleaner technology, considering that copper is removed from the effluent and it is simultaneously recovered to be recycled and reused in the WPO process, reducing dramatically the final disposal of copper into the environment. Notation C F

Concentration, mol/m^ Flowrate, m^/h

Superscripts e Effluent phase em Emulsion phase 0 Organic phase s Stripping phase

Subscripts Cu Copper Ex Extractant f Final time H Proton i Initial time opt Optimal

Acknowledgement Financial supportfromprojects PICT 1412729 ANPCyT and BQU2002-03357 is acknowledged. References A. Urtiaga, M.J. Abellan, J.A. Irabien and I. Ortiz, 2005, Membrane contactors for the recovery of metallic compounds. Modelling of copper recovery from WPO processes, Journal of Membrane Science 257, 161-170. A.M. Eliceche, A.I. Alonso and I. Ortiz, 2000, Optimal operation of selective membrane separation processes for wastewater treatment. Computers and Chemical Engineering 24, 2115-2123. A.M. Eliceche, S.M. Corvalan, M.F. San Roman and I. Ortiz, 2005, Minimum membrane area of an emulsion pertraction process for Cr(VI) removal and recovery. Computers and Chemical Engineering 29, 1483-1490. gPROMS Technical Document, 2004, The gOPT dynamic optimization tool. Process Systems Enterprice, Ltd. M. Harada, Y. Miyake, Y. Kayahara, 1989, Kinetics mechanism of metal extraction with hydroxioximes, J. Chem. Eng. Jpn. 22, 168-176. I. Ortiz, M.F San Roman, S.M. Corvalan and A.M. Eliceche, 2003, Modeling and Optimization of an Emulsion Pertraction Process for Removal and Concentration of Cr(VI), Ind. Eng. Chem. Res. 42, 5891-5899.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Data-Centric Demand Forecasting for Utilities Zdenek Beran, Karel Mafik, Petr Stluka Honeywell Prague Laboratory, Pod voddrenskou vezi 4, 182 08 Prague 8, Czech Republic Abstract Growing amounts of data being archived and maintained in process historians calls for ever more intensive use of data-mining technologies allowing to explore the data, extract useful knowledge, and turn it into business advantage. Utility producers and utility distributors can take significant benefits from application of database-intensive techniques. This paper describes how exploratory analysis of historical datasets can help utility companies to identify typical demand patterns, and how these can be further used for forecasting of friture consumption, and consequently for making decisions on optimal operation of the utility plant. The technical concept as well as all steps of the forecasting process are described along with summarization of the experience gained from development and practical implementation of a proprietary forecasting tool. Keywords: utilities, data-mining, exploratory data analysis, forecasting, load allocation 1. Introduction Accurate demand forecasting is a high priority for operators of power plants, heating plants, municipal utilities, and distribution companies. Using the forecasted consumption for a given time window, they can determine which configuration of machinery and associated set points are necessary to meet the demand at lowest operational cost. In this way, demand forecasts provide important information for making critical decisions on balancing supply and demand in the distribution network. Although all forecasts are uncertain by the nature, they can help to reduce risks of underestimating consumption peaks as well as overestimating the real demand, which may cause either penalization or wasting of resources. Many forecasting methods have been used successfully for forecasting demand for electricity, heating, cooling, and gas. The range of approaches includes semi-parametric regression (Bunn, 2000), time series modeling (Amjady, 2001), exponential smoothing (Taylor, 2003), neural networks (Ringwood et al., 2001), expert systems (Perchard et al., 2000), and decomposition techniques (Temraz et al., 1996). Each of these methods has its pros and cons. The approach designed and implemented in Honeywell Prague Laboratory uses methodology known from non-parametric statistics (as locally weighted regression) and machine learning (as memory-based learning). The differentiating feature is that the algorithm runs on top of a process history database, and the local regression models are built on the fly using only a fraction of the most relevant past data points. This type of data-centric solution has been implemented and successfully applied in a number of projects. The paper is organized as follows. Mathematical background of the forecasting algorithm is presented in Section 2, which is followed by the description of a typical forecasting workflow (Section 3), highlighting specific aspects of the data-centric approach. Section 4 outlines overall solution architecture and summarizes several issues faced in implementation projects.

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2. Bayesian locally weighted polynomial regression System definition. Consider a system on which a vector x of w independent (explanatory, regressor, input...) variables is used to predict the vector yofn dependent (response, output, ...) variables. In general, the explanatory and response variables can be continuous or discrete or mixed. Data set. Suppose that the sequences x^ = (XU...,XN) ajA = (yu...,yN) are observed and stored in a table that has (n + m) columns and Arrows. Weights. Each of the data points (yk, x^), k= l,...,N is assigned a weight 0 < w^: < 1, which expresses the relevance of the data point for prediction of response vector yo at a given point XQ. Model. Suppose that the dependence ofy on x is described through the stochastic functional relationship jj/k =A^k\ k= l,...JSf. The data vector x can be mapped onto a feature vector (/>=^Xk), possibly of much higher dimension. This gives a possibility to express the functional mapping J{.) as a parametric model, which in our case is the linear regression for one scalar response >^k y,=0'(/>,^e,

(1)

where ^^ is a vector of regression coefficients, ^ = ^Xk) is a regressor vector, and 8k is a stochastic component of the model. Bayesian prediction. The data set composed of explanatory and response variables is supposed to be a sample from the conditional density/7(y|x,^. The unknown parameters 0 can be eliminated using standard probability calculus rule that enable to evaluate the predictive probability density function of >^o given XQ and past data (x^, j ^ ) as follows:

Piyo\xo,x\y'')=

lpiyo\xo,y\x\0)p^iO)da,

PAO) - PoiO) n Piyi^h' ^y • ^k=K(||x - X,I)

(2) (3)

k=l

whQYQPQ(0) = p(0)is

the prior density and /?(J^QXQ,X , y , ^ ) i s the sampling density (local model likelihood) and K[.),K{0) = 1, is a suitable kernel function. As illustrated in (Kulhavy, 2003), in case of the polynomial regression and presumption of normal distribution of the stochastic component £k, the conditional density takes the form p(y,\x„0) = [im'Y^ e x p j - ^ (y, - d'{x,)f^

(4)

which allows an explicit evaluation of the unknown regression coefficients. Locality. The locality of the method is guaranteed by an appropriate choice of the neighborhood, which is defined by bandwidth vector h=(hj,..,hm)' All historical observations x with normalized Eucliedean distance D^ less than 1 are considered as neighbors of query vector XQ.

\x-x^\^=D^={\x-x^\lh)<\

(5)

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Normalized Euclidean distance D^ also defines weight of particular observation x applying kernel weighting function K. In cases when there is a lack of points, the bandwidths - and thus the size of the neighborhood - are enlarged in order to get sufficient number of points for reliable forecast. The bandwidths as well as the polynomial degrees must be known before the procedure for computation of the regression coefficients is applied. Selection of appropriate polynomial fits and bandwidths is a separate task of model selection and tuning. 3. Forecasting workflow Practical implementation of the presented predictive technology is usually carried out in a sequence of tasks, which corresponds to a broadly standardized forecasting workflow. 3.1. Collecting data Although collecting of relevant data seems to be the easiest and straightforward step of the entire workflow, it must be done very carefully because unreliable input data causes unreliable forecasts. Certain amount of historical data must be collected in advance to identify and tune the forecasting models appropriately. The demand data is usually available as a time series at the level of granularity ranging from 5 minutes to 1 hour. For purposes of the off-line analysis also the historical observations of influencing factors like ambient temperature must be in place. Then, the data collection process has to be automated in order to allow continuous 24/7 execution of the algorithm. Based on the specific model structure, the estimated future values of all relevant explanatory (input) variables must be available. This usually requires to ensure regular updates of the weather forecasts, which can typically be downloaded from a web service. Similarly, all future holidays and other calendar variables must be collected in advance and entered to the system tables. 3.2. Analysis of influencing factors After all relevant data has been collected, one can start analysis of individual influencing factors in order to identify the most suitable forecasting model. Popular approach relies on exploitation of various tools for graphical exploratory data analysis, which helps to identify hidden relations and trends, and enable to compare them with typical behavior. Number of specialized views has been designed and customized for the analysis of utility demand (see also figure 1), for instance: • Normalized monthly profiles showing averaged hourly consumption for each month. • Type of day check plot enable to recognize specific demand profiles for specific days (working days, holidays, etc.) • Demand vs. temperature scatter plot has all day types distinguished by different color. This plot requires the demand data to be aggregated on the daily basis. Based on the experience, all influencing factors can be divided into several groups: Weather and environmental conditions - Weather, and especially ambient temperature, usually has the key impact on demand. The other factors like humidity, wind speed, cloud cover, or sun irradiation can sometimes be used for better interpretation and finer modeling of the demand data but their effect is never so crucial. Calendar variables - Calendar-based variables can efficiently help with understanding the structure of demand patterns. The variable time of day, which is defined on closed interval (0;1) where 0 corresponds to 0:00 and 1 to the midnight, supports the analysis of daily demand profiles. The other important calendar variables like day of week.

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holiday or special day have categorical nature, therefore allow clustering of similar days into coherent groups. Seasonal effects - Having data for one year or more, possible significant changes can be recognized in the level of demand. Usually different volumes are required in winter compared to the summer season. The patterns of winter demand strongly correlate with

9

11

13

Hour

15

17

19

21

23

0

5

10

15

20

Ambient Temperature

Fig. 1. (a) Type of day check plot; (b) Demand vs. temperature scatter plot ambient temperature because the utilities are extensively used for heating purposes. In contrast to that the demand doesn't depend on temperature in summer, and if yes, this dependence has just reverse trend since it is usually caused by the need for cooling. Economic variables - Factors like market price for each type of the utility can improve the overall prediction accuracy. However, the key issue related to use of economics during the forecasting process is that they are frequently unavailable, their gathering is costly, or their quality poor, and therefore they are used quite rarely. Nominations - Including nominations is usually the only feasible way how to generate forecasts for industrial customers, which consume utilities for technological purposes. 3.3. Model tuning Decisive for satisfactory results is the right selection of influencing independent (input) variables whose categories have been described in previous paragraph. The following inputs assure satisfactory results for majority of problems that could be met in utility demand forecasting: • Serial time - enables to weight relevance of historical data. • Time of day - describes the demand dynamics during the day. • Holiday - can be defined in multiple ways. The simplest definition assumes distinguishing of holidays from working days. • Ambient temperature - usually the key influencing factor. The adjustable model parameters are bandwidths and polynomial fits, which are assigned to each input variable. After benchmarking multiple case studies it has been found that there are common robust settings of these parameters, however they don't work optimally for every specific case. Overcoming this difficulty may require application of some model selection algorithm like, for instance, the sequential Monte Carlo known also as particle filtering (Andrieu, 1999). The algorithm can be launched during the off-line phase to identify enhancements of the initial robust settings, or any time later during the 24/7 execution.

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3.4. Performance assessment Performance of the forecasting algorithm can be calculated in several ways. The standard approach employs the relative forecast error defined as follows: " {Forecasted, - Actual. I Error(%) = 100 V J '• ^ ~^ Actual^

(6)

This averaged error is calculated over H past data points where H corresponds to the forecasting horizon The prediction error can be easily calculated on-line, immediately as the actual data arrives, and these performance measures can be saved to the database together with forecasts. Performance data could be indicative of changes in the demand patterns that are not efficiently captured by given model. Then, the possible remedy may consist in the automated adaptation of model parameters. 3.5. Optimal resource allocation Demand forecasts can be naturally used as input to the tools for planning and scheduling, which compute optimal schedules and set points for each equipment. The schedules determine when each boiler, turbine, chiller, or other type of machinery shall be on or off (Schindler, 2004) The optimization objective is to minimize the total production costs, which requires the following three types of input parameters: • Unit parameters like overhead costs, startup and shutdown costs, efficiency curves • Forecasted demand of the consumers including all types of media • Real-time prices of energy and fuels that are purchased by the utility company From the mathematical point of view the scheduling problem is NP-hard and can be formulated as Mixed Integer Linear Programming (MILP). Suboptimal solution can be found applying standard techniques like sequential quadratic programming (SQP).

4. Implementation 4.1. Solution architecture Architecture of the data-centric forecasting solution comprises three important parts. Database layer uses infi-astructure provided by standard relational databases. Historical observations of each forecasted variable are stored and maintained in specifically designed tables, which allow for fast data retrieval, and are complemented by tables for storing results of computation. All database tables are continually fed by new data. Application layer includes forecasting engine that periodically computes new forecasts by fitting multiple local models to data points that have been identified as the most relevant and retrieved from the database. The structure and parameters of each model are coded and stored in a specialized repository. Execution of the engine in 24/7 mode is controlled by a task automation service. Presentation layer serves as an interface for operators and other users of the solution. It provides functionality for both analysis and configuration of the demand models, as well as for browsing and visualization of the computation results. 4.2. Implementation issues The presented forecasting methodology was proved in a number of projects, which however also revealed a couple of persisting issues that are frequently met in the practice (Piggott, 2003). Few examples:

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Quality of data is the most critical aspect of any forecasting application because the well-known principle "garbage in, garbage out" applies literally. Unvalidated errors in the metering system as well as unreliable weather forecasts may fundamentally damage the forecasting accuracy. Delayed consumption peaks are typically recognized when the ambient temperature reaches its daily maximum or minimum, while the peak demand is delayed by an hour or two. Behavior like this can be correctly modeled including lagged variables. Sunday afternoon effect. Usually Saturday's and Sunday's consumption profiles differs from the other days, however if the heating for Monday starts on Sunday evening it may require special treatment and model customization. Industrial customers, which use utilities for technological purposes, have either constant demand that corresponds to mutually agreed nominations, or their consumption follows actual production schedule, and thus cannot be efficiently forecasted. 5. Conclusion The forecasting solution that combines non-parametric statistics with database technology has been presented. Its major strengths can be summarized as follows: • Automatically adapts to changes in the process behavior as a result of weighting along the time axis. This allows an automated forgetting of the outdated knowledge. • The forecasting models require only a relatively small number of the most relevant points that are searched and retrieved from the historical database. This approach can be scaled up to really comprehensive datasets. • The concept of building multiple local models instead of one global model enables to handle strongly nonlinear behavior.

References Ch. Andrieu, N. De Freitas, and A. Doucet, 1999, Sequential MCMC for Bayesian model selection, In: IEEE Higher Order Statistics Workshop, Ceasarea, Israel. N. Amjady, Short-term hourly load forecasting using time-series modeling with peak load estimation capability, IEEE Trans. Power Syst., 16 (3), 498-505. D. W. Bunn, 2000, Forecasting loads and prices in competitive power markets, Proc. IEEE, 88 (2), 163-169. W. S. Cleveland, 1979, Robust locally-weighted regression and smoothing scatterplots. J. Amer. Statist. Assoc, 74, 829-836. R. Kulhavy, 2003, Bayesian Smoothing and Information Geometry, In: Advances in Learning Theory: Methods, Models and Applications, NATO Science Series III: Computers & Systems Sciences, 190, lOS Press, Amsterdam, 319-340. T. Perchard and C. Whiteland, 2000, Short Term Gas Demand Forecasting, 32"'^ Annual Meeting ofPSIG, Savannah, Georgia. J. Piggott, 2003, Accurate Load Forecasting - "You cannot be serious", 35^^ Annual Meeting of PSIG, Bern, Switzerland. J. V. Ringwood, D. Bofelli, and F. T. Murray, 2001, Forecasting electricity demand on short, medium and long time scales using neural networks, J. Intell. Robotic Syst., 31, 129-147. Z. Schindler, 2004, Efficient Load Forecasting and Optimal Resource Allocation, AICARR International Conference, Milan. J. W. Taylor, 2003, Short-term electricity demand forecasting using double seasonal exponential smoothing, J. Oper. Res. Soc, 54, 799-805. H. K. Temraz, M. M. A. Salama, and V. H. Quintana, 1996, Application of the decomposition technique for forecasting the load of a large electric power network, 1996, Proc. Inst. Elect. Eng. Gen., Transm., Distrib., 143 (1), 13-18.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A Screening Tool for Exploring Production Chains Lydia Stougie,^ Rob M. Stikkelman,^ Michiel Houwing^ ""Section of Energy and Industry, Faculty of Technology, Policy and Management, Delft University of Technology, P.O. Box 5015, 2600 GA Delft, The Netherlands Abstract This paper presents a new model for screening and comparing various production chains on a level that supports policy decision making. Two examples are presented, a centralized and a decentralized energy supply network in the Netherlands, of which we screen and compare the network with respect to key performance indicators like costs, energy efficiency, CO2 emissions, employment and capital outlay, in order to demonstrate the potential of the automated tool. Keywords: production chain modelling, policy decision making, energy infrastructures, distributed generation. 1. Introduction The search for sustainable energy carriers has increased significantly in recent years. Electricity producers use biomass and windmills, oil companies investigate the possibilities of bio-diesel and designer fiiels, car manufacturers test fuels cells, etc. Many routes for the production of sustainable energy carriers have been studied and described in many publications. An overview of the performance of the potential energy systems would be very useful to policy makers. However, the basic assumptions and therefore also the results of these studies quickly become outdated as a consequence of the continuous development of technologies and due to variations in price and availability of the primary energy sources. Moreover, it is often difficult to compare the results of these studies because they are not presented in a uniform manner. In this paper we present a new model for screening and comparing various production chains on a level that supports policy decision making. In the past we already modelled the methanol production chain as a chain of various independent operations (building blocks) (Herder and Stikkelman, 2004; Den Hengst-Bruggeling and Herder, 2004). In this study we expanded the model to make it possible to simulate branched chains, i.e. building blocks are connected to more than two other building blocks (see Figure 1).

Figure 1. Example of a branched chain. The branches A and B are part of the total chain.

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Furthermore, we generalized the description of the building blocks and added new blocks to cover the production chains of energy carriers like hydrogen and electricity. We have automated the process of assembling and adjusting the production chains. Section 2 gives a brief description of the screening tool. The centralized and decentralized energy supply network in the Netherlands are discussed in section 3, followed by conclusions and recommendations in section 4. 2. The Screening Tool The Chain Analysis Tool (CAT) enables assembling chains from individual processing steps, called building blocks. The construction and analysis of the various chains is assisted by an easy to use interface in which the user can select building blocks from the library, adjust its parameters to the specific chain at hand, and go straight to the chain analysis. The system warns the user in case there are any unspecified parameters. In addition, the user has the possibility to delete, replace or insert blocks including its sidechains without jeopardizing the overall calculations. Users can also modify building blocks or define their own building blocks (Table 1). Each building block is based upon mass and energy balances programmed into a spreadsheet. A building block requires input of data concerning the amount, physical properties and composition of the input flow. This information is extracted from the output section of the preceding block into the input section of the spreadsheet of the block. The major part of the spreadsheet of the block is used for the calculation of the output variables and performance of the block based upon the inputs. The output section, containing the calculated output including composition and conditions, provides the input of the subsequent block (Stikkelman, Stougie et al, 2003). Table 1. Use of the screening tool in brief Phase

Action

Input

Select building blocksfromthe library or build your own. Model various production networks as an assembly of independent building blocks in the click-and-drag interface.

Lego concept

Personal tuning

Tune the building blocks by adjusting their parameters to the specifics of any network.

Results

Assess the performance indicators that are instantaneously calculated by the software tool after any adjustment to the network.

In CAT two types of variables have been defined: local and chain variables. For each block, local variables can be defined and adjusted by the user. These variables, like a plant capacity, can only be used within the block in which they are defined. Chain variables, like a CO2 emission levy, are predefined and are used throughout the model. The user can adjust the global variables through a special block at the beginning of the chain, i.e. the "Begin" block. The values of the global variables are used in subsequent blocks without changing them, so they are available in all blocks.

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A list of indicators has been identified which assess the performance of an individual block on economic, environmental and social issues, e.g. costs (variable), efficiency, CO2 emissions, employment and capital outlay (investment costs). Calculations in the spreadsheet of each block result in the specific contribution of this block to the performance of the chain. These values are called incremental performance indicators or IPI's. Each block contains the same IPFs. A special block at the end of the chain, i.e. the "End" block, computes the overall values or chain performance indicators by adding, multiplying or maximizing the IPFs of the individual blocks, see Figure 2.

;,^-^'^-^-

Figure 2. Structure of a chain (Stikkelman, 2003). Each building block has its own variables (Var) and incremental performance indicators (IPI). The chain variables apply to each building block. The chain performance indicators are the overall IPFs of the chain. The expanded model enables the simulation of branched chains. We have maximized the number of input and output streams of building blocks to three, to preserve the user fi*om building very complicated production chains. There is no restriction to the number of utility and side-streams that are not connected to other building blocks. Another feature of the expanded model is the possibility to simulate production chains in which products without mass are produced, like heat and electricity. Also simultaneous production of chemicals and energy carriers, like the co-production of methanol and electricity can be simulated. The IPFs can be presented as absolute and relative numbers, the latter compared to one or all products of the network.

3. Modelling of Energy Infrastructures: Tool Application CAT is suitable for exploring many kinds of production chains, as far as it concerns continuous processes. In this paper we present two examples, a centralized and a decentralized energy supply network in the Netherlands (see Figure 3), of which we screen and compare the network with respect to key performance indicators, amongst others: costs, energy efficiency, CO2 emissions, employment and capital outlay, in order to demonstrate the potential of the automated tool. In both examples, the, approximately 7 million, Dutch households and their need for natural gas and electricity have been roughly modelled. The centralized network is a

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&M electricity MV

natural gas

a.

b.

Figure 3. Schematic representation of the centraUzed (a) and decentraUzed (b) networks. The red Hnes represent natural gas for heating, the blue lines represent high (HV), medium (MV) and low voltage (LV) transmission lines for electricity. simplification of the current Dutch configuration in which the Dutch households use natural gas for cooking (4%), heating (74%) and hot water (22%) (www.milieucentraal.nl, 2005). The required electricity is produced by large scale power plants fired on natural gas and coal. All households make use of a high efficiency boiler for heating and hot water purposes that uses natural gas as a fuel. In the other example, the decentralized network, the high efficiency boiler has been replaced with a small scale combined heat and power plant, a so-called micro-CHP. This micro-CHP takes care of the heating and hot water facilities and produces electricity; the ratio between the produced heat and electricity is fixed at 2,25 (Houwing, Heijnen and Bouwmans, 2006). As CAT is (not yet) suitable for the simulation of demand patterns, the demand for heat and electricity of the households has been cut into two extremes: summer and winter, and later on aggregated to a whole year. The amounts of heat and power produced by the micro-CHP are not in proportion to the household demands for heat and power. In winter the ratio between the demands for heat (heating plus hot water) and power is approximately 7, in summer this ratio declines to circa 0.9. When the demand for heat is always ftilfilled, in winter too much electricity and in summer too little electricity is produced. This problem has been solved by (a) the use of electric heaters in winter that convert part of the electricity produced by the micro-CHP into heat, and (b) the use of some electricity from the Dutch grid during summer. Table 2 presents some results of the simulation exercise, in which many data from various sources have been used (see References). It has to be emphasized that these numbers are rough numbers. A more detailed simulation exercise with the screening tool is possible. Also other variants of the decentralized network could be modelled, like the promising future option of "all-electricity" households that use electrical heat pumps in combination with low-temperature heating and aquifers for space heating.

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Table 2. Key incremental performance indicators of the energy supply networks (rough numbers). Costs represent variable costs and capital outlay represents investment costs. Incremental performance

"centralized network"

"decentralized network"

Costs [billion US$/y]

2

5

Energy efficiency [%]

87

85

CO2 emissions [million ton/y]

50

40

Employment [fte]

90

30

Capital outlay [billion US$]

20

60

indicator (whole chain)

Sources (see also References): www.cbs.nl, 2005; www.cogen.org/Downloadables/Publications/ FactSheet_MicroCHP_Netherlands.pdf, 2005; www.ecn.nl, 2005; www.energietransitiebeleid.nl/ energietechnologie, 2005; www.milieucentraal.nl, 2005. The rough numbers in Table 2 indicate that the centralized network is more advantageous w^ith regard to energy efficiency and costs, but the carbon dioxide emissions are higher compared to the decentralized netw^ork. In addition, Table 3 summarizes the household demands and the overall demands for natural gas and black coal of households in The Netherlands for both networks. Table 3. Household demands for heat and electricity and the overall use of natural gas and black coal of households in The Netherlands for both networks (rough numbers). Demand

"centralized network"

"decentralized network"

Natural gas use of one

1.7

2.3

Idem [GJ/y]

49

64

Electricity use of one

3.3

1

Idem [GJ/y]

12

3.6

Total energy use household [GJ/y] Overall natural gas use of the network [million GJ/y]

(61)

(68)

440

480

Overall black coal use of the network [million GJ/y]

88

26

Total overall energy use of the network [million GJ/y]

(528)

(506)

household [1000 m3/y]

household [MWh/y]

According to Table 3 the total energy use of a household in the centralized network is lower, but the total overall energy use of the network is higher. This can be explained by the higher demand for electricity and the losses occurring during electricity production.

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4. Conclusions and Recommendations The screening tool is useful for making a rough assessment of the viability of various production and transportation networks and offers policy makers an up-to-date and consistent overview of the performance of several energy networks. It enables the modelling of spatial and temporal variations of the network by adjusting parameters manually. The performance indicators are instantaneously calculated by the software tool after each adjustment to the network. The goal of the tool is not to obtain ever more accurate economic and environmental performance indicators. The scope of the tool is unlimited: additional building blocks can be added when desired. The rough numbers indicate that the decentralized network is more advantageous regarding the total energy use and carbon dioxide emissions, but is also more expensive. ft is recommended that the screening tool is used to model and assess both networks in more detail. ft is also recommended that other variants of a decentralized energy supply network in the Netherlands are taken into account.

References M. den Hengst-Bruggeling and P. M. Herder, 2004, '"Quick and Dirty' Modeling in a Decision Support Tool for Supply Chain Design," presented at 2004 IEEE Int. Conf. on Systems, Man & Cybernetics, The Hague, The Netherlands. P. M. Herder and R. M. Stikkelman, 2004, "Decision making in the methanol production chain A screening tool for exploring alternative production chains," presented at PSE2003, World Conference on Process Systems Engineering, Kunming, China. M. Houwing, P. Heijnen, and I. Bouwmans, 2006, "Deciding on Micro-CHP; A Multi-Level Decision-Making Approach," presented at IEEE, International Conference on Networking, Sensing and Control, Ft. Lauderdale, Florida, USA (to be published). R.M. Stikkelman, L. Stougie et al, 2003, Search for alternative production chains for methanol, report for the International Methanol Producers and Consumers Association, Volume I, 48 pages. L.J. de Vries, Section of Energy and Industry, Faculty of Technology, Policy and Management, Delft University of Technology, 2005, personal communication.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Cost versus Network Length Criteria in Water Network Optimal Design Petrica lancu^, V. Ple§u^, V. Lavric^'*' ^Univeristy POLITEHNICA of Bucharest, Chemical Engineering Department-CTTIP, Polizu 1-7, RO-011061, Bucharest, Romania ^Vrije Universiteit Brussel, Mechanical Engineering department, Pleinlaan 2, B-1050, Brussels, Belgium Abstract A cost-based optimization criterion is used to find the best water network topology which reduces both the investment and operating costs, when water sources with or without multiple contaminants are available. When data regarding the costs related to pipes, energy and pumps are not available, unreliable or could undergo large fluctuations, another optimization criterion could be the minimum active network length, including both the internal topology and the supply and discharge piping systems. The mathematical model of the wastewater network, assembling all the unit operations, is based upon total and contaminant species mass balances, together with the input and output constraints for each and every unit. The cost-based criterion includes the piping network cost, based upon the pipes' diameter, and the pumping cost, while the network length criterion is simply the sum of all the pipes through which contaminated water flows with a flow-rate higher than a threshold value. The optimal topologies found using these two criteria are compared against each other and also with the best topology acquired using supply water savings as criterion. In all cases, the optimization is carried out via an improved Genetic Algorithm variant, which uses one of the two aforementioned criteria and observes, in the same time, all restrictions. Keywords: wastewater network, genetic algorithms, cost based optimization, multiobjective function, threshold internal flow. 1. Introduction The problem of wastewater minimization is a challenging one at least three fold: a) lowering the fresh/supplied water consumption decreases not only its bill, knowing that low-contaminated water grows expensive, but also the operating costs; b) reusing as much water as possible not only decreases the pumping costs, but also increases the pollutant concentrations, thereby easing up the treatment operations; c) eliminating the internal loops not only saves pumping energy but also keeps the process within the bounds set by the principle of equipartition of the driving force, which ensures a lower entropy production. Ultimately, all these aforementioned factors determine a significant decrease of the water network pipes' diameter, thus lowering the investment costs. The optimization methods developed so far can be lumped into three broad categories: water pinch analysis, mathematical models of superstructure assembled systems and artificial intelligence based models and solving algorithms. Water pinch analysis embedded in wastewater minimization techniques offers simple, intuitive, geometric based methods and beneficial results, when applied to water using industries or wastewater treatment facilities (Thevendiraraj et al., 2003). Their advantage is that they provide valuable conceptual insights into the performance or

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behavior of the system under consideration (Bagajewicz, 2000; Hallale, 2002; Lavric et al, 2004). These methods were appHed for single contaminant, with very Umited extension to the multicomponent mixture, although many efforts are being made to overcome these limitations (Lavric et al, 2004). The methods based upon superstructures try to capture all possible reuse, regeneration and water internal mains opportunities, leading to non-convex NLP and MINLP models, with the drawback that there is no guarantee of global optimality (Feng and Seider, 2001; Feng and Chu, 2004). Using these concepts, Suh and Lee (2002), developed a robust optimal design strategy, considering the uncertain parameter variations in both economic and technical aspects. The objective function is the net present cost, consisting of the network piping and pumps cost and the freshwater usage cost. In the last years, the concepts emerged from artificial intelligence like neural network models or genetic algorithms gained a vivid attention, the first for on-line control, due to their self-improving ability, while the later due to their capacity of well solve any non-convex problem, without being trapped into local minima (Lavric et al, 2005a&b). 2. The mathematical model An optimal water network, supplied with fresh water only, is an oriented graph, starting with the unit operations free from contaminants at entrance. Every other unit operation i receives streams from possibly all previous operations y only (j = 1, 2 ... i-1) and sending streams to probably all next operations A: (k = i+1, i+2 ... N). When the supply water is slightly contaminated with pollutants, the associated graph remains oriented, but the starting unit operations are lumped according to their input concentration restrictions - see Lavric et al, (2004; 2005a). In the present paper, we consider that the supply water comes from four external sources, each resource having a different level of pollutants' contamination. The complete mathematical model of the wastewater network, as resulted from total and contaminant species mass balances, together with their constraints both at input and output, is given in Lavric et al (2004; 2005a) for the general case of multiple contaminated sources at different levels of contamination. 3. The objective function The same system can have multiple optimal states, each corresponding to a different objective fiinction, which encodes the peculiar performance criteria envisaged by the problem at hand. The optimality of the wastewater network can be sought with respect to either the fresh/supply water consumption (Lavric et al, 2004; Lavric et al, 2005a), or to some profit function which encapsulates the market uncertainties. In Lavric et al, (2005b), the sum of the pumping costs and the fixed charges for the piping system was used as an economic objective function. In the present study, we employ a multi-objective function which avoids the explicit use of any economic term/criteria, knowing that these factors strongly depend upon the market conditions; sometimes, correct economic figures and/or trends are hard to estimate, thus affecting the confidence level in the results. Our objective function is the weighted sum of the normalized fresh/supply water consumption and active network pipes' length:

Y fz +z

N

Z

(

/•min

F^^ = (o^^^i^

/•min \

/;=1 -» \ , F,>(i,Wi>^

in,i

Wy z out,i)

+ (1 - (o)'-^:^^^ ^ total

/

-•

^-^ ^total

i,j

(1)

Cost Versus Network Length Criteria in Water Network Optimal

1823

Design

In equation (1), A^ stands for the number of unit operations; Ftotal is the maximum supply water the network would employ in the absence of any internal reuse; Lin,i and Lout,i are the lengths of the supplying and discharging pipes, respectively, while Ltj is the length of the pipe from unit i to unit j ; Fi>0, Wi>0 and Xij>0 are the active pipe conditions, such that its length to be considered in the summation process; Ltotai is the overall length of the wastewater network's pipe system; co is the weighting factor. Details regarding the complete development of the flow term of the objective ftinction can be found in Lavric et al (2004; 2005a). Solving the associated optimization problems is not trivial, since the unknowns' number outcomes the equations' number. We employed an improved genetic algorithm which uses each internal flow as a gene, defining a chromosome from all these flows (Lavric et al 2004). The restrictions are coped with during the population generation eliminating these individuals outside the feasible domain. The individuals are interbreeding according to their selection frequency, using one-point crossover method, and then mutation is applied to randomly selected ones. 4. Results and discussions 4.1. Cost versus Network Length Criteria - no thresholdfor inner flows In order to verify if the Network Length Criterion (NLC), as resulting from (1) putting co = 0, gives comparable results with the economic objective function (EOF) used in Lavric et al. (2005b), the same runs were carried out, the results being presented in Table 1, for both criteria. These results are compared against those obtained when only fresh/supply water (F/SW) was employed as objective function, as resulting from (1) putting o) = l. We used two ranking methods, by fresh flow needed when no internal reuse is envisaged or by maximum pollutant load, to comply with the principle of driving force equipartition, as describe in Lavric et al. (2004). Irrespective of ranking procedure, NLC gives better results than F/SW but worse than EOF, when no threshold value for the internal flows is imposed, although using maximum pollutant load as ranking criterion lowers the significantly the active pipes' length. The drawback of using NLC instead of F/SW is a slight increase in the water consumption, with a maximum of 1.786 t/h for the worst case. But, this is compensated largely since the investments are lower (48.07 km against 54.42 km and 39.7 km against 53.88 km) and so does the pumping cost, since the frictions in a smaller network will be lesser. We present in Figure 1 the best NLC topology, against the best EOF one, both corresponding to the maximum pollutant load ranking procedure. The surplus of Table 2 Results of the design of a wastewater network with 15 unit operations and 6 contaminants; all four resources, as presented in Lavric et al. (2005a), are used to supply the network Objective Function

>>

.o o> CO

a:

lo Z3 •o tn 0)

"o CD

o> CD

Supply Water

Network Length & Supplyf Water

Total Cost

Inlet Pipe's flow, t/h length, km

N

54.42

Inlet Cost, Pipe's Inlet Pipe's Inlet Pipe's k$/year flow, t/h length, km flow, t/h length, km flow, t/h length, km 207.4

27.75

48.08 457.194

LI-

CO O "O _l

Y

N

co = 0

CO = 0.5

457.194

48.07 458.88

15.21

37.44

204.6

53.88

173.5

30.18

458.044

39.79

458.62

39.7

39.6

176.0

9.72

457.194

10.28

458.14

9.76

457.194

12.09

15.33

1824

P. lancu et al

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kl49.S5fi

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Figure 1 Optimal network topology using as objective function the total costs or the active pipes' length (bold-italic figures) - all flows are taken into consideration; Resources A, B, C and D are the same as in Lavric et al. (2005a) 9.52 km of pipes serves to a more uniform internal flow distribution, at the expense of operating and investments costs. But, in the same time, the NLC topology is quite independent to the market fluctuations. Combining both F/SW and NLC criteria into a single multi-objective function, putting CO = 0.5 in equation (1), made no improvement into the topology of the optimal network, irrespective of the ranking procedure we used. Still the cost based optimized network is at least 30% smaller. The network changed, in terms of active pipes and internal flows, but its total length did not. However, it is worth mentioning that the fresh/supply water consumption decreased either to its minimum value, whenfi*eshflow needed when no internal reuse was used for ranking, or close to this value, when maximum pollutant load was used instead. 4.2. Cost versus Network Length Criteria -threshold of 1 t/hfor inner flows A completely different behavior of the optimization results was observed when we imposed a threshold value of 1 t/h to the internal flows (the lines corresponding to Y in the second column of Table 1), even when only the F/SW criterion was used, although in this case, after reaching the minimum of 457.194 t/h fresh/supply water consumption, the algorithm ceases to search for a better topology. The lengths of 37.44 t/h and

Cost Versus Network Length Criteria in Water Network Optimal Design

1825

39.6 t/h, respectively, were obtained after several runs with the genetic algorithm optimization starting from random internal flows. It must be stated that the final active lengths were all in the vicinity of these values. Quite remarkably, putting a threshold value for the internal flows improved dramatically the topology of the EOF optimized wastewater network (see Figures 1 and 2 for details). In both these Figures, the EOF optimized topology corresponds to the normal written numbers. What is really surprising is the way the internal water reuse simplifies when there is this threshold value - see the conNLCtions between unit operations below 11 in Figure 2 in comparison with Figure 1. Although only two links have actually their flows below 1 t/h (0.964 t/h from 1^12 and 0.205 t/h from 9^12), their disappearance completely changed the topology. The internal flows grow bigger and its distribution between the first half units of the network changed such that the input pollutant concentrations approach to a greater extent the imposed limits, decreasing the need for internal reuse among the terminal units. This has as result a reduction of the active network length to 32.2% of the previously optimum value, at the expense of an increase with 1.44% of the costs. This increase is due to the raise of pipe diameters and fluid velocities, but the simplified network will imply lower maintenance costs, not included in the EOF.

•(8)375,008

^A) 32.436™-*4

• CD| 7232

-(CH2.457--"-*!

- f ^ 0 J4g.

Figure 2 Optimal network topology using as objective fiinction the total costs or the total pipes' length {bold-italic figures) - the flows under 1 t/h are disregarded; Resources A, B, C and D are the same as in Lavric et al. (2005a)

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P. lancu et al

When NLC objective function is used (with co = 0) to find the optimum wastewater network topology, disregarding the flows under 1 t/h, the changes are even more dramatic, the result obtained being very close to the one owed to EOF (see Figure 2 for details). Not only had the supplemental fresh water added to the network through unit 13 dropped with 33.71%, but also the internal network architecture simplified almost as much as when EOF was used, under the same circumstances. There are only one supplemental flow introduced (10.071 t/h from 4^'14) and one discharge instead of internal reuse (6.552 t/h from 11-^12 is directly sent to treatment from unit 11). The main differences come from neglecting several internal water reuses and modifying the internal flows, accordingly. But the important benefice is that we obtained the same simplification of the internal network for the last half units. Using a multi-objective optimization fimction does not improve the resulting topology. On the contrary, the beneficial effect of NLC is hindered partially by the use of F/SW, the system increasing slightly the active pipes' length, although the fresh/supply water consumption reaches the aforementioned lowest value. 5. Conclusions In this paper, we presented a multi-objective optimization criterion which can be successfully used to find the best network topology. The parameter of the function permits its use for composite demands, starting from the plain S/FW minimization and ending with optimal NLC. As expected from our previous researches largely presented in Lavric et al (2004; 2005a&b), ranking the network by the maximum pollutant load gives better topologies, no matter the parameter's value. Another important finding is that the resulting complexity of the network is heavily lowered when a given threshold value is imposed upon the internal flows, situation in which both EOF and NLC gave almost the same results. So, a straightforward continuation should be a thorough investigation of this threshold value upon the optimal topology. References M. Bagajewicz, 2000, A review of recent design procedures for water networks in refineries and process plants. Computers and Chemical Engineering, 24, 2093-113 X. Feng and W.D. Seider, 2001, A new structure and design methodology for water networks, Ind. Eng. Chem. Res., 40(26), 6140-6 X. Feng and K.H. Chu, 2004, Cost optimization of industrial wastewater reuse systems, Trans IChemE, Part B, Process Safety and Environmental Protection, 82(B3), 249-55 N. Hallale, 2002, A new graphical targeting method for water minimisation, Advances in Environmental Research, 6, 377-90 V. Lavric, P. lancu and V. Ple§u, 2004, Optimal Water System Topology through Genetic Algorithm under Multiple Contaminated-Water Sources Constraint. In Computer-Aided Chemical Engineering (Barbosa-Povoa A, Matos H, Editors), 18, Elsevier, 433-8 V. Lavric, P. lancu and V. Ple§u, 2005a, Genetic Algorithm Optimization of Water Consumption and Wastewater Network Topology, Journal of Cleaner Production, 13(15), 1405-15 V. Lavric, P. lancu, V. Ple§u, I. Ivanescu and M. Hie, 2005b, Cost-Based Water Network Optimization by Genetic Algorithm, Chem. Engng. Transactions 7, 755-60 M.-H.Suh and T.-Y. Lee, 2002, Robust Optimal Design of Wastewater Reuse Network of Plating Process, Journal of Chemical Engineering of Japan, 9, 863-73 S. Thevendiraraj, J. Klemes, D. Paz, G. Aso and G.J. Cardenas, 2003, Water and wastewater minimization study of a citrus plant. Resources, Conservation and Recycling 37,227-50 Y.H. Yang, H.H. Lou and Y.L. Huang, 2000, Synthesis of an optimal wastewater reuse network. Waste Management 20: 311-9

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Synergy Analysis of Collaboration with Biofuel Use for Environmentally Conscious Energy Systems Metin Turkay and Ahu Soylu^ College ofEngineering, Kog University, Rumelifeneri Yolu, Sariyer, 34450 Istanbul, TURKEY

Abstract The energy sector is a high-cost and high-emission sector. With the demanding environmental regulations, the energy producing companies must find new solutions continuously to decrease emissions while satisfying the energy demand. It was shown in previous studies that collaboration by exchanging steam among the energy companies can create synergy both in environmental and economical criteria. In addition to collaboration, transition to new technologies is necessary to satisfy environmental regulations. This paper presents an analysis of the expected gains in a collaborative setting with an environmentally friendly fuel alternative, biodiesel. The problem is modeled as a Mixed-Integer Linear Programming. The results of the solutions are analyzed indicating the benefits of collaboration. Keywords: Collaborative Planning, Energy Planning, Supply Chain Optimization, Biodiesel, Mixed-Integer Linear Programming 1. Introduction Energy is a fiindamental entity of modem life that has strong influence on social, industrial and economic activities in a society. Among the various forms of energy electricity constitutes a major proportion of the energy requirements. Fossil fiiel based energy systems are the dominating electricity production technologies in the world. An important concern with the fossil fiiel based electricity production systems is the release of large quantities of environmentally harmful substances such as, SOx and Green House Gases (GHG). Environmental protection must be treated as an important factor in the energy supply chain due to growing awareness of the environmental problems and enforcement of stricter environmental regulations. The Kyoto Protocol [1] demands for reductions in greenhouse gas emissions by the industrialized countries. The energy sector will be seriously affected with Kyoto Protocol since it requires countries to have an air pollution management strategy. The amount of substances released to environment will be restricted. This restriction can be achieved by reducing the energy consumption, changing energy production technologies to environmentally fi-iendlier ones or taking some remedy actions in energy supply chain management. With the increasing demand of energy by growing population and requirements for higher quality of life, the projections in EIA's report [2] suggest that without transition to new technologies, it is impossible to reduce the emissions to the year 1990 levels or less. Renewable energy technologies are promising in the sense that they release almost no emissions to the atmosphere [3]. They constitute a large group of technologies including, solar photovoltaic cells, wind turbines, and biodiesel usage in energy production. The weakness of renewable energy sources is that their high costs of ^ Current address: Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL,USA

1827

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M TurkayandA. Soylu

investment and unreliable output, changing with the shining rate, wind speed, etc. As a consequence, the solutions that integrate the renewable energy technologies together with conventional electricity production techniques are emerging. There are usually a number of energy production companies in an industrial area. Since these companies are in geographical proximity, process integration and collaboration among them is viable. It has already been shown that collaboration by exchanging steam among the energy producing companies may create synergy both in environmental and economical criteria by Tiirkay et. al.[4] and Oru9 [5], Soylu et al. [6] and Soylu [7]. Energy production is the major source of the environmentally harmful emissions and the emission reduction technologies have been studied within the context of energy systems interactions. Barreto and Kypreos [8] model the CO2 emissions trading in an energy systems model in a world-wide basis. They include technology learning in their model that affects the technology choice and emissions of regions. Hanson and Laitner [9] analyze the policies in order to select the most appropriate advanced energyefficient low-carbon technology. One of the environmentally friendly fuels is the biodiesel. Biodiesel is a nontoxic alternative fuel made from renewable fats and vegetable oils with a relatively very small fifference in performance than the petroleum-based diesel. Free of sulfur and aromatics, it can be used in engines and systems with few or no modifications. A biodiesel blend is pure biodiesel blended with petrodiesel. Blends up to 20 % biodiesel are compatible with all known oil tanks and systems. The compatibility of higher biodiesel blends depends on the properties of the materials of the tanks, pumps and fiiel lines. The usage of biodiesel as an urgent action in emission reduction by blending to existing fuels is examined and suggested in this study. 2. Analysis of Energy Production Systems A typical energy production system consists of storage tanks, boilers that convert fuel into steam at high pressures, turbines that expand higher pressure steam to lower pressure steam and convert the mechanical energy released during this expansion in the electricity and mixing equipment for mixing compatible materials originating from different sources in the system. Energy systems utilize fuel, air and other materials to generate electricity and steam. Companies can collaborate by exchanging High Pressure (HP), Medium Pressure (MP) and Low Pressure (LP) steam. There is an investment cost for such inter-company material exchanges, i.e. pipeline construction. The energy production systems that collaborate in order to improve their financial and environmental performance can exchange steam while satisfying the demand for HP, LP, MP steam and electricity. The model consists of MILP models for boilers, turbines, fiiel tanks, mixers, exchange structures and environmental constraints with an objective function of minimizing cost. The generation of HP steam is accomplished in the boilers by burning fiiel, which results in emission of harmful substances such as GHG or SOx. The boilers can be supplied with different fiiels as raw material with minimal adjustments in the operating conditions. This requires the selection of economically and/or environmentally attractive fuel among the available alternatives. The alternatives may be sulfurless oil, heavy oil, etc. which differ in calorie content, harmful emissions and cost. A model on energy production systems should include environmental limits. When environmental constraints appear, companies try to find new alternatives for producing energy with minimum emissions.

Synergy Analysis of Collaboration with Biofuel Use

1829

It is essential to include the possibility of using renewable energy technologies in the energy production systems in order to improve environmental performance. For that purpose, a new renewable energy source, biodiesel is introduced to the model with its own limitations and constraints. While modeling the following properties of biodiesel are taken into account: • The purchasing cost of biodiesel is a little higher than petrodiesel and holding cost is higher because of its material properties [10]. • The biodiesel can be mixed to only one type of the fuel and the other fiiels cannot be mixed to each other. The complete model is not given here for the sake of briefness. The constraints added for biodiesel blending are as follows: (1)

'-^^^ijkjuJ

^iJhuelL

(2) ArGFuel

z

(S)

^ijklco,

iteBioFuel

keVuel

XHF,.^<MxYFU,^^

(4)

E yFu,.,<\

(5)

Are Fuel ki BioFuel

The variable representing the HP steam production in a boiler {XykHpigent) is disaggregated into variables (XHFijkt) for the fuel type it has been produced. Eq. (1) states that the HP steam production from a fuel is proportional to calorific value of fuel, cck, and the boiler efficiency, (1////>). Eq. (2) models that the amount of HP steam produced in a boiler is equal to the sum of HP produced from different fuels in that boiler. Eq. (3) restricts the amount of biodiesel usage to maximum 20% of the blend used in that period. According to Eq. (4), if a fuel type is used in a boiler in that period YFUijkt becomes 1 where M is large number. Eq. (5) states that only one type of fuel can be used and mixed to biodiesel in a period. 2.1. Collaboration for Financial Performance In order to understand the model behavior, the model is solved for two energy producing companies whose schematic flowsheet is given in Figure 1. As can be seen, both companies have three fixel tanks, two boilers, two turbines and one mixer for each pressure level of steam. The continuous lines represent material flows within units and the discontinuous one represent steam exchanges between companies. For the sake of simplicity, the data used in the solution are not presented here. The problem is solved under non-collaborated and collaborated scenarios. The models are coded in GAMS [11] and solved with CPLEX solver [12]. The objective function consists of costs of fiiel (purchasing, ordering and holding), electricity, SOx penalty, unit operating, unit startup and construction of exchange structure. Figure 2 shows the changes in the portions of the total costs with collaboration.

1830

M Turkay and A. Soylu

Company 1

ifsZ: IP iXxER

LP I^XEI

Company! Eleciricity

I

Y

1

HP MIXER ET3

~' c

rg

E

P

Figure 1: Schematic Flowsheet of Energy Production Systems with Two Companies.

m Non-Collaborated • Collaborated

cf

0°

CP
(/

.<^

#

^+" o<^^

Figure 2 Changes in the Portions of Total Cost

It is seen that total cost of companies decreases with collaboration. The main contribution for this decrease comes from the fuel purchasing cost. Companies save from fuel cost when they collaborate, although they spend for construction of exchange structure. Companies make some operational changes in their schedules and usage characteristics in order to benefit from collaboration. Therefore, they can gain more than they spent for constructing exchange structures. Since the companies are still under their

Synergy Analysis of Collaboration with Biofuel Use

1831

environmental limits, we can conclude that it is possible for companies to decrease their costs by collaborating without worsening the environmental performance. 2.2. Collaboration for Environmental Synergy With the increasing environmental demands of the environmentalists, government and the nature itself, environmental consideration is no more a limit but an objective to be minimized. In this part, the question "can the companies benefit in terms of environmental performance, without worsening financial performance?" will be analyzed. For that reason, the objective fiinction is changed to minimizing GHG emissions. First of all, the problem is solved in non-collaborated setting. The total costs of the companies in the solution of non-collaborated setting are taken as limits on the total cost of the collaborated setting. The results are as at Table 1. Table 1 Benefit Analyses for Minimizing GHG. Non-collaborated Case

Collaborated Case

Percent Change

Total Cost

70,449.9

70,280.18

-0.24%

Total GHG Release

2,420,271.2

2,100,144.3

-13.23%

As can be seen from Table 1, by collaboration, it is possible for companies to benefit in environmental criteria without sacrificing from financial performance. It is possible for companies to benefit in environmental criteria without sacrificing from financial performance. With environmental objective fiinctions, it is interesting to examine the biodiesel usage and its effects on environmental improvements. Table 2 gives the usage and purchase amounts of biodiesel in collaborated cases with different objectives. The companies spend more biodiesel than they purchase because of the initial inventory in their fuel tanks. According to the table, both usage and purchase of biodiesel increase when the objective is to minimize GHG emissions. This shows us that, as a rapid action in emission reduction route, increasing the biodiesel usage is a possible alternative. Table 2 Biodiesel Usage and Purchase with Minimizing Emissions Objective. Collaborated Case with Minimizing Cost Objective

Collaborated Case with Minimizing GHG Emissions Objective

Total Biodiesel Purchase

0

108.66

Total Biodiesel Usage

22

130.71

3. Conclusion It was shown in previous studies that collaboration by exchanging steam among the energy companies can create synergy both in environmental and economical criteria. In this paper, a systematic approach is conducted to examine the synergy created by collaboration among the energy systems. Since the environmental regulations become stricter day by day, the companies must find new solutions continuously to decrease emissions while satisfying the energy demand. The minimization of environmental emissions therefore becomes an objective in management of energy systems. Biodiesel is a nontoxic alternative friel which can be blended with petrodiesel. In this paper the biodiesel usage is found to be usefril when emission reductions are aimed. There are also other emission reduction techniques that extensive research is conducted on. For example. Carbon Capture and Sequestration (CCS) is a technique mainly

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M TurkayandA.

involving capturing the carbon from the fuel and injecting it underground. Another technique is switching the boilers to bum natural gas which has less emission than oil and coal technologies. Biodiesel usage, CCS and switching to natural gas can be regarded as transition technologies that can be applied to fossil ftiel based cogeneration systems. As an urgent action, biodiesel usage is suggested in the short run of emission reduction targets. For examination of the long-run emission reduction, the authors are working on a system which has been developed for modeling the transition to new technologies for carbon reduction in energy systems.

References [I] UNFCC, 2004. Climate Change Information Kit, retrieved from http://www.unfccc.int/resource/iuckit/cckit2001 en.pdf. on April 15, 2005 [2] What Does the Kyoto Protocol Mean to U.S. Energy Markets and the U.S Economy, Energy Information Administration, retrieved from www.eia.doe.gov on 02.19.2005 [3] E. S. Cassedy, Prospects for Sustainable Energy: A Critical Assessment, Cambridge University Press, 2000, Cambridge, UK [4] M. TUrkay, C. Oru9, K. Fujita, T. Asakura, Multi-Company Collaborative Supply Chain Management with Economical and Environmental Considerations, Computers and Chemical Engineering, 28 (2004), 985-992. [5] C. Oru9, Analysis of Multi-Company Collaborative Supply Chain Management Using MILP, M. Sc. Thesis, K09 University, (2003) [6] A. Soylu, C. Oru9, M. Tiirkay, K. Fujita and T. Asakura, Synergy analysis of collaborative supply chain management in energy systems using multi-period MILP, European Journal of Operational Research, In Press, Corrected Proof, Available online 23 May 2005, [7] A. Soylu, Modeling and Analysis of Energy Production Systems for Environmentally Conscious Supply Chain Management, M. Sc. Thesis, K09 University, (2005). [8] L. Barreto and S. Kypreos, Emissions Trading and Technology Deployment in an EnergySystems "Bottom-Up" Model with Technology Learning, European Journal of Operational Research, 158 (2004), 243-261 [9] D. Hanson and J. A. S. Laitner, An Integrated Analysis of Policies that Increase Investments in Advanced Energy-Efficient / Low-Carbon Technologies, Energy Economics, 26 (2004), 739-755 [10] L.S Belyaev, O.V. Marchenko, S.P. Filippov, S.V. Solomin, T. B. Stepanova and A.L. Kokorin, World Energy and Transition to Sustainable Development, Kluwer Academic Publishers, Dordrecht, The Netherlands, (2002) [II] GAMS IDE, 2.0.26.8 License date Jan 19, 2004. Build VIS 21.3 138. [12] Hog, 2003. ILOG CPLEX 9.0 User's Manual.

Soylu

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Process Optimization and Scheduling of Parallel Furnaces Shutdown in Large-Scale Plants Erica P. Schulz, J. Alberto Bandoni and M. Soledad Diaz Planta Piloto de Ingenieria Quimica, PLAPIQUI (UNS-CONICET), Camino La Carrindanga Km7, Bahia Blanca (8000), Argentina

Abstract In this work, a multiperiod Mixed Integer Nonlinear Programming (MINLP) model is proposed for the scheduling of parallel furnaces shutdowns in an ethylene plant. The entire plant has been modeled through linear and nonlinear functions. Furnaces decaying performance is represented through coils internal roughness and heat load to furnaces, which are linearly dependent on operation time. Two sets of binary variables are associated to shutdowns and time dependent variables. The model determines not only the optimal shutdown schedule, but main plant operating variable profiles throughout the time horizon and the inventory management to satisfy time varying product demands. Keywords: Scheduling, MINLP, Ethylene Plant. 1. Introduction The problem of cyclic scheduling on continuous parallel units has been addressed by a few authors in the last years. Sahinidis and Grossmann (1989) proposed an MINLP model for multiproduct plants, with a continuous time representation and strategies for efficiently solving the resulting problem. Jain and Grossmann (1998) addressed the scheduling of multiple feeds to parallel units with decaying performance, solving the particular case of cracking furnaces in naphtha-fed ethylene plants. More recently, the problem of processes with decaying performance has been addressed by Houze et al.(2003); they formulate a multiperiod model for optimal catalyst replacement, incorporating an empirical correlation to model the deactivation of a catalyst over time. Ethylene plants have continuous reactors that operate in parallel, cracking ethane to produce ethylene and secondary products, presenting coke deposition on coil internal surface. Consequently, furnaces performance decays with operation time, requiring periodical cleaning shutdowns for cleanup. The determination of the optimal shutdown schedule constitutes a challenging problem in these plants. In the present work, a model for the scheduling of furnaces shutdowns is proposed, as a multiperiod MINLP model, based on a discrete time representation in a weekly basis. Furnaces decaying performance has been represented by the variables coil internal roughness and heat load to furnaces, which linearly increase with operation time, as coke deposition increases. The entire plant mathematical model has been included to take into account an ethane recycle stream fed to the furnaces and to obtain the main plant operating variable profiles throughout the time horizon. Time varying demands have been imposed and storage tanks balances have also been included. A fixed cycle length has been imposed for all ftimaces. The objective function is the maximization of the net profit. Numerical results provide detailed information on plant operating variables and inventory management, as well as the optimal shutdown schedule for the parallel continuous furnaces.

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2. Ethylene plant description A typical ethylene plant has several parallel ethane cracking furnaces, a cracked gas compressor, a separation system, a refrigeration system, a heat recovery network and a steam plant. The plant fresh feed, mainly ethane, is mixed with an ethane recycle stream and ethane fi*om storage and is then diluted with process steam to minimize coke deposition on furnaces coils. The furnaces outlet gas (ethylene and subproducts) is cooled and compressed to cryogenic conditions and it is afterwards fractionated in the separation train: demethanizer, deethanizer, depropanizer, debutanizer and two splitters to further separate ethane from ethylene and propane from propylene. 3. Mathematical model Cyclic scheduling of shutdowns for parallel ethane craking furnaces is modeled as a multiperiod Mixed Integer Nonlinear Programming (MINLP) problem, with discrete time representation. A fixed cycle length, based on plant historical data, has been considered for all furnaces. Furnaces decaying performance is described through coil internal roughness. This empirical variable has been correlated based on rigorous simulations of the plant and it has a linear dependence on furnaces operating time. In comparison with previous work, another distinctive characteristic of this paper is that the entire plant model downstream furnaces is included to capture the influence of an important ethane recycle, which is part of the frimaces feed. The objective is to determine the optimal shutdown period for each furnace throughout the time horizon, together with plant operating and production variables and determine inventory levels for raw material and products, for given product demands. Coil roughness time dependence (and time varying variables) prior to and after shutdowns is modeled through a set of binary variables, z^j, which are equal to 0 before furnace h shutdown and 1, otherwise, in time period t. They are defined by the inequalities below, in which TPh is the cleanup period (defined in Eq. (7)) for furnace h. t
z^j

t>{TP^+l)-BMl*(l-Zhf)

yhj

(1)

yh,t

(2)

The following Big-M formulations model roughness, Rught, behavior for each furnace h at time period /. Coil roughness increases linearly with operation time and lowers to a minimum roughness value (CIclean = 6.42 E -4) when the furnace is cleaned. After the shutdown period, roughness increases linearly again. All furnaces have different initial conditions (parameters Ch ) and slightly different slopes in the roughness correlations (parameters C2h), as it is shown in Fig. 1 (in Numerical Results Section). Rug,, >C\clean, +C2,^[t-{TP,+\)]-BM2^(\-z,,)

^^^^

(3)

Rug,,
^^'^

(4)

Rug,, >C\, + C 2 , U-BM2^z,,

^^'^

(5)

Rug,,
^^'^

(6)

+ BM2^(\-z,,)

Process Optimization and Scheduling of Parallel Furnaces Shutdown

1835

The heat load to each furnace depends not only on the furnace load but on the operation time (it also decreases after the fiimace has been cleaned and starts increasing as operation begins), therefore, it has been modeled with equations analogous to (3) - (6). There is a second set of binary variables associated to furnaces shutdown, yh,t, which is 1 when fiimace h is shutdown in time period t and 0, otherwise. Eqn. (7) defines the shutdown period for furnace h, TPh, and Eqn. (8) states that each furnace can be cleaned only once throughout the time horizon.

Jh,t=^

yh

(s)

Some assumptions have been made on the furnaces operating mode. At most, two fiimaces can be simultaneously shutdown during each time period. A fixed cycle time of 16 weeks and a shutdown period of one week are considered for all furnaces. As the same cycle length is assumed for all furnaces, their behavior has been considered identical. The cyclic nature of the problem is modeled by imposing that final conditions in coil roughness and inventory levels must be the same as initial ones. There are nonlinear correlations, validated with plant data, for fiimaces production iffhj.r^) as function of total flimace inlet flowrate (FfhJ"), inlet flimace pressure (PfhJ") and main optimization variables: dilution ratio (RDt) and flimace conversion (CONVhj). Inlet fiimace pressure is, in tum, a nonlinear function of preceeding variables and coil roughness, Eqn. (9). A detailed description of the ethylene plant model can be found in Schulz et al. (2005). Eqns. (10) to (15) ensure that the feed (ffhj.n and the production {ffh,j,r^) of component 7 in fumace h are zero when it is being cleaned. Mp nij and BM5j are Big-M parameters for each componenty. Pft

= f(Fft,RD„Pf^:;' ,CONV,„Rug,,]-BMA*y,_,

ffi,l,<Mj*[l-y,,,)

\fh,t

VA,?

(9) (10)

01) ffCi., ^ f(Ffh.t,ffL"j.,: RD,> Pft- CONV,,)-BM5j

*y,^,

ffrl

*y,_,

^ f[Ffh.fffh.j.t'RDfPfKt:

CONvJ+BM5j

^/hj,t Mh,j,t

ff,""j,>0-BM5,*{l-y,,,)

yhj,t

ff,""'j,<0 + BM5j*{l-y,_,)

yh.j.t

(12) (13) (14) (15)

Mass balances are formulated around splitters, mixers and ethane and products storage anks. The hydrocarbon stream, after being diluted with steam, is divided prior to the

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E.P. Schulz et al

entrance of ethane cracking reactors; there is another splitter to divide ethane recycle stream and ethane sent to storage. The model includes two main mixers: fresh feed with ethane recycle and ethane from storage, and furnaces outlet streams, at the entrance of the separation train. Economic penalties have been included in the objective function when ethane storage level does not meet a target one. Sales cannot exceed the demand forecast and the unmet demand is monitored by a penalty, also included in the objective frmction. This objective fiinction is the maximization of net profit, defined as the difference between the incomes and the costs (heating, cleaning, inventory and raw material costs) and the penalties related to the unmet demand and the unmet security level in ethane storage tank.

4. Numerical results The MINLP multiperiod model has 9703 constraints, 5049 continuous variables and 256 binary variables. Its resolution provides a cyclic schedule for the cleanup shutdowns of eight parallel ethane cracking frimaces, on a fixed cycle length of 16 weeks, as well as optimal profiles for the main operating and optimization variables and the optimal inventory management. Figure 1 shows coil roughness behavior for all furnaces. Furnace 6, which is the one with higher coil roughness value at the beginning of the time horizon (0.006), is the first one that is shutdown for cleaning at the fourth time period. In the fifth week, the fiimace starts operating and its roughness increases up to the initial value at the end of the time horizon. On the other hand, fiimace 1, which is clean at the beginning of the time horizon (initial roughness value=0.000645), is the last one being cleaned, in the fifteenth period. Each fiimace heat load shows a similar behavior, with a decrease of energy consumption after cleanup that results in a fairly uniform overall heat load toftxmaces.This behavior is shown in Fig. 2, where total heat duty decreases during shutdown periods, i.e., seven fiimaces in operation, instead of eight. However, the overall heat load remains uniform when all furnaces are in operation. Ethane conversion profiles in fumaces, which is the main optimization variable, is shown in Fig. 3 for fiimaces 1 to 4. It can be seen that ethane conversion is set to zero during shutdown periods (periods 15, 12, 9 and 8, respectively).

8

10

Time (weeks)

Figure 1. Coil internal surface roughness profiles forftimaces1 to 8

Process Optimization and Scheduling of Parallel Furnaces

Shutdown

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48000

30000

7

9

11

13

15

Time (weeks) Figure 2. Total heat load to furnaces 0.8

^

0.6

[11

>

JI 11 Hi fH

? 0.4 DFl

HF2

nF3

D F4 ri

0.2 0.0 1

3

5

7

9

11

13

15

11

13

15

Time (weeks) Figure 3. Ethane conversion in furnaces one to four. demand 2.E+05

" 1.E+05

O.E+00 1

3

5

7 9 Time (weeks)

Figure 4. Ethylene storage profile (continuous line), demand (dashed line), sales n and production

1838

E.P. Schulz et al 24000 21000 Sphere conteHl"'^""" v 18000 1

15000

^

\ /

Storage Target

^v_-^-^--^

''•y

\^ 12000

9000 1

3

5

7

9

11

13

15

Time (weeks)

Figure 5. Ethane storage level and storage target level. Figure 4 shows ethylene production and sales profiles, as well as ethylene demand and ethylene tank storage level. The model determines the production levels required to meet demands, which are satisfied in all but the last time period. Ethane storage level as compared to a target level is shown in Fig. 5. As deviations from the target level are economically penalized in the objective function, the storage level fluctuates around the storage target level. The problem has been solved in GAMS (Brooke et al., 1992) with DICOPT++ (Viswanathan and Grossmann, 1990) in four major iterations, with C 0 N 0 P T 3 and CPLEX, in 2016 sec. in a Pentium IV. 5. Conclusions A multiperiod MINLP model has been formulated for the optimal schedule of the cleanup shutdowns of parallel furnaces with decaying performance in an ethylene plant. Two continuous variables are linearly dependent on operation time: coil internal roughness and fimiace heat load. The model includes the entire plant description through nonlinear correlations and mass balances, as well as inventory management in ethane and products tanks. Two sets of binary variables have been introduced to model the effect of cleaning in the furnace performance. Despite the model has been run at academic level, numerical results show a good agreement with plant historical data. A model extension to allow for two or more fixmace shutdovms in a given time horizon is part of current work. References Brooke, A., D. Kendrick, A. Meeraus, 1992. GAMS: A users guide. Scientific Press, Palo Alto. Houze M.; Juhasz N.; Grossmann 1. E, 2003. Optimization model for production and scheduling of catalyst replacement in process with decaying performance. 4* Intemational Conference of Foundations of Computer - Aided Process Operations, Florida, EE.UU., 311 - 314. Jain v., Grossmann I. E, 1998. Cyclic scheduling of continuous parallel - process units with decaying performance. AIChE Joumal, 44, 1623-1636. Sahinidis, N. V.; Grossmann, I. E., 1991, MINLP Model for Cyclic Multiproduct Scheduling on Continuous Parallel Lines, Comp. Chem. Engng., 15,2,85 - 103 Schulz, E., Diaz, M.S., Bandoni, A., 2005, Supply chain optimisation of large-scale continuous processes, Comp. Chem. Engng, 29, 1305-1316. Viswanathan J.; Grossmann I. E., 1990. A combined penalty function and outer- approximation method for MINLP optimization. Comp. Chem. Engng., 14, 769 - 782.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Pubhshed by Elsevier B.V.

Operational Optimization of the Thermoelectric System of an Oil Refinery Sandra R. Micheletto^'^ Jose M. Pinto*''" " Petrohras - Petroleo Brasileiro S.A., Brazil ^Department of Chemical Engineering, University of Sao Paulo, Sao Paulo SP, Brazil ^Othmer-Jacobs Department of Chemical and Biological Engineering, Polytechnic University, Six Metrotech Center, Brooklyn NY, 11201, USA. Abstract The objective of this work is to develop a mathematical programming model applied to operational planning of the thermoelectric plant of the RECAP Refinery (Sao Paulo Brazil) as well as its interconnections with the process units. The problem is formulated as a Mixed Integer Linear Programming (MILP) model where the mass and energy balances, the operational status of each unit, and the demand satisfaction are defined in multiple time periods. The model determines the operational configuration of the plant by minimizing utility costs, and identifies steam losses as well as inefficient units by comparing the optimal solution with the current operation. The MILP is able to accurately represent the topology and optimize the operation of the real-world system under different utility demands and abnormal situations, achieving a 5% cost reduction. The MILP is currently integrated with the refinery database and used for the planning of the refinery utility system. Keywords: MILP, thermoelectric plant, utility planning, refinery, optimization. 1. Introduction Demand fluctuations are typical of thermoelectric plants in petroleum refineries. Among the main causes are changes in the oil feed properties, multiple operational campaigns, maintenance of process units, interruption of electric energy supply, and cost variations of fuels and electric energy. On the other hand, utility systems are essential for the operational feasibility of refineries and must continually adapt to satisfy such dynamic demands. There are important contributions that address similar problems within the optimization framework, both in academia and in industry (see Hobbs (1995) for a review). Papoulias and Grossmann (1983) proposed an MILP framework for the design and synthesis of chemical plants, including utility systems. An MINLP approach for the optimal energetic planning of chemical plants was presented by Kalitventzeff (1991). More recent applications include Iyer and Grossmann (1997), Papalexandri et al. (1998), and Strouvalis et al. (2000); an interesting industrial application of MILP planning to a petrochemical plant was developed by Hui and Natori (1996). The objective of this work is to develop a mathematical programming model applied to the thermoelectric plant of the Capuava Refinery (RECAP, Sao Paulo - Brazil) as well as its interconnections with the process units, which solves the structural and parameter optimization and satisfies demand and allocates energy at minimum cost. Several process constraints are imposed, such as mass and energy balances, operating limits of the units, demand constraints and electric energy imports. The expected results are the optimal operational level of each unit at minimal operating cost.

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S.R. Micheletto andJ.M. Pinto

2. Description of the Thermoelectric System Figure 1 presents the simplified flowchart of the RECAP central thermoelectric system in which each unit connected to header (collector) UT is represented by a rectangle, as well as its identification; this configuration considers only the systems that are subject to optimization. The initials J are used to identify pumps and ejectors; GV or L for furnaces and boilers; Mfor heat exchangers, O ox P for vessels, TF for valves and Ffor blowers and compressors. The utilities are superheated steam at 30 kgf/cm^ (V30S), saturated steam at 30 kgf/cm^ (V30) and its condensate (CV30) , 5 kgf/cm^ steam (V5) and condensate (CV5), 1 kgf/cm^ steam (VI) and condensate (CVl), ; there are also two other collectors for condensate that are CVV and CM, which originatefi*omthe vacuum system and firom vessels, respectively. The water streams are denoted by Ax, where x denotes B=brute, C=clarified, T^cooling to process, Q=cooling from process, D=dearated, G=make-up to boiler, A=feed to boiler. Some of the electric energy is produced by the refinery from the effluent gas of the catalytic cracking unit, in the turbo-expander/electric generator (S-571TE); however, some of the energy is acquiredfi*omthe local company (denoted by BE). Finally, the fuel gas and natural gas network, is denoted by GCGN; it receives natural gas from the desulphurization unit (UDS) and pipeline GASAN and fuel gas from the catalytic cracking unit For instance, the CL_V30S headers receive V30S from the GV-6301, GV-6302 and L402 boilers and from the L-572 the heat recovery boiler. The headers distribute the steam to counter-pressure turbines, condensation of the process units, URFCC, ejectors and flare system.

3. Mixed Integer Optimization Model The steam networks are designed so to allow headers to transfer utility to headers at lower pressure levels, through a pressure reduction valve. In this case, a controller regulates the water injection so to maintain the quality of the steam. Hence, pressure and temperature can be considered constant and equal to the values obtained from the data acquisition system; a MILP representation of the system is sufficiently accurate. It is important to note that design and synthesis of these systems would require an MINLP representation, as in Bruno et al. (1998). Although the model developed in this work is based on the approach of Papoulias and Grossmann (1983), who address the design of thermoelectric units in chemical plants, it relies on the conceptual modeling framework of Pinto et al. (2000) and Neiro and Pinto (2005) that was developed for production planning problems, in which the elements of the utility systems are represented according to figure 2. Moreover, the topology of the thermoelectric system is defined by the interconnection energy generator-collector and collector-consumer. In Figure 2, the utility stream ut that is generated in unit eq is sent to collector cl_ut at time t with flow rate F^^ ^^^ ^z ut,t • This collector receives all the streams of utilities that are produced in the same time period from all units [eq^ ,eq^, -e^ J that are mixed at the temperature and pressure conditions of the network. In the collectors in which pressure and temperature are variable, the resulting temperature is defined by the energy balance. Similarly, collector clut sends utility ut at time period t to consumption units. A decision variable y^^^ is associated with a flow rate variable F^^^^^^ ^^ .

Operational Optimization of the Thermoelectric System of an Oil Refinery

1841

The model relies on Mixed Integer Linear Programming (MILP) techniques where the mass and energy balances, the operational status of each unit, and the demand satisfaction are defined by mixed integer linear constraints, for each time period t.

\—[&M]

KEY:

•

CL_XX

Figure 1. Schematic representation of the RECAP thermo electric unit.

UNIT HEADER

S.R. Micheletto andJ.M. Pinto

1842 Vol ut,ut,eq_dl,t '^eql,ut,clut,t

^

eqi

Vel ut,ut,eq_d2,t f^ eq2,ut,cl_ut,t

eq2

Vcl ut,ut,eq_d3,t

^ eq_d] W ^

W

eq_d2

^

eqjds

^ W

eq_d

W

f^ eqn,ut,cl_ut,t

eqn

w

Vcl ut,ut,eq_dm,t

Figure 2. Schematic representation of collector of utility clut The following are the sets, indices, parameters and variables: Sets and Indices: CEeqjl

I

CSeqd

EQ: eq I eq_d EUiit,ut'

UT: ut I uf UTCeq UTPeq

T:t Parameters:

enthalpy of utility ut at the T-P conditions of the collector at time t price of utility ut in time period t heat received from external sources at time t (and it is only positive for the boilers; zero for all other units) lower / upper bounds of unit eqd

Put.t Qeqdj

nt

units that feed / receive streams unit eqd units / collectors units eqdihdii consume utility ut and generate utility ut' utility consumed / generated utilities consumed in unit eq utilities produced in unit eq time period

u^. u

Variables: continuous - flow rate of utility ut from eq to eqd in period t

" eq,ut,eq _d,t

binary - 1 if unit eq operates in time period t\ 0, otherwise

yeq,t

The objective is to minimize the overall utility cost at all periods that is given by the production costs, among others, of 30 kgf/cm^ steam, of fiiel consumed in the boilers as well as of electric energy. Moreover, there is an additional cost for the production of demineralized water in the reverse osmosis unit. MinC = Z ut

2-( 2^'^eq,ut,eq_d,t \^eq_d eq

(1)

The main constraints of the model are defined in general form in the sequence. Firstly, the material and energy balances are generated for each collector and for each unit in the system (2 and 3). Then the operational state of each unit is also established in (4)

Operational Optimization of the Thermoelectric System of an Oil Refinery

1843

and it relies on the binary variable yeqj.t, as well as lower and upper bounds Q.. Finally, utility demand satisfaction constraints are presented in (5). ^

eq,ut,eq_d,t

~

^

^eq_d,ut\eq,t

~^

y(ut,ut')G EU^,^^,.,Mieq_d,cl_ut\\/t 2J *^eq,ut,eq_d,t'^ut,t'^^eq_d,t~ eg^CE^q_d

^eq_dyeq_d,t

D,,, < Z eq_d

—

^

I

^eq,ut,eq_d,t

F,^,,,,, ,,

2J ^eq_d,ut\eq,t'^ut\t ^^^CS^q_d

^ <

(2)

~^

y(ut,ut')E EU^f^^f<,\/{eq_d,cl_ut)yt

(3)

dyeq d,t ^ut^UTC,^

(4)

^ut, yt

dyeq__d,\ft

(5)

eqeCE^^^

There are also constraints that are specific for the following classes of units: • Steam boilers: these comprise the purge streams based on the boiler feed stream, fuel gas consumption based on the boiler efficiency, production capacity of steam, upper bound on fuel consumption in the burners, consumption at the air pre-heater; • Vessels: steam flow rate that resultsfi-omthe vaporization that takes place after the pressure reduction valves; • Electric pumps: calculation of the electric energy consumption and bounds on the pumping capacity and on the energy consumption • Turbine pumps: calculation of the 30 kgf/cm^ steam consumption; calculation of the 5 kgf/cm^ production and bounds on the pumping capacity and steam consumption. • Number of pumps in operation for each of the units. The model is composed of objective function (1) subject to constraints (2) to (5) as well as to the specific constraints. Due to the presence of binary variables and to the linear form of objective and constraints, the model is a multiperiod MILP. 4. Computational Results The industrial plant operation was obtainedfi-oma typical selected day that contains all the available information from the process. Mass and energy balances were reconciled and this data set, denoted Base Case, was used as a reference to evaluate the performance of the proposed model. The data set was also used to the critical analysis of the simplifications introduced in the satisfaction of the real-world mass and energy balances. The MILP is solved with GAMS/CPLEX (Brooke et al, 1998). Different scenarios are considered. In the first one, the configuration and the operating conditions do not change with time, and the demand for utilities is constant. Table 1 shows the main results of the MILP model that contains 59 binary variables, 271 continuous variables and 335 constraints. The optimal solution was found in 0.4 s in a Pentium IV platform. More complex multiperiod scenarios are studied that consider the existing flexibility in the industrial plant. These correspond to changes in the operating conditions and in the configuration of the processes, import of electric energy, shutdown in equipment and units, and results in a variable utility demand along the time periods and generated an

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S. R. Micheletto and J. M. Pinto

average savings of 3-5%. Abnormal conditions are optimized, such as the shutdown in the Propylene Unit, interruption in the electric energy from the local supplier, shutdown of the turbo-expander/ electric generator set and of the heat recovery boiler from the Residue Catalytic Cracking Unit.

Table 1. Results from the optimal case and comparison to the base case. Utility

Base case

Optimum

superheated steam @ 30 kgf/cm^ (kg/h) saturated steam @ 30 kgf/cm^ (kg/h) electric energy (equivalent kg/h) fuel and natural gas (kg/h) water (kg/h) operating cost C ($/h)

117507.5 55297.2 11465.7 12934.8 43105.4 4201

112159.7 54912.2 11404.4 12597.9 43017.5 3967

Prices R$/kg 0.0116 0.0895 0.3480 0.5987 0.0179

-

Savings R$/h 1.02 5.49 117.25 13.99 96.42 234.17

5. Conclusions This paper presented a multiperiod MILP model for the operation of a refinery thermoelectric system with the objective of minimizing operational costs and loss of revenues. The MILP was able to accurately represent the topology and optimize the operation of a real-world system, achieving a 5% cost reduction. The model determines the operational configuration, identifies steam losses and inefficient units. It was applied to abnormal situations, indicating that the usual action of shutting down units is not really necessary and can be replaced with a new configuration. The MILP is currently integrated with the refinery data base for the planning of the refinery utility system. References Brooke A., Kendrick D. and Meeraus A., 1998, GAMS - A User's Guide. The Scientific Press. San Francisco, CA. Bruno J.C, Fernandez F., Castells F. and Grossmann, I.E., 1998, MINLP Model for Optimal Synthesis and Operation of Utility Systems. Trans. Inst. Chem. Engineers, 76, 246-258. Hobbs B.F., 1995, Optimization Methods for Electric Utility Resource Planning. Eur. J. Oper. Res., 83, 1-20. Hui C-W. and Natori Y., 1996, An Industrial Application Using Mixed-Integer Programming Technique: A Multi-Period Utility System Model. Comp. Chem. Engng, S20, S1577-S1582. Iyer R.R. and Grossmann I.E., 1997, Optimal Multiperiod Operational Planning for Utility Systems. Comp. Chem. Engng, 21, 787-800. Kalitventzeff B., 1991, Mixed Integer Non-Linear Programming and its Application to the Management of Utility Networks. Engng Opt., 18, 183-207. Neiro S.M.S. and Pinto J.M., 2005, Multiperiod Optimization for Production Planning of Petroleum Refineries. Chem. Engng Comm., 192, 62-88. Papalexandri K.P., Pistikopoulos E.N. and Kalitventzeff B., 1998, Modelling and Optimization Aspects in Energy Management and Plant Operation with Variable Energy Demands Application to Industrial Problems. Comp. Chem. Engng, 22, 1319-1333. Papoulias S.A. and Grossmann I.E., 1983, A Structural Optimization Approach in Process Synthesis. Part I: Utility System. Comp. Chem. Engng, 7, 695-706. Pinto J. M., Joly M. and Moro L. F. L., 2000, Planning and Scheduling Models for Refinery Operations, Comp. Chem. Engng, 24, 2259-2276. StrouvaHs A.M., Heckl I., Friedler F. and Kokossis A.C., 2000, Customized Solvers for the Operational Planning and Scheduling of Utility Systems. Comp. Chem. Engng, 24, 487-493.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

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Water Reuse: A Successful Almost Zero Discharge Case R. M. B. Alves*, R. Guardani*, A.E.Bresciani^, L.Nascimento** and C. A. O. Nascimento*'* *LSCP/CESQ - Department of Chemical Engineering, Polytechnic School University of Sao Paulo,Av. Prof Luciano Gualberto, n. 380, trav. 3, CEP 05508-900 Sao Paulo, SP, Brazil, e-mail address: [email protected]; [email protected] ^'Polibrasil Resinas S/A, Av. Airton Senna da Silva, n. 2700, CEP: 09380-090 Maua - SP - Brasil Abstract This paper presents a procedure to optimize a real problem of freshwater and wastewater reuse allocation. The case study is an industrial polypropylene unit and the solution achieved is an almost zero discharge case. The problem was decomposed into subsystems according the type of approach used to water minimization: process changes, regeneration reuse and regeneration recycling. For the regeneration approach, an innovative photochemical process capable to remove all the organic compounds contained in the wastewater in order to make it suitable to be reused in the process was used. For process changes approach, since the major water used in factory is for the cooling process system, a hybrid system composed by air cooler and wet cooling tower is been proposed to replace the wet cooling tower. The air cooler system is used first and the final temperature approach is achieved by the wet cooling tower. Thus, the main scope of the present work is to show that is possible to reach the "almost zero discharge" for an industrial case by using innovative wastewater treatment technologies together with optimization of water/air cooling systems. The results obtained prove that the water minimization techniques used can effectively reduce overall fresh water demand and the overall effluent generated, resulting in lower costs of fresh water and effluent treatment costs. Key words: Water reuse. Wastewater treatment, Optimization, Polypropylene 1. INTRODUCTION Water is an important resource for industrial activity, recreation and life in general. In most process industries water is vital in many operations and is utilized for different purposes (product formulation, cooling, high-purity water makeup systems, general plant service water, waste conveyance/transfer, potable/sanitary service and fire protection) [1]. However, processes and systems using water are being subjected to the increasing costs of wastewater treatment to meet the increasingly stringent environmental regulations, the growing demand for fresh water, the scarcity of good quality water around the world for different purposes, economic consideration, growing public concern for the quality of the environment, and restrictions on the expansion of * to whom correspondence should be addressed

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R.M.B. Alves et al

water use at many sites [2]. All these factors have created a powerful economic driving force to rationalize the water use and have increased the need for better water management and wastewater minimization, i.e., it deals with a water/wastewater allocation-planning problem. It consists of finding the minimum amount of fresh water that each water-using process needs, together with the maximum amount of water effluent from these processes that can be reused in other processes [3]. Water and wastewater minimization techniques have been widely researched, developed and applied to process industries. Bagajewicz [4] presented a review of recent design procedures for water networks in refineries and process plants. The basic concept of water minimization is through maximizing water reuse and the identification of regeneration opportunities [5]. In general, there are four approaches to water minimization [6]: process changes, reuse, regeneration reuse and regeneration recycling. The concept of zero discharge applies alternatively to the total elimination of the disposal of environmentally hazardous substances or to the concept of a closed circuit of water, such that water disposal is eliminated altogether, that is zero "liquid" discharge [4]. Closed circuits are appealing because end-of-pipe regeneration does not have to be conducted to the full extent required for disposal as water can be reused with higher level of contaminants [3]. In practice, the zero discharge process solutions are difficult, then the problem of optimization deals with the minimization of freshwater usage through re-use and proper allocation of wastewater, or series/parallel clean-up structures. Diepolder [7] and Goldblatt [2,3] discussed how realistic is this concept from the practical point of view. The industrial wastewater optimization is in many cases a tailor made problem. The present work demonstrate that innovative wastewater treatment technologies together with optimizing the water/air cooling system is able to show that is possible to reach the "almost zero discharge" under technical and economical point of view. 2. PROBLEM STATEMENT The case study is an industrial polypropylene unit. This process uses Ziegler-Natta catalyst in a loop of a pair of continuous polymerization reactors. The polymerization process is shown elsewhere [8]. The water system in this industry involves a number of fresh water sources available to satisfy the demands of each water-user: industrial process, utilities and administrative. A set of water treatment operations is available to reduce the freshwater consumption in the site and/or to achieve the environmental limits imposed on the wastewater discharge. Figure 1 shows the water system of this industry. The water used in this industry may be classified in three types: the tap water for administrative use, the water for the cooling tower to remove heat from the process (industrial water), and the water for the polymer-monomer separation process and in the extruder and pelletization process (demineralized water). The tap water has a high cost for the industry because it competes against the domestic use due the urban agglomeration nearby. The demineralized water is also expensive due the use of tap water as raw material and the demineralization process as well. All the wastewater is sent to physical and biological treatment and discharged. The task is to find the best configuration and specific wastewater treatment that will minimize the overall demand for freshwater (and thus minimize wastewater generation) at minimum total annual cost.

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Case

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Fig.l. The water system for the industrial polypropylene unit - current situation 3. Proposed WaterAVastewater Reuse 3.1. Wastewater Treatment There are three different wastewater streams deriving from the industrial process. Two of them were treated separately by photochemical process using H2O2AJV [9]. This technology was selected among several others advanced oxidation processes because it allows more simple reactors and no separation unit is needed. The process requires a previous treatment to remove solids in suspension, which, if present in the photochemical reactor, could cause light dispersion and consequently reduction in the process efficiency. Since this process is able to remove the organic compounds contained in the wastewater in a desirable level that is possible to reuse this treated water anywhere in the process. The treated wastewater from the extruder and pelletization process (2-2.5 m^/h) showed to be economical to reduce the organic concentration to be reused in the same system (substituting demineralized water) - regeneration recycling. The treated wastewater from the separation process (2-2.5 m^/h) will be reused as the raw water to produce demineralized water (until now tap water is used) - regeneration reuse. The last stream of the process wastewater is highly polluted compared with the previous streams, however the flow is only few liters per hour (around 25). This stream can be treated using reversal osmosys or even evaporation process. The administrative wastewater follows the biological treatment and will be used as washing water. 3.2. Water Consumption reduction by process changes Another way to minimize the fresh water consumption in an industrial plant is by process changes. Since the major water used in the factory is for the cooling process system, changing from wet cooling towers to a hybrid water/air cooling system is been proposed. The change of the cooling water system minimize the lost by evaporation, entrainment and purge and as consequence to minimize the fresh water (make-up water) consumption. The water is cooled first by the air cooler and the final temperature approach is achieved by the wet cooling tower. Figures 2a and 2b show schematically the base case (wet cooling tower only) and the hybrid system.

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R.M.B. Alves et al Demineralized Water I—Make-up—

Demineralized Water —Make-up-

^'urge^|' Make-ui Industrial Water

-Make-u| Industrial Water

(a)

(b)

Fig.2. Cooling Process System, (a) Base Case; (b) Hybrid System The task is to find an optimal solution for the total cost (operational and equipment costs) for the hybrid system relatively to the base case. Thus, the objective fiinction is defined as: OF = mm\^^^'

OC,

where OC is the total cost and the subscripts hs and he are hybrid system and base case, respectively. The ratio(OChs/OC^Jcan be computed also as ratio between areas or thermal duties. This work presents comparative data of costs concerning equipment, operation, acquisition and treatment of water for the wet cooling tower system and for the hybrid system. Some of these situations show a point of minimum cost and others show that is possible to replace completely the wet cooling tower by an air cooler system. The optimal solution depends on the process water temperature. Figure 3 shows the relative total cost (OChs/OC^J as a function of the relation between the air cooler and the base case thermal duties {QatrlQwater) f^^* different inlet temperatures. The curves decrease until to achieve a minimum point at Q^.^ iQyvater between 0.8 and 1.0. This minimum indicates that 80 - 100% of the total energy required could be removed by the air cooler, then the hybrid system is the more economical solution. Higher inlet temperatures favor this situation. For the industrial case presented, the optimal system will be the air cooler process as seen in the Figure 4. 4. RESULTS AND CONCLUSIONS The water system for the industrial polypropylene unit after the water and wastewater minimization as proposed in this work is shown in the Figure 5. Although the air cooler system is the better solution for the industrial case studied, on the real case, a hybrid system will be employed since the wet cooling tower is already installed and the capital cost of this equipment had been already paid off Moreover this wet cooling tower is also used for cooling some streams coming from the condenser in the purification process. As the

Water Reuse: A Successful Almost Zero Discharge

1849

Case

industry is planning, in short time, to increase the production, the water-coohng system will be a bottleneck. The solution will be to install an air cooler as proposed (including the existing water cooler). If all system is changed for air cooler (technically possible) the save will be 100%. Operational Cost - Ti„ = 55C

Operational Cost - Ti„ = 75C 1

^•-N^"'"""""'"'"'"'""""

" '""" "'"'

"" ""

"^"^-'^^ .

0.8 ^0.6

S 0.4

i

i

^ ^ - ^

O 0.2

i

\

0 J

3

0.2

0.4

0.6

0.8

1

Qair/Qwaterl

Operational Cost - Ti„ = 120C

0.4

0.6

0.8

Qair^Qwaterl

Fig. 3. Relative Total Cost (00^,/OC^,) - Tout = 40°C Operational Cost - Ti„ = 63C

Qair/Qwaterl

Fig. 4. Relative Total Cost for the polymerization reactor case study Table 1 shows the water consumption savings with the configuration proposed for the water system studied. The savings of industrial water is based on the cooling water reduction. From the process and administrative wastewater (6-7 w?/h) less than 0.5 m^/h will not be reused (cooling water purge, steam blow-down, residual wastewater process).

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R.M.B. Alves et al

Tap Water A Administration

I

• Service Water

Fig. 5. The water system for the polypropylene unit - "almost zero discharge" condition Table 1. Water Consumption Savings Savings Potable Water

5.0m^/h(50%)

- Demineralized Water

2.5 mVh (25%)

Industrial Water

100%

Wastewater reuse involves environmental benefits, because, it decreases discharge of pollutants and collection of high-quality water from ground and surface aquifers. Moreover, wastewater recycling permits industry to diminish costs for depuration processes and for freshwater availability.

REFERENCES: [1] R. M. Rosain, Chemical Engineering Progress, 89(4) (1993) 28. [2] M. E. Goldblatt, K. S. Eble and J. E. Feathers, Chemical Engineering Progress, 89(4) (1993) 22. [3] M. Bagajewicz and A. Koppol, AIChE Annual Meeting (2001) [4] M. Bagajewicz, Computers and Chemical Engineering, 24 (2000) 2093. [5] Y. P. Wang and R. Smith, Chemical Engineering Science, 49(7) (1994) 981. [6] L. Zbontar Zver and P. Glavic, Resources, Conservation and Recycling, 43 (2005) 133. [7] P. Diepolder, Hydrocarbon Processing, 71 (10) (1992) 129. [8] P.A. Melo, E. C. Biscaia Jr., J. C. Pinto, Chemical Engineering Science, 58(2003) 2805. [9] C. A. O. Nascimento, A. C. S. C. Teixeira, R. Guardani, F. H. Quina, O. Chiavonne-Filho and A.Braun, Journal of Solar Energy Engineering (in press, 2006)

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Model Development for the Optimal Water Systems Planning "E. Kondili/J.K. Kaldellis "" Optimisation of Production Systems Lab, Mechanical Eng. Dept., TEI of Piraeus, 250 P. Ralli and Thivon Av., Athens 12244, Greece ^Lab of Soft Energy Applications & Environmental Protection, TEI ofPiraeus, P.O. Box 41046, Athens 12201, Greece

Abstract A systems engineering approach is proposed for the optimal water supply chain management. The work introduces the idea of optimally allocating the existing resources quantifying the profitability of water use and eliminating inefficiencies rather than continuously seeking ways to expand the supply sources. The developed mathematical model takes into account the costs of each supply source and the benefits from the water allocation to each user, allowing the water availability to be less than the total demand. The variables of the model include the time varying water quantities supplied by different water sources and the time varying water quantities being delivered to various users. Model constraints include demands, limitations in the capacity of the various water sources and technical specifications that must be followed in the water allocation. Optimisation criteria for the water planning are proposed aiming to the identification of the most efficient operation of the integrated water system. Keywords: Mathematical Modeling and Optimisation, Water Allocation, Water Systems Optimisation

1. Introduction - Optimisation methods in Water Systems Planning Water is the most valuable natural resource and its shortage is a serious problem being faced by many areas of the planet. The water supply chain management and optimisation are evolving as the most difficuh and urgent problems [1]. Furthermore, in many areas, economic growth often brings expansion of various water demands and, thus, causes serious water shortages, either periodically of permanently. In this case, the efficiency of water allocation (i.e. what quantities are delivered to each user group) depends highly on the time period, since the existing water availability may not cover all the demands and priorities may need to be assigned to each user. In remote areas, water is normally supplied from various local sources, such as dams, desalination plants and ground reservoirs, or even may be transferred by boats. On the other hand, various users impose conflicting demands on the resources, requiring water quantities that may temporarily not be available. Therefore, the problem of the optimal water system design and planning is created. Several methodologies from systems engineering have increasingly been used over the last few decades for the design and operation of water resource systems. Optimisation models have also been applied for the solution of a number of problems related to the optimal planning of supply sources, or dealing with the total water resources management system [2-4]. However, limited research work has been carried out for the most difficult and urgent problem of the integrated water supply chain optimisation. The present work approaches the water systems planning problem taking into account the characteristics of both, supplies and demands. The methodology and the

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E. Kondili and J.K. Kaldellis

optimisation model proposed are generic and can be applied in any water system. For illustration purposes, special emphasis is given in the area of Aegean islands. 2. Overview of the water resources management problem in Hellenic Aegean Sea 2.1. Water demand Water is a constrained resource in many areas of the planet. Most of the Aegean islands suffer from severe lack of good quality fresh water, mainly because of the low precipitation and the specific geomorphology of the islands. Furthermore, these places accept many tourists; especially during the summer period their population may be five times more than the winter, thus resulting in more serious and acute water shortage problems. Classification of the water consumption includes urban users (commercial, permanent and seasonal domestic users), industrial and agricultural users. However, in the area under discussion, water demand originates from the agricultural and the urban users. Industry is not a significant water consumer in the islands. 2.2. Water supply The most common water supply sources in remote areas with limited water resources are the ground reservoirs and dams - associated with water treatment plants, the desalination plants, wells and boreholes, the transfer of water through ships, water recycle and reuse (not commonly used yet). For a long time, increasing water demand was covered by water transfer through ships in the Hellenic islands. However, there are very significant economic costs associated with this method, as well as the faith that it is completely unsustainable and does not create any infrastructure for the long-term problem solution. The resulting costs of water are different for each of the above sources. In practice, the cost includes a fixed term, associated to the depreciation of the capital investment and a variable cost term. The desalted water has a significant operating cost, while the water from ground reservoirs and dams has s serious fixed cost term, because of the high capital investment required. Indicatively, desalted water cost accounts for 3 €/m^, water transferred by ships almost 7€/m^ and from ground reservoirs almost 4,5 €/m^. 3. The Proposed Mathematical Model 3.1. Basic Characteristics and Structure of the Proposed Model The mathematical model that is proposed in the present work intends to identify the optimal solution in the operation of the water system, taking into account: • Various supply sources, each one with an associated water cost and a certain and possibly time varying capacity. • Various users, each one associated with a time varying demand and a benefit for the use of water (expressed as a monetary value per cubic meter of water). The objective of the model is the identification of the appropriate flow values from each supply source and of the quantities allocated to each user, keeping in mind that the total water availability may be less than the total demand. The allocation of the available water quantities will be made following the more sustainable principle that the real and most urgent needs must be satisfied [5]. At the same time, constant inefficiencies of the water system will be identified. Figure 1 shows a schematic representation of the system.

Model Development for the Optimal Water Systems Planning

1853

The supply sources provide water in a real or virtual storage tank; the storage tank has a specific capacity (upper limit) and a low limit that should never be violated. The usual supply sources taken into accoimt in the present work are the desalination units, the ground Reservoirs and dams, the water transfer by ship, possible own water resources (e.g. wells). In fact the model can accommodate any type of water supply. The information that is required is its cost, capacity and any existing operational constraints. 3.2. Water Supply The supply limits are determined from the capacity of each specific source. For the desalted water, the supply limit is the desalination unit's capacity, for the ground reservoir is its capacity - taking also into account the capacity of the water treatment plant, since in most cases the water from a reservoir needs to be treated before reaching the consumer. The ship transfers water into the storage tank at specific time periods. The capacity limit in this case is determined by the quantity that has been transferred by the boat. The supply costs may simply be considered as linear terms multiplying the corresponding water quantity or follow more complicated economic fiinctions. More specifically, the desalted water cost may be calculated as the sum of a fixed term, expressing the depreciation of the unit and a variable cost term or be expressed with a more complicated economic function, taking also into account various parameters of the unit's operation [6]; the same is valid for the ground reservoir and the dam. On the contrary, the water transferred by ships has only a rather high variable cost term. 3.3. Water allocation The users take water from the storage tank. The upper limits of the quantities being delivered to the various users are the corresponding time-varying demands. The water users are the agriculture (irrigation), the urban use (including permanent and seasonal domestic use and commercial use), the industry and other secondary uses. In case the required water quantity exceeds the available one, not all the requirements will be satisfied. This will definitely cause some consequences to the users (e.g. cancellation or limitation of expansion plans, losses etc.).

Demand:

Figure 1 Schematic representation of the water system The allocation of water to users will be determined by the optimisation. The model will allow the water demands to exceed the total availability, and, therefore, some users demands to be only partially satisfied, since the water allocation will be done following certain and predetermined priorities. In any case, the discrepancy between the allocated quantity and the demand should be penalised, in such a way that an extra 'cost' term is included in the objective function, caused by the water shortage for a certain user at a certain time period. The penalties

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E. Kondili and J.K. Kaldellis

should reflect in some way the losses caused by the water shortage and must be time varying, since the impacts of the water shortage are not all the times the same for a user. In addition, environmental considerations should also be taken into account in the water allocation. The simplest way to achieve that is to assign high costs in the most unsustainable water supply methods.

4. Mathematical Model Development 4.1. System parameters and variables The variables and the parameters of the system are shown in Tables 1 and 2 respectively. The optimal planning problem will be solved in a predetermined time horizon. The length of the time horizon depends on the specific problem and the desired use of the results. Actually, the length of the time horizon will also indicate the time interval that will be the basic step for the optimisation model. Parameter Magnitude Bjt Benefit for the use of the water from user j at time interval t (in €/m ) Djt Q^ MiN Sit Pjt Vmax Vmin Cj

3x

Demand of water from user j at time interval t (m ) Minimum water flow to user j at time interval t Capacity of Supply source i (m^) at time interval t Penalty for not satisfying the demand of user j at time interval t (€/m^) Maximum volume of water that can be stored in the storage tank (m^) Minimum volume of water that should be stored in the storage tank (m^) Cost of water from supply source i

Table 1: Model Parameters Variable Fit Qjt _Vt

Magnitude Flow of water from supply source i at the time interval t Water flow to user j at time interval t Water level in the reservoir at time interval t (m^)

Table 2: Model Variables 4.2. Optimisation Criterion The optimisation criterion that will try to maximise the efficiency of the water system is the maximisation of the water value, taking into account benefits and costs, i.e. Maximize Total Value of Water = Maximize (Total Benefit - Total Cost) Total Benefit = ^

^ B ^ - ^ *Q^.f

t

j

Total Cost = Supply Cost + a penalty for the discrepancy between demand and real supply to the users. Hence, the Total Cost term in the objective function is expressed as:

Total Cost = l i e , * /=;, + 2 t

i

t

Y.Pjt * (Dj, - Qj,) j

The Benefits reflect the productivity of the water allocated to each specific user and vary with time. In fact the Benefits are determined by the area under consideration, the time of the year and the profitability of each water use. Benefits should affect the water pricing, and this is currently being applied, however not consistently and rationally. Proper quantification of Benefits is an issue of further study.

Model Development for the Optimal Water Systems Planning

1855

Therefore, the optimahty criterion that maximises the total benefits and at the same time attempts to minimise as much as possible the costs and the differences between the quantities supplied to the users with their real requirements is expressed as follows:

Max X X^^t * Q . . - / I i c , * /^. + E HPJ * ^Djt - Qjt)J 4.3. Model Constraints The model constraints impose limits on the problem variables and include: The continuity equation in the water storage tank: Vf. = \/f_i "^ ^ ^ t " / ^Qit Upper and lower bounds ofthe water in the reservoir: V^-^^ <= V^ <= V^^^^ Capacity limitations of each supply scheme: F,f <= Sf^. Flows allocated to each user should not exceed the corresponding Demands. Furthermore, it may be desirable to assign a minimum water quantity to some users.

5. Application Results The above model is applied in a simple case study to illustrate the type of results that can be expected of this work. A typical island of Aegean Sea is considered. The basic parameters ofthe problem are shown in Table 3. Time Horizon Twelve months, 1 month Time interval Supply Sources Desalination (1), ground reservoir(2), transfer by ships (3) Urban (A), Irrigation (B) Users Shown in Figure 2 and Table 4 respectively Demand Profile and Bjt Q MIN 50.000 m^ jt 300000, 200000 and 500000 mVmonth respectively Si The same for both users, 10 €/m^ Pjt Cj (cost from sources 1, 2 and 3 respectively) Ci= 3 €/m^ €2= 4,4 €/m^ £3= 7 €/m^ Table 3: Data and basic assumptions for the case study

• Urban Use D Irrigation

1

2 3 4

5 6 7

8 9 10 11 12

Figure 2: Demand profile for the case study Months 1 2 3 4 5 6 7 8 10 9 11 15 15 10 5 5 BAI 5 5 5 10 15 15 5 5 5 5 5 Bet 5 5 5 5 5 5 Table 4: Benefits for the users ofthe Case study for twelve months (in €/m )

12 5 5

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E. Kondili andJ.K.

Kaldellis

600000 500000 400000

• Urban Use D Irrigation

300000 200000 100000 0

jjj,m^ 1 2 3 4 5 6 7 8 9 10 11 12

Figure 3: Model Results - Water distributed to users In the Case Study the Benefits differ for each the two users. For the urban use the benefit increases significantly during the summer months, while it remains the same throughout the year for the irrigation use. Figure 3 shows the water distributed to each user during the year. As shown, the distribution is completely directed to the urban use during the two summer months, when the demand is the highest and the corresponding benefits for user A the biggest.

6. Conclusions and Further Work An optimisation model has been proposed in order to carry out water systems planning in complex water systems with multiple supply sources and multiple users. The resulting mathematical programming model provides the water flows from each supply source and the flows allocated to each user, in order to optimise an economic / operation efficiency criterion. The model takes into account the costs of each supply source and the benefits from the water allocation to each user and allows the water availability to be less than the total demand. Thus, the work introduces the idea of optimally allocating the existing resources rather than continuously seeking ways to expand the existing sources. The present study, being part of an ongoing research, provides a basis for further formal modeling. This research has been conducted within the framework of the "Archimedes: Funding of Research Groups in TEI of Piraeus Programme ", co funded by the EU and the Greek Ministry of Education. References 1. Shah, N. (2005) Process Industry Supply chains. Advances and Challenges Computers and Chemical Engineering 29 (2005) 1225-1235 2. Voivontas, D., Arampatzis G., Manoli E., Karavitis C, Assimakopoulos D., (2003) Water supply modeling towards sustainable environmental management in small islandsL the case of Paros, Greece, Desalination 156(2003) 127-135 3. Juan Reca et.al. 2001, Optimisation model for water allocation in deficit irrigation systems: I. Description of the model, Agricultural Water Management, Vol. 48, Issue 2, pp. 103-116 4. Nishikawa T. (1998) Water Resources Optimisation problem for Santa Barbara California, J. Water Resources Planning and Management, 124(5), 252-263 5. Gleick P. H (2000) The Changing Water Paradigm. A look at Twenty-First Century Water Resources Development, International Water Resources Association, Water International, Vol. 25, 127-138 6. Kaldellis, J.K., Kavadias K.A., Kondili, E. (2004), Renewable energy desalination plants for the Greek islands—^technical and economic considerations, Desalination, Volume 170, (2), 187-203

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Pubhshed by Elsevier B.V.

Simulation of Electricity Production Systems in Autonomous Networks in order to Maximize the Wind Energy Penetration J.K.Kaldellis",E.Kondili^ ''Lab of Soft Energy Applications & Environmental Protection, TEI of Piraeus, P.O. Box 41046, Athens 12201, Greece ^Optimisation of Production Systems Lab, Mechanical Eng. Dept., TEI of Piraeus, 250 P. Ralli and Thivon Av., Athens 12244, Greece

Abstract Wind energy applications are recently characterized as an economic and attractive solution for the urgent electrification problem of most Aegean Sea islands. On the other hand, the fluctuation of daily and seasonal electricity load in almost all island grids, in combination with the stochastic behaviour of the wind, lead to substantial wind energy penetration limits, especially during the low consumption periods of the year. In this context, the present study is devoted to the development and implementation of a mathematical simulation model of the electricity production process in autonomous electrical networks, based on various types of power plants. For this purpose a new reliable and integrated numerical model is developed, using the available information of the corresponding electricity generation system (EGS), in order to calculate the maximum acceptable wind power contribution in the system, under the normal constraints that the system manager imposes. Keywords: Energy system, modeling and simulation, wind power, maximum wind energy penetration 1. Introduction-Position of the Problem Wind energy applications are characterized during the last decade as an economic and attractive solution for the urgent electrification problem of most Aegean Sea islands, especially in regions with high or medium high wind potential [1,2]. As a result, numerous wind turbines were installed between 1991 and 2001 in several islands, including Crete [2]. At this point it is important to note that during the last 25 years all these islands present a continuously increasing electrical power demand [3], which in several occasions approaches 500% compared to the corresponding demand in 1980. Up to now, the electricity requirement has been hardly fulfilled -at very high fuel consumption values- by the existing outdated autonomous power stations (APS) based on highly polluting internal combustion engines and low efficiency gas turbines. This choice results in an APS electricity production cost exceeding the amount of 200,000,0006, the fuel cost sharing more than 50%; figure (1). In an attempt to limit the extremely high electricity production cost of the entirety of autonomous electrical grids, one should definitely encourage the installation of new wind parks. Unfortunately, the fluctuation of daily and seasonal electricity load in almost all island grids, in combination with the stochastic behaviour of the wind, lead to substantial wind energy penetration limits, especially during the low consumption

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periods of the year. This obstacle results in the stagnation of new investments in wind parks erection during the last four years, since there is no guarantee of the electricity production absorbance of any new wind power addition, whereas the financial efficiency of the already operating wind parks will also decrease [4]. CRETE ISLAND ELECTRICITY SYSTEM MAIN PARAMETERS - B - Peak Electrical Power

mothers 500

m Fuel Cost

^400

•

Annual Electricity Demand

|

y^~~~-~-m'^^^'^^ \

•ti

§300 a. 2200 a. 100

Figure 1: Electricity production cost of Figure 2: Time evolution of Crete island Greek APS. electricity system parameters. In this context, the present paper describes a new reliable and integrated numerical model for the calculation of the maximum acceptable wind power contribution in an autonomous electrical system, on the basis of the available information concerning the main parameters of the corresponding electricity generation system (EGS). The proposed methodology may well be applied for the simulation of any autonomous electrical network and wind potential type, in order to estimate the optimum electricity production configuration.

2. Existing Situation in the island of Crete The Aegean Archipelago is a Hellenic area including hundreds of small or medium sized islands possessing excellent wind potential, since in many regions the corresponding annual mean wind speed exceeds the 8m/s. The biggest island of the area is the Crete island. Crete island EGS is based, since its foundation in mid-sixties, on oilfired thermal power units located either near Chania (west of Crete) or at Linoperamata (location outside of Heraklion). Recently, two internal combustion engines (2x5 IMW) started their operation in the new Atherinolakkos power station (NE Crete). The official capacity of the local EGS is 742.9MW, although the real power of the system is 693MW for winter and 652MW for summer operation. Using official long-term data (1975-2004) concerning the Crete EGS [3] several conclusions may be drawn, see also figure (2), i.e.: • There is a considerable annual increase of electricity demand approaching the 7% during the last decade, when the corresponding national average is 3.5%. • The corresponding peak power increase is even higher, since the official hourly mean peak load demand appearing during August 2004 is 543MW, almost ten-times the value of 1975. • Due to the development of the tertiary sector, i.e. services, commerce and primarily tourism, a high seasonal variation of electricity demand is encountered during the last years. For example, the (1998-2004) mean monthly electricity demand of summer (~250GWh) is more than 50% higher than the corresponding winter one (~150GWh). On the other hand it is widely accepted that Crete has a very high wind potential [5], while the wind energy exploitation activities have started since mid eighties. Taking also into account the size of the island (4* biggest of Mediterranean) and the rather

Simulation of Electricity Production Systems in Autonomous Networks

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good infrastructure situation, as well as the definitely positive attitude of local people towards wind energy applications [6], wind energy has been characterized as an advantageous option to meet the increased electricity demand requirements of local economy. As a result, a remarkable wind park installation activity started since 1992, leading by 2004 to the existence of sixteen (16) wind power stations of rated power 90MW (end of 2004). Up to now the centre of wind energy production is the East part of the island (Lasithi prefecture), although recently significant investment interest is expressed for the other parts of the island. However, the modest wind energy absorption of the already operating wind parks [4] strongly worries the new investors, thus delaying the erection of new wind energy based installations. 3. The Proposed Model 3.1. Simulation of the Existing Electricity Production Units For the simulation of the existing thermal power stations (TPS) one may use the following equation expressing the corresponding outlet power "Nf" in course of time, i.e.

Nf{t) = Y,S,{t)-N,{t)

(1)

where "Nd" is the exit power of each unit belonging to the existing EGS and "5i(t)" is the technical availability of the equipment. Generally speaking 5i=1.0, however in cases of technical problems or during the maintenance procedure 5i=0.0. Besides, it is important to note that for every power unit the following constraints should be respected in order to protect the equipment from increased wear and maintenance requirements, i.e. A^^" < N^^ < N"^^

(2)

Similarly, the energy production of the existing wind parks of the island can be estimated as:

A^.(0-ZAf„^(0

(3)

J

and depends on the wind speed "V(t)" and the ambient density "p(t)" of the area as well as on the power curve (i.e. N=N(V)) and the technical availability "5j" of the wind turbines constituting every wind park. More specifically, one may write: N„,{t) = ^^-Sj{tyNj{Vjit))

(4)

Finally, taking into consideration the local network load demand "NL(t)" and the precondition of zero load rejection or zero loss probability operation (i.e. the local network should cover the electricity demand at any time) the following relation should apply at each time point:

N,{t)
+ NSt) + NXt)

(5)

where "Nr" is any power addition from other existing sources, primarily small hydroturbines and photovoltaics. 3.2. Wind Energy Penetration Limit For grid protection reasons, in case that the wind energy production is suddenly zeroed, the remote island electrical networks manager defines an instantaneous upper wind energy penetration limit ">-" [7]. This empirically chosen value permits the operating thermal power

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J.K. Kaldellis and E. Kondili

units to replace the wind power contribution without overloading problems or electrical system voltage and frequency fluctuations and is directly related with the current back-up power of the operating thermal engines. More specifically, the operator maintains full spinning reserve of the existing thermal power units to avoid loss of load in cases of an unexpected loss of available wind power. On top of this, an additional dynamic penetration limit is necessary, directly related with the rate (kW/sec) that the thermal units in operation can replace any wind power decrease without jeopardizing the local grid stability. In this context, the maximum approved by the local EGS wind energy production "Nw*(t)" can be estimated according to the following equations, i.e.: //

N,{t)
thenNl=0

(6)

where "Nmin" is the power output of the existing TPS operating at their technical minima. In this case there is no wind energy absorption by the local network, hence all the wind energy production is rejected.

/ / N^^it)
thenN:^N^(t)-N^^(t)

(7)

The "^" limit is empirically estimated, resulting by the "m operation'' thermal power units' technical characteristics (including the constraints related with the system reactive power) and the time depending electricity load profile "ND(t)". Usually, this value is set almost arbitrarily less or equal to 30%, i.e. X.<0.3.

//

N^it)>(l + A)-N^^(t)

thenNl
(8)

In this last case the wind energy penetration is bounded by the upper wind energy contribution limit "A," and the instantaneous load demand of the system. Recapitulating, taking into consideration the equations (6) to (8) one may estimate the wind energy amount that cannot be absorbed by the local network "ANw(t)" using the following relation:

AiV., (0 = K (0 - K, it)) if (K, (0 ^ K (0) otherwise

(^)

4. Numerical Algorithm For the implementation of the proposed model, the developed algorithm progresses along the following steps for every time point (t): Step 1: Read the corresponding load demand Step 2: Define the operational characteristics of the "in operation" engines, including the technical minima and the maximum wind penetration limit "X,". At this point a preselected dispatch order of the existing units is applied by the local network operator, based on fuel consumption, engine age/history, maintenance requirements, etc. Step 3: Apply equations (6) to (8) in order to define the maximum wind energy absorbance by the local network, in accordance to the existing conditions Step 4: Estimate the electricity production of the existing wind parks (optional, only in case wind energy rejection is required) Step 5: Select the optimization criterion to be applied, e.g. maximization of wind energy penetration

Simulation of Electricity Production Systems in Autonomous Networks

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Step 6: On the basis of the decision of Step 5, calculate the corresponding wind energy absorbance and the necessary wind energy rejection, if required. Step 7: Proceed to the next time step, until the time-period is fully analyzed. Subsequently, the proposed methodology is applied for the simulation of the autonomous electrical network of Crete island. 5. Application Results As already stated, the above-described algorithm can be used for the calculation of the maximum wind energy penetration in the island of Crete for year 2003, where official data are available [3]. According to equations (6) to (8) during a typical low electricity demand period the following wind penetration scenarios may appear, see also figure (3): Figure 3: Detailed simulation of Crete island EGS operation. Load Distribution and l\^axinnum Wind Power Penetration, Crete Island

i. Zero wind energy absorption (3h
i.120 \ I 100 Q.

H80

48

72

96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720 744

Time (h)

Subsequently, the proposed algorithm is applied for a year long time-period. Selected results are presented for a low (February) and a high (August) electricity consumption period. On

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the basis of the calculation results, the maximum wind energy penetration during August is almost equal to the rated power of the existing wind parks, i.e. approximately 90MW by the end of 2003, figure (4). In most cases another 30MW could be absorbed during the most of August. However, this is not the case during February, since 50% of the installed wind power cannot be absorbed for almost the two thirds of the hours of the month, figure (5). Similar situations may appear from October to April, limiting thus the existing wind parks financial efficiency [2]. On top of this, one should not disregard that, due to the stochastic behaviour of the wind, quite often the wind energy production is lower than the maximum permitted wind energy absorbance, see also figures (4) and (5). Figure 5: Wind energy penetration in Crete island EGS, low load demand period. Maximum Wind Power Penetration, Crete Island, February 2003

- Maximum Wind Power Penetration - Installed Wind Power

0

24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672

Time (h)

6. Conclusions An integrated numerical model is developed for the simulation of the electricity production systems of existing remote islands in order to maximize the wind energy contribution in the corresponding energy balance. According to the results obtained for the Crete island, one can realistically analyze the local network behaviour for a selected timeperiod. The proposed methodology may well be applied for the simulation of any autonomous electrical network and wind potential type, in order to estimate the optimum electricity production configuration, given the desired optimization criterion. Hence, by using the proposed model one can define the maximum wind energy penetration in a given autonomous EGS. Finally, only by using a properly dimensioned energy storage installation it is possible to increase substantially the wind energy penetration in the autonomous electrical networks at a reasonable electricity production cost.

References 1. European Wind Energy Association (EWEA), 2003, "Record growth for global wind power in 2002", available in: http://www.ewea.org, 2003. 2. Kaldellis J.K., 2004, "Investigation of Greek Wind Energy Market Time-Evolution", Energy Policy Journal, Vol.32/7, pp.865-879. 3. Greek Public Power Corporation, 2003, "Annual program of autonomous power stations", Athens, Greece: Greek Public Power Corporation, Dept. of Islands; 2003. 4. KaldeUis J.K., Kavadias K.A., Fihos A., Garofallakis S., 2004, "Income Loss due to Wind Energy Rejected by the Crete Island Electrical Network: The Present Situation", Journal of AppHed Energy, Vol.79/2, pp. 127-144. 5. Centre for Renewable Energy Sources, 2005, http://www.cres.gr, CRES, Athens, Greece. 6. Kaldellis J.K., 2005, "Social Attitude Towards Wind Energy Applications in Greece", Energy Policy Journal, Vol.33/5, pp.595-602. 7. Greek Regulatory Authority of Energy, 2005, http://www.rae.gr, RAE, Athens, Greece.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Heat Integration in Micro-Fluidic Devices Toshko Zhelev", Olaf Strelow^ ^Stokes Research Institute, U niversity of Limerick, Ireland ^University of Giessen-Frieberg, Germany Abstract Presented paper addresses some problems of energy consumption minimisation in portable micro-total-analysis systems (|i-TAS) and more precisely in micro-polymerase chain reaction systems (MPCRS) used for on spot DNA analysis. Applied methodology takes on board two well-established heat integration concepts such as earlier developed deterministic method and the conceptual design approach, adapts them and applies them in the challenging area of micro-reactors. The focus of presented procedures is on a range of designs, proven as promising by earlier researchers. It considers three heating/cooling options: (a) a classical resistor heating and cooling case; (b) a three fluids case, when the sample droplets moving in a carrier fluid are being heated and pumped by a third fluid, and (c) a new proposed design claiming to overcome major drawbacks of mincrofluidic devices. The main optimisation objectives are the energy conservation and the minimisation of the time for DNA amplification. Additional design requirements are the high throughput and flexibility (versatility) improvement. The major control parameters are the cycle time, the ramp rate and the number of cycles. Advanced steps towards development of a computer code for temperature distribution simulation using deterministic heat transfer model are also reported. Future steps towards more precise problem formulation including the consideration of an enzyme-catalysed bio-chemical reaction and its impact on the fluidic properties are discussed. Keywords: bio chips, heat integration, micro systems, PCR 1. Introduction Today's chemical industry is constantly searching for controllable, high throughput, and environmentally friendly methods of products generation characterised by high degree of chemical selectivity. For both synthesis and analysis, integrated chemical microdevices are now attracting great interest from many research groups. Novel microdesigns are made to perform many standard operations opening up new ways of carrying out chemical transformations. Since transport phenomena are scale-dependent, micro reaction systems, as a new type of reaction systems, possess some unique characteristics. Heat transfer coefficients exceed those of conventional heat exchangers by an order of magnitude. Micro-mixers can reduce mixing time to milli- or nano- seconds. The increased surface to volume ratio in micro reaction systems has implications for multi-phase surface-catalysed reactions (Ehrfeld et al., 2000). The benefit - ability to maintain high level of control and selectivity; elimination of problems associated with the conventional scaling up procedure, high throughput, rapid reaction, improved conversion and many others. Busy with the competitive game of cheap manufacturing, smaller sizes, challenges with surface phenomena and contamination, the researchers still battle with the separate components of the micro-systems and do not pay serious attention to grass-root

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optimization. Our touch with the intensive activities in the area of micro-medical devices design, exercised by the members of Stokes Research Institute - Ireland (www.stokes.ie), helped to identify some optimization objectives that have important impact on the design and efficient operation of micro-reactor systems for DNA analysis, that will be revealed after a short introduction of the process.

2. The Process (PCR) PCR is an enzyme-catalyzed amplification technique, which allows any nucleic acid sequence to be generated in vitro (McPherson et al., 2001). Since its introduction in 1983, PCR has been playing a central role in the field of molecular biology. It is the most powerful technique, by which a DNA segment may be amplified exponentially, for applications like DNA fingerprinting, genomic cloning, genotyping for disease diagnosis, etc. PCR typically requires thermo-cycling with three-temperature steps to sequentially subject template DNA to denaturation (spliting the DNA molecule) at approximately 95°C, primer annealing at 50-60°C and extension at 72 to 75°C. A typical PCR reaction thus goes through 20-40 cycles so as to prepare sufficient DNA material for analysis by hybridization. The number of molecules or copies of segmented molecules varies from 10^ - for cancer detection, to 10^ fingerprinting and genetic disorder predisposition. 3. The Micro-System A micro polymerase chain reactor system (MPCRS) is generally defined as a series of interconnected micro-channels in the range of 10 to 300 microns in diameter etched into a solid substrate. The channel network is connected to a series of reservoirs containing chemical reagents, products and/or waste to form the complete device with overall dimensions of a few centimetres. PCR reactor (cycler) is usually silicon, glass, siliconglass, quartz or plastic device. The MPCRS in general case consists of three main components - (a) sample preparation subsystem; (b) polymerisation reactor; (c) product separation; (d) detection subsystem. In general the MPCR system comprises of integrated micro-unit operations such as channels and fluidic connections (piping), pumps, dosing and injection devices, reactors, mixers, valves, filters, heaters, coolers, physical and chemical sensors, separation and extraction units, detectors, centrofuges, feedback and control loops, etc. As one can see, the MPCRS appears to be in its character very similar to a typical large-scale processing technology and therefore its optimal design and operation would be expected to be very similar to the major chemical production processes. The current ability to design highly efficient, controllable and reliable industrial processes is mainly based on modem achievements of chemical and process engineering. The major approaches assisting the design process of such complex systems are the process modelling, process simulation and process optimisation. They intensify the design process providing ability to screen large number of options avoiding the danger of safety character, environmental hazards and waste of materials, time and money. Interesting enough, the application of current advances of process modeling, simulation, design and integration in the area of micro reactor systems are far from satisfactory.

4. Design Implications The system design (the architecture), is perhaps the most important feature of a microbio-chip as it defines the function and sequence of processes taking place in the device. Miniaturisation of the PCR system provides significantly improved thermal energy

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transfer compared to macro-scale systems, thus enabling an increased speed of thermal cycling. The design implications are quite different compare to the macro scale. For instance, the reserves (over-design, design margins) accounting for possible deviations from the nominal operation condition, can not be used as flexibility compensation in the case of micro reactors, because any over-sizing can substantially affect the function of the device. As stated by Hasebe (2003), the shape factor plays much bigger role in the design of micro devices compare to conventional unit operations. Terms such as perfect mixing, overall heat transfer coefficient, plug-flow, steady state operation, etc. as usual assumption in micro-systems are not applicable in micro-scale. The shape of the device has a large degree of freedom and needs to be constrained. New types constraints are to be introduced in the design problem formulation, such as average residence time; residence time distribution and temperature distribution. 4.1. Design Optimisation At present the design of PCR systems is based on intuition and trial-and-error approach. Lots of knowledge is palled in the area of fluidics (hydrodynamics of samples movement in small channels), heat transfer and samples preparation. The optimisation criterion targeting increased PCR efficiency is, according to Gad-elHak (1999), the cycle time minimisation, leading further to ramp-rate optimisation. As suggested by Walsh and Davies (2004), this consequently leads to minimisation of sample fluid volumes (in order to minimise the lumped heat capacity and transition times between temperature zones allowing for controllable residency times in each of the stages of the cycle and user controlled temperatures in each of the zones of the device). In summary, the following factors influence the efficiency of MPCRS: (a) temperature of denaturing, annealing and elongation; (b) duration of these thermal processes (heating and cooling); (c) ramp rate (trajectory) of heating and cooling; (d) cycle number to generate reliable amount of product. Some additional efficiency related factors should be considered, such as (e) transfer rate from a zone to zone (related to the design specifics), catalyst nature (enzyme), primer length, reaction buffer, preparation conditions (temperature), etc. It is interesting to note that most of the efficiency related factors are time-dependent. 5. Integration Nguyen (2002) refers to the density of component, concluding that the degree of integration in micro devices follows Moore's law, doubling integration density every 18 months. This growth currently is limited by the photolithography technology, slowing the forecast and doubling integration density every 24 months. The new paradigm for process-systems on chip has analogy with the Systems on chip announced by Benini (2003). It brings the problems of micro-network synthesis, micronetwork integration and resources management onto the domain of optimal process design and optimal process operation. 5.1. Architecture of Micro-system (structure, topology) As it can be expected, the architecture of micro-system plays substantial role in the total efficiency consideration. The production rate can be improved by increasing the number of micro-units operating working in parallel. The structure varies from aggregated micro devices, through a combination of conventional and micro devices, to a hybrid system. The production rate can be changed by changing the number of parallel reactor units. Following these principle we recognise the needs of a new design combining the qualities of traditional and micro-devices {paragraph 6.4.).

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5.2. "Horizontal" and "Vertical" Integration The development of systems featuring "horizontal" integration by building parallel lanes for high throughput applications, and "vertical" integration by implementing several functions on a single device is the most exciting trend in the microchip world. Often the major push towards integration is the portability and decreased reliance on external infi'astructure. This brings the problems of heat integration, energy recovery and conservation higher in the agenda.

6. Heat integration We support the predictions for emerging market of portable DNA analysis devices. Decreasing the risk of contamination, sample mismanagement, rapid diagnostic and quick treatment support the concept of "on spot analysis". In this case energy conservation problems are to be of important concern together with the systems flexibility (ability to tackle different diseases). Let us analyse the possibility for energy conservation and energy recovery in some of the known designs. Due to different geometric design parameters, the contribution of heat conduction within the channel wall is significantly different from that of macroscopic devices. As reported by Punch et al (2003), the Biot number (the ratio of conductive to convective thermal resistance) is a key parameter in the thermal analysis of micro-PCR device. It was found that the velocities changes within 0.1 - 1.0 mm/s have little effect on the temperature profile along the micro-channels. 6.1. Classical meander-type PCR The design solution for controlling the temperature and the cycle-number in the three PCR's temperature zones is through solid resistors fixed under the meanders of channels. The following obvious sequence of heating/cooling zones assists the efficient energy management: The sample after being heated to 93''C passes bypassing meanders trough 72°-zone, assisting the beginning of the cooling and is ftirther cooled in the meanders of the 58° zone. Next, passing quickly through the 72° zone the sample is preheated helping to reach the required temperature in the adjacent 93° zone, and so on. The length of the channel in each zone/the wide of the heater, the volumetric flowrate/pumping abilities and the minimum size of the channel are in strong relation, where the pressure drop and contamination in parallel with the rigid control options are of major concerns. Temperature and cycle number control is possible through manipulating the number of heating sections or the meanders; 6.2, Liquid-liquid heating and pumping Due to the demands of highly-efficient handling of large groups of samples, methods of high-throughput PCR are considered. Serial processing of many samples in small time intervals is nowadays the most popular strategy used in laboratories, and therefore the generation of a continuous flow is a crucial factor. Since in microchannels extreme velocity gradients exist over the cross-section of streaming fluids and chemical guttering of templates and enzymes by reaction concentrates at the wall surfaces, the cross-talk leading to contamination between samples in serially working devices is a ftmdamentally important issue. Therefore, microheterogenous phase systems are applied in micro PCR systems, where a carrying fluid can insulate the reacting liquid volumes from other reacting samples. An example of such a system is the rotational device of Walsh & Davies, (2004). It arranges the droplets of sample fluid to float in a carrier immiscible fluid to prevent contamination. A third fluid comes in direct contact with the carrier fluid performing two fimctions - pumping and heating/cooling and leaving the channel in particular point. The control is much more flexible and easy, but the system

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stability is a bit of a problem. The task of heat exchanger network synthesis has here its specifics. It is required to find network structure which ensures minimum heating and cooling requirements adjusting the supply and target temperatures of one service stream (utility) having two functions (heating and cooling). The minimum temperature approach was 0.2°C; As pointed out earlier, the heat exchanger units are of parallel flow type because the heating/cooling fluid is in direct contact with the carrier fluid and has to perform pumping service as well. The mass flow rate in this case is restricted and supports the slow motion of the carrier fluid (in the range of 0.1-1.0 mm/s). The temperature difference at the "hot end" of the exchanger can be used to control the ramp rate. The optimal energy management structure was found using the Pinch concept. Because the cost is not of a concern and the maximum energy recovery is a goal driving us closer to portable device, the minimisation of Zir^/„ would be the only option and the trade-off in the tendency to allow for very small AT^nm would be offset by the quality/efficiency of DNA amplification related to the time to produce the require minimum of amplified DNA segments. The suggested energy recovery option requires more than twice less external energy supply (exactly 53.6%) and twice less cooling compared to the case when no energy recovery is considered. 6.3. Transferring Knowledge from Deterministic Plate Heat Exchanger Systems Design As it is well-known, the biggest advantage of plate heat exchangers is their flexibility. Strelow, (2000), proposed a generalised deterministic calculation method that can be utilised for the case of micro heat exchanger systems. The method considers the complexity of streams' flow patterns in plate heat exchangers. It accounts for parallel, serial and subdivided stream passages; covers the heat transfer between two and more process streams, spiral passages and other deviations from the classical counter-flow fluids pattern. The proposed method allows exact and iteration fi-ee determination of steady-state temperature profiles in all variations listed above and provides easy calculation of heat flow along the plates' walls and temperature distribution in plate passages. Iterations are unavoidable only in cases of phase changes and consideration of nonlinear temperature dependent fluid properties. Presented model allows adequate simulation of heat exchanger network operation based on generalised operating conditions. It is an approximation of the exact solutions of the precise system of differential equations describing accurately the system. In our case it permits the consideration of viscosity change and heat capacity change related to polymerisation process. The universal deterministic model of a heat exchanger system of an arbitrary structure/topology (as proposed by Strelow, 1997), consists of five matrices (input /, output O, structure 5, matches of thermal passages K and heat resistance L) and two vectors (of temperature T and heat capacities C). The matrix form of the plate heat exchanger model is: 7^ = ^f, or

(1)

T° =[OiE-Y[@,SrY{e,J]T' i-n

where

(2)

i=n

^-[(E±S'r'

0,,. = 2 ^ {

O'OjE ± Sy IC]K,LKf ^

}

The presented approach was used to model and predict the temperature distribution in a range of classical micro-fluidic devices - the meander type PCRs. A prototype of

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software for interactive simulation of the heat integration was developed by Professor Strelow. 6.4. Revolving type cycler (new design) The reason to propose a new type of reactor was the desire to overcome the difficulties and inherited drawbacks of recent microfluidic devices. The biggest advantage of the micro-fluidic bio-chips is the size of the micro-channels and the continuous process allowing for rapid heating and cooling at a high throughput tackling large number of samples. The challenge in this case is the contamination, which has to be ignored at 100%. The second challenge is the flexibility and controllability of MPCRS. All known devices suffer drawbacks in these areas because of their rigid design. The new design is based on batch principles and has a revolving principle of operation and design. The samples are injected in a circular disposable cartridge, when the core (internal) of the system contains a revolving cylinder divided in three heating/cooling sections. The controllable speed of rotation allows for time controlled cycling; number of rotations will control the number of cycles and dynamic temperature control of heating/cooling zones control the ramp-rate. The samples are rejected to the separation and analysis section after efficiency test in vitro, the revolver is decoupled from the preparation section and the micro-channel cartridge is disposed. The process of manufacturing of disposable cartridges is still under design, but gives potential for increasing the number of micro channels (samples throughput), minimisation of wall thickness and decreasing the production cost. 7. Conclusions Putting together the results of this study we realise that we just started to scratch the surface of the problems of the optimal design and operation of efficient, high throughput micro polymerise chain reaction systems. We realise that the results are not fully conclusive and the main reason is the large variety of designs under constant and rapid development at present. When attempting structure synthesis and regimes optimisation, the process engineering practice usually steps on standard processes and unit operations. Here, in the micro reactor systems, the principles of design are still not settled. It will take quite some time to establish well proven efficient reactors, separators and heat transfer unit operations. This should be followed by reliable and adequate mathematical models helping simulation predictions and sensitivity analysis. This process should be guided by principles of process systems engineering, process integration and optimisation. Our work in this direction continues. 8. References J. Punch, B.Rogers, D.Newport & M.Davies, 2003, IMECE-41884, Washington, 1. L. Benini, G.De-Micheli, 2003, Networks on Chip: A New Paradigm for Systems on Chips Design, http://akebono.stanford.edu/users/nanni/research/net/papers/date02.pdf. M. Gad-el-Hak, 1999, The Fluid Mechanics of Microdevices, J.Fluid Engng., 121, 5. M.J. McPherson and S.G. Moller, 2001, The Basics PCR, BIOS Sci. PubHshers Ltd. N-T. Nguyen, 2002, Fundamentals & Application of Microfluidics, Artech House. O. Strelow, 1997, Eine Allgemeine Berechnungsmethode fiier Warmeubertragerschaltungen, Forsch Ingenieurwes, 63, 255-261. & 2000, Int.J.Therm.Sci., 39, 645-658. P. Walsh, and M. Davies, 2004, 7^ Annual Sir Bernard Crossland Symposium, 1. S. Hasebe, 2003, Process Systems Engineering, Elsevier Science, 89. W. Ehrfeld, W., V. Hessel, and H. Lowe, 2000 Microreactors. 1 ed., Wiley-VCH. 288.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Network synthesis for a district energy system: a step towards sustainability C.Weber% I. Heckl^, F. Priedler^, F. MarechaP, D. Favrat^ ^Laboratory of Industrial Energy Systems, Ecole Polytechnique Federale de Lausanne, LENI-ISE-STI, Bat. ME A2, Station 9, 1015 Lausanne, Switzerland ^Department of Computer Science, University of Veszprem, Veszprem-Egyetem u. 10, 8200 Veszprem, Hungary In this paper, the first results of a new method for the configuration of district energy systems are presented. District energy systems are believed to help decreasing the C02-emissions due to energy services (heating, cooling, electricity and hot water), by implementing polygeneration energy conversion technologies, connected to a group of buildings over a network. The synthesis of the network is an important but not trivial task, mainly because the problem involves a large number of integer variables and results in an mixed integer linear programming problem (MIL?) that needs to be optimised. Keywords: District heating. Network synthesis, MILP, C02-emissions. 1. Symbols Roman letters A: Set of arcs in the maximal structure Ao'. Fixed investment cost of operating unit o [CHF] Bo'. Proportional investment cost of operating unit o [CHF] Cb'. Investment costs of the boiler [CHF] Chp' Investment costs of the heat pump [CHF] Cm'- Price of raw material m [CHF] Hbi Nominal power of the boiler [kW^/i] Hhp'. Nominal power of the heat pump [kW^/^] M: Set of materials M: Arbitrary large number O: Set of operating units P: Set of products Pm'- Minimum amount of required product m [kg] R: Set of raw materials Ro,m'- Amount of material m produced or required by operating unit o if Xo=l Sm'- Maximum amount of available raw material m [kg] Xo'- Size of operating unit o

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yo'. Binary variable: 1 if unit o is present, 0 otherwise

2. Introduction The reduction of C02-emissions is a challenge for the coming decade, especially with the implementation of the Kyoto protocol. Besides transportation, energy services (heating, cooling, electricity and hot water) are responsible for a large share of the total greenhouse gaz emissions. For example in Switzerland, heating generates over 40% of the total emissions (all energy sectors considered, including transportation) [l]-[2], making it a priority candidate among energy services when considering ways to decrease the overall emissions of Switzerland. To decrease the emissions generated by the energy services, one way is to increase the efficiency of the different energy conversion technologies that provide these services, by combining them in a polygeneration energy system. A polygeneration energy system is a system that generates more than one single energy service. Advanced systems allow to save over 60% of the energy resources and emissions compared to conventional solutions [2]. However, to ensure that polygeneration systems operate as often as possible at or near their optimal load, they should be implemented so as to meet the requirements of more than just one building. By doing so, one can take advantage of the various load profiles of the buildings by compensating the fluctuations and having therefore a smoother operation. Besides, because these systems are complex and defacto difficult to operate, there are usually not justified in an individual building where no continuous professional control can be guaranteed. It is much more advantageous to implement them in a small plant that serves several buildings, and that is managed by an energy service company. The resulting energy system with one (or more) polygeneration energy conversion technologies, together with the network connecting the technologies and the different buildings, is called district energy system.

3. M e t h o d for the configuration of district energy systems The optimization of the network synthesis for district energy systems is combinatorially complex, for several reasons. First, the number of the various combinations of different locations and sizes of energy plants is extremely high. Second, there are usually a lot of different ways to link the buildings together. Third, the diameters of the pipes are usually defined by a given, non continuous set of possible diameters. Finally, the number of constraints related to a retrofit problem is usually larger than for a blank-sheet design. In this paper, we present the first results of an algorithm based on the graph theory approach, to synthesize and optimize networks for district energy systems. A single-period, single-service, multiple-ressources network synthesis problem has been considered. Since the focus is on the synthesis phase, considering single period instead of multiple period is acceptable. Besides, although the aim is to synthesize networks with polygeneration energy conversion technologies, the focus has been set on heating only (single-service), in this first step. Finally the network synthesis algorithm has the choice to implement a single or multiples ressource(s) (energy conversion technologies) in the network.

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4. Graph theory approach The proposed algorithm is based on the graph theory approach developed by Friedler et al. [3]. The graph theory approach allows to generate a mathematical representation of a superstructure, a superstructure being the unity of all production units and materials (raw materials, intermediates and final products) involved in the production of the required output. In the case analysed here, the raw materials are the inputs of the energy conversion technologies (for instance natural gas in the case of boilers and electricity in the case of heat-pumps), the final product is the heat delivered to each building, and the operating units are the energy converison technologies (heat-pumps and boilers) as well as the pipes transfering the water from the energy conversion technologies to the consumers and back. The mathematical representation enables the development of efficient algorithms for the synthesis and optimization of an optimal solution structure. The optimal solution structure is the network, among all the possible networks, that minimizes for instance the costs or the C02-emissions. To compute the optimal solution structure, the graph theory approach is combined with the accelerated branch-and-bound algorithm (ABB) [4]. The main equations of the ABB algorithm are given in fig.l. The first term of the objective function corresponds to the costs of the operating units, the second to the cost of the raw materials. The first constraint states that the leaving material m (on arcs or as product) is less or equal to the -, material entering the node (on arcs or as raw material). The second constraint ensures that the size of an operating unit is 0 if it is not present in a solution structure. Fig.2 shows a very simple example network including one heating plant (1), several possible connections (2-15), and the buildings to be heated, as well as the corresponding superstructure in the so-called process graph (P-graph).

o€0

meR y

{oeO:(m,o)eA}

S.t.

{oeO:(m,o)eA}

{oeO:(o,m)eA}

Figure 1. Main equations of the ABB algorithm

5. Mathematical model for the single-period, single-service and multiple-resources network Following characteristics have been considered to build the superstructure:

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Energy source (sun, wind, waste, natural gas,...) A Material i a Operating unit ^ Material "*15

Figure 2. Left: District with several possible connections; Right: Resulting P-graph of the superstructure (Q^: power delivered to building i): Operating unit 1 is a plant, and the others are pipes.

1. The energy conversion technologies for heating are boilers and heat-pumps. They can be located in one or more building(s) belonging to the district. 2. The geographical distance between two points is fixed. This distance is computed using GIS (Geographical Information System). It does not necessarily correspond to the shortest geometrical distance between the two points (Fig. 5). 3. The return pipes are parallel to the ongoing pipes to the buildings, and have the same diameters. The ongoing and return pipes are represented by a single operating unit on the P-graph. (Ongoing pipe: pipe carrying the hot water from the heating plant to the buildings; return pipe: pipe carrying the cold water back from the buildings to the heating plant.) 4. Different constraints, e.g. spatial constraints in a technical gallery/rack, constraints on the size of pipes (availability on the market), can be easily implemented. 5. In the resulting optimal solution sturcture there can be spHtting, but no mixing, between the pipes going to the buildings, except if one of the two pipes comes from a plant. 6. The temperature level at which the heat needs be delivered and the heat losses have not yet been taken into account in the optimization.

Network Synthesis for a District Energy System: A Step Towards Table 1 Heat requirement in each building

Building 1 2 3 4 5 6 7 8

Consumption [kW] 526 745 254 95 367 289 846 103

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Table 2 Selected connections: diameters of the pipes and power transfered

Pl_2 P3_l P5_4 P5_7 P6_3 P6_5 P7_8

Diameter [mm] 150 150 50 150 150 150 50

Power transfered [kW] 745 1271 195 949 1525 1411 1103

6. R e s u l t s The optimization has been done to minimize the total annual costs (investment and operation). The model (102 continuous and 102 integer variables) was written so that the algorithm had the choice to implement or not a boiler and/or a heat-pump in each building. The choice of the size and number of technologies was left over to the algorithm. Following costs were applied, based on [5] and [6] : Energy conversion technologies: Costs for the boiler: Cfo= 27Hhp+ 10000 CHF Costs for the heat-pump: Chp= 405///,^+ 340000 CHF Costs for the pipes as a function of the diameter in mm: 50 mm: 1200 CHF/m; 80 mm: 1350 CHF/m; 150 mm: 1750 C H F / m Costs for the utilities (raw materials): Electricity: 0.13 C H F / k W h Natural gas: 0.05 C H F / k W h For the boiler(s) an efficiency of 90% was chosen and for the heat-pump(s) a coefficient of performance of 4 is selected, assuming that a low temperature heating system is available in the building. Fig. 3 and table 2 show the optimal solution structure for a test network. On this figure, the broken lines represent the possible connections between the buildings, the thick and thin continuous lines show the connections that have been selected by the algorithm and that are part of the optimal solution structure, as a function of the diameter of the pipe. One can see that the optimal solution comprises one heat-pump but no boilers, although boilers have lower investment costs. This is due to the higher efficiency of the heat-pump when compared with a boiler. The total annual costs for the optimal network is 868'000 CHF (313 days, 24 hours per day). (On the figure H P means heat-pump, P X _ Y is a pipe from building X to building Y). Fig. 4 shows the optimal solution structure assuming that between buildings 1 and 3 one cannot implement pipes with a diameter larger than 50 mm (costs: 868'000 CHF/year).

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Figure 3. Optimal network (no constraint)

etal

wr^

Shortest distance — —Geographical geometrically distance

Figure 4. Optimal net- Figure 5. Difference between the work with constraints fixed distance and the shortest distance geometrically

7. C o n c l u s i o n A method that applies the P-graph approach has been developed to synthesize district energy systems. The method developed allows to consider constraints like diameters, restricted paths and the possibility of decentralised production of heat with multiple units. The system developed is integrated with a GIS system and technology data base systems. The network synthesis method will be further developed to be integrated in a method that will consider the optimal management issues multi-period problems. REFERENCES 1. Swiss Federal Office of Energy, Statistique globale Suisse de I'energie, 1997 2. F. Marechal, D. Favrat, J. Eberhard, Energy in the perspective of the sustainable development: The 2000 W society challenge. Resources Conservation and Recycling, Vol. 44, No. 3 (2005), pp. 245-262 3. F. Friedler, K. Tarjan, Y.W. Huang, L.T. Fan, Graph-theoretic approach to process synthesis: polynomial algorithm for maximal structure generation, Computers and chemical engineering. Vol. 17, No. 9 (1993), pp. 929-942 4. F. Friedler, J.B. Varga, E. Feher, L.T. Fan, Combinatorially accelerated branch-andbound method for solving the MIP model of process network synthesis, (Eds: C. A. Floudas and P. M. Pardalos), pp. 609-626, Kluwer Academic PubUshers, Dordrecht, 1996. 5. M. Buerer, Multi-criteria optimization and project-based analysis of integrated energy systems for more sustainable urban areas, Ph.D. Thesis n°2842, Ecole Polytechnique Federale de Lausanne, Switzerland, 2003. 6. V. Curti, Modelisation et optimisation environomique de systemes de chauffage urbains alimentes par pompes a chaleur, Ph.D. Thesis n°1776, Ecole Polytechnique Federale de Lausanne, Switzerland, 2003.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Close Loop Supply Chains: Managing Product Recovery Portfolio A.C.S. Amaro^ and A.P.F.D. Barbosa-Povoa*'^ ^ISCAC, Quinta Agricola, 3040 Coimbra, Portugal ^CEG-IST, 1ST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

Abstract In this paper a general model formulation is developed that accounts for the close loop supply chain structural and dynamic characteristics. Two sequential decision levels are considered: (i) planning and (ii) scheduling. The former defines the assignment of operations and associated amounts (supply, production, packing, transportation and distribution) to the suitable chain structures (plants, factories, transportation fleets, warehouses) so that an aggregated level of demand is satisfied (e.g. three months). The latter is achieved by extending the previous model to account for the operational events at a more detail time level (e.g one week). Close loop operation is attained by systematically and jointly tackling recovery tasks and transportation flows (forward and reverse), while guaranteeing general customers and market requirements within a given global profit criteria. A Mixed Integer Linear Programming formulation (MILP) is obtained for both decision levels. The two models are sequential but integrated where the optimal solution is reached using standard a Branch and Bound (B&B) procedure. The applicability of the developed formulation is illustrated through the solution of a real industrial pharmaceutical chain Keywords: Planning, Scheduling, Supply chain. Optimisation, Recovery. 1. Introduction A supply chain is usually defined as an operational structure that produces and distributes a defined set of suitable materials (intermediate, final products, etc) to a set of market places geographically disperse, using internal resources (production, storage and transportation facilities) and some external resources (raw-materials, utilities, etc). With the recent increasing environment concerns there is growing recognition that the supply chain management should incorporated the return of used products with the forward operational aspects (Shah, 2004). Different studies have appeared that span fi-om the definition of the close loop supply chains structure (Fleishmann et al, 2001) till the study of integrated logistic operational systems (Krumwiede and Sheu, 2002). Recovering operations become a major chain event that allows the fulfilment of environmental, legislation and safety concerns using an efficient strategy. This is mainly focused on cost savings and operational improvements (Fleischaman et al, 2001). The contemplation of these aspects lead to a more complex supply chain management where the sharing of information amongst partners is crucial at all decision levels. The To whom correspondence should be addressed: [email protected]

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operational level is not an exception and the planning process of such structures must reflect all the possible interactions that exist amongst partners (Shah, 2004), both in the forward and reverse chains. In this paper, this problem complexity is addressed at both a planning and scheduling levels. An integrated approach of both levels is developed. Supplying of products as well as return of the non-conformed ones are considered. At the planning level, decisions related to the role of each partner (e.g. production, customization, distribution, transportation, etc) in the chain performance are defined. On the other hand, the scheduling level involves the definition of the operational details of each partner. Two integrated models are developed each one being described by a Mixed Integer Linear formulation. The GAMS/CPLEX was used as implementation/solution method. The models are characterized below and some considerations are performed on the assumptions made. A real case-study is solved showing the model applicability. 2. Model Characterization The recognized importance of recycling and remanufacturing operations in the supply chain environmental, economical and operational performance introduces the need of looking into the supply chain management in a broad manner. The models developed in this paper take this into account and consider that the supply chain operability is characterized by two major set of events: transforming and transportation events. The former represents general processing operations involving a material "transformation" and are referred as tasks, while the later describes the materials transportation and are called transportation flows or simply flows (Amaro and Barbosa-Povoa, 1999). These account for both, forward and reverse material flows whose main difference relies on the materials transported and on the associated paths across all the supply chain instances. The former, is related to the supply of a set of clients with origin in the raw materials suppliers passing through factories and distribution centers. The latter, are flows originated by non-conform final or intermediate products usually collected at distribution centers (closer to the end users) and ending at remanufacturing, disposal or other purpose. For the reverse material flows two options are considered: recycling flows and recovery or remanufacturing flows. Recycling flows consist of reusable materials (e.g. reusable bottles) that can recover their former properties, through a pre-recycling operation (as cleaning). These are afterwards incorporated into the supply chain and give rise to new forward flows. Recovery or remanufacturing material flows consider all the materials that must be reprocessed in order to achieve an utility purpose such as: (i) being associated with other product portfolios (e.g. wood wastes transformed into different wood agglomerates); (ii) acting as burning materials or energy sources (e.g. burning of different food wastes to produce energy) and (iii) safety recovery flows (e.g. returned medicines with time expired). Using the above characteristics a planning/scheduling tool is developed that involves two models running in sequentially levels. The first model considers the planning level of decisions where the role of each partner in terms of production, customization, distribution and transportation is defined. Market issues as the products customization or operational criteria imposed by recycling or remanufacturing operations are explicitly accounted. The integration of reverse material flows is judged based on the economical and operational improvements achieved on the global supply chain performance. An aggregated level of detail is assumed where an aggregated time is used (macro time), typically a period of three to six months. The

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model involves a new set of events and instances that take advantage of the macro time modelling scale. These are the macro tasks/flows and the macro units. The former may involve a single task or a specific cluster of sequential tasks/flows. Each of the macro tasks is linked to a defined set of suitable equipments/facilities/transports used to perform it. For a macro task/flow where a set of tasks/flows is considered its duration can be defined in two different ways. Based on the cycle time of the enclosed tasks/flows (overlapping mode) or alternatively based on the summation of the set of single task/flows duration (non-overlapping mode). The macro units are an equipment unit/facility or a specific ordered set of equipment units, facilities or transportation resources where an aggregated capacity is considered. This is defined based on the minimum set of capacities available within a macro unit. The planning horizon is divided into discrete time intervals of equal duration (e.g one week) where multiproduct campaigns are defined with duration equal to the discrete time unit considered. The materials to be supplied/produced/distributed and the associated amounts are calculated. The later are defined within the available minimum and maximum capacities of the suitable macro units existing within a site. The second model describes the scheduling of the supply chain in detail where a time basis horizon is defined equal to the minimum time defined at the planning level (e.g one week). The planning results such as materials to supply/produce/transfer are now taken as constraints within each scheduling problem and the operational details of each partner in terms of production, distribution and reprocessing are calculated. The supply chain operability is optimised by exploring the detailed resource capacities (e.g transforming, storage and transportation) and resource sharing policies based on equipment/tasks suitability's, economical performances and operational restrictions. In Figure 1 a scheme for the planning/scheduling approach described is drawn. From the planning level different scheduling problems are generated where a common time basis is considered allowing a closer approximation between both levels. Scheduling Planning Level

Time Hori: (e.g. 1 weel

\given Time H o r i z o n , / / p (e.g. 3 , 6 months)

^

Discrete tim e int( rvals, A (e.g. 1 week)

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and

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Events - Macro tasks and flows - Operability r equirem< nts ion etc ) Material State - Bounds and Requirements (e.g. raw-medicine A, white medicine B , final m edicine C etc) Markets - A m o u n t s /Due dates - Delivers

and Receipts

Detailed Supply Chain Schedule - Assignment of tasks and flows to the suitable units (e.g. Plant 1 equipment U 1 produces a batch of white medicine B from 2 to 6 hr, C from 6 to 8 hr; vehicle vl of fleet cl goes from plant II to warehouse 2 at...with ...amount of product B and C , etc)

rder to achieve the supply cha

\determine |

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\in order to achieve Global Planning

the supply

Hs

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iven I Time H o r i z o n , Hs, {Hs„-Atp=n )

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Figure 1 - Planning/Scheduling Methodology The approach developed allows the study of tactical decisions at an aggregate level of planning before entering on the study of the operational details. In particular, and due to the characteristics of the model where reverse flows are considered the management of product portfolios recovery can be analysed. Situations of non-recovery versus recovery of products are studied. This will be explored later on in the case study section.

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3. Case-Study A pharmaceutical supply chain is considered where different injection drugs, tablets and oral suspensions are produced and distributed for different Portuguese and international markets, figure 2(a). The supply chain is characterized by a production cluster involving three medicine producers (II, 12 and 13). The production process, independently on the medicine considered, has two main steps: (i) the formulation step that involves operations processing the raw-medicines and additive substances and producing the non packed medicines - intermediate products and (ii) the packing and labelling step that accounts for the final medicines customization.

Figure 2 - Supply chain: (a) structure, (b) Close Loop Recovery. Four major intermediate medicines, IPl to IP4, can be produced at II and 13 plants, while 12 produces exclusively IP3 and IP4. Medicines IPl and IP2 are injection drugs produced fi-om raw medicines RMl and RM2 respectively. Medicine IP3 is a soft tablet drug produced from RMl while IP4 is an oral suspension produced from RM3. Based on market and legislation requirements three type of medicines customisation are defined: (i) PF - exclusively for the Portuguese market and African customers, (ii) SF exclusively for Spanish market and associated customers, and (iii) EF - for a set of EEC market positions. The customisation process can be performed at any industrial plant directly fi-om the intermediate materials produced at that production site or fi-om any intermediate material formulated at another production partner. The raw medicines and general raw-materials (e.g. packing materials, glass bottles) are receipt at the plants warehouses accordingly to a pre-defmed supply timetable considering a fortnightly supplying period. Their transport is not a chain concern. The road materials distribution amongst partners is performed by three transportation fleets. The former is a set of vehicles mainly planned for long range travels or higher capacity charges, while the latter two are intended for lower charge capacities. Concerning the storage policies, it is assumed that at the beginning of the planning period all supply chain sites have a storage level of 25% of the storage capacity dedicated to each suitable material state. Also, no storage level less than 5% is allowed whatever the site, the material state or the scheduling time. The supply chain market requirements are defined for both decision levels studied. The demands involve both pre-defined orders and an historical data. All the prior commercial arrangements are attained by defining weekly supply chain minimal production requirements for all the customized medicines. Weekly historical delivering data colleted all across the supply chain sites are used to define expected demand profiles. This accounts for minimal and maximal delivering capacities. The commercial deals are treated as a chain concern without any a priori partner assignment assumption.

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At the planning decision level global minimal delivering requirements are defined, over the planning horizon, for the Portuguese and Spanish market positions fulfilled directly by plants or distribution sites (WHl to WHS or WSHl and WSH2, Figure 2a). This minimal delivers must be satisfied while accounting for a fortnightly maximal delivering capacity at each supply chain partner. Delivers to the African and European customers (fulfilled by sea and air Portuguese ports and by road through Spanish distribution sites) considers also a minimal amount for the planning horizon and a maximal fortnightly capacity for AP, SP, WSHl and WSH2 sites. The close loop supply chain operation (Figure 2b) is evaluated by comparing the supply chain economical performance for a given aggregated planning period where two independent operational scenarios are studied. The former represents a non product recovering scenario where all the non conformed products are send to burning centers (while removed from the market places). The latter, considers a scenario where the medicines are clustered based on the non conformity. Two main groups of returned medicines are considered: (i) B-Medicines (BFi) and R-Medicines (RFi). The former considers all non-recoverable medicines (e.g. date expired) that must be send to burning centers (BCl and BC2) while the later represents recyclable, RC, or remanufacture medicines, RM, (e.g. RC glass, RM repacked medicines). These can be received either, directly at the plant warehouses or through distribution sites. R-Medicines receipts within the chain are sent to II plant where a recovery facility is installed. The RC are processed at an U-facility (II) and generates low added value materials as well a percentage of non-recoverable materials. Instead the RM medicines processed at Ufacility enter the plant production lines in order to produce new final medicines. In this latter scenario two situations are evaluated: (2a) where no minimal requirements for the recovery of products are imposed and (2b) where minimum values are imposed. Based on the above characteristics the supply chain planning is performed for a planning period of three months with a weekly discretized unit with the objective of evaluating the best recovery portfolio scenario. Due to the lack of space only some of the results are presented. Table 1 presents the economical results for the three studied planning scenarios. Scenario 1 (non-recovery option) presents a smaller supply chain profit while compared to scenarios 2a or 2b. The best profit is obtained when recovery is considered with no minimal recovery percentages of products (scenario 2a). Table 1- Economical Planning results for scenarios 1, 2a and 2b. Scenario 1 Scenario 2.a Scenario 2.b Burning No Min Rec. Min Rec. (80 % RF) CASE I - Non Conforms = 5% of a Prior Trimester Production 17712983.7 " 1 Global Profit (m.u) | 18093060.0 18072930.0 A (Sc2.a-Scl) = 380076.0; A (Sc2.b-Sol) = 359946.0 ; A(Sc2.a--Sc2.b) =20130.0 37.3-35.3 36.32-34.90 36.35-34.91 Production, F-C % 9.54 9.45 9.65 Transportation, % -2.08 -0.37 -0.53 Reverse Profit % CASE II - Non Conforms = 10% of a Prior Trimester Production Global Profit (mu) | 17296465.4 | 17882151.4 | 17837039.4 A (Sc2.a-Scl) = 585686.0; A(Sc2.b-Scl) =540574.0; A(Sc2.a - Sc2.b) =45112.0 37.02-35.14 37.85-36.21 36.48-35.15 Production, F-C % 9.70 9.40 9.66 Transportation, % -0.94 -4.30 -1.26 Reverse Profit % m.u - monetary units; F- formulation; C - Customization; % item value over net profit.

Using the best planning scenario (2a) the scheduling was performed. In order to illustrate the scheduling solution a single week is analysed. This corresponds to a weekly operation that is characterised by a constrained production and transportation

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requirements (8^ week). The obtained results are compared to the results of the corresponding week at the planning level (Table 2). When analysing the global planning economic values it can be seen that the delivers are the most important source of income while the formulation operations are the most costly activities. This remains true when comparing the results of the planning and scheduling for the 8^^ week. In terms of differences between the scheduling and the planning it is observed that the final existences amounts are higher in the scheduling. This is not translated into an increment of the economic values due to the fact that the final existences are essentially of lower added value products. On the other hand, the scheduling is able to deliver a higher amount of products that result into a higher amount of income. Globally, the incomes for scheduling are higher than the ones of planning. The same is true for the costs. The latter are explained essentially by the transports and production operations (formulation and customization). The transport differences are explained by the integration of compatibility constraints for the equipment sharing as well as to the constrained distribution requirements detailed at the scheduling decision level. In terms of production the customization is more costly and occurs in larger scale due to the processing of intermediate materials received from the previous scheduling period. Table 2.- Planning and Scheduling at 8*" week Decision

Global S2a Planning

8* Week Values Planning Scheduling

A Economic A(Ecsch-Ecp,) (m.u) Aei= 399908 -92519.0 492427.0 Aec=222863. 103573.8 -1950.6 37752.5 83487.5 35400.0

A Amount A(Ecsch-Ecp,) (r.u.) Aai= 979352. 811702.0 167650.0 Aac=1044600 249308.0 347646.0 13765.0 433881.0 16000.0

Eco% Eco % Econ. (m.u) 55804008.5 Incomes 38.9 23923780.0 30.2 Existence Pp 31880228.5 61.1 69.8 Deliv., Pp+Pr 37921857 Costs 1681059.4 12.3 18.1 Transp.Pp+Pr 1.9 126073.9 1.5 Store, Pp+Pr 6620982.5 52.1 46.1 Formul., F Custom., C 6284278.7 33.7 34.3 35400.0 741843.8 Recover, P/*^ 17882151.4 1343124.6 1520169.5 Profit (m.u) PF /Pr - Product ponfolio Forward/ Reve rse; m.u - monetary imits; r.u.- reference; unit; ^ ^ Value of Recovered Products.

The model statistics at the planning level involve 6682 continuous variables, 2760 integer/binary variables and 13930 equations. It took to solve on average 1000 CPU seconds for a gap of 1.5% (Pentium III). Each module of the scheduling is characterised by 5960 binary variables and 177232 equations and it takes about 8755 CPU seconds. 4. Conclusions This paper presents a sequential but integrated approach to the planning and scheduling of supply chains. The models developed address very detailed characteristics such as production, storage, distribution and recovery of products. The latter allows for the evaluation of different recovery portfolio scenarios at a planning level that are then optimized in detail at the scheduling level. A real case study taken from a pharmaceutical industry was studied and the results obtained were promising. Further studies are under development to improve model generalization. References 1. Amaro, A.C and Barbosa-Povoa, A. P. (1999), EJOR, 199, 461-478. 2. Fleischmann et al. (2001), POM, 10(2), 156-173. 3. Krumwiede, D.W. and C. Sheu (2002), Omega, 1.5, 325-333. 4. Shah, N. (2004), CACE, 18, Eds Barbosa-Povoa and H. Matos, Elsevier.

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Simulation based optimization for risk management in multi-stage capacity expansion Xiaotao Wan," J. F. Pekny,^ G. V. Reklaitis^ ''GE(Chma) R&D Center Co. Ltd, 1800 Cailun Road, Shanghai, 201203, P.R. China ^School of Chemical Engineering, Purdue University, West Lafayette, IN, 47906, USA Abstract Risk management for multi-stage capacity expansion optimizes average return and risk simultaneously. None of the existing algorithms for stochastic dynamic programming can accommodate general risk measures. Algorithms based on simulation based optimization are proposed in this research to address arbitrary risk measures for multistage risk management in capacity expansion. These algorithms utilize multi-stage a back-propagation scheme and function approximation techniques. Their effectiveness is demonstrated by applying them to a pharmaceutical product pipeline case study. Keywords: simulation based optimization, dynamic risk management, stochastic dynamic programming, multi-stage capacity expansion. 1. Introduction Strategic capacity decisions are positioned at the top of the hierarchy of supply chain management decisions. Under the risk management framework, the expected return of a capacity decision is simultaneously optimized with the risk of the decision weighted by a risk aversion parameter, where the risk is quantified in terms of variance, semi-norm, value-at-risk etc. It has been proven that stochastic math programming is limited to nondecreasing risk measures, and not applicable to general risk measures (Takriti and Ahmed, 2004). Cheng et al. (2003, 2004a, 2004b) solve the multi-stage risk management problem in capacity expansion through exploring the special property of a separable risk measure; However, the back-propagation scheme for dynamic programming used therein cannot accommodate non-separable risk measures. In this research, new techniques based on simulation based optimization are proposed to address multi-stage capacity expansion problems for arbitrary risk measures and are applied to a pharmaceutical product pipeline case study. 2. Algorithms for risk management in dynamic optimization main text The proposed algorithms extend the Bellman equation of stochastic dynamic programming. Two underlying fundamental components are multi-stage backpropagation and fiinction approximation of simulation results. For a dynamic problem with T stages, let s be the state space, x be the decision, co be the random event, a pseudo-utility function is defined as follows: U(^,) = max[E(/(^„x,6;,) + V(^,,,(^,)))-ARK,x)]

(1)

X

where E{f{s^,x,co^) + V{s^^^{s^))) is exactly as in the Bellman equation w i t h ^ ) being the immediate return and V() being the value fimction, R(5'^,x) denotes the risk of making decision x at state Sj and ^ is the risk aversion parameter. The pseudo-utility

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function is a natural extension of the value function to incorporate risk associated with each state. It represents the desirability of states in the risk management context. In the Bellman equation, the value function at stage t can be obtained by observing the value function at stage /+7, through the typical one stage back-propagation scheme. However, the pseudo-utility function at stage t generally cannot be calculated this way unless the risk measure in Eq. (1) is separable (Li, 1990). For non-separable risk measures, calculation of the pseudo-utility function at stage / requires all the information at the final stage T. The corresponding computational scheme is called multi-stage back-propagation in our study in view of the one stage scheme used to solve the Bellman equation. For most capacity expansion problems, it is impossible to solve Eq. (1) analytically: simulation based optimization (Fu, 2002) is the only feasible method. At each state in the simulation, a stochastic optimization problem (Eq. (1)) must be solved to obtain the optimal actions. As the number of states is usually extremely large, function approximation is necessary to mitigate the prohibitive computational burden. The strategy is to generalize the observed simulation results of a sample to the whole space by building an approximation function using the observed sample. Such a strategy has been used to approximate the value function in the Bellman equation to overcome the curse-of-dimensionality (Bertsekas and Tsitsiklis, 1996). 2.1. Revised back-propagation algorithm The pseudo-utility function in Eq. (1) is not amendable to ftinction approximation as no capacity decisions can be derived fi*om knowledge of the function. For purpose of reducing the computational difficulty, we define a state-action pseudo-utility function as U(^, x) = Q(s, x) - AR(s, x)

(2)

where Q() is the expected return for making decision x at state s. The optimal decision for a state s can be found through X* = arg max U(s, x)

(3)

X

Define a random vector ^^ = {co^,",a)^), i.e. ^^ represents the random events that occurred between x and T. Then the revised back-propagation algorithm for dynamic risk optimization is as follows: at the decision time T 1. Sample m state-action pairs and obtain (s^,x^),z = l,'--,m . 2. For each pair (s^, x ^ ) , generate realization ^^ ,/ = 1 ,•••,«; simulate the n realizations from T to T. For each realization, at the decision point T'>T, given the state is s^,, take action x^. according to x^, = arg max U ^, (s^,, x)

(4)

X

Where \j ^,{s^,,x) is the approximated state-action pseudo-utility function for time T'. After compiling the results of the n simulations, compute the expected retumQ^(^^,x^),the riskR^(^^,x^),and the pseudo-utility \J^{s^,x^) with U(^, ,xJ = Q(s,,xJ-

AR(s^, X,)

(5)

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3. Fit an approximate state-action pseudo-utility function U^ (•,) based on the m points \5^{s^,x^),z = \,"',m. The multi-stage back-propagation principle is reflected in step 2 where simulation is conducted from T to T instead of from x to x+l as in the one stage back-propagation. This algorithm builds T-1 approximation state-action pseudo-utility functions, one for each stage except for stage T. Those approximate functions greatly reduce the computational overhead since a deterministic optimization problem represented by Eq. (4) is solved instead of the much more complex stochastic optimization problem represented by Eq. (1) at each state in the simulation. 2.2. Optimal policy approximation algorithm The number of deterministic optimizations in the form of Eq. (4) in the revised backpropagation algorithm is proportional to the number states visited in the simulation, thus the curse-of-dimensionality persists. The following optimal policy approximation algorithm tackles the dimensionality issue with another level of function approximation. In stochastic dynamic programming, a policy is a fiinction returning optimal actions for any state. Let n^(^) denote the approximation optimal policy at the decision time x, then for the decision time x 1. Sample m state-action pairs and obtain {s^, x^), z = 1, • • •, m . 2. For each pair {s^ ,x^), generate realization 4^^, z = 1 ,•••,«; simulate the n realizations from x to T. For each realization, at the decision point x'>x, given the state iss^., take action x^. according to

x,.=il,.K.) After

(6)

compiling the results of the n simulations, compute the expected

return ^ri^^^^^) \J{s^,x^) =

^ nsk ^ ^ ^-^^' ^^ \ and pseudo-utility ^ ^ (^^, x^) ^^ .^^ Q_{s^,x^)-XK{s^,x^)

3. Fit an approximate state-action pseudo-utility function U^ (•,) based on the m points U^(^^,x^),z = l , - - , w . 4. For each states^,z = \,"',m, find its optimal action x^{s^) via x^(^^) = argmaxU^(^^,x)

(7)

Fit an approximation optimal policy function n^(^) with the m points (^^,x^(5^)),z=l,-,m. In this algorithm, T-2 approximate functions for optimal actions are built from stage 2 to state T-1. The number of deterministic optimization performed in the form of Eq. (7) is O(Txm), which is independent of the number of states visited in the simulation. As a result, the curse-of-dimensionality is avoided. In our research, least squares support vector machine (LSSVM) (Wan et al., 2005) is adopted to build all the approximation functions.

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arrives at 0

1

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P2 arrives;^tt2

r

~ ^

J

V

A competitor arrives

\

t Demand

t+1 1 ^

t

Produce & fill demand

Figure 1: Scheme of the case problem

3. Risk management in pharmaceutical capacity expansion When a pharmaceutical company expands its manufacturing capacity upon new drugs exiting its development pipeline, it may increase the capacity just enough to meet the forecasted demand or it may purchase more capacity for future drugs to reduce the setup cost. However, purchasing capacity for future drugs inevitably incurs risk: the capacity may not match the demands of the future drugs, extra-capacity will reduce the return of investment, while a capacity shortage will necessitate another purchase which incurs an additional undesirable setup cost. The uncertain exit time of future drugs may also make it cost effective to perform additional capacity expansion. Other important factors include competitors: there exists the possibility that competitors will enter the market in the future to take away part of the demand. The right capacity level can only be identified through solving a multi-stage risk management problem. 3.1. Case Study: Capacity expansion in a pharmaceutical company A pharmaceutical company A has a new drug (PI) exiting its development pipeline at the beginning of the horizon, and the initial available capacity is 0. The demand for the drug is stationary, following a normal distribution N (20, 9) in each period. The total horizon considered is 40 periods. Within the horizon and with a probability 0.5, a second new drug (P2) will exit the pipeline. The exit time follows a triangular distribution Tri (10, 20, and 30). The demand for the second drug is also stationary; with normal distribution in each period with mean N (20, 9) (i.e. the mean demand is uncertain) and the coefficient of variation the same as that of the first drug. Assume the second drug is similar to the first drug: they have the same production cost, market price, etc. A single competitor B exists whose product will share the demand of the first drug if it enters the market but does not affect the demand of the second drug. The arrival of B's product follows an exponential distribution with expected arrival time 45. If B enters the market, its product will take away normal distributed market share N (0.4, 0.01)fi-omA's first product. 3.2. Implementation of the optimal policy approximation algorithm As shown in Fig. 1, this case problem is a dynamic optimization problem with capacity decision at 0 and t2, and contingent production decisions at each period. In accordance with the proposed optimal policy approximation algorithm, the problem is approached as follows: sample the state-action space at t2, build the state-action pseudo-utility function and consequently the state-optimal action function (i.e. the policy) after simulating the sampled points; sample the state-action space at 0, simulate and build the corresponding state-action pseudo-utility function; obtain the optimal capacity decision at 0 by optimizing the corresponding state-action pseudo-utility function. The default number of sampled points is 40 for the first stage and 650 for the second stage; the

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15

20

30

Figure 2: NPV and semi-norm of the first stage capacity decision

0,004

0.008

0.012

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risk aversion parameter

Figure 4: Optimal first-stage capacity decisions under different risk aversion parameters

Figure 3: NPV and semi-norm efficient frontier

0.008 0.012 risl< aversion parameter

0.016

Figure 5: The effect of the demand variance on first-stage capacity decisions

default number of sample path simulated for each sampled point is 4000. Those default numbers are chosen such that they provide satisfactory results for this case, and there is no significant improvement with larger values. The implementation of the revised back-propagation algorithm is similar to the above procedure except that the state-optimal action surrogate model is not constructed and a deterministic optimization problem is solved at t2 while simulating a sample path from 0 toT. S.3. Results and discussion For the non-separable risk measure semi-norm, Fig. 2 shows that the optimal decisions for NPV and semi-norm are 26 and 22 respectively when A is equal to 0.001, indicating

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that the optimal decision for pseudo-utility must lie between 26 and 22 to balance their trade-off. For other values of ^ , there will be similar relations between the first stage capacity and the NPVs as well as the semi-norms, from which the corresponding optimal decisions together with the NPVs and semi-norms under these decisions can be calculated. The results are presented as the efficient frontier in Fig. 3. This figure simply states that capacity expansion under dynamic conditions demonstrates the NPV and risk trade-off, well known in stock portfolio management: higher NPVs are necessarily associated with higher risks. Fig. 4 shows that the optimal first-stage capacity level decreases as the risk aversion parameter increases to avoid the risk of lower demand either due to possible arrival of the competitor or non-materization of expected fiiture drugs. Fig. 5 studies the effect of demand variance under different risk aversion parameters on the first-stage capacity level. Clearly, larger demand variances lead to higher capacity levels to avoid the cost of missing demand. 4. Conclusions Two algorithms are proposed for risk management in dynamic optimization based on multi-stage back-propagation scheme and function approximation. The algorithms are the first of their kind to be valid for arbitrary risk measures. Their effectiveness is illustrated by computing the NPV vs. risk efficient frontier for a dynamic capacity expansion case problem. References Bertsekas, D. P., Tsitsiklis, J. N., 1996. Neuro-dynamic programming. Athena Scientific. Cheng, L., Subrahmanian, E., Westerberg, A. W., 2003. Design and planning under uncertainty: issues on problem formulation and solution. Comp. Chem. Engng. 27, 781-801. Cheng, L., Subrahmanian, E., Westerberg, A., 2004a. A comparison of optimal control and stochastic programming from a formulation and computation perspective. Comp. Chem. Engng 29(1). Cheng, L., Subrahmanian, E., Westerberg, A., 2004b. Multi-objective decisions on capacity planning and production-inventory control under uncertainty. Ind. Eng. Chem. Res. 43, 21922208. Li, D., 1990. Multiple objectives and non-separability in stochastic dynamic programming. International Journal of System Science 21 (5), 933-950. Fu, M. C, 2002. Optimization for simulation: theory vs. practice. INFORMS Journal on Computing 14 (3), 192-215 Takriti, S., Ahmed, S., 2004. On robust optimization of two-stage systems. Mathematical Programming 99, 109-126. Wan, X., Pekny, J. F., Reklaitis, G.V., 2005. Simulation-based optimization with surrogate models—application to supply chain management. Comp. Chem. Engng 29 (6), 1317-1328

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A decision support tool for process optimization of sulphur free diesel production Z. Lukszo^, M. Salverda,^P. Bosman^ ''Delft University of Technology,P.O. Box 5015, 2600 GA Delft, The Netherlands ^Shell NederlandRaffinaderij BV, Rotterdam, The Netherlands Abstract This work is a contribution to the Sulphur Free Diesel (SFD) production. The research presented in this paper is performed in one of the Dutch refineries, which is going to be one of the front producers of SFD from high sulphur crude. The challenge is to adapt the process operation to changes in the specification requirements without capital investments, i.e. to find optimal process settings which maximize the margin taking into account blend values, deactivation costs of the catalyst, shutdown margin losses, product specifications and quality margins and considering the organizational complexity, too. The decision support tool, called SFD optimizer, determines production settings on a weekly basis to decide on the weekly schedule by Economics and Planning Department and in case of disturbances. The NLP-optimization problem with two types of decision variables (desulphurization depths for three HDS and five quantities of components sent to product pools) and with 22 (non)-linear (in)-equality constraints is solved with generalized reduced gradient method. The optimal solution results in maximal SFD margin, i.e. the total blend value minus the deactivation costs. The SFD decision support system presented in this paper was proven to effectively assist decision makers at the Economics and Planning Department. Keywords: sulphur free diesel, process optimization, decision support tool 1. Introduction Since the seventies the European Union, together with the petroleum industry, aims to reduce the negative environmental impact of the use of hydrocarbon fuels [European Standard, 2004]. A part of this effort focuses on the reduction of sulphur in friels. In 2005 fiscal incentives are imposed to stimulate the production of sulphur free diesel (SFD), a diesel grade which contains at most ten sulphur parts per million. One of the Dutch refineries that are going to produce sulphur free diesel for the Dutch market is Oil & Co Refinery (OCR). To be capable to produce SFD some plants need to be driven to more extreme operational conditions [Torrisi, 2004]. Besides operational aspects the introduction of SFD also has some economic consequences. Revenues drop as many former diesel components cannot be used for the diesel production anymore and should be downgraded. Moreover, the operational costs go up as the costs of desulphurization rise. To cope with the critical operational conditions and minimise the loss of margin the refinery's Economics and Planning department would like a decision support tool that optimizes the diesel production. To be successfiil the process optimization should, besides concentrating on the optimization aspects, also take into account the organisational complexity and consider the chance of successfiil incorporation of the decision support tool. The decision support tool, called SFD optimizer, determines the production settings that maximise profit, including catalyst lifecycle economics in the decision-making process and assuring that at all times all product specifications are met.

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2. Sulphur Free Diesel production At present the European norm for sulphur in diesel corresponds to the ULSD specification, being 50 ppm, but in some European countries the SFD specification of 10 ppm is already operative. In the beginning of 2005 OCR has chosen to import SFD to satisfy these demands rather than producing it themselves. Currently the full diesel production capacity of OCR is used to produce SFD, as presented in Figure 1. Kerosene

HDS/4

JetA1 3000 ppm S

HDS/1

Light gas oil

HDS/2

J

SFD

RBLAGO

10 ppm S

blender HDS^

SBP Kero 2ppmS

HCU gas oil

IGO 2000 ppm S t'-

LO feed

Figure 1. Present diesel production with four Hydro Desulphuisers (HDS) For the SFD production three input flows are processed (see Figure 1), i.e. Kerosene (Kero), Light Gas Oil (LGO) and Hydro Cracking Unit Gas Oil (HCU gas oil) to make the following products: Jet Al: fiiel for airplanes Sulphur Free Diesel (SFD): light gas oil (LGO) is desulphurized by the HDS's. The desulphurization of kerosene has a 12-day cycle: during 10 days the, by HDS/4, desulphurized Kero (Des-Kero) goes to the Jet Al pool, the two following days the Kerosene is desulphurized to 7 ppm. This Des-Kero is stored in tanks. SBP (Special Biling Product) Kero: HDS/3 is, besides for the desulphurization of light gas oil, also utilised to desulphurize kerosene to 2 ppm. HDS/3 is alternately used for the production of diesel and SBP Kero. The number of days a month that HDS/3 is utilised for SBP Kero production varies, but is generally speaking smaller than the number of days this unit is utilised for the desulphurization of light gas oil. Low Olefins (LO) feed: depending on the value of HCU gas oil as feed for the Olefins plant, it can be profitable to run down the HCU gas oil to LO. Also in case the HCU gas oil cannot go to the diesel pool, HCU gas oil can be used as LO feed, rather than downgrading it to industrial gas oil. Industrial gas oil (IGO): the price of industrial gas oil is normally lower than the prices of all other products. That is the reason why downgrading to the IGO pool is something to be prevented.

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3. SFD Optimizer The site-wide diesel optimizer, from now on called the SFD optimizer, aims to support the operation of diesel production. It takes in blend values and properties of the diesel components and then, taking into account the deactivation costs and SFD specifications, it determines the production settings for the Des-Kero production (HDS/4), the desulphurization of light gas oil (HDS/1/2/3) and the blending (RBL AGO blender; RBL stands for Refinery Blending and Logistics). This optimizer is important for two reasons. Firstly, it can optimize the economic performance regarding the diesel production, as both costs of the diesel production, e.g. desulphurization, and the revenues are included in the decision-making. Secondly, the run length can be expanded if the level of desulphurization on the HDS's is adjusted to the situation, rather than a steady state operation. Below important optimizer aspects are mentioned. 3.1. Model goal The main goal of the optimization model is to support the decision-making concerning the diesel production by calculating the solution that maximises the economic performance (SFD optimizer margin) i.e. the total blend value of diesel components minus the costs related to the desulphurization of light gas oil. 3.2. Model requirements The main functional model requirements are functionality and reliability. Functionality is determined by the degree to which the model generates optimal and practical production settings (blending quantities and desired levels of desulphurization). The model is considered reliable if the experts involved in the sulphur free diesel production have confidence in the SFD optimizer and its outcome. The main non-functional requirements are user-friendliness and feasibility. 3.3. Level of aggregation The model has a high level of aggregation, as the overall OCR performance needs to be optimized, satisfying the cost-focus of the production units and the margin-focus Economics and Planning department. 3.4. Model scope The model scope is determined by which factors are included in the model as model variables and which factors are not included. The factors that are included are: Properties of diesel components (i.e. sulphur content, density and flash point) Product specifications (i.e. sulphur content, density and flash point) Desired level of desulphurization on HDS/1/2/3 Quality margins for sulphur content, density and flash point Blending quantities. Run lengths HDS catalysts SFD optimizer margin: total blend value minus deactivation costs Total blend value of diesel components Deactivation costs HDS/1/2/3: replacement costs plus loss of income Blend values of diesel components (kerosene, light gas oil and HCU gas oil) Below the reasoning for excluding some other factors is given: Deactivation: there is already a model in place that predicts the HDS catalyst run length. This model, mastered by the HDS technologist, extrapolates the previous degradation to estimate the remaining run length. HDS settings: the Economics and Planning Department determines the desired properties of the HDS effluent and communicates these to the production unit responsible for desulphurization of light gas oil, Refinery Treating and Conver-

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sion (RTC). To respect this operational freedom the decision support tool should indicate the desired HDS effluent properties, rather than the operational settings. 3.5. Time horizon Although market prices change continuously, the refinery apphes estimates on a weekly basis. Given that, the time horizon of the model is a week. 3.6. Desired output The model should provide the optimal solution for next week's SFD production. Moreover, it should be transparent how the solution is formed and the user should be supported to convert the optimal solution into a feasible one applicable in the real plant, e.g. rounding of the optimal solution. Hence the user can change the decision variables found by the optimizer. 4. Optimization problem This chapter contains the formulation of the optimization problem as described above. 4.1. Optimization criterion [Edgar, 2001] 5

^ {

C •

-\-R LGO MsFD Pi Cj

SFD optimizer margin (thousands of US dollar per week = k$/w) value of product stream i (US dollar per ton = $/ton) deactivation costs, i.e. replacement costs plus loss of income, for a change of the catalyst in HDS/j (k$) - RLj(Xj) run length of the HDS/j catalyst, if the coming week the desulphurization depth on that HDS is going to be x ppm (weeks) RLGO revenues of the light gas oil (LGO) stream (k$/w) 4.2. Decision variables Eight decision variables are identified: qi quantity of Kero to Jet Al pool (kt/w) q2 quantity of Des-Kero to SFD pool (kt/w) qs quantity of HCU gas oil to SFD pool (kt/w) q4 quantity ofHCU gas oil to LO feed (kt/w) qs quantity ofHCU gas oil to IGO pool (kt/w) Xi desulphurization depth on HDS/1 (sulphur parts per million = ppm S) X2 desulphurization depth on HDS/2 (ppm S) X3 desulphurization depth on HDS/3 (ppm S) The decision variables are constrained by upper and / or lower limits and system constraints as: SsFD 5 SsFD spec" ^s (Sulphur content of SFD is smaller than the SFD sulphur specification minus the quality margin for sulphur) psFD spec min + CJp < psFD < PsFD spec max" CJp (Dcusity of SFD is bctwcen the upper and lower SFD density specification taking into account the density quality margin). The sigma a represents the quality margin necessary to guarantee that the product is within specification at the gas station. This quality margin is determined by the inaccuracy of the measurement equipment and the chance of the contamination after production. FPsFD > FPsFD spec + c^FP (Flash poiut of SFD is higher than the SFD flash point specification plus the quality margin for flash point)

Decision Support Tool for Process Optimization

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C, = r

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"^

^replacement,;

The type of the optimization problem is determined by the following: The criterion is a nonlinear continuous function with eight decision variables Fourteen linear constraints on decision variables (lower and upper limits) Four nonlinear equality and four nonlinear inequality constraints. The resulting Non-Linear Programming (NLP)-optimization problem with two types of decision variables (desulphurization depths for three HDS and five quantities of components sent to product pools) and with 22 (non)-linear (in)-equality constraints is solved with generalized reduced gradient method [Floudas, 1995]. The optimal solution results in maximal SFD margin, i.e. the total blend value minus the deactivation costs. 5. Decision support by SFD optimizer To be successful the decision support tool should generate additional margin and improve the decision-making process of the Economics and Planning department. To estimate the monetary gains of the SFD optimizer a reference scenario is used without the SFD. In this case the HDS's would desulphurize in a steady state, to a sulphur content of 7 ppm. The production of Des-Kero would be characterised by a 12-day cycle. Ten days the HDS/4 would be utilised for Jet Al production and two days for the desulphurization of kerosene for the diesel pool. Given the maximum throughput of HDS/4 the weekly available quantity of Des-Kero that can be blend into the diesel pool would be X kton. The SFD optimizer does not consider these two factors fixed but rather flexible depending on economics and component properties. Scenario

HDS depth HCU gas oil in Des-Kero in (ppm) SFD (kt/w) SFD (kt/w) Max X 5ppmS Ref 7 HCU gas oil Opt. Max 9,6 X+1 Max Ref 7 X 10 ppm S HCU gas oil Opt. Max 8,3 x+1 17 ppm S X-0,4 Ref. 7 Max-3 HCU gas oil Opt. Max X+1 6,6 Max-9,4 33 ppm S Ref 7 X-3,7 HCU gas oil Opt. 5 Max-5,1 X-1,5 50 ppm S 7 X-4,4 Ref Max-10,8 HCU gas oil Opt. 5 Max-8,2 X-3,1 Table 1. Optimized advice for the HDS depth for five scenarios

Delta margin (k$/w) 0 117 0 105 , 0 233 0 329 0 180

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To obtain a better idea of the potential gain of the SFD optimizer the reference case above, i.e. fixed HDS depths and Des-Kero production, is compared with the advice generated by the SFD optimizer. Five scenarios were run with ascending levels of sulphur content in the HCU GO, see Table 1. On the basis of these scenarios, we can conclude that the SFD optimizer maximises the margin, which is the total blend value of the diesel components minus the deactivation costs of the desulphurization catalysts. The decision support tool is going to be used by the refinery's Economics and Planning department once a week for the formation of the week schedule and in case of disruptions in the diesel production process. Taking the average over the four year run length of the HCU catalyst, the estimation of annual gain is approximately 9 million dollars a year. 6. Final remarks Conclusions regarding the outcome of the SFD optimizer are: Desulphurization in HDS/3 is more expensive than in HDS/2; and desulphurization in HDS/2 is more expensive than in HDS/1. Decreasing the quality margin on sulphur leads to significant gains. Additional margin due to the SFD optimizer varies between 100 and 300 thousand $/w, depending on diesel properties and the desulphurization depth. The SFD decision support system was proven to effectively assist decision makers at the Economics and Planning Department. It was not only proven to be effective in decision support on process settings and the schedule, but also to be effective in improving conmiunication between the relevant company departments. It should be mentioned, that the level of communication and cooperation between the production units and the Economics and Planning Department become more important. The SFD optimizer leads to less steady state operations, smaller margins for compensations for process disruptions, shared responsibility for the diesel quality, and potential conflicts in the HDS utilisation. This imposes a communicational challenge. The clarity of the graphical presentation and the quantitative results of the optimization model and its user friendly interface evidently contributed to breaking down information and communication barriers between the involved production units, planning and logistics departments. Moreover, the refinery realized that not only the actor performance itself but also the way this performance is achieved should be considered. A kind of flexibility is built in the performance assessment system that takes into consideration the sacrifices some production units make to maximise the site-wide performance. At the moment the implementation phase of the decision support system in the refinery has started. It should be stressed, that although the SFD optimizer is developed for a specific refinery, the approach aimed at the formulation of the optimization problem can be applied to a wide variety of production processes for fuels. References European Standard EN 590:2004, Automotive feuels. Diesel-requirements and test methods. Technical Committee CEN/TC 19, Petroleum products, lubricants and related products, 2004 Torrisi, S., M.P. Gunter, Keyftmdamentalsof ultra-low sulphur diesel production: The four C's, Cataly Catalysts & Technologies, 2004 T.F.Edgar, D. M. Himmelblau, L.Lasdon, Optimization of Chemical Processes, McGraw Hill, 2001 Floudas, C.A., Nonlinear and mixed-integer optimization: fundamentals and applications, Oxford University Press, 1995

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An Attainable Region Approach for Effective Production Planning Charles Sung, Christos T. Maravelias Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI53706, USA Abstract A novel approach for the solution of production planning problems is presented. A detailed scheduling model is analyzed off-line to obtain a convex approximation of the feasible production targets, i.e. the production attainable region (PAR) of each manufacturing facility. The PAR is expressed via simple inequalities that involve only planning variables, lending itself to effective integration with production planning formulations. Moreover, PAR contains all the relevant scheduling information necessary to solve the planning problem with high quality. Keywords: Production Planning, Scheduling, Production Attainable Region. 1. Introduction To remain healthy in today s competitive global environment, chemical firms must have an integrated view of all their operations and use advanced modeling and optimization methods to achieve enterprise-wide optimal solutions. At the tactical and the operational levels, decisions are integrated and should be simultaneously optimized due to interconnections between the various nodes of the supply chain (SC) and the interdependence of the decisions at the various planning levels and geographical locations. In this paper we develop a novel approach for the integration of medium-term (3-6 months) production planning with short-term (1-2 weeks) scheduling. In the production planning problem, we seek to satisfy the (stochastic) demand at the customer-facing nodes of the SC (Fig. 1) at minimum cost. The SC is usually represented as a time-extended network with nodes i,JEN, arcs (i,j)eA, products keK and time periods t ET, and the production planning problem is solved as a multi-period min-cost network flow problem. The decision variables include the inventory levels (likt) and the shipments between nodes (F^,^). Sources, sinks and intermediate nodes are modeled via constraints for supply availability (eq. A), demand satisfaction (eq. E), and material conservation (eq. D), respectively. Since planning problems result in complex multi-period, multi-product (stochastic) mixed-integer programming (MIP) formulations, manufacturing facilities are often simplified via aggregate capacity (eq. B) and material conversion (eq. C) constraints. This simplified representation, however, does not capture the complexities of chemical process networks, leading to infeasible or suboptimal production targets. Several researchers have developed integrated planning-scheduling schemes, where a detailed scheduling model provides information for the planning decisions (Kallrath, 2002; Shah, 2005). The proposed integrated planning-scheduling schemes, however, are hard to solve and used only for problems with short planning horizons, few manufacturing facilities, and simple production recipes. In this paper we develop an approach that allows us to effectively solve large planning problems without compromising the quality of the solution.

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Supply availability: Z^*'-•'^* v/es,v/t,v/ (A)

Suppliers

Manufacturing facilities: ZZ^«^*^<^"A V/,V/

Manufacturing Facilities

(B)

Z i = 1 I^',«'^;,r, V/,V^,V? (C)

Material conservation: Z ^>* + '^*-. = Z ^* + A*, V/ e TV, VA:, V? (D)

Demand satisfaction: E^y*-A*, Vzer,VA:,Vr (E)

Customers (a) Supply Chain of a Chemical Firm

(b) Min-cost network flow problem

Fig. 1: Supply chain planning problem as network flow model.

2. Proposed method 2.1. Production Attainable Region The main idea of the proposed approach is the development of the production attainable region (PAR) of a manufacturing facility via the off-line analysis of detailed scheduling model. Consider the production network in Figure 2, where raw material RM is converted into intermediate INT, which in turn is converted (no storage for JNT) into final products A and B (production rates are in kg/day). 150
200
O

4CT


<)INT

R,

--150 (kg/d)

-&}

OB Fig. 2: Process network of motivating example. If an aggregate capacity constraint is used to describe weekly production levels, the set of feasible production targets is overestimated leading to either infeasible or suboptimal solutions (Fig. 3a). If a detailed scheduling model is incorporated, production targets are accurate because all the production details (capacities, storage policies, etc.) are taken into account (Fig. 3b); note that the feasible region cannot be described based on the capacities of the individual units. Since the integrated planning-scheduling model is hard to solve, however, we would like to obtain the true feasible region without incorporating complex scheduling formulations. This can be achieved if we develop a set of constraints involving only production variables F that describe the set of feasible production amounts (Fig. 3c).

1.5F,+0.5F„<2,100

1,000

2,000 F^ (kg)

(a) Aggregate capacity constraint

1,000

2,000 F^ (kg)

(b) True production via scheduHng

i,000

2,000 F^ (kg)

(c) PAR constraints

Fig. 3: Production attainable regions of process networks for one week.

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2.2. Construction of Production Attainable Region In the proposed approach we develop a set of constraints to describe which quantities of final products are feasible to be produced in a manufacturing facility (process network) in one planning period. Our goal is to off-line develop a convex approximation of the true production attainable region. Our approach is outlined in Table 1 and is based on search vectors (SV-Opt) and the iterative convergence of an underestimated convex hull (UECH) with an overestimated convex hull (OECH) (Director and Hachtel, 1977; Goyal and lerapetritou, 2002). We iteratively solve (SV-Opt) to find vertices of the UECH and inequalities of the OECH. The UECH set of inequalities is constructed by inputting UECH vertices into the Quickhull algorithm (Barber et al., 1996). The OECH is already expressed as a set of inequalities, but may be filtered at any time to remove redundant inequalities. Our measure of convergence is the maximum perpendicular distance (MPD) between the UECH and the OECH, found by solving (MPD-Opt). Table 1. Outline of proposed algorithm for the construction of the PAR. 1. Choose initial K-dimensional search vectors w = [wj, W2, w fj^ 2. Solve scheduling model (SV-Opt) for each initial search vector: (SV-Opt)

maxO = ;^w,F, k

s.t. Scheduling Formulation

3. 4.

5. 6.

to obtain a vertex for UECH and an inequality for OECH Run Quickhull to convert UECH vertices into inequalities Iterate until MPD stops improving: Solve (MPD-Opt) to get MPD and new search vector Solve (SV-Opt) for new search vector; obtain new vertex and new inequality Run Quickhull to update UECH Filter OECH to remove redundant inequalities Set PAR equal to OECH

2.2.1. Off-line Scheduling For a given search vector W*^''=[wi^^\ W2^^\ ^K^I^ at iteration iter, we solve the MIP scheduling model (SV-Opt) maximizing Ij, Wk'^^Tk. Each run yields the best solution {Fi , F2 , FK} (with objective value 0*) which is added to the UECH as a new vertex and a best bound 0^^ which is used to develop an OECH inequality: k

2.2.2. Quickhull A Igorithm Quickhull is used to develop the convex-hull of a set of points (i.e. UECH vertices). For a set of two-dimensional vertices, it returns a set of doublets [L, R/"^^: each doublet represents a different face. By convention for this paper, L refers to the left point when viewed from outside the convex region. The inequality associated with each face (2) may be recovered from the coordinate information of L and R, where coi = (L y Ry), 0)2 = (Rx LJ and f2 = L^Ry LyR/. k

2.2.3. Finding New Search Vectors Until convergence is reached, gaps will exist between UECH and OECH. As a measure of convergence, we use the maximum perpendicular distance (MPD) between UECH

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and OECH. We take advantage of the fact that if the coefficients of UECH inequaHties are normaHzed, then the slack of each inequality is equal to the Cartesian distance by which that inequality is violated. To find the MPD, we set up a target point restricted to lying within the OECH. The optimization MPD-Opt finds the maximum slack (distance) by which target is able to violate any one UECH face. 2.2.4. Iterative Improvement of UECH and OECH Approximations In each iteration, the MPD (length of dotted line in Fig. 4a) is found. Based on the new search vector (Fig. 4b), (SV-Opt) is run to generate a new UECH vertex (dot in Fig. 4c) and a new OECH inequality (dotted line in Fig. 4c). Quickhull is then run to update the UECH (two new dotted lines in Fig. 4d). The vertex and inequality are tangent if and only if (SV-Opt) is solved to optimality, but this is not always practical. Quickhull is run to update the UECH (two new dotted lines in Fig. 4d). If (SV-Opt) is solved to optimality, the UECH and OECH converge and MPD is goes to zero, otherwise a termination criterion for MPD is used.

(a)

(b)

(c)

(d)

Fig 4. An iteration of PAR construction.

The PAR of process network/? is described by the inequalities of the OECH:

I k=\..K J where Fk,p is the total out-going flow (production) of product k from process network/?. Not only does PAR/eq. (3) provide a good approximation of feasible production targets, it involves only planning variables (flows) Fk,p and is simple to implement in the production planning formulation. In addition, PAR is only generated once for each distinct scheduling submodel. 2.3. Integration ofProduction Planning and PAR Eq. (3) replaces the scheduling submodel (and/or eq. (B)) in the production planning formulation. If detailed schedules are required for the first few weeks of the planning horizon, a schedule can be recovered by reconciling individual scheduling subproblems using the flows Fk,p predicted by the planning problem as production targets. Multiple subproblems may be reconciled in a rolling-horizon manner. 2.4. Remarks Our PAR is a convex approximation of the true production attainable region. If this is not an appropriate assumption, then special-ordered-set (SOS) variables may be introduced to add non-convex characteristics to our PAR (Chu & Xia, 2004). Although SOS variables are mixed-integer and will increase the complexity of the production planning problem, the resulting complexity will still be significantly less than if the entire scheduling submodel were integrated. Second, the proposed approach can be extended to include cost information. If the production cost is approximated well by a convex function Cp =f(Fip, F2p, F Kp) of the produced amounts, then the PAR is amended by a set of L linear inequalities for cost:

An Attainable Region Approach for Effective Production Planning

Cp^

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Z 4 ^ . , , + / /=u...,L \/p

(4)

k=\..K

where Cp is the production cost in network p, Fk,p is the production of product k in p, and ;^^ is a constant. If production cost cannot be approximated by a convex function, then SOS variables can be used to develop piece-wise linear approximations. Finally, planning periods of variable duration can be handled by modifying the RHS of the constraints in eq. (3) to scale with time: (5) V

k=\..K

J

where T is the duration of the planning period and 0/^'^(T) is a piece-wise linear function of T. This feature would be helpful when long horizons are considered and decisions involve planning periods of different duration.

3. Example Consider the supply chain in Fig. 5 that involves two products, three process networks, four intermediate nodes and six customers. The objective is to meet the demand at minimum cost. Three production planning problems, with time horizons of 8, 12 and 16 weeks, are solved using the following two methods: 1) Full-space planning-scheduling model: An STN-based scheduling model is integrated with the network planning model: The solution is feasible but suboptimal because the problem cannot be solved to optimality. 2) Proposed approach: We first develop the PARs of the three process networks (Fig. 6) and then solve the planning problem with eq. (3). The CPU time to offline generate all PARs was 8,356 seconds; a resource limit of 300 CPU seconds for each iteration, and a tolerance of 2% for the MPD were used as termination criteria. Process Networks

Customers Fig. 5: Supply chain for production planning problem.

o - - - - 0 ^ o - - 0 - - o — ^ ^ P2 500

2000 r

\P2

P2

000-

.-

1000-

. '--.:;.'"''->

500-

PI 1000

2000

(a) PAR of PNl (MPD=42)

1000

2000

(b) PAR of PN2 (MPD=20)

0

PI 1

500

1

1000

1 1500

(c) PAR of PN3 (MPD=0)

Fig. 6: Production attainable regions of process networks PNl, PN2 and PN3 (in kg/week).

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Since the PAR is an overestimation, the bound provided by the integrated planningPAR model provides a lower bound to the objective function. A detailed schedule can be generated by solving a scheduling subproblem for each process network in each time period using the flows of the planning problem as production targets. The bounds and the feasible solutions of the full-space (FS) planning-scheduling approach and the proposed approach are shown in Fig. 7. The full-space approach provides a good bound, but feasible solutions deteriorate as the problem size increases. In contrast, the proposed method scales very well as the time horizon increases, and can be reconciled to yield significantly better solutions. Cost ($)

1 0

Planning-PAR

n Planning-PAR «fe Scheduling B FS Planning-Scheduling ^ FS Planning-Scheduling (bound)

Fig. 7: Comparison of proposed approach with full-space planning-scheduling model The solution statistics of the two methods are given in Table 2. In all cases the planning problem using the PAR is solved in less than 20 CPU seconds, i.e. we obtain a very good bound and approximate planning solution immediately. If a detailed schedule is required for the entire horizon, then the proposed method provides a detailed schedule in less than 4,000 CPU seconds. The full-space method cannot be solved to optimality within 7,200 CPU seconds. Table 2. Computational requirements (CPU-seconds). Without production cost 8-week 12-week 16-week 15.05 15.06 Planning-PAR 15.06 1,927.95 2995.14 + Scheduling 4,001.27 1,943.00 3010.20 Proposed Method 4,016.33 7,200.0* 7,200.0* Full-space 7,200.0* * Not solved to optimality in 7,200 CPU seconds.

With production cost 8-week 12-week 16-week 15.05 15.05 15.08 1033.92 1718.07 2239.90 1048.97 1733.11 2254.98 7,200.0* 7,200.0* 7,200.0*

4. Conclusions To solve complex production planning problems we propose to develop the PAR of a process network, i.e. a convex approximation of the feasible production targets. PARs can be used to obtain approximate planning and detailed planning-scheduling solutions of very good quality in a fraction of the time required by current methods.

References J. Kallrath. (2002). Planning and Scheduling in the process industry, OR Spectrum, 24, 219-250. N. Shah. (2005). Process industry supply chains: Advances and challenges, Computers and Chemical Engineering, 29, 122-1235. S.W. Director, G.D. Hachtel. (1977). The simplicial approximation approach to design centering. IEEE Trans, on Circuits and Systems, CAS-24, 7, 363-372. C.B. Barber, D.P. Dobkin, H. Huhdanpaa. (1996). The Quickhull algorithm for convex hulls, ACM Transaction on Mathematical Software, 22 (4), 469-483. V. Goyal, M.G. lerapetritou. (2002). Determination of operability limits using simplicial approximation, AIChE J., 48 (12), 2902-2909. Y.Y. Chu, Q.S. Xia. (2004). Generating benders cuts for a general class of integer problems. Lecture Notes in Computer Science, 3011, 127-144.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A Planning Support System for Biomass-Based Power Generation N. Ayoub, K. Wang, T. Kagiyama, H. Seki, Y. Naka Chemical Resources Laboratory, Process Systems Engineering Division, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku Yokohama 226-8503, Japan

Abstract When planning biomass-based power generation, planners should take into account the different stakeholders along the biomass supply chains, e.g. biomass resources suppliers, transport, conversion and electricity suppliers etc. Also the planners have to consider the social concerns about environmental and economical impacts of establishing the biomass systems. Accordingly, in order to overcome these struggles in sustainable manner, we should take into account the new environmental institutions, e.g. RPS (Renewable Portfolio Standard) to promote them as well as the bioenergy production systems. To address the problems mentioned above, a Planning Support System (PSS) for biomass-based power generation is developed. This paper introduces the general structure for the PSS at both national and local levels of the biomass planning process. The PSS is a user-oriented system which employs data visualization, data analysis and simulation methods in interaction with the knowledge and intentions of the users to provide them with understandable results. A case study on planning forestry residues utilization at the national level is presented. Keywords: Planning Support Systems (PSS); Geographical Information System (GIS); Bioenergy; Data Mining; Fuzzy C-means Clustering [Shen et al., 2005] 1. Introduction Nowadays, bioenergy production from renewable resources is of great importance in keeping the level of emissions under control. The literature reports many practical experiences in realizing this potential [Ushiyma, 1999; Pari, 2001; Akella et al., 2005]. Quantitative analyses of strategies for utilizing biomass energy sources have been performed to evaluate the potential resources of bioenergy in different countries (forestrich regions, nations with agricultural-land surplus) [Hall et al., 1998] and matching the woody biomass demand and supply with forestry industries in Europe [Kuiper et al., 1998]. The same works, however, fail to propose a systematic approach to define the actual availability of energy from biomass. At present, a comprehensive approach to biomass exploitation is required for both nations and regions where other kinds of energy are difficult to exploit or where the use of biomass could decrease environmental pollution and enhance regional welfare, e.g., by providing employment opportunities or improving environmental preservation. In this paper. Planning Support System (PSS) is proposed to assist in planning for bioenergy generation from biomass. There are many models in the bioenergy field that have been developed over the last few years with a decision support system capability [Mitechell, 2000]. However, PSS specifically provides the general frmctions of data visualization, data analysis and simulation methods for the bioenergy generation decisions support. The PSS divides the bioenergy generation decisions into national and

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Local (Regional) Level

Fig. 1. The two levels with different objective in the PSS. regional levels respectively. Different methods of data visualization, analysis and simulation are provided for the two planning levels. In section 2, the planning problem description and the possible interaction between two levels are discussed. In section 3, the PSS architecture is explained. The system development, national & local target users and system applications are explained in sections 4 and 5. In section 6 a case study for electricity production from forestry residues in Japan is introduced. In the case study section, a description of a systematic approach for data handling within the proposed system is given. Conclusions and future work are stated in section 7. 2. Description of Planning Problems The objectives of a national planner who proposes a biomass exploitation strategy for the whole country, are different from those of regional planners and executers who are applying the policy in a small scale with specific requirements and limitations of using biomass, e.g. resource availability, topological restrictions, energy needs etc. as shown in Fig. 1. Such differences in the scope must be taken into account when developing systems for decisions and planning support for the exploitation of biomass as a bioenergy resource. For national planners, they take into account the following: the available biomass materials in the whole country; the available energy conversion technologies for converting different biomass materials to different bioenergy; economic effects, such as the cost of energy production, or the potential legislative benefits which means that a rough cost model is fitting such macro level of planning. Another concern of planners at this level is the need to apply emission prevention institutions, e.g. Emission Taxation, Renewable Portfolio Standard and Fossil Fuel Taxation etc. Based on the decisions of the planners at the national level, the regional executers and designers will determine the detail unit processes to meet these decisions: the detail optimal size of each biomass plant in terms of both energy production and feeding; the percentage of electrical energy with respect to the total energy produced; the quantity of biomass that must be collected; and the detailed location to collect the biomass. Specifically, in the regional level, more detailed information is provided, so a rigorous

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Technological information base

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y^. UP Simulation module

Optimizer software

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Fig. 2. PSS Architecture. model considering the biomass and its logistic costs, the energy conversion costs produced by the whole system and labor cost should be applied for its whole life cycle. To control biomass exploitation in a sustainable way, users need to determine the quantity of biomass that is feasible for collection, the collection points and, consequently, their position in the territory. According to the decision made by the regional planners and executers, the decisions of the national level may be applied, rejected, or modified, as a result of typical regional inputs. 3. System Architecture The PSS is based on the technological information infrastructure concept [Naka et al., 2000]. The technological information base includes: the basic information for the whole life cycle of biomass and bioenergy; the model information for the decisions of biomass and bioenergy; and the detail decision information. Scenario database is also saved in the system, which is used for instructing new user and also for case base reasoning by the user. Communication with the different databases is managed by a proper ODBC (Open DataBase Connectivity) interface. Fig. 2 shows the PSS architecture that involves an integrated framework integrating Unit Process (UP) simulation, ArcGIS9 [ESRI] for data visualization, data optimization using Matlab Software [The MathWorks] and the inline domain knowledge for evaluating the user's input. These aspects of the system make it highly suitable planning biomass energy systems.

4. Development of the Planning Support System The PSS consists of the four parts shown in Fig. 3.; common database, simulation module, in line domain knowledge evaluation module and user interface. The common database includes relevant data about: biomass types, biomass logistics, biomass conversion and GIS database. The planners define the feasible paths, through defining the biomass material life cycle using UP simulator, from the common DB by the support of in-line domain knowledge evaluation module, e.g. the selection of sewage sludge cannot match with the gasification without sever drying. After the feasible path selection, further data analysis, i.e. Fuzzy C-means clustering algorithms and GAs

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Fig. 3. The conceptual structure of the PSS. [La'zaro et al., 2005;Shen et al., 2005] are performed in matlab and envisioned to users via ArcGIS data visualization for evaluating the supply chains for the decision-making.

5. System Users and Application When designing any planning support system one should consider vs^ho will use the system and the types of functionality the system will provide. In following sections summarize these two important issues. 5.1. User classification The target users of the system include governmental policy designers of energy systems in the national level, individuals related to eco-management (environmental protection, environmental accounting and reporting) at the enterprise and governmental level, and university and government researchers. 5.2. Functionality The proposed PSS is designed to provide the following functions • GIS data visualization methods to identify the geographic distribution of the economically exploited biomass potential; • data analysis methods (e.g. Fuzzy C-means clustering methods and decision trees) for the assessment of biomass potential as theoretical, available and economically exploitable respectively; • the determination of the economical biomass collection points, storage points and bioenergy conversion plants positions; and • simulation methods based on cost modeling, thereby providing optimal decision-making for bioenergy generation at the national and local levels. 6. Case Study: Forestry Residues Exploitation in Japan Planning for forestry residues supply chains for power generation in Japan has different characteristics than in other countries e.g. European countries. These differences originate from the nature of the land, forest residues density, forestry population and weather. For example, Japanese weather is mostly wet and the rain occur at any time, which creates a need for covered material storage. Also, the mountainous topography prevents high capacity equipments from being employed. Limited access distances and

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low-density forests results in difficulties in estimating the feasible amount of forestry residues that used as a source of bioenergy. In this case study the Japanese Geographical Information System Data Base (GISDB) for forestry residues is applied to demonstrate that the PSS is efficient in planning for bioenergy production. With the support of PSS, feasible amount, and environmental, economical and social impacts of forestry residues are estimated as follows: 1- The resource quantity in tons, area in square meters and centroid in Cartesian coordinates of each polygon, which represents the city where the resource is located, were determined using the ArcGIS 9 software [ESRJ]. 2- Using Matlab® Software, Version 7.0.1 with service pack 1 [The MathWorks], the (x, y) coordinates in meter of the collection points were generated randomly around the centroid of the polygon, assuming the area of each polygon is a circle. 3- The system asks the user to define; (a) the average access distance inside the forest from the road in the country, (b) total requirements fi^om electricity in the country, (c) efficiencies and expected number of power plants in the country, (d) maximum traveling distance between the storage location and the collection points and (e) the maximum allowable distance between the storage and the power plant. Default values are provided for system users who do not have sufficient information for the required inputs. 4- Once the inputs are defined, the system omits all collection points located at distances greater than the access distance. This resulted in the feasible number of collection points. Employing the resulting feasible collection points locations, the system calculates the suitable number of clusters (storage houses) based on the Validity Index (VI) [Shen et al., 2005]. The VI is defined as the ratio between the intra (average distance between the storage points and the collection points) and the inter (minimum distance between the storage houses). 5- A table of clusters (storage house) numbers versus the VI is automatically generated by the system and displayed for the users thus allowing them to select the desired clusters number or to assist the system in automatically defining an optimum number of clusters. 6- The collection points are then clustered to the pre-defined number of clusters in step 5 using fuzzy C-means clustering method in a fuzzy toolbox provided in Matlab Software.

Fig. 4. Economic locations (cities) of plant installation defined by PSS.

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N.Ayoubetal The storage locations are mapped to Japanese map and visualized for the user to define the locations of the desired power plants or help the system to locate them via clustering the storage points. Once the power plants locations are defined, the system start calculating all information about costs, emissions, number of labors required for establishing the business and total investments required for realizing the biomass-based electricity. All calculated information then included in the power plants layer in the ArcGIS9 where the data attributes can be visualized. Fig. 4 shows the power plant layer viewing the city names where power plants of forestry residues conversion should be located in the Japanese case.

7. Conclusions and Future W o r k This paper proposes an explanation of the concepts of the Planning Support System for Power Generation. The PSS helps in solving the real biomass planning problems at the national level. The authors are currently developing system support for local level planners and executers. At that level a more robust model is required which can support detailed information about biomass types, properties and technical data evaluating the most suitable technology for the local resources. This work also is going to be expanded to a Web-based Decision Support System called general Bioenergy Decision System (gBEDS) supporting the maintenance of the system's technological information base. This paper is restricted to only the procedure for the system application. A more complete research description and associated results will be published as a journal paper. Acknowledgments The authors wish to thank the Ministry of Education, Culture, Sports, Science and Technology of Japan for the financial support of this work and the Central Research Institute of Electric Power Industry in Japan for providing the biomass GIS database. References A. K. Akella, M. P. Sharma, R. P. Saini, 2005, Optimum utilization of renewable energy sources in a remote area, Renewable and Sustainable Energy Reviews, In Press, 1-15. ESRI GIS and Mapping Software, www.esri.com. D. O. Hall, J. I.Scrase, 1998, Will biomass be the environmentally friendly fuel of the future?, Biomass and Bioenergy, 15, 6, 451-456. L. C. Kuiper, S. R. Sikkema, 1998, Establishment needs for short rotation forestry in the EU to meet the goals of the commission's white paper on renewable energy, Biomass and Bioenergy, 15,4-5,367-375. J. La'zaro, J. Arias, J. L. Marti'n, C. Cuadrado, A. Astarloa, 2005, Implementation of a modified Fuzzy C-Means clustering algorithm for real-time applications. Microprocessors and Microsystems, 29, 8-9, 375-380. C. P. Mitchell, 2000, Development of decision support system for bioenergy applications, Biomass and Bioenergy, 18, 4, 265-278. Y. Naka, M. Hirao, Y. Shimizu, M. Muraki and Y. kondo, 2000, Technological information infrastructure for product life cycle engineering, Proc. of 7th International Symposium on Process Systems Engineering, Keystone, Colorado, USA, 665-670. L. Pari, 2001, Energy production from biomass: the case of Italy, Renewable Energy, 22,1, 21-30. J. Shen, S. I. Chang, E. S. Lee, Y. Deng, S. J. Brown, 2005, Determination of cluster number in clustering microarray data. Applied Mathematics and Computation, 169, 2, 1172-1185. The MathWorks Inc., www.mathworks.com I. Ushiyma, 1999,Renewable energy in Japan, Renewable Energy, 16, \-A, 1174-1179.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubHshed by Elsevier B.V.

Semantic Analysis for Identification of Portfolio of R&D projects. Example of Microencapsulation Andrzej Kraslawski Lappeenranta University of Technology, Department of Chemical Technology, P.O. Box 20 53851 Lappeenranta, Finland [email protected] Abstract The paper introduces a method for building portfoUo of R&D projects. The proposed method consists in comparison of the technological problems addressed in the patents with those referred in the papers. The problems are identified using the semantic analysis for the determination of the structures subject-action-object. As an illustration, there is given an example of the identification of problems related to research and application studies of microencapsulation. Keywords: R&D management, technology forecast, semantic analysis, microencapsulation Introduction Answering the questions about future technology developments supports the decision makers in administration and business communities with rationale to allocate resources in order to better address emerging social, environmental and economic issues. The substitutive resources, directly connected to capital, like equipment or information repositories are accompanied by the non-substitutive ones like highly qualified research and technical staff. The wrong allocation of any resources is always a loss but a misuse of non-substitutive resources is especially painfull as top-class specialists are usually in a very short supply. In consequence, the wrong decisions lead to the lost of time which makes the gap between the needs and the technical possibilities of their satisfying to grow very fast. Therefore the systematic methods of technology forecast are of paramount importance for the decision makers. One of the major goals of the technological forecast is to build a portfolio of R&D projects. It means to identify a group of projects which should be funded in a given period of time. There are used various methods of portfolio creation but usually there are three objectives to be fulfilled by an optimal portfolio of R&D projects, effectiveness: it means to identify the projects contributing to reahsation of the strategic goals of the organisation efficiency: it is to identify a group of the projects able to ensure, with the highest degree of probability, the fulfilment of the performance measures like shareholders value, long-term profitabilify, return-on-investment etc. diversification: it is to identify the group of the projects for which there is a balance ensured between the possible gains and losses. When analysing those objectives, there is clearly visible a lack of method for the identification of the most promising technologies in terms of their innovative potential.

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The objective of this paper is to propose a method for the support in building portfolio of R&D projects basing on the analysis of the scientific publications and patents. Information analysis There is a broad spectrum of technology forecast methods. For example a group of authors (Technology Futures, 2004) list 50 methods divided into nine classes. The most common methods used are: analysis of citation of papers, patents, web pages etc., visualisation of the knowledge domain, statistics-based analysis of the texts (words occurrence), scenarios and Delphi method. They have serious drawbacks strongly limiting their usefuhiess. For example, the citation and co-citation is strongly criticised as it is not related to the context of the document, e.g. some very often cited references are given as an example of wrong or erroneous interpretation of data. It is a generally accepted conclusion that there is actually a lack of broadly acknowledged forecasting method. Moreover, the growing complexity of technologies and their interactions with the social and natural environment lequires the development of new methods for forecasting of technological developments. The characteristic features of the contemporary research are high complexity of the studied issues, their interdisciplinary character and the overwhelming amount of the generated information. The high complexity is related to the high cost of research in financial and non-financial terms and hence a must of a very carefiil selection of research subjects. The interdisciplinarity requires the analysis of the reference materials which is based on the information context and not only on the mechanical application of the citation indexes or occurrence of the words as the different fields use their specific vocabularies to represent the identical entities. The amount of information forces the researchers to look for new methods of their analysis. As a result the methods able to handle three above mentioned issues are of great interest to decision makers. The problems of the subjectsD complexity, their interdisciplinary character and amount of the available information make the task of the automatic text analysis an important research issue. The orthographic, semantic, statistical, syntactic and usage analysis are the basic methods used to handle this issue, (Losiewicz et al. 2000). An approach, based on the semantic analysis of the text, leading to the identification of the structure subject-action-object has been applied in this work. Identification of portfolio of R&D projects The proposed method for identification of the development trends of given technology, and finally the proposed portfolio of R&D projects, consists in comparison of the problems addresses in the patents with those referred in the papers dealing with the technology under consideration. The research problems encountered exclusively in the papers are subject of actual great scientific interest but do not have any practical application In this paper they are referred as set A problems. The problems existing only in the patents seem to be less interesting for the research. They are named as set B problems. The problems common for the patents and scientific papers are specified as set C problems. The discovery of the issues common for the patents and the papers presents the areas which are extensively studied scientifically and have the great practical interest. The set B problems are usually the issues of the minor research interestfi-omthe point of view of planning the allocation of R&D resources as they are right now practically solved The start of the research for solving of the set C problems requires a considerable analysis in order to position own skills and resources. The wrong decision of starting such research

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in the organisation without the sufficient experience and resources would lead to the situation called "me-too strategy". It means the satisfactory level of expertise and interesting results could be obtained too late. The competitors would much earlier successfully address the problems and patent the methods of their solving. The set A problems are generally the issues to be addressed. However, a careful analysis is required to study the cost, required duration of the research and eventual applicability of the obtained results. The presented method is a preliminary step in the determination of the portfolio of R&D projects. It is composed of the following steps: 1. Determination of the set PP of problems in patent literature 2. Determination of the set RP of problems in the scientific literature 3. Identification of the sets A, B and C by the comparison of sets PP and RP 4. Discovery of the frequency of occurrence of the problems in sets A, B, C 5. Identification of the Hst of the interesting R&D subjects and the inventory of the subject which should not be further investigated The determination of the sets PP and RP is realised by use of semantic analysis of the texts aimed at the determination of the structures subject-action-object. The identification of the sets A, B, C is realised by the analysis of PP and RP sets using the strings con^arison and filters for the identification of the synonyms. The identification of the frequency of the problems occurrence is done using the standard tools used in the databases (e.g. ACCESS). The most commonly encountered problems are used for the automatic generation of the list of problems. The rarest problems are analysed individually by the experts. The lists of the problems for the detailed consideration are next visualised (Pasteur's quadrant, Fig. 1) in order to facilitate the selection process. Analysis Subject-Action-Object The structure subject-action-object is a universal template of all sentences of any natural language. The use of the specialised knowledge bases allows to organize the concepts into a problem-solution relationships. The semantic analysis of the text leading to the determination of the structure subject-action-object (SAO), and finally to the identification of the pair problem-solution (PS), is based on the method presented in the set of the patents and patent applications (e.g. Tsourikov et al.2000). The coinputer implementation of the method has been realised as the program Knowledgist ^ . Its concept has been presented in several patents (e.g. Batchilo et al. 2003). The outline of the analysis performed by Knowledgist is presented in Fig. 2. The linguistic knowledge base is composed of the dictionaries, classifiers, and database for linguistic models recognition (text-to- word splitting, rule for determination of cause-effect relationship). application inspired I 1 yes 1 curiosity inspired no

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Fig.l Pasteur's Quadrant. The pure basic research is called Bohr's approach; purely applied research- Edison approach and use-inspired R&D is called Pasteur's method.

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The knowledge base is used to perform the following operations: Preformatting- the document is divided into small parts for the purpose of analysis. It is split into the sentences and words. linguistic analysis - consists in tagging, parsing and extraction of knowledge bits, corresponding to objects, facts and rules of the knowledge domain. sentence weighting - used to quantitatively evaluate importance of information contained in each sentence of the analyzed document. summary generation is used to produce the digest in the form of a list of keywords, topics, or SAOs Linguistic Knowledge Base

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Fig. 2. The structure of Knowledgist TM Example The determination of the portfolio of R&D projects in the field of microencapsulation has been studied in this example. An introduction to mircoencapsulation techniques and applications has been given by Gouin (2004). The research of patents addressing any aspect of microencapsulation has been performed on US Patent and Trademark Office Full Text Database. The search performed with Knowledgist ^^ resulted in 2606 structures subject-action-object identified in 95 patents. The pairs action-object are treated as problem while subject corresponds to the solution. The identified 2606 pairs problem-solution constitute the PP set. The screen of Knowledgist^^ presenting the results of search in the patent database is given in Fig. 3. The analogical search in the Elsevier database ScienceDirect of the scientific papers, years 1995-2005, have resulted in 1656 papers. The 5529 structures problem-solution have been identified. They constitute the RP set. The comparison of the both sets has resulted in the identification of the sets A, B. and C. The problems of pure basic research (classified here as mentioned only in the papers) are given as a set A They are related to the following subjects: production of chitosan microspheres, treatment of heavy metal contaminated sites, application of microcapsules to copper plating, studies on toxicity related to use of microencapsulated materials, improvement of loading capacity of microcapsules, magnetic properties of microcapsules, bioadhesion, stability of encapsulated pigments, pulmonary delivery, immobilized ruthenium catalysts, photoluminescence of microcapsules, thermal stability, application to wound dressing, microcapsules for the cells entrapment studies of swelling stress, behaviour in electric field and use of ultrasonic atomization for the production of micro capsules. The given-above list is not

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here due to the lack of space. The problems of the purely applied research (classified here as mentioned only in the patents) are given as a set B. They are related to the following subjects: encapsulation of explosives, anesthetics and cements and use of light for hardening of microcapsules. The problems common for the research papers and patents are given as the set C. They are visualised and specified in Fig 4. The problems to be considered for R&D portfolio are given as the elements of the set A and possibly C. The problems of the set B should not be considered for the further research. m^\m\y\

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Fig. 3. The screen of Knowledgist. On the right, there is a list of the problems addressed in the patents dealing with the microencapsulation and on the left an example of the problem presentation. Summary The presented method for building a portfoUo of R&D projects is based on evaluation of the published information and allows to avoid the subjectivity in the experts judgement characteristic for many popular approaches e.g. Delphi method. The use of semantic analysis allows to identify problem-solution pairs. It is a conceptually different approach to the text analysis which usually is based on the keyword search or studies of the word frequencies. The identified problem-solution pairs can be used not only in the building of the projects portfolio but also may be applied in creativity enhancement tools based on use of analogies.

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Fig 4. The comparison of the most common problems encountered during the analysis of patents and papers related to microencapsulation (1-stability and size distribution ; 2controUed release; 3-use of polymers; 4-oils encapsulation; 5-biodegradation; 6nanoparticles 7-drug encapsulation; 8-formulations; 9-emulsification; 10-insulin; 11double layer microcapsules; 12-agglomeration; 13-durability; 18-reference point). References Losiewicz, P, D.W.Oard, R.N. Kostoff, (2000) Textual Data Mining to Support Science and Technology Management Journal of Intelligent Information Systems, v.l5, pp99 n i l 9 . Tsourikov, V.M., Batchilo, L.S. and Sovpel, I.V. (2000) Document semantic analysis/selection with knowledge creativity capability utilizing subject-action-object (SAO) structures. US Patent 6,167,370. December 26,2000 Batchilo L.,V.Tsourikov,L Sovpel (2003) Computer based summarization of natural language documents US Patent Application no US 2003/0130837 Al. July 10, 2003 Technology Futures Analysis Methods Working Group, 2004, Technology futures analysis: Toward integration of the field and new methods Technological Forecasting and Social Change, v 71, pp 287-303 methods of forecasting because Stokes D., (1997) Pasteur-s Quadrant, Brookings, Washington. Gouin S., (2004) Microencapsulation: industrial appraisal of existing technologies and trends Trends in Food Science & Technology 15, 330-347.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 PubHshed by Elsevier B.V.

Dynamic Rule-Based Genetic Algorithm for Large-Size Single-stage Batch Scheduling Yaohua He, Chi-Wai Hui Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, P. R. China. E-mail address: [email protected] Tel: +852-23587137 Abstract Single-stage multi-product scheduling problem (SMSP) with parallel units has been widely studied, very often solved by using mixed-integer linear programming (MIL?). As the problem size increases linearly, the computational time of MIL? increases exponentially, so that it is very difficult for MILP to obtain an acceptable solution to large-size problems within reasonable time. Hence, the preferred method in industry is the use of scheduling rules. However, due to the constraints in SMSP, the simple rule-based method may not guarantee the feasibility and quality of the solution. In this study, random search (RS) based on heuristic rules was first proposed. By exploring a set of random solutions, better feasible solutions were acquired. To improve the quality of random solutions. Genetic algorithm (GA) based on heuristic rules was then proposed. The heuristic rules are crucial to cut down the solution space; but there is no single universal rule, and the effectiveness of the rules depends on the scheduling objective and the prevailing shop or plant conditions. Hence, a dynamic rule selection method based on GA was finally proposed. To increase the search speed in solving the high-constrained problem, a penalty method was adopted. Through the comparison of the computational results of MILP and the new methods, the proposed methods have demonstrated their effectiveness and reliability to solve large-size scheduling problems. Keywords: Parallel Unit Scheduling; Mixed-Integer Linear Programming; Random Search; Genetic Algorithm; Heuristic Rule. 1. Introduction The complexity of scheduling problems can easily exceed today's hardware and algorithm capacities. Scheduling problems are usually NP-hard, no standard solution techniques are available. Hence, in many cases, feasible solutions to the problems are more considered and practical, rather than optimal solutions [1]. Single-stage multi-product scheduling problem (SMSP) with parallel units has been widely studied [2, 3, 4], very often solved by using mixed-integer linear programming (MILP). As the problem size increases linearly, the computational time of MILP increases exponentially, so that it is very difficult for MILP to obtain an acceptable solution to large-size problems within reasonable time. This paper presents a novel genetic algorithm that for large-size SMSP. This method will dynamically select suitable rule to synthesize evolved order sequences into high quality schedules. The algorithms in this paper were implemented in C language and the computation tests were run on a PC with Intel (R) Pentium (R) M 1.5GHz CPU and 768M memory.

1911

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Y. He and C.-W. Hui

2. Problem Definition In SMSP, there are M production units available to process N customer orders. Each order involves a single product requiring a single processing stage, has a predetermined due date, and can only be processed in a subset of the units available. The production capacity of a unit depends on the order processed. The size of an order may be larger than the size of a batch, requiring several batches to satisfy an order. Batches of the same order are processed consecutively in the same unit. A production unit processes only one batch at a time. The batch-time of an order is fixed and unit dependent. When one order changes over to another order, time is required for the preparation of the unit for the changeover. The changeover time is sequence-dependent. Forbidden changeovers and processes may exist in the problem, named as CP constraints. Scheduling objective is to minimize the makespan Cmax, which is the duration for completing all the orders. Table 1 is the data of a simple example (the former 10 order of Example 1 for case study), which is used to illustrate the following algorithms. Table 1 Changeover times, due dates and process times of Example 1 (the former 10 orders) Changeover times (%)

il

di

Process times (p,„)

ori

jl

J2

J3

J4

J5

J6

J7

J8

J9

jlO

-

1.00

0.15

1.10

2.00

0.65

0.30

1.20

0.85

0.40

10

0

ul

u2

u3

u4

10.20

3.60

4.20

10.80 4.50

12

1.80

-

1.10

1.30

1.40

0.90

0.20

1.20

0.40

0.30

22

0

7.20

10.50

4.50

13

1.00

0.15

-

1.20

1.50

2.10

0.30

1.80

1.60

0.20

25

0

6.25

5.20

5.50

5.00

14

1.20

0.02

0.10

-

0.05

1.60

1.20

2.00

1.20

0.50

20

0

11.20

13.60

15.40

12.00

15

0.10

0.20

0.30

0.30

-

0.70

0.90

0.60

1.00

0.90

28

0

9.00

8.40

4.50

3.20

16

1.40

0.80

0.30

0.70

2.00

-

0.90

1.20

1.20

1.60

30

0

9.60

4.00

10.80

5.50

17

1.20

1.80

1.30

0.90

0.85

0.80

-

0.45

1.20

1.30

17

0

7.60

6.00

3.00

6.40

18

1.30

1.40

1.50

1.40

1.20

1.30

1.65

-

1.30

0.80

23

0

14.00

14.70

16.80

16.80

19

2.10

2.00

1.25

1.35

1.45

0.80

1.60

0.80

-

0.65

30

0

4.80

3.00

6.30

3.60

110

1.50

1.20

0.60

0.75

0.50

0.40

0.90

0.60

0.70

-

30

0

7.80

5.70

4.80

7.20

UTu

0

0

0

0


orf. order release time;

«r„: unit release time

3. Random Search Combined with Prefixed Rule Due to the difficulties of MILP, the preferred method in industry is the use of scheduling rules, such as shortest process time first (SPT), earliest due date first (EDD). According to a scheduling rule, the customer orders are sequenced in decreasing priority order, and then one by one assigned to the units according to a unit selection rule. However, due to the constraints in SMSP, the simple rule-based method may not guarantee the feasibility and quality of the solution. In this study, random search (RS) based on heuristic rules was first proposed: the customer orders are randomly sequenced, and then one by one assigned to the units according to the same unit selection rule. Through exploring a set of random solutions, a number of feasible solutions can be obtained, among which much better solutions may be found. 3.1 Heuristic Rules To minimize the makespan, some unit selection rules are presented in Table 2. Table 2 Unit selection rules (forming a candidate rule base SR) Rule 1: earliest completion time (ECT);

Rule 4: shortest changeover time (SCT);

Rule 2: earliest start time (EST);

Rule 5: random Assignment (RA);

Rule 3: shortest process time (SPT);

Rule 6: shortest changeover+process time (SCPT)

Dynamic Rule-Based Genetic

1913

Algorithm

3.2 Schedule Synthesis by Heuristic Rule Get a random Calculate the The natural numbers 1, 2, 3, ..., N aiQ used to denote order string n objective value/f>r>) of the schedule the A^orders. An order sequence ;r= (TTJ, Ttj, 713, ••, ^N) is produced randomly, m ^ {1, 2, 3, ..., N } , i=l, 2, i=l 3, ..., N. And then, from TTI to TTN, one by one, each order will be assigned over the units according to a certain unit selection rule. As a result, a schedule is Assign the order ;^ to a unit according formed with an objective value/f>r:): / W = C.ax=max{Q,

Q

C,},

(1)

to the heuristic rule

Fig. 1 Schedule synthesis by one rule where Q is the completion time of ordery. Fig. 1 is the procedure of synthesizing an order sequence into a schedule according to one selected rule. Assume that a random order sequence is ;r = (3, 2, 7, 6, 4, 5, 9, 10, 1, 8) in the simple example. After the ten orders are sequentially assigned over the four units according to Rule I, SL schedule with a makespan of 24.8 is available as shown in Fig. 2. Fig. 2 A random schedule

3.3 Random Search Procedure Among the six rules, Rule 1 is pre-selected and used in the random search. The procedure of the random search is simple: a number (popsize) of feasible order sequences are generated and evaluated with their makespans. The order sequence with the best makespan is the solution we need. For the 10-order simple example, through random search, a solution is obtained: ; r = (8, 4, 7, 1, 6, 5, 3, 2, 9, 10), with a makespan of 18.15. The schedule is shown in Fig. 3. 4. Genetic Algorithm Combined with Prefixed Rule To evolve the random solutions, GA is applied. GA combined with the same unit selection rule in the random search. At the beginning of the algorithm, an initial generation of random solutions (named as chromosomes) is generated. Due to GA's evolutionary mechanism, near optimal solutions can be obtained at the end of genetic search. The procedure of GA is shown in Fig. 4, in which permutation-based representation [5] (see Fig. 5), tournament selection method [6], partial-mapped crossover [7] (PMX) and reversion mutation [7] are adopted. Rule 1 is still utilized in GA. The computational results are presented in Table 4.

Fig. 3 A better schedule

Initial generation and evaluation

Output the best solution

N

^ ^ " ' ^ ^

Selection

'r

'

^-^lerminaier^^

Crossover

^r . llUll

Muta

Evaluation and

''

new generation

Fig. 4 Flow chart of GA 6 3 5 4 8 2 7 1 9 10

5. Automatic Rule Selection by GA Fig 5 A sample chromosome In section 3 and 4, RS and GA utilize a pre-selected rule to synthesize the order sequences or chromosomes into schedules. However, even for the same scheduling objective, the effectiveness of the rules depends on the prevailing shop or plant conditions, such as the problem size and the constraints. To select a suitable rule for a specific problem, traditionally.

1914

Y. He and C.-W. Hui

great simulation experiments are required. Here we propose an automatic rule selection (ARS) method. GA is still used for this purpose. The basic procedure of GA for the automatic rule selection is the same as that in section 4, but the following components are changed. (1) Mixed chromosomes are originally adopted. Assume that Order sequence Rule sequence the number of rules in the candidate rule base SR is R, and the |3 1 6 5 4 2 6 3 5 4 8 2 7 19 10 1 natural numbers 1, 2, 3, ..., R are used to denote the R rules. A mixed chromosome P=(Pi, Pi) consists of two parts: Fig. 6 A mixed chromosome Pi=(yL yi, 73, '•', JR) is a rule sequence, yk^{l, 2, 3, ..., R}, Parents: k=l, 2, 3, ..., R; and P2={^j, 7t2, Tts, ..., m) is an order 4 2i5 3 6;i 6 2 5:4 8 3 911 7 10 sequence, 7ti^{l, 2, 3, ..., TV}, i=l, 2, 3, ..., N. For the chromosomes in the initial generation, each part of the 2 5!6 3l!4 3 16;8 7 5 4;2 9 10 chromosome is produced randomly. Fig. 6 is a sample mixed chromosome. The evaluation of a chromsome P is to use the Offspring: Rule yi in Pi to synthesize P2 into a schedule with an 4 2|6 3l!5 6 2 3;8754;i9 10 objective value/(P), 211516 [4 5 16|4 8 3 9!2 7 10 / ( ^ ) = ^max=max{C„ C „ ..., C^}. (2)

I

(2) Crossover is conducted respectively for each part of two parents (see Fig. 7). Mutation is also conducted respectively for each part of the chromosome (see Fig. 8). The Rule yi will change with different chromosomes. Due to the evolution mechanism, GA will dynamically selected a suitable rule to synthesize evolved order sequences into high quality schedules.

Fig. 7 Crossover respectively 536214

4 2 8 3 9 5 6 1 7 10 (Reverse the chosen part)

526314

4 2 8 5 9 3 6 1 7 10

Fig. 8 Mutate respectively

6. Case Study The following three examples are used for case study. Example 1 and Example 2 were widely used in literature [2, 3, 4], but the authors often solved small size problems (limited to 20 orders over 4 units) by MILP. In this paper, the problem size is enlarged up to 50 orders over 4 units. Example 3 is a very large randomly generated problem, very difficult to MILP. Table 3 Changeover times, due dates and process times of Example 2 (the former 10 orders) Changeover times (c//)

il

12 13 14 15 16 17 18 19 110

jl

J2

J3

J4

J5

J6

-

-

-

0.15

-

0.65

1.00

-

-

-

1.40

-

1.10

1.50

-

0.70

0.90

0.60

-

-

-

-

1.20

0.45

-

23

0

-

0.65

30

2

4.80

-

0.70

-

30

6

7.80

UTu

0

-

0.85

1.25

-

-

0.80

0.60

0.75

0.50

-

1.80

2.10

-

0.70

df. Due date;

orf. order release time;

1.65

-

-

jlO

0.85

0.40

10

-

-

1.60

Process times (p/„)

0.30

0.30

-

J9

ort

-

0.30

0.05

di

J8 -

-

J7 -

ul

u2

u3

u4

0

10.20

-

22

5

-

0.20

25

0

6.25

-

0.50

20

6

13.60

-

28

0

-

-

30

2

9.60

9.00

17

3

«r„: unit release time

4.50 5.50

-

3.40

3.15

6.60 16.80

5.70

-

3

2

3

8.40

-

-

Dynamic Rule-Based Genetic

1915

Algorithm

Example 1: There are 4 units to process up to 50 orders. The order release times and unit release times are null. The changeover times, due dates and process times are presented in Table 1. Table 1 only includes the data of the former 10 orders. No CP constraint exists in Example 1. Example 2: There are 4 units to process up to 50 orders. The order release times and unit release times are finite. The changeover times, due dates and process times are presented in Table 3. Table 3 only includes the data of the former 10 orders. CP constraints exist in Example 2. Example 3: There are 16 units to process up to 200 orders. The order release times and unit release times are randomly generated, 0 ^ o r / < 6 , 0 ^ w r „ < 4 . The changeover times and process times are also randomly generated, C/,G(0.10, 2.00), ;?/„e(5.0, 20.00). No CP constraint exists in Example 3. Due to the CP constraints in Example 2, infeasible chromosomes will surely be generated in these steps: "Initial generation", "Crossover", "Mutation". In this case, new feasible chromosomes should be re-generated to replace the infeasible ones. It will take a lot of CPU time for GA to generate the feasible chromosomes. In order to increase the computing speed of GA for problems with CP constraints, a penalty method has been adopted: replace the forbidden changeovers and processes with a very large penalty numerical value, e.g. 200. Consequently, every random chromosome becomes a feasible one. As a result, the CPU time reduced greatly; furthermore, GA obtained similar solutions as before. Example 1 and Example 2 were solved by MILP, RS_R1 (RS+i?w/^ 7), GA_R1 {GA+Rule 1) and ARS. Due to the difficulties for MILP and RS, Example 3 is just solved by G A R l and ARS. The results are presented in Table 4. Fig. 9 is a Gantt chart of a schedule for Example 3 by ARS. Table 4 Results of Example 1, 2 and 3 by MILP, RS__R1, GA_R1, ARS Ex.

MILP

GA_R1

RS_R1

ARS

Orders

1

2

CPU time

Makespan

CPU time

Makespan

CPU time Makespan

CPU time

Makespan

8

1.75

15.20

<1

14.10

<1

Rule

14.00

<1

14.00

10

18.90

18.20

<1

18.15

1

<1

17.35

<1

17.35

15

27.53

27.75

<1

1

23.62

<1

22.59

<1

22.05

20

41.99

35.45

1

<1

30.05

<1

28.05

1

27.95

50

3224.57

1

103.95 1

<1

78.65

2.00

74.95

1.00

70.75

6

10

0.51

26.25

<1

26.25

<1

26.25

<1

26.25

6 6

20

34.28

59.24

<1

39.39

1

39.05

1.00

36.92

50

158.62

137.65

107.00

107.14

2.00

101.10

2.00

96.50

1

100

N/A

N/A 1

N/A

N/A

4.00

48.96

5.00

47.40

6

200

N/A

N/A

N/A

N/A

1252

97.33

1134

92.83

6

3

From Table 4, it is seen that from MILP to ARS, the solutions become better and better, and the CPU times turn to be shorter and shorter. For the same example, with the increasing of problem size, the effective rule selected by ARS also changes. For the problems with the same size, if the constraints changed, the effective rule selected by ARS is different. GA and ARS can solve large problems within very short time, achieving near optimal solutions.

1916

Y. He and C.-W. Hui

j5 • j93 JIK 7. Conclusions jl64 j200 jll8 J147jl77 jl94 jl87 ^-^:s^^^r+4^ j72 jlO? jl60 jl52 j29 'jSr For the small-size problems, j69 . j54. jl20 . j92. jl39 .jl89 jl j74 jl24 j.128 . j24 Hh-HI r l> the random search based on + jl62 > trH^ j6 • j l 4 jeS M 6 4 ; jlYb -H HhH^ ^l?T J 5 2 jl83( j35J : 1 9 0 ^ J 8 8 : jl70 . J I M Jl? j67 ,;, j6g heuristic rule can also acquire !^^^ jl82 J 1 2 6 ,^.jl96.,, j75 , j86 .jl50 jlOl . j7i j20 J I B B ! jl22 j90 optimal solution as MILP do. '^'f^^ , jl38 • jlOO j H O j25 • jl92 j99 ill4J41 • jl59 i82 j77 • jl93 j73 For the large-size problems, the Hh-H|i j53 •jl97- jl95 , j 105 J 3 6 J ^ 1 7 1 : J 1 2 9 . r . j r ..J 1^^" random search based on Jl65 'j45 jl85. j23 J167. H&S , ^iii+i •jl57 jlO J-•145 jl72- j l 4 2 ' J 1 2 f j57- J3 il jl41 j96 heuristic rule outperforms HI... I 11^ J32 „ j 4 2 . .jl23; J191 J188 jl21 ;, jl76-j59_jl69,, jl MILP. Actually, the initial step H r . j . ,• 3 8 ^ ^ jl44 j l 5 3 jl04: j27 (R^ j58 jl.73 •j4i •47 • • " • ^> of GA to generate random |jl99.^ j9t _jl30J.135 ^.jim j 3 9 . jl78 .j34 p 156. ^j 163^ i & ^ j33 , „j2 jue- j i ; • , j65 ^ j50 j solutions is a process of 4HHI . I . I • j i n ji( j l 9 ' j97 :J37 .J149J61 . random search. jll3 .jl51. jll2 .j87. jl74 jl32 . jl31 . jlO! .jl2 .jl33 j46 .1154. II 1 ^ I 11 11 11 1' 11' |.|J—H H - • I I GA was able to get acceptable solutions for the large-size 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 problems within moderate time. GA is a process to evolve a set Fig. 9 A schedule of Example 3 by ARS (C„,ax = 92.83) of random solutions. Hence the CPU time of GA is longer than that of the random search. However, compared with MILP, it takes much shorter time for GA to obtain much better solutions to the problems. The effectiveness of rules depends on the scheduling objective and the prevailing shop or plant conditions. A good heuristic rule raises the quality of the solutions and reduces the search time of the algorithms. ARS is able to automatically and dynamically select a suitable rule to synthesize the evolved order sequences into high quality schedules. The novel methods are also suitable for large-size high-constrained problems. Due to the penalty method adopted, the search times of GAto the problems are reduced greatly.

""imr..

Acknowledgement The authors acknowledge financial support from Hong Kong RGC grant (No.614005). References [1] J. Kallrath, Planning and scheduling in the process industry. In: Gunther H. -O., Van Beek P. (Eds.), Advanced Planning and Scheduling Solutions in Process Industry, Springer, 11-42, 2003. [2] J. Cerda, P. Henning, I. E. Grossmann, 1997, A mixed integer linear programming model for short-term scheduling of single-stage multiproduct batch plants with parallel lines. Ind. Eng. Chem. Res. 36, 1695-1707. [3] C. W. Hui, A. Gupta, 2001, A bi-Index continuous time MILP model for short-term scheduling of single-stage multi-product batch plants with parallel line. Ind. Eng. Chem. Res. 40, 5960-5967. [4] C. Chen, C. Liu, X. Feng, H. Shao, 2002, Optimal short-term scheduling of multiproduct single-stage batch plants with parallel lines. Ind. Eng. Chem. Res. 41, 1249-1260. [5] M. Gen, R. Cheng, Genetic algorithms and engineering optimization; New York: Wiley, 1997. [6] D. E. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms. In: Rawlins G (Eds.), Foundations of genetic algorithms, Morgan Kaufinann, 1991. [7] P. W. Poon, J. N. Carter, 1995, Genetic algorithm crossover operators for ordering applications. Computers & Operations Research. 22, 135^7.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Application of multi-stage scheduling Peter M.M. Bongers, B.H. Bakker Unilever Food and Health Research Institute, O. van Noortlaan 120, 3133 AT Vlaardingen, The Netherlands

1, Abstract A simplified food factory model has been derived from a medium size ice cream manufacturing plant. This is done by building a multi-stage scheduling model which describes the infra-structure of the factory, which products are being produced and how the plant is operated. Key has been to translate complexity of the plant (and the operations) into a simplified, but realistic, multi-stage scheduling model. This model has been implemented in commercial available software. The production schedule could not be derived automatically, but needed manual intervention. Scheduling the factory as a whole, the available overall capacity of the factory can be increased significantly. The integrated factory schedule squeezes 10-30% additional capacity out of the factory. 2. Introduction 2.1. Background Most of Unilever's food processes consist of (1) storing a large number of ingredients; (2) a small number of process plants; (3) a large number of intermediate product storing; (4) a smaller number of packing lines. The practical scheduling inside the vast majority of food factories is focussed on scheduling the packing lines on the main production floor only. This schedule is then 'thrown over the wall' to the process department, in which a schedule is being made to satisfy the packing demand. This schedule is also "thrown over the wall" to the incoming materials department, in which a schedule is made to order/receive the materials. 2.2. Problem formulation The way factories are being scheduled at present is posing two problems: • Each department will strive to ensure that their department is not to blame for not packing products, hence less available capacity will be communicated to the plant management. • Any change in the packing schedule might lead to an infeasible schedule in the upstream departments. For example, as a result packing lines may not run due to lack of intermediate products or wrong intermediate products being made in the process plant. The above problems are frequently observed in the factory environments. The challenge is to reduce the impact of these problems in order to increase the capacity of the factory and reduce the cost/tonnes of final products. For a daily use inside the factories, the recent theoretical developments such as [1,2] are too distant from the factory environments. The aim therefore is to use a commercial available package that can deal with the complexity of our food factories and help solve the above challenge.

1917

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P.M.M. Bongers andB.H.

3. Typical factory layout 3.1. Process description As an example, a typical medium scale ice cream factory will be used throughout this paper. All ingredients for the daily use are stored in the local warehouse. From here, the ingredients are transported to the mix department where according to the recipe, all ingredients are mixed, pasteurised (indicated by PAl and PA2) and stored in so-called ageing vessels. After a minimum required ageing time, the mixes are fi-ozen in continuous freezers (indicated by Fxxx). A number of freezers are needed to assemble the final products on the packing lines (indicated by for example Rollol, ILF2). After the product assembly, the products are frozen in a hardening tmmel. During the product assembly, a number of constraints have to be satisfied, like a limited number of fruit feeders to dose inclusions in the final product.

Constraints on Packing • Hardening Tunnel • Slue machines • Fruit feeder • Secondary packing machine

Figure 1 Factory structure The complexity is generated by a combination of (1) high utilisation (>70%) of each stage; (2) the capacity bottleneck can be located in all stages of the manufacturing process; (3) shared resources; (4) intermediate buffers with minimum standing time and maximum shelf life. The above factory infra-structure including the amount of various end products and all the practical constraints are too difficult to solve at this moment in time. Therefore, a partial factory model has been derived, considering one pasteuriser and two packing lines. Apart from the constraints, this simplified model contains all complexity. The model will be used to evaluate the capabilities of scheduling software and solvers. A detailed description of the simplified model is given in the appendix.

Bakker

Application ofMulti-Stage

Scheduling

1919

4. Implementation The simplified model has been implemented in the INFOR advanced scheduling software [3]. Within the INFOR software, stages are being defined as processes coupled with tanks. The behaviour of the stage is determined by the characteristics of both the process as well as the tank. Products can be seen as the material flow from one stage to another stage. A process step describes how a product is made (which stage, process, tank and characteristics). The simplified plant can be modelled as a two-stage production (see figure 2). processing

packing ^

^

1 ^

SKUs

Figure 2 two-stage production model The processing stage consists of the process line as a 'process' coupled to the vessels as 'tanks'. For each mix (A-H) a process step is defined, incorporating the specific resources (i.e. process line and the appropriate vessels), the production rate, maximum standing time and batch size on these resources coupled with a characterisation to calculate the change-over times. The packing stage consists of the packing lines as a 'process' (because this is an end product no tanks are necessary. For each sku (A-H) a process step is defmed, incorporating the specific resources (i.e. packing line), the production rate on this resource coupled with a characterisation to calculate the change-over times. To complement the model, routing constraints are added such that all products on process line 1 can be stored in all vessels, but the two 8000kg vessels are only coupled to packing line 1 and the four 4000kg vessels are only coupled to packing line 2. Next to the description of the plant, the desired amount of products (orders) to be scheduled is provided to the system. 5. Results In the INFOR implementation, batches on the packing lines are scheduled by creating inflow for the orders. Scheduling the batches "just-in-time" and minimising the "makespan" puts the SKU's in the following sequence D^^C-^B-^A and H^'G^'F—>E, as is expected from the change-over matrix. Scheduling the up-stream batches on the vessels and process line leads to an infeasible schedule ( vessels are being over filled, or are being emptied and filled at the same time). Applying the available solvers did not deliver a feasible schedule.

P.M.M. Bongers andB.H.

1920

Manual intervention was needed to create a feasible scheduled. During the 'manual intervention', the following heuristics where used: • The high speed products on line 1 need to be matched by the low speed products on line 2 (and the other way around), whilst obeying that the total makespan on line 1 has to be within one working week. • Creating slack between the batches on the packing lines will create more flexibility on the process line. • Observing the demand in the beginning of the week, by moving the filling of some vessels on the same packing line to 'as-soon-as-possible' and others to 'just-in-time' sufficient time can be created to fit the mix processing of the other line between them. It is shown in figure 3 that the required (challenging) demand can be manufactured in one week of production by combining the automated scheduling of the batches and the manual inverventions. The total lapse time to obtain the solution, including the manual intervention (as shown in figure 3) was appr. 5 minutes.

9m^\ 'M Inbox' f^kMrn^^d^^ '\m^^Mi^y9^*^f^::^ W^MMM

Figure 3 weekly production schedule Visual inspection of the feasible production schedule also shows that the schedule is not optimal with respect to shortest production times and/or lowest change-over times. The process line is utilised 90% of the time (for production and cleaning), compared to a food industry standard of 70%. The increased utilisation can be explained by the fact that the process line scheduling is integrated with the packing line scheduling.

6. Conclusions and future work A simplified food factory model has been derived. This model has been implemented in commercial available software. The production schedule could not be derived automatically, but needed manual intervention.

Bakker

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Scheduling

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By scheduling the factory as a whole, the available overall capacity of the factory can be increased significantly. The integrated factory schedule squeezes 10-30% additional capacity out of the factory. The next step is to validate the model by producing in the factory according to the schedules. References 1 Sand, G., S.Engell (2004), Modelling and solving real-time scheduling problems by stochastic integer programming. Computers and Chemical Engineering 28, 1087-1103. Mendez, C.A., J. Cerda (2004), An MIL?frameworkfor batch reactive scheduling with limited discrete resources, Computers and Chemical Engineering 28, 1059-1068 Agilisys (2003), Advanced Scheduling: Users Course and Modelling course, Rijswijk, The Netherlands 7. Appendix 7.1.1. process description The basic configuration is a two step manufacturing process with intermediate storage. The mix (mix A-H) is produced on the process line (process line 1) and stored in a vessel (vessel 1-6). After a minimum standing time, the mixes (mix A-H) is used in product (SKU A-H) on the packing line (packing line 1-2) 7.1.2. equipment description Process line 1 has a capacity of 4500kg/hr and can feed all vessels. Packing line 1 is fed by 2 vessels of 8000kg capacity, whereas packing line 2 is fed by 4 vessels of 4000kg capacity. All the equipment is available for 120 hours a week (a 48hr weekend). The mixes can be made and kept over the weekend, but the maximum shelf life has to be obeyed Vessels can not be emptied and filled at the same time. The product composition, packing line and packing rate is given below: product composition Packing rate [kg/hr] packing Line SKUA SKUB SKUC SKUD SKUE SKUF SKUG SKUH

mix A mixB mix C mixD mixE mixF mixG mixH

T750 1500 1000 1500 1750 2000 2000 2000

1 1 1 1 2 2 2 2

product

minimum standing time [hr]

maximum shelf-life [hr]

mix A mixB mixC mixD mixE mixF mixG mixH

~\ 3 3 0 2 2 2 2

72 72 72 72 72 72 72 72

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For the packing lines, the following change-over table applies (for example going SKU C to SKU D a 60 min change-over time is needed). to idle SKUA SKUB SKUC SKUD SKUE SKUF SKUG from idle 0 ^20 ^20 "120 ^20 ~120 120 ~m SKUA 120 0 60 60 60 0 0 0 SKUB 120 30 0 60 60 0 0 0 SKUC 120 30 30 0 60 0 0 0 SKUD 120 30 30 30 0 0 0 0 SKUE 0 0 120 0 0 0 60 60 SKUF 120 0 0 0 0 30 0 60 0 0 0 SKUG 120 0 30 30 0 SKUH 120 0 0 0 0 30 30 30

from SKUH

"120 0 0 0 0 60 60 60 0

For the process line, the follovs^ing change-over table applies (for example going from mix C to mix D a 30 min change-over time is needed). mixB mixC mixD mixE mixF mixG MixH Mix A to idle from 0 120 120 120 120 120 120 ~m ~m idle mix A 120 0 30 30 30 30 30 30 30 mixB 120 30 30 30 0 30 30 30 30 mixC 120 30 30 0 30 30 30 30 30 mixD 120 30 30 30 0 30 30 30 30 mixE 120 30 30 0 15 30 30 15 15 mixF 120 30 5 30 30 30 0 15 15 30 30 30 5 mixG 120 30 5 0 15 5 mixH 120 30 30 30 30 5 5 0 The amount product SKUA SKUB SKUC SKUD SKUE SKUF SKUG SKUH

of products to be produced is given below: Quantity fkg] 80000 48000 32000 8000 112000 12000 48000 24000

Bakker

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Slot-Based vs. Global Event-Based vs. Unit-Specific Event-Based Models in Scheduling of Batch Plants Munawar A. Shaik, Stacy L. Janak, Christodoulos A. Floudas Princeton University, Princeton, NJ, 08544, USA Abstract During the last two decades, the problem of short-term scheduling of multiproduct and multipurpose batch plants has gained increasing attention in the academic, research and manufacturing communities, predominantly because of the challenges and the high economic incentives involved. In the last ten years, numerous formulations have been proposed in the literature based on continuous representations of time. These continuous-time formulations have proliferated due to their established advantages over discrete-time representations and are often compared with one another in terms of computational complexity. The various continuous-time models can be broadly classified into three distinct categories: slot based, global-event based, and unit-specificevent based formulations. In this paper, we compare and evaluate the performance of these models, based on our implementations, using a benchmark example problem from the literature. Two different objective functions: maximization of profit and minimization of makespan are considered. Keywords: short-term scheduling, continuous-time model, multiproduct batch plants, multipurpose batch plants, mixed-integer linear programming. 1. Introduction The problem of short-term scheduling for multiproduct and multipurpose batch plants has received significant attention from both academic and industrial researchers in the past years, primarily due to the challenges and high economic tradeoffs involved. Recently, Floudas and Lin (2004, 2005) presented state-of-the-art reviews comparing various discrete and continuous-time based formulations used for short-term scheduling. A variety of continuous-time models have been proposed in the literature and can be broadly classified into three distinct categories: slot based, global-event based, and unitspecific-event based formulations. One of the first methods used to formulate continuous-time models for the scheduling of network-represented or sequential processes is based on the concept of time slots. Time slots represent the time horizon in terms of ordered blocks of unknown, variable lengths, or slots, as presented by Pinto and Grossmann (1995), Karimi and McDonald (1997), and recently by Sundaramoorthy and Karimi (2005). Alternate methods that define continuous variables directly to represent the timings of tasks and avoid the use of time slots can be classified into two different representations of time: global-event based models and unit-specific-event based models. Global-event based models use a set of events that are common across all units, and the event points are defined for either the beginning or end (or both) of each task in each unit. Research contributions following this direction include those presented by Castro et al (2001) and Maravelias and Grossmann (2003). Unit-specific-event based models, originally developed by Floudas and co-workers (lerapetritou and Floudas, 1998a, b; lerapetritou et al, 1999; Lin and Floudas, 2001; Janak et al, 2004), define events on a unit basis.

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allowing tasks corresponding to the same event point but in different units to take place at different times. This representation is considered the most general, compact, and "true" continuous-time model used in short-term scheduling. Another continuous-time model, developed by Giannelos and Georgiadis (2002), is based on a unit-specific-event representation; however, due to special sequencing restrictions on the start and finish times of tasks consuming or producing the same state, it is effectively transformed into a global-event based model. Their formulation is similar to lerapetritou and Floudas (1998a), however, due to the special sequencing constraints, leads to suboptimal solutions. In this work, we compare and evaluate the performance of the slot-based models (Sundaramoorthy and Karimi, 2005) versus global-event based models (Maravelias and Grossmann, 2003; Castro et al, 2001) versus the unit-specific event based models (lerapetritou and Floudas, 1998a; Giannelos and Georgiadis, 2002), and study the computational effectiveness of each. The rest of the paper is organized as follows. In section 2, we describe the different performance metrics and in section 3, the formulations for the different models compared in this study are briefly discussed. The benchmark example is presented in section 4 followed by computational results and discussion in section 5 and conclusions in the last section. A detailed comparative study with several benchmark problems will be provided elsewhere (Shaik et al, 2005). 1.1. Description ofPerformance Metrics As mentioned earlier, there are numerous formulations proposed in the literature, often claiming superiority over each other, for short-term scheduling of batch plants using continuous-time representations. Hence, for a fair and legitimate comparison of the different models with respect to their computational effectiveness, the following precautions are used. First, the example chosen is the standard benchmark example fi-om the recent literature, used by many of the researchers in short-term scheduling of multipurpose batch plants. Both the STN and RTN based process representations are considered with variable batch sizes and processing times. The resource constraints considered are only those related to the raw material and equipment availability and the storage limitations. The various models are evaluated with respect to two dissimilar objective fianctions: maximization of profit and minimization of makespan. In addition, all the problems are solved to zero integrality gap, except in some cases where excessive computational times are required. Also, for each model and for each instance of the example, we increase the number of events or slots until there is no fiirther improvement in the objective function. For a valid comparison, all the models are solved on the same computer and under similar conditions where only default solver options are used. Finally, the various formulations are compared based on our own implementation of the above mentioned models where the number of binary variables, continuous variables, and constraints, the total number of nodes explored and computational time (CPU time) to reach zero integrality gap, the number of nonzero elements in the resulting coefficient matrix and constraints, and the relaxed LP solution are all used to compare the different models.

2. Description of Different Continuous-Time Models The key features and the differences among the various continuous-time models compared in this work are briefly discussed below. 2.1. Unit-specific-event Based Model oflerapetritou and Floudas (1998a) The authors presented the original concept of event points which correspond to a sequence of time instances located along the time axis of each unit, each representing

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the beginning of a task or the utiUzation of the unit. The location of event points is different for each unit, allowing different tasks to start at different times in each unit for the same event point. The timings of tasks are accounted through special sequencing constraints involving big-M constraints. No resources other than materials and equipment are considered. The resulting model requires less event points compared to the corresponding global-event or slot based models, yielding better computational results, although big-M constraints are used. This model was later extended by Janak et al (2004) to allow tasks to extend over multiple events to accurately account for the utilization of different resources and storage poHcies. 2.2. Global-event Based Model of Castro et al. (2001) The authors proposed an improved formulation using the RTN representation for shortterm scheduling of batch plants. The time horizon is divided into several global-events that are common across all units. Binary variables are defined for assigning both start and end times of different tasks to the corresponding global events. Because of the unified treatment of various resources in the RTNfi-amework,no special sequencing constraints are enforced. All the balances are written in terms of a single excess resource constraint, which implicitly includes the balances on the status and batch amounts of each unit. This model has no big-M constraints except for those that relate the extents of each task to the corresponding binary variables. 2.3. Unit-specific-event Based Model ofGiannelos and Georgiadis (2002) The authors proposed an STN represented, unit-specific-event based formulation for short-term scheduling of multipurpose batch plants. This is a slight variation of the model proposed by lerapetritou and Floudas (1998a), wherein the authors relaxed the task durations using buffer times and implicitly eliminated the various big-M constraints. However, the authors introduced special duration and sequencing constraints that effectively transform the nonuniform time grid to a uniform one (global events) for the purposes of material balance and storage constraints. The start times (end times) of the tasks producing/consuming the same state were respectively forced to be the same, leading to suboptimal solutions, as observed by Sundaramoorthy and Karimi (2005) and also as demonstrated later in this paper. 2.4. Global-event based Model ofMaravelias and Grossmann (2003) This is amongst the recent global-event based models using STN process representation. The model accounts for resource constraints other than equipment (utilities), various storage policies (unlimited, finite, zero wait and no intermediate storage), sequence dependent changeover times and allows for batch mixing/splitting. Global event points are used that are common across all units, and tasks are allowed to be processed over multiple events. Different binary variables are used to denote if a task starts, continues, or finishes processing a batch at a given event point. Also a new class of tightening inequalities is proposed for tightening the relaxed LP solutions. The model we implemented is based on the reduction to no resources and it is included in this comparative study since it has been compared to other continuous-time models. A comparison of this model with a unit-specific-event based formulation that includes resources and mixed storage policies can be found in Janak et al. (2004). 2.5. Slot-based Model of Sundaramoorthy and Karimi (2005) Amongst the various slot-based formulations proposed in the literature, this is the recent model available for the short-term scheduling of multipurpose batch plants. The authors claim superior performance amongst all the models, including those based on global events and that of Giannelos and Georgiadis (2002). They use generalized recipe

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diagrams for process representation. No resources other than materials and equipment are considered and transfer and set-up times are lumped into the batch processing times of tasks. The time horizon is divided into multiple time slots of varying lengths, and tasks are allowed to continue processing over multiple time slots. Several balances are proposed based on the status of each unit, material and storage constraints, and a new way of writing the balance on the time remaining on each unit leading to a compact model. 3. Description of Benchmark Example The following benchmark example, which has been studied by many authors, is considered from the short-term scheduling literature for multipurpose batch plants with variable processing times. Two different products are produced through five processing stages: heating, reactions 1, 2, and 3, and separation, as shown in the STN representation of the plant flow sheet in Figure 1. The relevant data for the example can be found in Maravelias and Grossmann (2003) or Sundaramoorthy and Karimi (2005). Product 1

Heating 3=1

FeedA Separatiod

/ Qo l ^ J Reaction ll

oFeedBn '=2,31 FeedC

Figure 1. State-task network representation for the example. 4. Computational Results and Discussion The five short-term scheduling models are implemented and solved for the above example with respect to two diverse objective functions: maximization of profit and minimization of makespan. All the resulting MILP models are solved in GAMS distribution 21.1 using CPLEX 8.1.0 on the same computer (3 GHz, Pentium 4 with 2 GB RAM). In all the subsequent results reported for the model of Sundaramoorthy and Karimi (2005), we show 'n' event points to represent 'n-T slots for a valid comparison with the other global-event and unit-specific-event based models presented in this work. 4.1. Maximization ofProfit The example problem is first solved for the maximization of profit using a time horizon of H=16 hours. The model statistics and computational results are shown in Table 1. In the paper of S&K, they report the results for this problem for the first feasible solution of 7 events only. In this work, as already mentioned, we solve the problem to its global optimal solution and for zero integrality gap. For this case, the slot/global-event based models require at least 11 event points compared to the model of I&F which requires only 7 events. The model of I&F yields the global optimal solution ($2658.5) with 7

Slot-Based vs. Global Event-Based vs. Unit-Specific Event-Based Models

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events. The model of M&G and Castro are not solved down to zero integrality gap as they take excessive computational time compared to other models. The integrality gap for these two models is 6.72% and 23.87% respectively for 11 events. The model of G&G gives a suboptimal solution ($2564.6) for 6 to 11 events. Thus, the model of S&K performs better than the other two global-event based models of M&G and Castro with respect to the computational time and the number of binary variables. The model of I&F takes only 7 seconds compared to the model of S&K which takes 31776.6 seconds for solving to the global optimal solution, even though the latter has no big-M constraints. Table!. Model statistics and computational results for the maximization of profit. Model

Events CPU (s) Nodes

RMILP MIL?

($) S&K

($)

Binary Var.

Cont. Var.

Constr. Nonzeros.

10

4119.10

1764539

3315.8

2646.8

108

622

596

2188

11

31776.6

13085318

3343.4

2658.5

120

631

663

2371

10

32081.5

7154176

3315.8

2646.8

144

775

1786

6782

11

>67000

9568000

3343.4

2658.5

160

858

1978

7815

Castro

10

45038.8 20972898

4770.8

2646.8

347

489

959

3620

11

>56000

13900000

5228.7

2646.8

426

582

1147

4437

G&G

6-11

0.31

666

3190.5

2564.6

48

208

348

1238

I&F

7

7.00

16958

3788.5

2658.5

42

165

318

1046

M&G

4.2. Minimization of the Makespan In the literature, finding optimal solutions for this objective is generally found to be the hardest even for simple examples. Although S&K claim that their model performs the best for this objective function, it can be seen that their model sometimes does not find the global optimal solutions (even at higher event points) that are obtained by the unitspecific-event based model of I&F. The data are the same except that we consider fixed demands and the time horizon (H) is not fixed. For all the models that have big-M constraints (M&G and I&F) we need to specify the horizon time as an upper bound on the makespan. For fair comparison, we consider the same value of H=50 hours used in S&K. Using a demand for states SB and S9 of 200 units, the model and solution statistics are shown in Table 2. All of the models are able to find the optimal solution of 19.34 hr, except for the model of G&G, which did not show any improvement in the objective from 8 to 10 event points. The model of M&G performs weakly in terms of the computational time and number of continuous variables, constraints, and nonzeros. Table 2. Model statistics and computational results for the minimization of the makespan . RMILP (hr)

Model

Events

S&K

10

599.49

175760

MILP Binary Cont. Constr. Nonzeros. Var. Var. (hr) 18.685 19.340 108 622 597 2188

M&G

10

3881.46

432461

18.685

19.340

144

776

1790

6850

Castro

10

715.20

469515

9.889

19.340

347

490

966

3693

G&G

10

1775.22

1408762

10.475

19.789

80

340

589

2093

I&F

9

324.55

444182

10.785

19.340

58

215

421

1427

CPU (s)

Nodes

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M.A. Shaik et al.

5. Conclusion In this paper, we compare and assess the performance of various continuous-time models proposed in the Uterature for short-term scheduhng of multipurpose batch plants. These models are broadly classified into three distinct categories: slot based, global-event based, and unit-specific-event based formulations. It is observed that both the slot based and global-event based models always require the same number of event points. The unit-specific-event based models require less event points for solving to the same global optimal solution obtained by the slot based or global-event based models. Thus, the unit-specific-event based model of lerapetritou and Floudas (1998a) performs the best both in terms of faster computational performance and having smaller problem sizes due to the requirement of relatively less event points.

References P. Castro, A.P.F.D. Barbosa-Povoa and H. Matos, 2001, An Improved RTN Continuous-Time Formulation for the Short-term Scheduling of Multipurpose Batch Plants, Ind. Eng. Chem. Res., 40, 2059-2068. C.A. Floudas and X. Lin, 2004, Continuous-Time versus Discrete-Time Approaches for Scheduling of Chemical Processes: A Review, Comp. Chem. Eng., 28, 2109-2129. C.A. Floudas and X. Lin, 2005, Mixed Integer Linear Programming in Process Scheduling: Modeling, Algorithms, and Applications, Ann. Oper. Res., 139, 131-162. N.F. Giannelos and M.C. Georgiadis, 2002, A Simple New Continuous-Time Formulation for Short-Term Scheduling of Multipurpose Batch Processes, Ind. Eng. Chem. Res., 41, 21782184. M.G. lerapetritou and C.A. Floudas, 1998a, Effective Continuous-Time Formulation for ShortTerm Scheduling: 1. Multipurpose Batch Processes, Ind. Eng. Chem. Res., 37, 4341-4359. M.G. lerapetritou and C.A. Floudas, 1998b, Effective Continuous-Time Formulation for ShortTerm Scheduling: 2. Continuous and Semi-continuous Processes, Ind. Eng. Chem. Res., 37, 4360-4374. M.G. Ierapetritou,T.S. Hene and C.A. Floudas, 1999, Effective Continuous-Time Formulation for Short-Term Scheduling: 3. Multiple Intermediate Due Dates, Ind. Eng. Chem. Res., 38, 34463461. S.L. Janak, X. Lin, and C.A. Floudas, 2004, Enhanced Continuous-Time Unit-Specific EventBased Formulation for Short-Term Scheduling of Multipurpose Batch Processes: Resource Constraints and Mixed Storage Policies, Ind. Eng. Chem. Res., 42, 2516-2533. LA. Karimi and CM. McDonald, 1997, Planning and Scheduling of Parallel Semicontinuous Processes. 2. Short-Term Scheduling, Ind. Eng. Chem. Res., 36, 2701-2714. X. Lin and C.A. Floudas, 2001, Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-Time Formulation, Comp. Chem. Eng., 25, 665-674. C.T. Maravelias and I.E. Grossmann, 2003, New General Continuous-Time State-Task Network Formulation for Short-Term Scheduling of Multipurpose Batch Plants, Ind. Eng. Chem. Res., 42, 3056-3074. J.M. Pinto and I.E Grossmann, 1995, A Continuous Time Mixed Integer Linear Programming Model for Short Term Scheduling of Multistage Batch Plants, Ind. Eng. Chem. Res., 34, 30373051. M.A.Shaik, S.L. Janak and C.A. Floudas, 2005, Comparative Study of Different ContinuousTime Models for Short-term Scheduling of Multipurpose Batch Plants, in preparation. A. Sundaramoorthy and LA. Karimi, 2005, A Simpler better Slot-Based Continuous-Time Formulation for Short-Term Scheduling in Multipurpose Batch Plants, Chem. Eng. Sci., 60 2679-2702.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A Unified Approach for Knowledge Modeling in Pharmaceutical Product Development Chunhua Zhao^, Ankur Jain^, Leaelaf Hailemariam^, Girish Joglekar^, Venkat Venkatasubramanian^, Kenneth Morris^ and Gintaras Reklaitis^ '^ Laboratory for Intelligent Process System, School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA ^Department ofIndustrial and Physical Pharmacy, Purdue University, West Lafayette, IN 47907, USA Abstract Development of a drug product is a complex, iterative process consisting of selection of a dosage form, excipients, processing route, operating equipment and so on. At each stage, knowledge in various forms, including heuristics, decision trees and mathematical models, is used in making decisions. Typically knowledge is modeled specifically for the tool that uses it, such as expert systems and mathematical modeling software. This makes it very difficult to share the knowledge across different tools and among development teams, and integrate various forms of knowledge to assist in making pharmaceutical product development decisions. To provide easier access to available knowledge and better decision support, we propose an open and unified approach to systematically model the different forms of knowledge. Keywords: Knowledge Modeling, Ontology, Pharmaceutical Product Development, Decision Support System 1. Introduction To increase the speed to market and robustness, pharmaceutical product development calls for a quality-by-design approach which requires deep understanding of materials, equipment, processes and their interactions. The knowledge required for implementing this approach can be classified into two categories: implicit knowledge which is in the mind of a domain expert and hence usable only by the expert; and explicit knowledge, such as decision trees and procedures, mathematical models of operations, guidelines from FDA or ICH, and so on. Explicit knowledge can be shared and used by all. As knowledge and information grow, however, it becomes difficult to manage and use them effectively. To get maximum benefit, we believe that information/knowledge should be modeled in a form which can be directly interpreted by computers. Tools can then be developed to utilize the computer interpretable information/knowledge to support decision making. While information modeling approach is discussed in (Zhao et al., 2006), in this paper, we concentrate on knowledge modeling. The conventional ways to model knowledge are programming based or rule based which have severe limitations. In programming based methods the logic is hard coded using a suitable programming language and therefore it is not accessible to a user. To make any changes a user needs access to the source code, which sometimes may not be available or the user may not have the understanding of the particular programming language used. In rule based expert systems, unorganized collection of rules are used to model the pieces of knowledge. Often, it is difficuh to determine the purpose of individual rules, and to

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envision potential interactions between rules. These issues greatly limit the scalability and maintainability of rule based systems. To address the deficiencies of the current techniques, we propose an ontology based approach to model the knowledge. Ontology (Gruber, 1993) is an explicit description of domain concepts; properties and attributes of concepts; constraints on properties and attributes; and individuals. In this approach, the concepts to represent knowledge are modeled in the ontology. A particular set of knowledge is modeled as an instance of the ontology (Noy and McGuinness, 2001). Furthermore, since ontology could also be used for information modeling (Zhao, et al., 2005), the ontology based approach provides a natural link between the knowledge and the information it utilizes, as well as the knowledge and other tools. In this paper, we illustrate the use of two forms of knowledge, guidelines and mathematical models. Guidelines model the procedural knowledge used in a decision making process, and mathematical knowledge which ranges from simple equation sets like mixing rules to complex discrete element models provide a way to characterize the behavior of the underlying phenomena. Section 2 discusses the details of modeling the guidelines. Section 3 discusses modeling of mathematical knowledge. Ontology has been developed to model the knowledge and support the integration of the knowledge with information and other tools.

2. Modeling a Guideline A guideline models procedural knowledge, which mainly consists of decision logic, information look up and evaluation of decision variables. For example, to determine whether direct compaction is appropriate for a particular drug substance, values of several properties, such as flowability and compressibility are examined. To determine what types of excipients are needed in a formulation, a specific sequence of decisions must be made. Systematically modeling this type of procedural knowledge makes it possible to provide a standardized approach for product development, and could be easily reused to ensure the quality of product development. 2.1. Structure of a Guideline The Guidelines presented in this study were derived from GLIF (GuideLine Interchange Format), a specification developed mainly for structured representation of clinical guidelines (Peleg et al., 2004). GLIF was developed to facilitate sharing of clinical knowledge and was designed to support computer-based guideline execution. GLIF guidelines are computer interpretable as well as human readable and are independent of computing platforms which enables sharing of guideline. The guidelines discussed in this paper were developed using Web Ontology Language (OWL) (W3C, 2004). Protege (see URL) was used as ontology editor. Based on the GLIF specification, each guideline is represented as an instance of the guideline concept. The main objective of a guideline is specified as free text using the intention attribute. The steps involved in decision making are modeled as an algorithm of that guideline. An algorithm is represented as a flowchart of nodes connected by directed links. Each node represents a guideline step, and a directed link denotes the order of execution of the steps. A step can be one of the following five step types. An action step represents a domain specific or computational action; a decision step represents a decision point; a state step is used to specify the state of product development in the specific context of a guideline's application; a branch step is used to initiate multiple actions in parallel; and a synchronization step is used to coordinate concurrent steps or steps with arbitrary execution order. The decision making process can be nested using

A Unified Approach for Knowledge

1931

Modeling

subguidelines, and thus multiple views to the process with different granularities can be defined. Figure la shows the different components of the Initial_Selection_of_Processing_Route_Based_on_Mechanical_Properties gmdtXmQ.

»3rtJMti3l„S^ectumjarj^oi»ssiiiaitoiite,B3S8d_(m^^^ Action_Step

Branch_Step

Ba^ic Ste^s

State„Step

Synchronization_Step

DirectCompressionJfttetGraniriation.RolierCompaction

WetGramila«ion_RoflerCon]«)action_NoDirectCompression

next_step

next_step

En(tiJiritiaLSdleG«i<m_of_Pr<M:es^ite„ito«^8j3a^d_enJW^hatric^_Prai)^

Figure la. Details of a specific guideline

fm losJmdwl #

A R - 1 : Hauisner Ratio..

8«l»n^|8j©.,.*iateral

^

[^

^=

I^API-I

Foi DC the flowatoility of API nsi^cls to be qoocJ Poor flow may cause two pi oblems 1 Content I Inifoi mity and 2 Handling Addition of flow aids may improve the flowability however if the flow is extremely poor and the dose is high granulation may be needed.

^ #

j ^ r

iCriterion j 1 ^ Get_Data_Hausner_Ratio

iE

» #

y * |

^ |

iA »

#

Directly Linked to Property

Figure lb. Details of a decision step in a guideline. A guideline always starts and ends with the state step. The state step points to the development state instance which contains the information about the current state of pharmaceutical product development. To use the guideline, first the state must be defined by creating an instance of the development state. To populate the development state instance, the API for which the pharmaceutical product development is desired is selected. Other details which are known at this stage are also specified, for example dose (250.0 mg). As the guidelines associated with the drug development are executed, the development state is updated. Every decision point is specified using a decision step (Figure lb). The main attribute of a decision step is expression. An Expression defines the underlying logical expression consisting of a link to the variable used in the expression, logical condition and threshold. The value of the decision step can be either true or false. For example, the decision step Is_Flowability_Very_Bad is based on the expression Hausner_Ratio > 1.75, where Hausner_Ratio is the property of the material under consideration. Once the material is known, the property value can be retrieved from the material ontology.

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Decision step also specifies different options one of which will be selected based on decision criterion. 2.2. Guideline Development Several guidelines have been developed for the preformulation stage in pharmaceutical product development (Figure 2). These guidelines were based on the knowledge gathered from detailed discussions with domain experts in pharmaceutical product development at Purdue University. Drug_Product_Development SelectionofDosageForm DevelopmentforlRSolidOralDosageForm Selection_of_Processing_Route Excipient_Selection_Jbr_Roller_Compaction •M EvaluateCompressionAidforRC Evaluate Flowaid for RC M

EvaluateDisintegrantforRC

Excipient_Selection^or_Wet_Granulation

-<—>\

Excipient_Selection_for_Direct_Compression

Development_for_Liquid_Dosage_Fomi

Figure 2. Collection of guidelines developed for pharmaceutical product development. Each box represents a guideline. If a guideline invokes a subguideline, a link is shown to represent the hierarchy. Thus, the guidelines can be developed as modules and used in hierarchical fashion. Also, new guidelines can be added as more domain knowledge is accumulated. 2.3. Execution of Guideline A guideline is implemented by an execution engine. The engine uses the knowledge in the guideline and the information stored in an ontology-based information repository to provide decision support. The execution of engine can be interrupted at any step and the current developed state can be saved if necessary. The guidelines can be executed in two modes: continuous and batch. In continuous mode all the steps of the guideline (including the subguidelines) will be executed without interruption whereas in batch mode one step of the guideline is executed at a time. When executing in batch mode, the user decides whether or not to execute the next step in the guideline.

3. Mathematical Knowledge A considerable amount of knowledge used in pharmaceutical product development is in the form of mathematical equations, collectively referred to as mathematical knowledge. Typically, mathematical knowledge is either embedded in software tools or specified in a specific syntax understandable by the general purpose tools like Matlab or Mathematica, thus reducing its access and adaptability. In this work, we have developed an ontology to represent mathematical knowledge (Figure 3). The main attributes of a mathematical model are the following: independent variables, dependent variables, parameters and the model equations. Each model equation is defmed using Mathematical Markup Language (MathML), an XML based standard for describing

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mathematical equations. Equation editor in the system can automatically generate MathML for the equations entered in a natural way. The variables are denoted by the symbols which are used in the equations, and are linked to a Universal Resource Identifier (URI). With this, tools can access the description and establish the link between the variables and the information resources at runtime. Each model in turn is an instance of the model class of the ontology. For example, the fluidized bed drying model (Kunni and Levenspiel, 1977) in the ontology, solves the moisture content in the output of the dryer as a fimction. The two equations, one for the heat transfer controlled regime and one for mass transfer controlled regime are shown in Figure 3. A model ontology allows representing mathematical equations naturally in a form that is independent of the solver. The solution of the model equation is governed by the context in which that model is used. For example, suppose the drying model is used in a stand alone mode to predict the moisture profile in a specific dryer under given conditions, and Mathematica is used for solving the model equations. An engine must be written to translate all model equations and to create Mathematica statements to retrieve values for parameters describing the given dryer and operating conditions from information repository, into a syntax required by Mathematica. With this information, the engine invokes the Mathematica kernel which solves the equations and returns the values to the engine. The engine then displays the computed values in the desire form. A similar engine must be developed if a different solver is used. In this work, an engine has been written to interact with Mathematica. The engine is used as a medium layer between the Mathematica kernel and the web pages from which information on a model instance is presented as well as output generated by Mathematica in tabular or graphical form is displayed. Applications (Web or Rich Client)

Ontology Ontology for Other Information Model Instance Model Ontology Equations in MathML Variables linked throuah URI Natural Representation of Model Equations Solver Independent

J

Engine

Automatically generated Mathematica Statements Solver

H:

Mathematica Kernel Solver Specific Translation of Model Equations

Figure 3. Model mathematical know^ledge

4. A Case Study Reformulation of a MDRTB (multi-drug resistance tuberculosis) drug was used as a case study to demonstrate the applicability and benefits of the proposed approach. The guidelines (Figure 2) were used for recommending processing route and route dependent excipients to manufacture the drug product as immediate release oral solid dosage form. The choice of processing routes for manufacturing the dosage form was limited to direct compression, roller compaction and wet granulation. The properties of the API were experimentally measured and stored in an information repository in the form of material ontology. Some of the important propreties are Hausner ratio, angle of

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repose, compressiblity, density, stability and so on. Also, the excipient properties, available in the literature, were stored in the repository. After the guidelines are used to determine the feasible processing route(s) and excipients, other knowledge (for example other guidelines, or mathematical knowledge) can be used for detailed process design. The Selection_of__Processing_Route guideline selected the roller compaction as the feasible processing route. It eliminated direction compression because of poor flowability of the API. Since, API is the dominant component of the final formulation, the guideline concluded that excipients would not be able to influence the flowability to the extent that it will be within the allowable range. It eliminated wet granulation because of poor chemical stability of API. To select the excipients for selected route, Excipient_Selection_for_Roller_Compaction guidelines identified that flow aid, filler and lubricant were needed, based on the properties of the API. Specific guidelines (EvaluatejCompression_Aid_for_RC,Evaluate_Flowaid_Jbr_RC,Evaluate_DisintegrantJ'or_RC) were used for selecting a set of excipients for each role. To select excipients these guidelines used mixing rules to predict the mixture properties. The mixing rule that computes the mixture properties as the weighted average of pure component properties was used. Using the approach discussed in section 3, mixing rules and other mathematical knowledge were modeled and accessed directly by the guideline execution engine. 5. Summary We argue that the various types of knowledge should be modeled and used in decision support tools to better support pharmaceutical product development. An ontology-based approach has been proposed to model knowledge in forms of guidelines and mathematical models. This approach provides an open and easy way to modify the knowledge and supports the integration of the knowledge with information resources and other tools. An execution engine, independent of the domain ontologies, was developed to execute the guidelines and provide the necessary decision support. Ontology based approach also provides semantics to mathematical equations as well as the variables involved in the equations. The fully described mathematical knowledge can be shared and used directly by software tools. This approach could also be used to effectively capture the knowledge in different domains including process development and operations in chemical and pharmaceutical industry. References Gruber, T. R. (1993), A Translation Approach to Portable Ontology Specification, Knowledge Acquisition, 5, 2, 199-220. Kunii, D., and Levenspiel, O. (1977), Fluidization Engineering, Robert E. Krieger Publishing Co, 424-428. Noy, N. F., and McGuinness, D. L. (2001). Ontology Development 101: A Guide to Creating Your First Ontology, Stanford Knowledge Systems Laboratory Technical Report KSL-01-05. http://www.ksl.stanford.edu/people/dlm/papers/ontology-tutorial-noy-mcguinness.pdf/ OWL (2004), Web Ontology Language Overview, W3C Recommendation, http://www.w3 .org/TR/owl-features/ Peleg, M., Boxwala, A., Tu, S., Wang, D., Ogunyemi, O., Zeng, Q. (2004), Guideline Interchange Format 3.5 Technical Specification, http://www.glif org/ Protege (Version 3.1), http://protege.stanford.edu/ Zhao, C, Joglekar, G., Jain, A., Venkatasubramanian, V., Reklaitis, G. V. (2005), Pharmaceutical Informatics: A Novel Paradigm for Pharmaceutical Product Development and Manufacture, Proceedings of ESCAPE15.

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A Multistage Stochastic Programming Approach with Strategies for Uncertainty Reduction in the Synthesis of Process Networks with Uncertain Yields Bora Tarhan^, Ignacio E. Grossmann^ "^Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh PA, 15213, USA

In this paper we consider multistage stocliastic programs with endogenous parameters where investment strategies are considered to reduce uncertainty, and time-varying distributions are used to describe uncertainty. We present the proposed ideas in the context of the planning of process networks with uncertain yields. We propose a new mixed-integer/disjunctive programming model which is reformulated as a mixed-integer linear program. The model can be solved through an LP-based branch and bound for smaller instances or with a duality-based branch and bound for larger problems. 1. INTRODUCTION In many real world problems, the problem data cannot be known accurately due to insufficient information about the future (e.g. demand) or to the uncertainty in the technical parameters (e.g. yield). Stochastic programming is an approach to model these problems by taking these uncertainties into account (Birge & Louveaux, 1997). The basic idea in stochastic programming consists of making decisions today while anticipating corrective (recourse) actions tomorrow when the uncertainty in parameters reveals or becomes certain. According to Jonsbraten (1998), uncertainty in planning problems can be divided into two classes: exogenous uncertainty (e.g. demands) and endogenous uncertainty (e.g. yields). Problems where stochastic processes are independent of project decisions are said to have exogenous uncertainty, whereas problems where stochastic processes are affected by project decisions are said to possess endogenous uncertainty. In this paper we focus on the class of problems which possess endogenous uncertainty. For the literature about the class of problems that deal with endogenous uncertainty, see Goel & Grossmann (2004) Another aspect that is addressed in this paper is gradual reduction of uncertainty over time (e.g. see Jonsbraten, 1998).

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In this paper, we focus on problems where the time horizon is discretized into time intervals and the probability distributions of the uncertain parameters are discrete. These specifications allow us to represent the stochastic process by scenario trees. Figure 1 shows the standard scenario tree representation and an equivalent alternative tree representation (Ruszczynski, 1997) in which the original nodes are disaggregated so that each scenario is represented by a unique path which facilitates the modeling. ^^

t=l t=2

t=3 Figure 1. Standard and alternative scenario tree with uncertain parameters ^land ^2-

2. PR The problem that we consider is the multi-period synthesis of process networks under gradual uncertainty reduction in the process yields and with possible investments in pilot plants for reducing uncertainties. Given a process network with availabilities of raw materials, intermediates and demands for final products over T time periods, the problem is to determine in each time period t whether the capacity of specific processes should be expanded or not (including new or existing processes), whether specific processes should be operated or not, and whether pilot plants for new specific processes should be installed or not. In addition, other decisions include selecting the actual expansion capacities of the processes, the flowrates in the network, and the amount of purchase and sales of final product. The objective is to select these decisions to maximize the expected net present value. The time-varying uncertainty reduction is contingent on the operation and pilot plant installation decisions. Uncertainty is reduced in two different ways. One way is installing the plant directly and observing how the yields reveal over the different time periods (Figure 2a). We assume for simplicity that this occurs only over two time periods. Another way is investing in a pilot plant in order to gain more accurate knowledge of the uncertainty (Figure 2b). One can then invest in a plant, starting at a reduced level of uncertainty. In the second option, we shield the process fi'om a larger uncertainty but delay large scale production for one period. Figure 3 shows the relation between steps and

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periods, concepts that are used interchangeably in the paper. Step 1, beginning of time period t, is the one where no investment is made for reducing uncertainty. Step 2, end of time period t and beginning of time period t+1, is the step where uncertainty is revealed partially. Step 3, end of time period t+1, is the last step where uncertainty is folly revealed. i^ a) Install actual Uncertainty y plant without (Variance) ^^ pilot plant in yields i OR \ j j b) Install actual \ 1 j / plant after \ 1 y i investing in pilot 01

11

21

Step 1

Step 2

Step 3

i

-^ Period t

Y Period t+1

1 1^

Figure 2. Options for uncertainty reduction.

Figure 3. Relation between steps and periods.

0

Pilot Plant

Process I

Process III

Figure 4. Schematic representation of process networks problem.

Figure 4 shows an example of a process network that can be used to produce a given product. Currently, the production of A takes place only in Process III which consumes an intermediate product B that is purchased. If needed, the final product A can also be purchased so as to ensure demands are always satisfied. The demand for the final product, which is assumed to be known, must be satisfied for all periods over the given time horizon. Two new technologies (Process I and Process II) are considered for producing the

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intermediate from two different raw materials C and D. These new technologies have uncertainty in the yields which is reduced over time. The scenario tree representations for gradual uncertainty resolution for the two processes are given in Figure 5. It is assumed that at step 2 the only realizable yields are the highest and the lowest of all possible values. At step 3, when the uncertainty is totally revealed, all possible yields are realized. In the figure there are two possible realizations for yields in step 2 and four possible yields in step 3. For modeling the problem, the scenario trees will be considered according to the representation given by Ruszczynski (1997), as discussed in the background section. Stepl t+1

Step 2

Step 3 O 0.69

O 0.73

O

O 0.77

O O O O ^ " ^ ^ 0.81

0.6

0.7

0.8

0.9

Figure 5. Scenario tree representation for process 1 and 2.

3. MODEL The variables in the model are classified as decision, state and recourse variables. Decision variables (>^ f ) are related to decisions that are taken at the beginning of each time period t and scenario s (e.g. capacity expansion, pilot plant, operation and input flowrate). State variables (W^') are the variables that are automatically calculated when decision variables are set (e.g. output flow rate). Recourse variables {xf) are decisions taken at the end of each period to satisfy feasibility (e.g. inventory levels). Decision variables (jF'f) are implemented at the beginning of time period t. This is followed by the resolution of exogenous and gradual resolution of endogenous uncertainty. At the end of the period recourse action {xf) is taken to satisfy feasibility. In the general model the objective function maximizes the expected net present value.

Max =Z/'^{Z^//(4v'^''^^"'^)}

(1)

The mass balance equations for each time period t and scenario s are given by,

g(j;«;;,w«-%x,"'%^,)<0

^seSyteT

(2)

The inequalities in (3) specify the assumptions that are problem specific. h{y^(^^;)<0 \/seSyteT (3) Finally, the expression (4) represents the non-anticipativity constraints for scenario pairs at each time period in order to define the structure of the scenario

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tree. The idea of non-anticipativity is that decisions at time t can only be affected by the decisions made before time period t. These conditional constraints that are modeled in the form of disjuctions, state that if two scenarios are indistinguishable in time period t (i.e. they are the same), then decisions for s and 5 in t should be identical.

\/s,/eSyteT,


4. SOLUTION STRATEGY The multi-stage stochastic optimization problem (l)-(4) with time varying uncertainty corresponds to a generalized disjunctive program which can be reformulated as a mixed integer linear program (MILP). Since the size of the MILP may become large, especially because of constraints (4), this will prevent finding the optimal solution in reasonable time. Therefore, we propose a solution strategy (Figure 6) SetUB< that relies on the successive calculation of lower and upper bounds of the expected net ^ Find a LB (feasible solution) by using present value. The basic idea W a heuristic for finding upper bounds is to Add reformulate the multi-stage Logic Find an UB by using Lagrangean model having time varying Cuts relaxation of the model gradual uncertainty resolution as smaller problems having Yes ^ . ^ . - - - - ' ^ ^ - ^ ^ ^ immediate resolution. <— < r (UB-LB>8 ^ ^ Combining the decisions from these smaller problems by considering the decisions given in the previous time periods, gives a feasible solution which is a lower bound for the model. Upper bounds are generated using Lagrangean relaxing the disjunctions and adding the first period non-anticipativity constraints to the objective function with Lagrange multipliers. The resulting model can be rewritten as independent subproblems for each scenario. The overall objective is to find the minimum upper bound by updating the multiplier by subgradient method (Fisher, 1985).

t

5. EXAMPLE Capacity expansion and operation decisions for the problem presented in Figure 4 are optimized over a time horizon of 10 years. Process III is already

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operational with an existing capacity of 3 tons/day and a known yield of 70%. All possible realizations of the yield for Capacity(tons/day) process I at step 3 are 69, 73, 77 and 81% where only 69 and 81% are realizable in step 2 of the uncertainty resolution. Similarly for process II, 60 and 90% are two realizations in step 2 with 60, 70, 80 and 90% as possible realizations at step 3 (Figure 5). For simplicity, demand, the only exogenous uncertainty in the problem, is assumed to be known. Therefore, we only focus on the endogenous and gradual uncertainty resolution. The example model has 7360 binary, 8841 continuous variables and Figure?. Capacity of Process I (best solution) 85139 constraints. The problem was solved within 2% tolerance of the upper and lower bounds with the proposed method. The solution proposes expanding Process I up to a capacity of 10 tons/day and making an additional expansion of 4.93 tons/day at time period 3 if the yield turns out to be 69%). If the yield for Process I is found to be 81% then an expansion of 2.98 tons/day is made at the time period 4. This solution does not involve the use of a pilot plant and yields an expected net present value of $8,050,500. Acknowledgement The authors would like to acknowledge financial support from ExxonMobil. References 1. Birge, J.R., Louveaux, F. (1997). Introduction to stochastic programming. New York, NY: Springer. 2. Fisher, M. L., 1985. An AppHcations Oriented Guide to Lagrangian Relaxation. Interfaces 15, 10-21. 3. Goel, v., Grossmann, I. E., 2004. A stochastic programming approach to planning of offshore gas field developments under uncertainty in reserves. Computers and Chemical Engineering 28 (8), 1409-1429. 4. Goel, v., Grossmann, I. E., 2005. A class of stochastic program with decision dependent uncertainty. (Accepted for publication) 5. Grossmann, I.E., 2002. Review of non-linear mixed integer and disjunctive programming techniques. Optimization and Engineering 3, 227-252. 6. Jonsbraten, T.W., 1998. Optimization models for petroleum field exploitation. PhD thesis, Norwegian School of Economics and Business Administration. 7. Ruszczynski, A., 1997. Decomposition methods in stochastic programming. Math. Programming (Ser. B) 79, 333-353.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A framework for capturing the impact of resource allocation policies in the selection of a new product portfolio Juan Camilo Zapata^, Vishal A. Varma'', and G. V. Reklaitis^ ^ School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA ^Air Products and Chemicals, Allentown, PA 18195, USA

Abstract The high development costs, low probability of success and intensive competition faced by pharmaceutical companies make management of their product pipelines a high risk undertaking. The strategic decision involving the selection of the particular set of drugs to be developed has implications that affect the behavior of the pipeline for years. While recently reported research has captured the stochastic character of the pipeline, to date no methodology has explicitly included the impact of operational policies in the selection process. In this work, a multi-level Sim-Opt strategy is used to assess the effect of resource allocation on risk and rewards. Product sequences generated by a GA are statistically evaluated using a probabilistic network model. The model includes all the tasks that have to be accomplished in order to release a new drug into the market. The resources assigned to each drug in each task are rebalanced by an optimal policy every time a project fails and at the end of each year. Based on the results a reward-risk frontier is constructed and compared to the one generated when no reactive allocation is considered. Results show that the inclusion of this additional degree of freedom in the decision process causes a significant change in the portfolio mix. Keywords: Strategic and tactical decisions; Sim-Opt; efficient frontier. 1. Introduction Portfolio management is a dynamic decision process, that provides the framework in which senior management operationalizes its business strategy. The future directionality of such a strategy is translated into the R&D portfolio. However, the presence of uncertainty, multiple objectives and decision makers, project interdependencies and a constantly changing environment makes that translation a very difficult process. A wide variety of methodologies has been proposed to facilitate this process (Cooper et al., 1999). In spite of the differences, all of them share one characteristic: the implicit use of a decomposition philosophy. The problem is broken down into different independent hierarchical levels, where each level uses a set of data and models whose degree of aggregation depends of the scope of the corresponding level. Intuition suggests that such an approach is valid when the project execution process tends to be deterministic. . However in the highly dynamic and constrained environment of R&D (projects fail, uncertainties are reshaped by internal changes and the surrounding environment, resources are not easy to replace, etc.) the answer is not so clear. Under these conditions, optimal strategic decisions may require the integrated consideration of key aspects of the tactical and strategic levels. Sensitivity analysis is sometimes implemented to assess the impact of using aggregated data. However, the results obtained by this strategy present two major pitfalls. First, the interdependencies between the projects are not captured. And second, completely different portfolio realization paths at the tactical level, with their corresponding rewards and risk levels, can be

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obtained for the same set of aggregated values. This study explores the implications of the choices at the tactical level on the selection and prioritization of new products in the R&D portfolio of a pharmaceutical company. 2. Tactical vs. Strategic Decisions The development of decision support strategies and systems for managing an R&D portfolio date back to the 60s. Since then it has become evident that R&D portfolio management is about minimizing risk while maximizing an objective or a set of objectives in the presence of constraints (Baker, 1975). In order to accomplish this goal many decisions have to be made at different levels in the organization. Depending on their scope they can be classified in two groups: strategic or tactical. Both of them imply the allocation of resources; the only difference is that the first group determines the objective, while the second one leads to it. At the strategic level some techniques were developed to support the selection of projects and their priority in the portfolio (Blau et al., 2004; Lin and Hsieh, 2004; Raynor and Leroux, 2004; Rogers et al, 2002). Others concentrated on the selection of one project from a group of candidates (Calantone et al., 1999; Loch and Bode-Greuel, 2001). At the tactical level, the focus has been on scheduling and allocating resources to activities within the projects (Maravelias and Grossmann, 2004; Subramanian et al., 2003). Regardless of the scope, all these methodologies are based on one of two paradigms, quantitative or qualitative. Real options, decision trees, discrete event simulation, mathematical programming, etc are at the core of quantitative decision support systems (Loch and Bode-Greuel, 2001; Maravelias and Grossmann, 2004; Raynor and Leroux, 2004; Subramanian et al., 2003). On the qualitative side the focus has been the direct translation of the decision makers' knowledge into portfolios and priorities. For that purpose, a wide spectrum of techniques that range from checklists to fuzzy theory have been used (Cooper et al, 1999; Lin and Hsieh, 2004). In spite of all the methodologies developed, quantitative and qualitative, none of them consider both decisions levels at the same time nor provide evidence to support the validity of the decomposition strategy. 3. Portfolio Optimization The pharmaceutical industry provides one of the most challenging areas in terms of R&D portfolio management. Regardless of the type of product, small molecule chemical compound or complex protein, the industry faces long development times, low success rates, very high investments and considerable uncertainty in sales revenue estimates (Blau et al., 2004; Loch and Bode-Greuel, 2001). It is thus a perfect testing ground for any stochastic decision support methodology. There are three major stages in the lifecycle of a new drug: discovery, development and commercialization. The discovery stage is highly unpredictable and case specific, while the other two generally follow a well defined activity path. Also, the typical situation in the pharmaceutical industry is that there are seldom enough renewable and nonrenewable resources available to develop all the lead compounds in the pipeline at the same time. Therefore, all the attention from a portfolio management perspective is given to the development and commercialization stages. They are divided in the following sequential activities: First human dose preparation, clinical trials I, II and III, first submission for approval, prelaunch, ramp us sales, and mature sales. Paralleling these activities are all the engineering and marketing related tasks. For a thorough explanation of each of the activities the reader is refered to Blau et al (2004).

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3.1. The portfolio management problem In this study a multi-level optimization version of the SIM-OPT architecture developed by Subramanian et al (2003) is used to solve the portfolio management problem under constrained renewable resources. SIMOPT combines discrete event simulation and optimization (Fig. 1). The inner loop contains a model of the process and an optimizer that is activated every time an event (i.e. project failure) takes place during the simulation. The outer loop optimizer makes higher order decisions based on the information collected from multiple runs of the inner loop. In the realization of the SIM-OPT architecture employed in Blau et al (2004) only the outer loop is included. A genetic algorithm (GA) is used to optimize the selection of drug candidates, while the discrete event simulation model, which is a complete representation of the probabilistic pipeline network, is used to evaluate the candidate selection and sequencing alternatives generated by the GA. In their work Blau et al (2004), which we call the base case, used the project sequence generated by the GA to determine the order in which projects are started in the development pipeline, as well as the priorities of the activities within projects and the resources allocated to them, regardless of the specific realizations of the uncertainties in the system. It follows the aggregation concept characteristic of the reported decision support methodologies. In our study Blau's work is extended by including an inner loop to reallocate the renewable resources every time a project fails, and refining the GA. In the extended model the portfolio and the priorities generated by the GA are used as starting point to allocate resources and schedule activities. However, those decisions are dynamically updated by an optimizer according to the resolution of uncertainties at the key termination points in the activity network. The optimizer is a decision support system developed by Varma (2005) that maximizes the expected economic return by collecting the most up to date system data and processing it through a series of control policies learned by running the model multiple times using a short time window 3 mode (upper, most likely and lower) resource allocation MILP.

wl - 1 -

•

>| 1 rigger Event | Process Simulator|

lOuter OptimizationI 1

'

|lnner Optimization|

Fig 1. Sim-Opt Architecture

3.2. Outer loop (GA) The same GA strategy is used for the base case and the extended model trial runs. Following Blau et al (2004) the portfolios are encoded in such a way that each gene contains the number of a drug candidate (with 0 indicating that a project was not selected), and its position in the chromosome represents the priority given to the project. The fitness fimction, Z^, is given by: Z,=6^

EPNPV.-EPNPV. K EPNPV^-EPNPV^^

/ +(1- -or) Risk^-Risk^, + r. V

min

+ r)

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Where EPNPVfnin and EPNPVmax are the minimum and maximum expected positive net present values, respectively, in the current population; Riskmm and Riskmax are the maximum and minimum risk levels in the current population, measured as the probability of losing money; y is a small positive number that prevents division by zero, and a weights the present value vs. the level of risk in a convex linear combination. The X+ /it selection strategy for new generations is also used. However, it was found that Blau's reproduction strategy within the GA algorithm caused the optimization to be trapped in a specific area of the search space. Therefore, an additional mutation operator was included to overcome this problem. The operator randomly changes some of the genes in the following way: If the gene holds a value, the corresponding project is abandoned, and if it is empty a project is randomly reentered into the portfolio. In addition to correcting the myopic behavior of the original GA, this adjustment also makes the algorithm very robust to changes in the initial population. Therefore, the risk of biasing the algorithm in the wrong direction from the start, due to the inadequate selection of the initial population, is considerably reduced. 33. Inner loop (Scheduling and resource allocation) Once the portfolio and the priorities of the projects have been identified, it is necessary to schedule the activities within a project and allocate resources to them. In the base case it is assumed that the priority sequence obtained for the projects applies to every activity, and the level of resources assigned is the most likely (ML) one according to the degree of difficulty of the project. The multi-level SIM-OPT strategy, refered as the extended case, on the other hand, follows an adaptive approach in which activities are scheduled based on the GA sequence, but resources assigned to a particular task are dynamically increased or decreased to speed up or slow down a project in response to events such as terminations or launches according to certain policies. The resource allocation control policies were obtained following the SIM-OPT framework conceived by Varma,(2005), which used an architecture in which there are two loops. The inner loop contains the discrete event simulation of the development and commercialization activity network and a resource allocation MILP. The outer loop includes an observer that learns the optimal static policy, while the inner loop is run hundreds of times with the MILP as the resource allocation decision maker. The state space for each drug in the policy is defined as: {DS(i), NLEV(i), NHEV(i)}, where DS(i) = Development Stage of Drug i, NLEV(i) = Number of drugs having Lower Expected Value than drug i in the same development stage, and NHEV = Number of drugs having Higher Expected value than drug i in the same development stage. The control space is a vector that has as many elements as there are projects in the portfolio. Each element can take only one of 3 values, which correspond to upper, ML and lower resource allocation; that are associated with three different levels of duration for each task. Finally, it is relevant to mention that Varma (2005) explored the use of a MRCPSP MILP in the inner loop to consider the impact of a reactive schedule on top of the dynamic allocation of resources. However, it was shown that the difference between this more computational expensive approach and the allocation only MILP was not significant. 3.4. Case study The same 9 drug portfolio case study reported in Blau et al (2004) is used. Most of the statistical distributions and parameters for the model were replicated. The only changes were in the duration and cost distributions, which were fixed at their most likely values. This was necessary due to the lack of data for the same distributions at higher or lower

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resource allocation levels. The interactions between projects, as well as the simulation philosophy were also retained. Finally, the same financial, technical, manufacturing cost and resource dependencies for drugs aimed to diseases I and II were included. It is important to highlight that in spite of the simplifications in the discrete event simulation, the results are not significantly affected. The influence of fixing the duration and cost distributions is strongly dominated by the uncertainties retained in the model. The base and extended case results for this study were obtained using the improved version of the GA with multiple a weights (0, 0.5, and 0.8), and a population size of 10. However the values of some of the parameters of the mutation operators were individually adjusted to prevent the algorithm from converging prematurely. The resource allocation levels for the inner loop in the extended case were identified fi'om suggestions provided by managers in the industry. The upper and lower levels correspond to using ±15% resources than the ML, while inversely changing activity duration by ±7.5%. Those flexibility values were doubled in a second run to better understand the implications of decisions made at the tactical level within different managerial frameworks. 4. Results The return as measured by the EPNPV and the probability of losing money (portfolio risk) for the base case are presented in Fig. 2. All the points corresponding to the maximum EPNPV for a given level of risk are linked to form an approximate returnrisk frontier. At first sight it looks like its shape reflects the general form found by Markowitz in financial portfolios (Luenberger, 1998), but a closer look reveals that the direct correlation between return and risk is violated in the middle section. The number of projects in the portfolios in that area is considerably higher than the number of those on the rest of the frontier. That demonstrates that the base case can not capture the trade off between the inclusion of more projects to reduce the level of risk due to failure, and the reduction in returns due to developmental delays, caused by exceeding the limits of available resources. Fig. 3 (only the dominating portfolios are plotted) shows that the addition of resource allocation flexibilities mitigates the effect of this trade off and pushes the frontier to higher returns for the same risk levels. This result can be explained from a conceptual point of view as follows. The lower or higher level of resource allocations constitute real options to delay or to expedite, which means that some of the flexibilities in the decision process where captured with a consequent rise in the value of the portfolios. However, the composition of the portfolios on the frontier in the base case and the extended one is remarkably different in the area where the depression is found. The extended case efficient portfolios can not be obtained by simply adding projects to the base case results. This demonstrates that it is not possible to decouple the strategic and tactical decision making processes at certain levels of risk. It was also found that the inclusion of flexibilities does not guarantee the improvement or sustainability of the performance of a specific portfolio. It was even found that the results of some interior portfolios from the base case completely dominate those observed in the extended cases. Most of the time the portfolio chosen at the strategic level is in the interior region, which means that its behavior is completely unpredictable based on aggregated quantitative or non quantitative methods. Finally, it is important to mention that the GA converges much faster in the extended cases than in the base case. Based on the progression of the algorithm we believe that such a behavior is due to the reduction in the search space. The inclusion of flexibilities decreases the importance of the position of the projects in the sequence. Therefore, what really matters in the initial

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generations is the presence of the projects that in later generations will be reordered to obtain the optimal solution. 6000 5000

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5. Conclusions The study demonstrated the use of a multi-level Sim-Opt strategy to determine an optimal portfolio of new drug candidates. It has shown that the use of aggregated strategic decision models in the presence of uncertainties, project interdependencies and constraints can be misleading. It also has shown and quantified the importance of considering flexibilities in the valuation of projects. Although the approach is computationally demanding, it is an important tool to help management understand how tactical and strategic decisions are inter-related in order to maximize portfolio returns at specific levels of risk.

References Baker, N.R., 1975. R + D Project Selection Models - Assessment. R & D Management, 5: 105111. Blau, G.E., Pekny, J.F., Varma, V.A. and Bunch, P.R., 2004. Managing a portfolio of interdependent new product candidates in the pharmaceutical industry. Journal of Product Innovation Management, 21(4): 227-245. Calantone, R.J., Di Benedetto, C.A. and Schmidt, J.B., 1999. Using the analytic hierarchy process in new product screening. Journal of Product Innovation Management, 16(1): 65-76. Cooper, R.G., Edgett, S.J. and Kleinschmidt, E.J., 1999. New product portfolio management: Practices and performance. Journal of Product Innovation Management, 16(4): 333-351. Lin, C.H. and Hsieh, P.J., 2004. A fuzzy decision support system for strategic portfolio management. Decision Support Systems, 38(3): 383-398. Loch, C.H. and Bode-Greuel, K., 2001. Evaluating growth options as sources of value for pharmaceutical research projects. R & D Management, 31(2): 231-248. Luenberger, D.G., 1998. Investment science. Oxford University Press, New York, xiv, 494 pp. Maravelias, C.T. and Grossmann, I.E., 2004. Optimal resource investment and scheduling of tests for new product development. Computers & Chemical Engineering, 28(6): 1021-1038. Ray nor, M.E. and Leroux, X., 2004. Strategic flexibility in R&D. Research-Technology Management, 47(3): 27-32. Rogers, M.J., Gupta, A. and Maranas, CD., 2002. Real options based analysis of optimal pharmaceutical research and development portfolios. Industrial and Engineering Chemistry Research, 41(25): 6607-6620. Subramanian, D., Pekny, J.F., Reklaitis, G.V. and Blau, G.E., 2003. Simulation-optimization framework for Stochastic Optimization of R and D pipeline management. AIChE Journal, 49(1): 96-112. Varma, V., 2005. Development of computational models for strategic and tactical management of pharmaceutical R&D pipelines. Ph.D Thesis, Purdue University.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Multi-Period Capacitated Lot Sizing with Variable Batch Sizes Yoong Chiang See-Toh", Stephen P. K. Walsh^ Nilay Shah' ""Centrefor Process Systems Engineering Department of Chemical Engineering, Imperial College London, SW7 2BY, UK ^ICI Paints, Wexham Road, Slough, Berkshire, SL2 5DS, UK Abstract In the production planning for strong seasonal demand products, it is uneconomical to configure the supply chain for throughputs equivalent to the demand peaks. Instead, a holistic approach to supply chain optimisation is adopted where forward demand forecasts drive the production planning process. There is a considerable amount of research literature available for the various types of lot-sizing models involved in production planning. It is generally assumed in these models that a continuous production system is employed for the relatively high volume, low value products. However, these models are not directly applicable for a large number of specialised products operating in batch mode for increased flexibility on multi-purpose equipments. This research addresses the batch sizing operation within a lot-sizing model in order to derive a simultaneous batch sizing and production planning optimisation. The degrees of freedom in this combined problem are the monthly batchsizes of each product, integer number of batches of each product produced each month, amount of overtime working and outsourcing required in each month as well as the time-varying inventory positions across the chain are manipulated. Values for these are selected to balance the trade-offs in batch costs (each batch produced incurs a fixed charge associated with set-up and cleaning), stock costs (these are proportional to the product batchsizes and the amounts of inventory^ carried) as well as the overtime and outsourcing costs. In greater detail, the multi-item lot-sizing problem has been extended to incorporate the batch-sizing complexity, where the production costs are aggregated to the batch level, with an additional discrete dispersion cost proportional to the batchsize. The operational requirements for overtime work and outsourcing are presented as vector constraints of each product. Through linearisation of the inherent non-linear product of the integer number of batches and its corresponding batchsize, a mixed-integer linear programming (MILP) model is formulated with a minimum cost objective function. For a large number of products with consumer paints, this single-stage deterministic model is intractable due to the large number of variables involved, despite effective reformulations for tighter linear programming relaxations. Methods have been developed in the research to disaggregate the vector constraints from the batch sizing and production planning model for each product. These implicitly or explicitly utilise decomposition algorithms to reduce computational complexity and solution times. They are able to solve the applied industrial problem. Keywords: Batch-Sizing, Lot-Sizing, Batch Production, MIP.

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1. Introduction The lot sizing problem arises from material requirements planning systems, where inventory lot sizes are time phased to when they are needed. The key decision is to determine how much of each product to produce in each period. This tactical planning decision explores the traditional trade-offs between higher inventory costs associated with larger production lots and higher setup costs associated with lot for lot production. The different types of lot-sizing problems are derived from the characteristics of the production process and the planning detail required. These can be broadly classified based on the number of items, number of production stages (levels), and capacity constraints. Other extensions to the lot-sizing problem include backlogging, start-up costs, start-up times, excess demand sales and safety stocks [1]. Mathematical formulations for these problems can be found in Drexl and Kimms [2] as well as in Belvaux and Wolsey [3]. For a large number of specialised products manufactured in batch mode, the models in the literature do not consider the further division of lot sizes into smaller batches in order to meet the multi-purpose equipment capacity constraints. The closest example is the lot-sizing with constant batches as a deviation to the constant capacity problem [4]. In this paper, we shall address both batchsizing and stock-build problems simultaneously through mathematical optimisation models applied to a real industrial study. The aim is to derive the optimal production batch sizes and quantity for each product with the lowest costs incurred, whilst fulfilling the demand forecasts and resource requirements. 2. Modelling Considerations For the multi-product batch paints plant in consideration, there are two product families present in the manufacturing facility, emulsion and gloss paints. Each product family utilises independent equipment and lines, thus they shall be treated separately for modelling purposes. For a minority of products, a high-value ingredient is required which is available in fixed volume containers. Due to its manual loading routine, it is uneconomical to use part containers. Therefore, these product batchsizes are constrained to multiples of the raw material supply batchsize, which is obtained from the paint formulation recipe and the high-value ingredient container volume. An additional operation requirement is the fixing of each product batchsize for the planning period. This is to reduce error during the charging of raw materials for the numerous product batches. In the paint manufacturing process shown in Fig. 1 [5], the only production steps captured in the production planning are the mixing and dispersion. Firstly, mixing is explicitly detailed since it is identified as the process bottleneck. There are two capacities of mixers available and production batches are specifically allocated to each 00 C5

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mixer size, with the exception that small batches may also be processed in a large mixer. There also exists a minimum production batchsize which coincides with the minimum liquid volume to cover the mixer impeller blades during operation. The manufacturing time and production costs are specified a fixed value regardless of the production batchsize for modelling simplicity. For dispersers with smaller volumes than mixers, the number of dispersion batches in excess of mixer batches is required. This figure is derived from the ratio between the production batchsize and the dispersion limit, as obtained from shear viscosity calculations for each producOt. The manufacturing time-resource availability is segmented into normal and overtime operation. Production resources are costed at 150% during overtime working. Any manufacturing requirements in excess of overtime are outsourced with a 400% cost penalty. The last scenario is usually avoided in view of the detrimental effect to the company's reputation, but shall be included in the mathematical model to guarantee a feasible production plan. The physical flow of inventory is addressed in an inventory balance equation, where the amount of inventory in the next time period must equal the current inventory plus what is produced less what is consumed by demand. Its financial costs are presented in three forms: the carry-over inventory at the end of each period, the average stocks on hand and the disposal costs. In our big bucket model, the carry-over inventory is easily determined, however the holding stocks within each period can only be approximated by a saw-tooth inventory profile. Assuming the full batch is available immediately on production and each batch is consumed in its entirety at a fixed rate upon the completion of the next batch, the average inventory on hand is equivalent to half the production batchsize. This places a bias for small batches and conflicts with the manufacturing time-resource availability. Disposal cost by volume is charged on holding inventory beyond the planning horizon, despite demand being zero. This is to accommodate cases when the minimum production batchsize exceeds demand. Based on these data, a mathematical optimisation model is developed to determine the mixer equipment usage, extra dispersion requirements, monthly batchsizes and number of batches for each product, inventory profiles and the overtime and outsourcing requirements. All of these must satisfy the time-varying demand forecasts to give an optimum value for total annual production costs.

3. Solution Approaches The chief non-linearity in this problem is that the product of batch size and number of batches yields the total demand. Model reformulation and separable programming techniques are thus employed to linearise the nonlinear batch quantity-size reciprocal relationship to give a Mixed Integer Linear Programming (MILP) mathematical formulation and solved using GAMS-CPLEX 7.5. However with a large number of products present, this single-stage deterministic model is intractable due to the large number of variables involved, despite effective reformulations for tighter linear programming relaxations [6]. Multi-Period Batchsize Optimisation Batchsizes Batch Inventory Storage Extra Dispersions Raw Material Supply

Periods Carry-Over Inventory

Products Overtime /Outsourcing Operation

Fig 2: Decomposition Domains for Multi-Period Batchsize Optimisation

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Fig 6: Total costs and computational times comparison For period decomposition, the tedious single iteration search resulted in a solution with lower computational requirements. However, the best solution obtained was still poorer than the benchmarks. An improved decision for costing the stock build inventory is necessary prior to acceptance of this quick approach. In the final product decomposition strategy, there is up to 41% reduction in emulsion manufacturing costs compared to the historic batchsizes. For gloss paints, the results are less prominent at 27% for the fewer sub-problems and this is only a slight improvement over the annual optimised batchsizes. This may be attributed to higher resource availability for gloss paints, thus a lesser improvement in the solution with optimisation. 5. Concluding Remarks In this work, decomposition techniques were explored to solve the multi-period capacitated lot sizing problem with variable batch sizes. This is to allow solution feasibility for the additional model complexities associated with batchsize variable costs. In particular, the product decomposition domain has been identified for the successful decomposition of this huge integer program.

References 1. 2. 3. 4. 5.

L.A. Wolsey, Management Science, 48 (2002) 1587. A, Drexl and A. Kimms, European Journal of Operational Research, 99 (1997) 221. G. Belvaux and L.A. Wolsey, Management Science, 47 (2001) 993. Y. Pochet and L.A. Wolsey, Mathematics of Operations Research, 18 (1993) 767. R. Lamboume and T.A. Strivens, Paint and Surface Coatings, 2^^ ed., Woodhead Publishing, Cambridge, United Kingdom, 1999. 6. N.V. Sahinidis and I.E. Grossman, Computers and Chemical Engineering, 15 (1991) 255. 7. A.R. Clark, Computers and Industrial Engineering, 45 (2003) 545.

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Update virtual demand with stockbuild

^

Initialise virtual demand with monthly forecasts

For each period, solve PBM for product batchsizes

Solve F5PM for stockbuild requirements

Optimal monthly Yes> batchsize and number of batches

Fig 3: Batchsize Decomposition Algorithm As shown in Fig. 2, the original multi-period batchsize optimisation problem may be decomposed into the three functions of batchsizes, periods and products. Each strategy is described in greater detail in the following sections. 3.7. Batchsize Decomposition In batchsize decomposition, the batchsizing decision is separated from the production planning stage. The proposed strategy involves iterating between the batchsizing step and lot-sizing with fixed batchsizes stages to determine the monthly product batchsizes and stock build sequentially. The aim of this method is to omit calculations for batch inventory storage, raw material supply and extra dispersions from the production planning stage, thereby improving computational efficiency. The algorithm is summarised in Fig. 3. The information exchanged between the models is the monthly product batchsizes and the corresponding virtual demand. The latter is equivalent to the real demand corrected for stock build after each iteration. To obtain the monthly batchsize for each product, the Period Batchsizing Model (PBM) is employed in each period to fulfill the virtual demand. In cases when the virtual demand is zero, the minimum batchsize is relayed to the planning model. The stock build requirement is then calculated with these batchsizes using the following Fixed Batchsize Planning Model (FBPM). 3.2. Period Decomposition The second decomposition approach relaxes the stock build calculations and obtains the monthly batchsizes in each period independently. The proposed strategy is a modification of the backward-then-forward pass heuristic in Clark [7] to include the monthly batchsizing operation. In the procedure outlined in Fig. 4, the production is initially optimised one period at a time in a backward pass from the last period, where the target stock levels necessary to fulfill the future demand in the adjacent month is identified. Commencing with the last period, the month-end inventory for each product is initially set to zero and the Backward Pass Batchsizing Model (BPBM) is optimised. The determined inventory level of the preceding month replaces the parameter for the subsequent optimisation. This process is repeated until the stock build requirement for period 1 is computed. Adjust carry -over inventory cost factor

^'

From last period, solve 5 P 5 M for stockbuild requirements

From first period, solve FPBM for product batchsizes

Fig 4: Period Decomposition Algorithm

Solution \ _ Y e s ^ xQonvergedK^

Optimal monthly ^^^^^ sizes and number of batches

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In the next stage, a forward pass then optimises production one period at time in order to meet these stock build requirements. The target inventory level for each product is assigned the monthly stock build requirements previously determined. Beginning with the first period, the residual inventory is initially equated to zero and the Forward Pass Batchsizing Model (FPBM) optimised. The month-end inventory level obtained then replaces the residual inventory and this process is repeated chronologically for the remaining periods. 3.3. Product Decomposition / Column Generation The final decomposition approach is to segment the overtime and outsourcing operation computation from the individual product batchsizing procedure. The solution strategy is to create multiple feasible production plans through the multi-period batchsize optimisation for each product individually. This generates a set of possible solution columns. Integer cuts are applied during the column generation phase to exclude a previous integer solution. Then a combinatory optimisation is performed to select the set of product batchsizes that yields the lowest total cost with the timeresource vector constraints. As outlined in Fig. 5, the Rigorous Single Product Planning Model (RSPPM) commences with an exhaustive search in the column generation phase to derive the global or near-global optimum within computational allowances and tolerances. This defines the extent of cost reduction of the problem. The concluding step of the singlepass routine is the selection of the individual production profile that incurs the lowest overtime and outsourcing penalties. This is achieved with the Combinatory Overtime Constraints Model (COCM).

4. Computational results The motivation for the current work is to provide a systematic and effective method to address the batchsizing and stock build problem. Current industrial practice is based on heuristical adjustments of product batchsizes to meet demand within capacity limits. These historic batchsizes will be used as a basis for comparison with the optimisation models developed, as shown in Fig 6. Another benchmark is the annual batchsizes as derived from a naive lumped model without the time-period complexities. Each product batchsize are then transcribed for multi-period comparison in a just-in-time manufacturing system and overtime costs are incurred where excess capacity is required. In the batchsize decomposition approach, a solution is only obtained for gloss paints, as the larger emulsion stockbuild model did not achieve a solution within a reasonable CPU time, even with a reduced number of batchsize discretisations up to a single batchsize. This is expected since the capacitated lot-sizing problem is known to be NPhard and a large number of products are simultaneously considered in this model. As the total cost for gloss paints fared poorer than the annual optimised batchsizes, thus the two sub-problems in the batchsize decomposition method is inadequate for solving this huge integer program. Add intger cuts on r— number of mixer

batches

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16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Integration of discrete-event simulation and optimization for the design of value networks M Schlegel, G. Brosig, A. Eckert, K. Engelke, M. Jung, A. Polt, M. Sonnenschein,

C. Vogt

BASF Aktiengesellschaft, Conceptual Process Engineering, D-67056 Ludwigshafen, Germany Abstract The design and restructuring of value networks in the process industry requires decision making in a space of numerous diverse constraints. A mathematical problem description leads to dynamic models involving continuous and discrete decision variables; furthermore, stochastic effects have to be taken into account. Instead of formulating an overall optimization problem, which would cover all possible decision variables at the price of being difficult to formulate and impossible to solve, we employ an approach based on discrete-event simulation. Mathematical optimization is rather applied to dedicated subproblems, such as planning, scheduling and resource allocation. The benefits of using optimization of subproblems arising in the discrete-event simulation of value networks are discussed in this contribution. Keywords: value chain design, discrete-event simulation, mathematical optimization 1. Introduction In an environment of increasing competition, margin erosion and commoditization of specialty products, the process industry is facing new challenges. As a consequence, the scope of projects searching for further cost reduction and business optimization is broadening from single plant optimization to the restructuring of entire value chains. For the planning of strategic investment projects, this requires to intertwine the conceptual design of processes (batch, campaign or continuous operation) with supply chain considerations addressing storages, logistics, raw material supply etc. Enterprise and supply chain optimization has been identified as one of the major research challenges for the process systems engineering community (e.g. Grossmann (2004)). The handling of uncertainty is one of the key issues here. In this paper, we describe a simulation-based approach (using discrete-event simulation) for value chain design, which employs underlying mathematical optimization for the solution of subproblems such as planning and resource allocation. The feasibility of this approach in the industrial practice will be illustrated. 2. Problem statement Shah (2005) has classified a supply chain problem space spanned by the dimensions "time" (consisting of the strategic, planning and operational level) and "space" (ranging from suppliers via production and storages to customers). The problem class addressed in this paper covers the whole space dimension (depending on the project), but is mainly concerned with strategic decisions. We consider projects during the conceptual planning phase, which have either to deal with the restructuring of existing value networks, or

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with the conceptual design for investment projects. Typical tasks arising in these type of projects are (Vogt et al., 2005): • evaluation of debottlenecking measures for plants and value chains • evaluation of design proposals for storage capacity within complex and strongly coupled value chains • evaluation and proposal for maintenance/shut down sequences • check design options for robustness under uncertainty • capacity analysis for early-stage business concepts • support change management with transparent results The processes treated in such projects cover the full range from one-product continuous plants to multi-product batch plants. An example for a value network is shown in Figure 1.

Drum Filling

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Supplier Figure 1. Example of a value network The problem of value chain design can be expressed in form of mathematical models, which are characterized by several important features: 1. The problem formulation contains many decision variables, which are at least partially of discrete nature, for example the decision, whether a certain piece of equipment shall be installed or not. Additionally, especially in case of multi-product plants, planning and scheduling questions arise, which require an ordering of the production sequence and the allocation of products to the process equipment. 2. Furthermore, we have to deal with stochastic influences such as fluctuating customer behavior (demand uncertainty), uncertain marketing forecasts as well as unplanned failures or shut-downs. 3. Tight project schedules and frequent changes of project objectives increase the need for cost and time efficient modeling in order to facilitate quick decisions.

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This list of challenges could certainly be extended. In principle, model-based optimization techniques are suited to provide decision support for value chain problems. How^ever, a careful comparison of the state-of-the-art with the aforementioned requirements for practical problems has to be undertaken: Optimization involving discrete decision variables is usually treated by mixed integer modeling and solution techniques. Thereby, it has to be distinguished between process design problems like the structural optimization of process flowsheets, and operational problems such as planning, scheduling and resource allocation. As observed by Shah (2005), there is still no connection between process design and supply chain operation. Although capacity (e.g. of the two plants shown in Figure 1) is a key quantity in supply chain network design, it is often represented in a fairly simple way. However, especially for multi-product plants, a better representation of capacity in supply chain models is required (Shah (2005)), and this issue forms the link between the design and operational level. Handling stochastic influences in optimization problems falls into the category of optimization under uncertainty. According to Sahinidis (2004), today there still do not exist any widely-available general-purpose software implementations for optimization under uncertainty (ftizzy mathematical programming or stochastic dynamic programming), although some applications even to real-time scheduling problems can be found (e.g. Sand and Engell (2004)). Summarizing, it can be stated that rigorous optimization for the solution of value chain design problems is still a very challenging task. For real-world problems there is the need for practical alternatives (see also Jung et al. (2004)). 3. Discrete-event simulation BASF's Global Competence Center Engineering is addressing these challenges by means of discrete-event simulation, which is well suited to handle dynamic problems strongly affected by stochastic constraints. Discrete-event simulation has a long and successful track record in the improvement of production processes. Though being more prevalent in the area of manufacturing and automotive industries, this simulation technique can also be applied to answer questions arising in the complex, coupled production networks of the chemical industry. We use the in-house developed tool eSCET, which is based on the discrete-event simulation platform eM-Plant^^ (Technomatrix (2002)). eSCET comprises predefined elements such as e.g. units, plants, filling lines, tanks etc., but also allows the customization of a model in order to include project-specific particularities like certain operation rules etc. Furthermore, it is crucial to have the means for a flexible modeling of the time domain. The time horizon for the simulation typically lies in the area of several months up to a year or longer. On the other hand, if detailed answers to feasibility questions are asked, the occurrence of events must be modeled on a quite fine level of granularity, say minutes or at least hours. This allows to model and simulate accurately effects such as shift operation, lab sample times, check tanks etc. Generally speaking, it is important to take dynamics into account as soon as simultaneity, interdependence and randomness of events play an important role. This simulation approach, enhanced by a modular, object-oriented software setup, is applicable to a broad range of value chain tasks. The model granularity can be adapted depending on the problems to be solved (see above). The software has been successfiiily appHed in many projects.

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As mentioned above, an accurate modeling of plant capacity is important. Naturally, it has an impact in considerations about the process layout of the plant itself, but it also strongly affects the other links of the value chain under consideration, e.g. upstream or downstream plants, storages and the logistic infrastructure (cf Figure 1). This holds especially for multi-product plants, where the throughput is depending on an often complex set of product-dependent production recipes and operation rules. On the other hand, we have talked above about stochastic influences like uncertain customer demand and failures. They may have a dominant impact on key performance indicators like service level and customer satisfaction. Under these circumstances, production planning is essential in order to capture the available production capacity with sufficient accuracy. 4. Integrating optimization into discrete-event simulation Simple, e.g. rule-based planning might be sufficient for situations with only a few products. However, if the product portfolio is more complex and shows a significant seasonality, more sophisticated planning methods have to be considered. The integrated simulation and optimization run is depicted in Figure 2 and will be described in the following. Essentially, the procedure consists of three loops. A discreteevent simulation run of the time period of interest (the operating horizon, e.g. one year) forms the core of the method. Given a customer demand for the products (with seasonal fluctuations), fixed process equipment and other infrastructure parameters, the simulation allows to evaluate key performance parameters like service level, equipment utilization etc. Now, in order to take the aforementioned stochastic influencing factors into account, the simulation run of the entire operating horizon is divided into smaller time periods (with a length of, say, one month). In each of these planning periods, the demand uncertainty is used to generate a demand forecast for the current period. Based on this input, an LP (linear programming)-based planning and a subsequent GA (genetic algorithm)-based scheduling are carried out resulting in a feasible product-equipment allocation and production plan. Given this schedule, the production is simulated for the particular planning period. Subsequently, the horizon is shifted to the next period, and a new planning and scheduling problem is solved. This procedure is continued until the end of the operating horizon is reached, similarly to the moving horizon concept in model predictive control. Since the customer demand is a stochastic quantity, the aforementioned simulation of the entire operating horizon (with underlying moving horizon planning and scheduling) is carried out several times by varying the fluctuating customer demand. This is shown as the Monte-Carlo loop in Figure 2. Finally, the decision variables of the overall value network (e.g. process equipment, buffer sizes, ...) need to be varied to evaluate their impact and determine the optimal setup. This is done in the outer loop by defining and varying scenarios. It should be noted that this procedure is certainly not striving for the (global) optimum of the value chain design problem. The planning and scheduling optimization is rather used to guarantee a feasible operation of the plants in the value chain and, hence, a realistic modeling of their capacity. The results for supporting strategic investment decisions are generated in the outer loop through simulation of different scenarios.

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Figure 2. Flow diagram for discrete-event simulation with underlying moving horizon planning and scheduling. The suggested approach has been successfully applied in several projects. The integration of mathematical optimization of subproblems in discrete-event simulation turns out to be essential especially for multi-product plants w^ith tight resource constraints in order to obtain a realistic model of the production capacity. This is a prerequisite to draw^ meaningful conclusions from scenario simulation and thereby to provide the required decision support for investment projects. 5. Example For illustration purposes, we will briefly discuss an example project, where the concept has been successfully applied. The aim of the project was an optimal investment for capacity extension and logistics optimization for a value network consisting of 4 different plants. Besides the production and consumption, the model also included the internal distribution between the plants. The core plant of the network produces about 80 intermediate products on 9 production resources; up to 300 products are modeled. The production is characterized by multi-stage operation, product-dependent capacities, fluctuating demands and failures or shut-downs. The number and capacity of the intermediate buffer tanks has been identified as a bottleneck for the production.

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It has been decided to build a dynamic model of the value network in order to support the search for the best concept by simulation of different investment scenarios. For this purpose, it is essential to achieve a good representation of the core plant's capacity. In the given setting w^ith many products and tight tank constraints this can barely be achieved by implementing a rule-based production. However, applying optimization for capacity planning and tank allocation, as described above, enables an appropriate modeling of the multi-product plant. Scenario simulation then allowed to determine the minimum investment required for the capacity extension. Additionally, significant savings in operating cost could be identified.

6. Conclusions In this paper, we have discussed several challenges in the design and restructuring of value networks. A full mathematical optimization of the resulting models is still very challenging and hardly a realistic option in the industrial practice. The evaluation of scenarios by means of discrete-event simulation serves as a practical alternative. In this framework, mathematical optimization can be used to cope with uncertainties in customer behavior, plant availability etc. The solution of planning and scheduling subproblems is required to obtain accurate models for plant capacity, which is particularly important in case of multi-product plants and tight resource constraints. Then, the simulation of scenarios allows to determine the best option with respect to investment and operating costs for the conceptual design or the restructuring of value networks.

References Jung, J.Y., G. Blau, J.F. Pekny, G.V. Reklaitis and D. Eversdyk (2004). A simulation based optimization approach to supply chain management under demand uncertainty. Comp. Chem. Eng.,2^,10^1-2X06. Grossmann, I.E. (2004). Challenges in the new millenium: product discovery and design, enterprise and supply chain optimization, global lifecycle assessment. Comp. Chem. Eng., 29, 2939. Sahinidis, N.V. (2004). Optimization under uncertainty: state-of-the-art and opportunities. Comp. Chem. Eng, 2S, 911-9S3. Sand, G. and S. Engell. (2004) Modeling and solving real-time scheduling problems by stochastic integer programming. Comp. Chem. Eng, 28, 1087-1103. Shah, N. (2005). Process industry supply chains: Advances and challenges. Comp. Chem. Eng, 29, 1225-1235. Technomatrix (2002). eM-Plant 6.0 User Manual. Vogt, C, T. AUers, G. Brosig, A. Eckert, K. Engelke, M. Jung, A. Polt, H. Schultz, M. Sonnenschein (2005). Paradigm shift and requirements in enhanced value chain design in the chemical industry. Chem. Eng. Res. Des., 83(A6), 759-765.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

A CP Method for the Scheduling of Multiproduct Continuous Plants with Resource Constraints Luis J. Zeballos and Gabriela P. Henning INTEC (UniversidadNacional delLitoral -CONICET), Guemes 3450- 3000 Santa FeArgentina Abstract This contribution presents advances in relation to a prior Constraint Programming (CP) formulation that addresses the scheduling problem of a multiproduct plant that has continuous stages and a limited number of intermediate storage tanks. The proposed approach is able to handle production rates of intermediate products which can be greater or lower than the rates of their corresponding consumption campaigns. In addition, the formulation includes a domain specific search strategy which aims at speeding-up the solution process and making problems tractable. The search strategy focuses on those campaigns that seem to be more demanding in terms of storage and equipment requirements and resorts to a domain reduction specific procedure. The proposed approach has rendered very good results for various test problems. Keywords: Scheduling, Multiproduct continuous facilities. Constrained resources. Constraint Programming. 1. Introduction The short-term scheduling of multiproduct continuous plants, having storage tanks of finite capacity between consecutive stages is a problem of practical interest. Tanks allow increasing plant productivity and operational efficiency. This type of problem has received some attention during the last decade. Early publications have addressed it by means of mathematical programming formulations (Zhang and Sargent, 1998; Mendez and Cerda, 2002). However, these approaches exhibit some disadvantages; specially when industrial problems are considered. First, large-size models, which become hard to solve, are obtained. Moreover, the large number of campaigns generated for each production task gives as result non-realistic schedules. On the other hand, some formulations introduce hard assumptions about the problem. For instance, if two consecutive stages are considered, certain approaches assume that the rate at which a given intermediate product (IP) is produced is always greater than the rate at which it is consumed. This assumption does not always occur in practice. This paper presents advances into a prior Constraint Progranmiing (CP) formulation proposed by the same authors (Zeballos and Henning, 2005), that addresses the scheduling problem of multiproduct continuous plants. CP (Brailsford and Potts, 1999) is a relatively new approach, originated in the computer science and artificial intelligence communities. However, it has already been adopted in the Process Systems Engineering field (Maravelias and Grossmann, 2004). The specific goal of this contribution is to extend such previous CP formulation by both including an extra set of constraints and incorporating a search strategy (SS) that allows reducing the computational load. In order to evaluate the performance of the improved formulation, different case studies were successfully addressed. Results allow to point out the benefits of the proposed approach.

1961

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LJ. Zeballos and G.P. Henning

2. Problem Statement The scheduling problem refers to a resource constrained multiproduct plant involving two continuous stages, in which the set of processing units able to execute each production campaign is known. The plant also involves tanks of limited capacity for the temporary storage of IPs. The problem objective is to find: (i) the optimal sequence of production runs to be executed in every continuous unit; (ii) campaign starting and completion times; as well as (iii) the assignment of IPs to tanks and the sequencing of those products allocated to the same tank, in order to minimize makespan. The model assumptions are the following: (i) final product (FP) requirements are known in advance, (ii) the number of production runs (campaigns) for each intermediate and final product is predefined and taken as problem data, iii) any campaign manufacturing an IP supplies it to several campaigns demanding such product; iv) production rates depend on the material to be produced; v) IPs can be stored in tanks or directly transferred to the following stage; vi) any IP must be stored in just one tank (no split is allowed); vii) changeover-times associated to processing units and storage tanks are sequence dependent; viii) the initial inventory of any IP is equal to zero, ix) the available storage capacity for FPs is unlimited; x) production rates of IPs can take any arbitrary value in relation to the consumption rates of such IPs by means of a set of FP campaigns. 3. Constraint Programming Model In order to address the problem described above, a CP model has already been proposed (Zeballos and Henning, 2005). It is based on the OPL constraint programming language supported by ILOG Solver (2004) and employs some specific scheduling constructs available in the ILOG Scheduler (2004). The model relies on the representation of several types of activities: (i) tasks modeling the manufacture of intermediate and final products (which are referred as intermediate and final runs), as well as (ii) tasks modeling the storage of IPs (that are referred as storage tasks). These task modeling elements are characterized by means of starting, duration and completion time variables, which are dependent among themselves. This contribution extends such model by including a new set of material balance constraints, that does not require the definition of new variables. The new constraints are posed for each final run consuming an IP. Each constraint enforces the amount of the IP stored in a given tank, at the starting time of each associated final run that consumes such product, to be less than or equal to the associated tank capacity. In other words, at the moment a FP campaign begins, the amount of its associated IP that has been manufactured up to such point, minus the amount that has been transferred to other consuming campaigns up to such moment, must not exceed the capacity of the allocated tank. Unfortunately, due to lack of space the whole model cannot be presented in this contribution. Readers are referred to Zeballos and Henning (2005) for further details.

4. Search Strategy Most of the CP systems allow users to easily define SSs. This functionality can lead to performance increases by implementing search schemes adapted to particular domains. This work proposes a SS for the scheduling problem described in Section 2, which was previously tackled (Zeballos and Henning, 2005) with the default strategies included in the ILOG Solver. The one presented in the rest of this section attempts to improve the computational performance by resorting to a variables' domain reduction (DR) approach and also to an intelligent assignment of equipment to production campaigns, that guides the movement through the search space. One of the characteristics of the

A CP Methodfor the Scheduling of Multiproduct Continuous Plants

1963

proposed search scheme is that it focuses on those sets of associated campaigns that "a priori" seem to be more demanding in terms of resources; i.e., the ones that demand more usage time to equipment units and tanks. Therefore, "the most constrained first" principle is appUed during the SS. Moreover, taking into account that the pursued objective is makespan, the proposed scheme performs a DR approach on the completion times of those campaigns that manufacture FPs. Since one of the assumptions of the addressed problem is that the manufacturing rates of both intermediate and final products do not depend on the assigned equipment items, these two components of the SS (DR and equipment-campaign assignment phases) are almost decoupled. In order to focus on those sets of campaigns that are more demanding in terms of productive resources a campaign ordering procedure is done as the first step of the proposed strategy. This procedure first generates clusters of campaigns. Each cluster contains a campaign that manufactures an IP as well as all the FP campaigns that consume such IP. For each cluster, the ordering procedure calculates its associated total processing time just by adding the times required by all the campaigns. Afterwards, clusters are organized in a decreasing order of processing times; moreover, for each of the identified clusters, campaigns manufacturing FPs are also arranged in a decreasing order of processing times. Having performed such arrangement of campaigns, the BoundingJProcedure (see Fig. 1) can start. It is a recursive procedure that resorts to other procedures. BoundingJProcedure operates on the domains of the completion time variables associated to the FP campaigns. The domains of such variables are characterized by two extreme points: the earliest end and the latest end times (EET & LET). The proposed DR approach performs the pruning by adopting lower values for the EETs. It begins by assigning the least possible value to the completion time of the FP campaign that belongs to the first cluster and has the highest production time. If such variable instantiation succeeds, it proceeds likewise with the next FP campaign in the first cluster ordered set. It continues in a similar way until all the domains of the FP campaigns associated with the first cluster have been tried. Then, the procedure continues in the same fashion with the remaining clusters by taking them in the order that was previously found. If an infeasible solution is found, backtracking takes place and domains are relaxed. In order to verify if the values adopted for the completion times of the FP campaigns are feasible, Bounding_Procedure resorts to the PartialSatisfiahle procedure. Assuming that the DR step has been successfiil, the Equipment_Assignment (see Fig. 2) strategy can start. It is a recursive procedure that guides the assignment of equipment to FP campaigns. It again resorts to the PartialSatisJiable one, that verifies if the proposed equipment assignments are feasible. The Equipment_Assignment strategy performs the assignment of campaigns in the same order that was previously established. If a given partial assignment becomes infeasible, backtracking takes place. As a result, a new unit would be checked. If all the available units associated to a given campaign fail during this assignment step, the backtracking step will be more significant. The procedure will jump back to the previously assigned campaign, will break such assignment and will try a new unit. If during the first assignment step a set of feasible assignments cannot be found for the whole set of FP campaigns, the backtracking step will be even more severe and will reach the DR phase. It will result into to a domain enlargement of the completion time variable that was last pruned. If all FP campaigns have successfully been assigned, the Tank_Assignment (see Fig. 3) procedure begins. It is another recursive procedure that guides the assignment of storage

1964

LJ, Zeballos and G.P. Henning

units to IP campaigns. This second assignment step is carried out under the same philosophy. IP campaigns are organized in a decreasing order of processing times. Thus, the procedure will attempt to assign the processing run having the longest time to the first available storage tank. Once again, after each campaign-tank assignment is done, model constrains are checked by the PartialJSatisfiable procedure. If during this second assignment step a set of feasible assignments cannot be found for the set of IP campaigns, the backtracking step will move fijrther and reach the previous assignment stage. In such situation, the last assigned FP campaign will be freed of its associated unit and a different one, not tried yet, will be attempted. On the other hand, if this second step of the assignment phase succeeds, a new feasible solution is found. Its associated makespan will be taken as a new bound on the latest end times of the FP campaigns. The procedure will continue in an iterative fashion by attempting a further reduction on the EETs associated to the completion times variables corresponding to the FP campaigns. Thus, the whole cycle will start again by trying a new DR phase and afterwards, a new assignment one. The strategy will finish when a further DR step turns out to be infeasible. fioanding-Fzocedare {} if {all FP-campaign.completion-times are fiKed) then if (Eguipment.Assignment 0 ) then return feasible solution else return f a l s e endif else choose FP_campaign without fixed coiipletion time FP-cawpaign.completion-time = lower value of EET if ( P a r t i a l - S a t i s f i a b l e f j ) then if (Bonnding-Procednrefj) then return true e l s e relax last FP-campaign.completion-time endif e l s e relax last FP-campaign.completion-time endif return f a l s e endif

Fig. 1. Bounding procedure == f Eqaifment.AssigmBent () It [all FP-cai^aigns are assigned) then If {Taitk-AssignmentO) then return true else return false endif else choose FP-campaign without processing unit for a l l processing units FP-ca^)aign r* processing unit if [Faxtial.SatisfiableO) then if (£guip]nent-As5ignzBent(j) then return true e l s e not FP-campaign r* processing unit endif e l s e not FP-campaign r* processing unit endif endfor return f a l s e endif

assignment

TanJr-Assignment () I f (all IP-cai^aigns are assigned) then r e t u r n true else choose IP-canpaign without tank assignment for a l l storage units IP-campaign r* storage unit i f (Faztial-ScitlsfiableO) then i f {Tamk-AssigmaentO) then r e t u r n true e l s e not IP-campaign r* storage unit endif e l s e not IP-campaign r* storage unit endif endfor return false endif

Fig. 3. Tank assignment procedure 4= Fig. 2. Equipment assignment procedure == j ^ For simplicity reasons, procedure arguments are not included

5. Computational Results In order to examine the effectiveness of the proposed CP model and of the corresponding SS, a set of test problem instances based on a fast moving consumer

A CP Methodfor the Scheduling of Multiproduct Continuous Plants

1965

goods manufacturing plant (Schilling and Pantelides, 1996) was solved. The plant has two processing stages (mixing and packing) and finite intermediate storage facilities between them. Specifically, the plant comprises three parallel mixers operating in a continuous way, three storage tanks of limited capacity and five continuous packing lines. The mixing stage produces seven IPs from three different base materials and the packing stage manufactures fifteen FPs from the seven intermediate ones. The adopted scheduling criterion was the makespan {Mk) minimization. Compared with a previous test problem based on the same plant (Zhang and Sargent, 1998), the problem instances that were adopted in this work have some differences. First, the production rates of the IPs can be less than, equal to or greater than the rates of their associated consumption campaigns. Moreover, there are no campaign-machine forbidden assignments, though the model can handle this type of situation. Therefore, the problem instances that were chosen in this contribution are more complex than the original one and exhibit a higher dimensionality. Problem instances were split in three classes. The first one (Problem Class I) includes case studies having rates of manufacture of IPs that are lower than the sum of the rates of the associated consumption campaigns. The second class (Problem Class II) considers case studies with production rates of IPs which are greater than the sum of the rates of the associated consumption runs. Finally, the third case (Problem Class III) involves a mixed case in which production and consumption rates can exhibit any arbitrary relationship. This group includes problem instances in which IPs are manufactured with production rates that are smaller or greater than the sum of rates of their associated consumption runs. This classification was made because the different classes imply different production patterns, that might have an effect on the performance of the SS. Several SSs were tested before accepting the one that was described in the previous section. The adopted one was the strategy that exhibited the best performance for the three classes of problems. In fact, it was the one that presented the best computational time-solution quality relationship. Tables I to III include some results obtained with both the proposed approach and the prior formulation (Zeballos and Henning, 2005) for Problem Classes I to III, respectively. Problem instances within each of the classes differ among themselves only on the values of the production rates and on the capacity of the three tanks. Therefore, all examples have the same dimensionality. Their associated models have 665 variables and 572 constraints. The time limit to obtain solutions was fixed into 500 seconds of CPU time. All the reported results correspond to good quality feasible solutions; in other words, in none of the cases an optimal solution could be obtained in less than 500 seconds. An analysis of the results reflects the fact that the proposed approach allows reducing the computational load. In addition, the CP approach, as well as the mathematical programming one, is quite sensitive to problem data. Solutions belonging to the same class of problem may demand CPU times that differ in two orders of magnitude (see Tables I and II). It may also be observed that good quality solutions are reached in low CPU times. Results were found with ILOG OPL Studio 3.7.1 and the embedded Scheduler Optimizer 6.1 release (ILOG, 2004). The computations were carried out on a Pentium IV 2.8 GHz, PC with 1 Gbyte of RAM memory. Fig. 4 schematically shows the best solution found (in less than 500 seconds of CPU time) corresponding the case study # 7. It depicts the Gantt diagram corresponding to the units' workload and, in its lower part, the associated inventory levels. It shows the sequence of IP campaigns assigned to the storage tanks and the temporal variation of the intermediates' inventory during the scheduling horizon.

LJ, Zeballos and G.P. Henning

1966 Table I. Results for Problem Class I This Approach Prior Approach Makespan / Makespan CPU Time* /CPU Time* 996/12.6 1046/32.3 Case Study # 1 948 / 342.0 995/353.7 Case Study # 2 996/166.0 1021/334.7 Case Study # 3

TaskI[I6)

Problem Instance

Table II. Results for Problem Class II Problem This Approach Prior Approach Makespan / Instance Makespan CPU Time* /CPU Time* Case Study # 4 972/193.8 — 919/138.6 Case Study # 5 .._ 927 / 349.6 Case Study # 6 ...

1

1

TaskI[I2] TaskI[Il]

TaskI[I5l

1

r

TaskI[I7]

1 1

_J TaskST[I2]

il

i

TaskST[I6]

\Asmn\

IaskF[P5|

TaskF[P3] J

TaskF[F4]

i

TaskST[I5]

TaskF[Pl] 1

TaskF[15]

TaskST[13]

i

\ \

ToskST[I4]

1

TaskF[P91

I

TaskF[10]

M^t*"!!

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TaskSTlIl] 1

)

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1

laskFIHI TaskF[13]

1

1

| I

TaskF[P71

!

1 TaskF[ll)|P6j

ZJ TaskF(P21

j

Table III. Results for Problem Class III Problem This Approach Prior Approach Makespan Makespan / Instance /CPU Time* CPU Time* 1130/2475 Case Study # 7 1014/11.44 996/199.8 Case Study # 8 ... Case Study # 9 1320/484.86 ...

* Time required in order to obtain the best solution within 500 seconds of CPU time. — No solution was found

Fig. 4. Diagram showing the best solution obtained for problem instance # 7

6. Conclusions An improved CP approach that addresses the scheduling of a resource constrained multiproduct plant having continuous stages and storage facilities has been proposed. The problem formulation comprises a CP model and also a domain specific SS. The model admits any type of relation between the rates of production and consumption of the intermediate and final product campaigns. The SS resorts both to a variables' DR approach and to an intelligent assignment of equipment to production campaigns. The proposal allows obtaining good quality suboptimal solutions with a low computational load. Future work includes extending the model in order to consider a higher number of processing stages of both batch and continuous types. Acknowledgments. Authors acknowledge financial support from ANPCyT under grant 11-14717, from CONICET under grant PIP 5915 and "Universidad Nacional del Litoral", under grant PI 003-14 (CAI+D 2005). References Brailsford, S.C. and C.N. Potts, 1999, Europen Journal of Operational Research, 119, 557-581. ILOG, 2004, OPL Studio 3.7.1- Scheduler 6.0. User[^ Manuals. Maravelias, C.T. and I.E. Grossman, 2004, Computers and Chemical Engineering, 28,1921-1949. Mendez, C. and J. Cerda, 2002, Computers and Chemical Engineering, 26, 687-695. Schilling, G, and C.C. Pantelides, 1996, AICHE Annual Meeting, paper no. 171d. Zeballos, L.J. and G.P. Henning, 2005, In proceedings of ENPROMER, Rio de Janeiro, Brasil, August 14-18. Zhang, X. and R.W.H. Sargent, 1998, Computers and Chemical Engineering, 22, 1287-1295.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

1967

Stochastic integer programming in chemical batch scheduling: evolution strategies vs. exact algorithms Jochen Till, Guido Sand, Sebastian Engell* Process Control Lab (BCI-AST), Department of Biochemical and Chemical Engineering, Universitdt Dortmund, Emil-Figge-Str. 70, D-44221 Dortmund, Germany Abstract Two-stage stochastic integer programming with discrete scenarios is a promising modeling approach to deal with uncertainties in chemical batch scheduling. However, the resulting optimization problems are of large scale and the application to real world problems using standard solvers is computational prohibitive. Thus, there is a need for algorithms that take advantage of the specific problem structure. The authors have recently proposed a stage decomposition based hybrid evolution strategy for two-stage stochastic integer programs. In this paper, this algorithm is compared to an exact scenario decomposition based branch-and-bound algorithm. A real world scheduling problem with uncertainties in demands and plant capacities serves as a benchmark example. The evolution strategy provides good solutions for a range of test examples with default settings. The performance of the branch-and-bound algorithm strongly depends on the choice of the underlying heuristics. The performance of the hybrid algorithm can be improved further by a tailored evolutionary algorithm. Keywords: stochastic integer programming, chemical batch scheduling, uncertainty, evolution strategies, hybrid algorithms 1. Introduction When optimization is applied to problems from process systems engineering, one is almost always confronted with uncertainties [1]. A promising approach to deal with these uncertainties is the use of stochastic integer programming [2]. Chemical batch scheduling is an important application for optimization and has received much attention in the past two decades [3]. Most of the scheduling models assume deterministic data. However, in reality chemical batch scheduling problems are governed by uncertainties in the process parameters (e.g. equipment failures) or in external data (e.g. demands). Sand and Engell [4] proposed a model predictive scheduling (MPS) framework for realtime scheduling. The framework considers uncertainties by the use of a two-stage stochastic integer program in a moving horizon scheme. However, for real-world applications the stochastic program results in a large-scale mixed-integer linear program (MILP). The size of the MILP makes the application of standard solvers computational prohibitive. An important prerequisite to apply stochastic programming to real-world problems is an algorithm which is able to provide good solutions in a limited time. Till et al. [5, 6] recently proposed a new hybrid evolution strategy based on stage decomposition, which has the potential to fulfill this requirement. In this paper, the hybrid algorithm is compared to an exact decomposition based algorithm by means of a benchmark example. Author to whom correspondence should be addressed: [email protected]

1968

J.Tilletal

2. Problem statement and solution approaches 2.1. The two-stage stochastic integer programming model In a stochastic integer program (SIP, see Eq. 1), the uncertain parameters are represented by a finite number of reahzations (scenarios) co with corresponding probabihties 71^ [2]. This leads to an optimization problem of the form min c^x + Y,^^^ly. ^'^« ^

^'t. Ax
+ W^y^
The parameters of each realization enter into the matrices 7©, W^a and the vectors h^^, q&. The decisions are assigned to the first and second-stage vectors x and jco» which belong to the polyhedral sets X and Y with integrahty requirements. The jc-vector represents "here and now"-decisions which are applied regardless of the future evolution and thus have to be identical for all scenarios. In contrast, the joj-vectors denote scenariodependent recourses under the assumption that the respective scenario realizes. The objective is to minimize the first-stage costs plus the expected second-stage costs calculated using the weighting-vectors c and q^^. 2.2. The hybrid evolution strategy based on stage decomposition The stage decomposition separates the variables and the constraints of the SIP (1) according to the stages into the master-problem (2) and Q subproblems (3). Due to the integrality requirements in 7, the implicit function QJx) is in general nonconvex. Q

min f,(x) = c''x + Y,'^MJx) QJx) = min f,jyj=qly^

^-^' Ax
eIR"''xIN"'",

(2)

sYJsIR"^'xIN"^". (3)

The main idea of the hybrid algorithm is to use an evolution strategy (ES) [7] to address the master-problem (2). An ES works on a population of individuals (pool of solution candidates). Here, the ES interprets a certain x as an individual and the corresponding fi(x) in (2) as the individual's fitness. The fitness is evaluated by solving the independent subproblems (3) using a standard MILP solver (e.g. CPLEX [8]). The ES iteratively applies the following operators: The variation operator generates X offspring from a population of ju parents by mutation and possibly by recombination. The mutation operator varies a single individual by adding values from a random function scaled by the adaptive mean mutation step size. The recombination operator generates an offspring by taking each of the offspring's parameters from one of two randomly selected parents with equal probability. The selection operator chooses the best individuals from the set of parents and offspring as parents of the next generation. When a parent individual exceeds a maximum age K, it is not further considered in the selection and thus deleted. The distinguished features of the ES used here are that the same representation is used for fitness evaluation and for variation (called natural representation), and that the mutation strength is evolved together with the optimization variables. The ES handles the feasibility constraints in (2) and (3) by a modified objective that prefers feasible to infeasible solutions. If jc is infeasible in (2), a high penalty is assigned

Stochastic Integer Programming in Chemical Batch Scheduling

1969

to/} without solving the subproblems. Else, if jc is infeasible at least in one subproblem (3),// cannot be evaluated and a low penalty is assigned. This allows to prefer solutions with feasible first-stage solutions within the infeasible solutions. The main algorithmic parameters of the ES are the number of parents |a, the offspringparent ratio X/ja,, the specification of the mutation and recombination operators, and the termination criterion for the solution of the subproblems. 23. The branch-and-bound algorithm based on scenario decomposition For the scenario decomposition, the SIP (1) is extended by copies of the first-stage variables for each scenario. To force the first-stage variables to be identical for all scenarios, nonanticipativity constraints are added. When these constraints are relaxed, the problem decomposes into subproblems corresponding to the scenarios. When the scenarios are decomposed, their first-stage solutions possibly vary over the scenarios. Caroe and Schultz [9] use the scenario decomposition based on the Lagrangian relaxation of the nonanticipativity constraints. They proposed a branch-and-bound (BB) algorithm to re-establish these constraints. At each node, a tight lower bound is provided by the Lagrangian dual. Each calculation of this bound requires the solution of the Q scenario subproblems. A candidate solution for the upper bound is generated by simple rounding heuristics from the Q possibly different first-stage solutions of the scenario subproblems. The candidate solution is evaluated in the primal problem (2)-(3). In contrast to the hybrid ES, this algorithm uses lower bounds to cut off parts of the search space.

3. Chemical batch scheduling example Fig. 1 shows the real-world multi-product batch plant for the production of expandable polystyrene (EPS) [4]. Two types (A, B) of the polymer in five grain size fractions each are produced from raw materials (E). The preparation stage does not limit the production process and is not considered here. The polymerization stage is operated in batch mode and controlled by recipes. Each recipe defines the product (A or B) and the grain size distribution such that each batch yields a main product and four coupled products. The batches are immediately transferred into the semi-continuously operated storage tanks of the finishing stage A or B. The finishing lines separate the product by grain sizes and are operated continuously. Each finishing line can be stopped and restarted after a certain period of time. The medium term scheduling problem the optimal choices of recipes and their starting times such that the profit is maximized are considered. The profit is calculated from sales revenues, production costs, storage costs, and penalties for lateness and for finishing line start-ups and shut-downs. L<^

K

1

i}

H

1

Ec>-

^0000 ®

%%\ Preporation

soti on

Fig. 1. Theflowsheet of the multi-product batch plant.

1970

J.Tilletal

Table 1. Properties of the SIP test models. model

optimal solution

Ist-stage vars. cont.

integer

Ist-stage constraints 4

I

-9.25

0

24

II

-14.21

0

12

0

2nd-stage vars. cont. 106 118

scenarios

integer

2nd-stage constraints

16

117

8

28

121

8

a

Two test models are investigated. Both are based on a five-period MIL? model. In model I, the uncertain demands are represented by 8 scenarios. The first-stage decisions are the production decisions of the first four periods. In model II, the 8 demand scenarios are extended by uncertain capacities which are due to a failure of one reactor at different times within the scheduling horizon. In this model, the first-stage decisions are the production decisions of the first and the second period. As a result of the uncertain capacity, not all first-stage feasible decisions have a feasible completion in the second-stage. A SIP is called a SIP with relative complete recourse when each first-stage feasible solution is second-stage feasible. Model I has a relatively complete recourse whereas model II does not. The problem dimensions are shown in Tab. 1. The global optimal solution was calculated by solving the SIP by CPLEX. 4. Numerical evaluation The performance of the hybrid ES and of the BB algorithm was compared and the effects of algorithmic parameters and model properties were investigated. For the ES, parameter combinations from )LI=[5, 10,20], X/|i=[3.5,7,14], K=[5,OO], each with and without recombination operator, were investigated. For the BB algorithm, 30 settings including different termination criteria for the Lagrangian dual and for the subproblems as well as different choices of the rounding heuristics were considered. The BB algorithm is initialized by the first-stage solution from the expected value (EV) problem. In the EV problem the uncertain parameters are replaced by their expected values. The initial ES population comprises the EV solution and the Q solutions corresponding to the scenarios. For both algorithms, the termination criterion of the MILP subproblems is a relative integrality gap of 1%. The performance of both algorithms is evaluated by its the best objective found at a certain CPU-time. However, since the ES is a randomized algorithm, its evolution over time is a random variable. This randomness is considered by testing each ES setting five times. Then the minimum and the maximum best objective at each point of time over the five runs is taken to define the expected range of the best ES solutions. It is highly probable, that further runs of the same setting will also stay within that range. The most sensitive parameter of the ES is the structural decision to apply or not to apply the recombination operator (see Fig. 2). Without recombination, the expected range of the best ES solutions stays within a relatively small band and the convergence rate is approximately reproducible; with recombination, some runs converge much faster while others get stuck at the initial objective and the convergence rate is rather unpredictable. This behavior does hardly depend on the model. A highly sensitive parameter of the BB algorithm is the rounding heuristic used to generate upper bounds. By using heuristic A (average of the first-stage solutions of the subproblems rounded to integer) no other than the initial solution was found for model I and no feasible solution was found for model II (see Fig. 3).

Stochastic Integer Programming

in Chemical Batch

Scheduling

1971

—optimum CZlreco. on ^Sreco. off

50

100 CPU-time (sec)

150

50

100 CPU-time (sec)

150

b) Model 11.

a) Model I.

Fig. 2. The expected range of the best objective found by the ES (|LI=10, XI\X=1, K=OO) in 5 runs. -5

[

J

J

—" heuristic A

-6

—optimum

-9|

""'optimum

|

0,-11

1-7 '^ o -8

— heuristic A

-io|

^1

'••••>•.

-13

i...

:

-14

-9

50

100 150 CPU-time (sec)

0

a) Model I.

50

100 150 CPU-time (sec)

200

b) Model II.

Fig. 3. The best objective found by the BB algorithm. The latter is caused by the incomplete recourse property: The expected value solution is not feasible for all scenarios because it violates the polymerization capacity in some scenarios. With heuristic B {take the first-stage solution of the best subproblem) the performance of the algorithm is significantly improved. However, the performance of a certain heuristic for a given model seems to be unpredictable. Another important parameter of the BB algorithm is the termination criterion of the Lagrangian dual, which is solved in the root node. When the termination accuracy is too low, the lower bound is not very tight and the BB algorithm converges slower. If the accuracy is too high, the solution of the Lagrangian can easily consume the total time available for the calculations. 5. A problem specific evolutionary algorithm To exploit the full potential of the hybrid approach, an evolutionary algorithm was designed, which is specifically tailored to the scheduling of the production of polystyrene [10]. A decision tree is used which represents only the feasible solutions. Specific metrics-based mutation operators with scalable mutation strength were specifically designed. The more powerful search operators and the reduced search space complexity lead to a significant reduction of the computation time.

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6. Conclusion In this paper, two decomposition-based algorithms for stochastic integer programs were compared and evaluated. One algorithm is based on stage decomposition and uses an evolution strategy as master algorithm and a standard MILP-solver for the subproblems. The other algorithm is based on dual scenario decomposition and uses an exact branchand-boimd algorithm. A real world scheduling problem with uncertainties in demands and resource availabilities was used as a test case. It was found, that the ES without recombination converges for the test cases using its default settings. The convergence is predictable and not much related to the properties of the stochastic model. In contrast, the performance of the exact approach depends on the choice of rounding heuristics. For well adapted parameters, both algorithms have a similar efficiency. The ES is more general to use, whereas the BB algorithm requires specific tuning. The evolution strategy presented constitutes a further step towards the application of stochastic integer models in a model predictive scheduling fi-amework. A further improvement of the ES is expected firom combining its metaheuristics with the ability of the branch-and-bound algorithm to discard large parts of the search space without explicit enumeration.

References 1. Sahinidis, N. V., 2004, Optimization under uncertainty: state-of-the-art and opportunities. Computers and Chemical Engineering 28, 971-983. 2. Birge, J. F.; Louveaux, F., 1997, Introduction to stochastic programming. Springer. 3. Floudas, C. A. and Lin, X., 2004, Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review. Computers and Chemical Engineering 28, 2109-2129. 4. Sand, G.; Engell, S., 2004, Modelling and solving real-time scheduling problems by stochastic integer programming. Computers and Chemical Engineering 28, 1087-1103. 5. Till, J., Sand, G., Engell, S., Emmerich, M., Schonemann, L., 2005, A hybrid algorithm for solving stochastic scheduling problems by combining evolutionary and mathematical programming methods. European Symposium on Computer Aided Process Engineering 15, Puigjaner, L. and Espuna, A. (Eds.), 187-192. 6. Till, J., Engell, S., Sand, 2005, Rigorous vs. stochastic algorithms for two-stage stochastic integer programming applications. International Journal of Intelligent Computing (IJIT) 11, Special Issue ICIC 2005, 106-115. 7. Beyer, H.-G.; Schwefel, H.-P., 2002, Evolution strategies - a comprehensive introduction. Natural Computing, 1, 3-52. 8. CPLEX, 2002, Using the CPLEX callable library. ILOG Inc., Mountain View, CA. 9. Caroe, C.C.; Schultz, R., 1999, Dual decomposition in stochastic integer programming. Operations Research Letters 24, 37-45. 10. Emmerich, M., Urselmann, M., Till, J., Sand, G., and Engell, S., 2006, Hybrid EA/MIP method for solving two-stage stochastic programming problems in chemical batch scheduling. 7th Intemational Conference on Adaptive Computing in Design and Manufacture (ACDM 2006), Springer, in press. Acknowledgement We gratefully acknowledge the fruitful cooperation with R. Schultz and E. Clostermann of the Department of Mathematics, Universitat Duisburg-Essen, Germany as well as the financial support by the German Research Foundation (DFG) for the Collaborative Research Center Design and Management of Complex Technical Processes and Systems by Means of Computational Intelligence Methods (SFB 531) at Universitat Dortmund.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Scheduling and Planning with Timed Automata Sebastian Panek^, Sebastian EngelP, and Olaf Stursberg^ ^Process Control Lab (BCI-AST), Department of Biochemical and Chemical Engineering, University of Dortmund, 44^^^ Dortmund, Germany. Abstract This paper reports on the successful appUcation of a recently proposed method for solving scheduling problems by reachability analysis of timed automata. The jobs and resources are modeled by a set of synchronized timed automata, and the scheduling problem is solved by a cost-optimal symbolic reachability analysis. It aims at finding a path of states and state transitions from an initial to a terminal state in which all jobs have finished. The appeal of the approach is the intuitive and decomposed graphical modeling, and an efficient solution, as is illustrated here for a real-world case study from the chemical industry. K e y w o r d s : supply chain, scheduling, timed automata, reachability analysis. 1. I n t r o d u c t i o n Research efforts in the area of batch plant scheduling (batch sizing, resource allocation, sequencing) have produced powerful and general modeling schemes such as STN [7] and RTN [6], which lead to optimization problems t h a t can be solved by mixed-integer programming (MIP) tools. The application of these approaches is limited, however, since building mathematical models is tedious and the solution of large scale problems is still computationally intractable [5]. Timed Automata (TA) were proposed in [2] as an extension of finite automata by clocks in order to model timing constraints. The original purpose of TA was to model and to analyze properties of timed systems, such as real-time software, embedded systems, controllers, and network protocols. The algorithmic verification of TA comprises reachability analysis to determine the reachable subset of states, and to check if formal properties (as safety or liveness) are satisfied for the latter. Relatively efficient and user-friendly tools as e.g. Uppaal [4] or IF [8] have been developed for this purpose. Recent research revealed that reachability analysis of TA is also suitable for optimization, and scheduling. In the context of scheduling, the objective is to find a path from an initial state to a target state in which all operations are finished while satisfying all timing requirements. If such a path is found and fulfills an optimization criterion, the solution to the scheduhng problem is obtained. A systematic approach to job-shop scheduhng with TA was discussed in [1], and promising results on hard job-shops benchmark instances were reported in [4]. Furthermore, specialized software tools like Uppaal-CORA [4] and TAopt [10,11] were developed for cost-optimal reachability analysis of Priced TA (PTA). In the PTA modeling scheme, transitions and staying in locations cause costs, such t h a t the overall cost of a path (or schedule) is equal to the sum of the transition costs and the integral

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costs for the residence in states. Cost-optimal reachability analysis for PTA aims at finding the cost-optimal path from an initial to a target state. 2. A B r i e f I n t r o d u c t i o n t o T i m e d A u t o m a t a We only give a short and informal definition of TA here, and refer for complete definitions and for PTA to [3]. A Timed Automaton is a tuple (L,/o,C',/, T) in which: L is a set of locations with an initial location /Q G L, and C is a set of clocks; / is a set of invariants, i.e. conjunctions of simple constraints c < /c, where c ^ C and A; G R. A location can only be active if the invariant is true. T is a set of transitions (/, p, a, r, /'), each of which leads from / G L to /' G L; the guard ^f is a conjunction of constraints c ~ / c o r c — c ' ~ A : with c,c' G C, A: G R, and ~G { = = , 7^, < , > , < , > } ; a is an action label and r C C is a set of clocks to be reset after taking the transition. A transition can only be executed if the guard evaluates to true. TA models are created in a modular fashion as sets of interacting TA. The system evolves by taking synchronized transitions, i.e. two or more automata change their locations simultaneously. A common way of interaction are binary synchronizations which are established by synchronization labels attached to transitions. Parallel composition combines a set of TA into one composed TA by considering synchronizing transitions. (Some implementations of TA support also interaction by shared variables.) Once a composed TA is constructed, a (cost-optimal) reachability analysis can be performed to derive schedules. This enumerative technique starts with an initial location of the composed TA and stepwise evaluates the so-called successor relation^ i.e. successor locations of an already reached location are determined. This is repeated until a termination criterion is satisfied, e.g. the target location is reached. Due to the concurrency of the single TA, the composed automaton is often nondeterministic, and the reachability analysis produces a diverging reachability graph while searching for an evolution that reaches a target location. If models in form of P T A are chosen, the objective is not only to find some path that satisfies all timing constraints, but to find the cheapest one with respect to the costs. 3. E x a m p l e : Scheduling of a Lacquer P r o d u c t i o n For illustration, we consider the scheduling for a lacquer production in a pipeless plant [9]. The plant facilities consist of 5 mobile mixing vessels and 9 stationary processing units which are a pre-dispersion line, a main dispersion line, a special pre-dispersion unit, a dose distributor, two identical dose spinners, two identical filling stations, and a laboratory for quality checks. Three basic recipes for lacquers, each with 6 to 8 operations are given, as shown in Fig. 1. The recipes involve timing constraints between individual operations, and parallel allocations of stationary and mobile units. Each of these constraints establishes a link between two operations and forces them to either start or to end within a time window defined by the constraint. During the processing steps, no material fiows from and to the vessels are considered such that the amount of product in the batch remains constant. Raw materials and the storage of end products are unlimited and available at any time. The market demand is represented by 29 production orders for different products and with irregular release dates and due dates ranging over

Scheduling and Planning with Timed pre-dispersion

dosing 1

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Automata quality check 1

dosing 2

quality check 2

filling

^K) mixing vessel main dispersion'^ 1

Figure 1. Recipe structure: the dashed pre-processing steps are not present in all recipes. several months. The objective is to meet the due-dates, i.e. to compute delay-free schedules. We assume mixing vessels of equal size with sufficient capacities to hold one batch of each product. For quality checks contained in the production sequence, we furthermore assume that the first quality check always fails and the second one always succeeds. Since the lab is an unlimited resource, quality checks may take place at any time and arbitrarily often, but require a fixed amount of time. This allows us to neglect the quality checks and the lab in modeling, but appropriate timing constraints must be established to ensure a required minimum time between the dosing, correction and filling operations. In total, we consider 29 batches of different lacquer types in various colors for which 144 operations have to be scheduled. 4. T h e T i m e d A u t o m a t o n M o d e l of t h e E x a m p l e P r o b l e m The basic principles of modeling with timed automata can be seen in the simple model in Fig. 2. The left automaton models the operation and the right one models the required resource. First, the automaton waits until the resource is available and takes the first transition to allocate the resource. The resource automaton transitions simultaneously from idle to busy triggered by the synchronization label a received from the operation automaton. The clock c is reset to measure the duration of the operation, modeled by the invariant c < d, and the next transition is enabled by the guard condition c > d. After d time units, the first automaton changes from exec to done, modeling that the operation is finished. At the same time, the resource is released, i.e. the corresponding automaton changes back from busy to idle by taking the synchronized transition labeled by 0. Following these principles, the TA model for the lacquer production problem is derived by constructing interacting TA for jobs and resources. A part of the model with one job and two resource automata is shown in Fig. 3. In general, the modeling scheme involves the following components: A clock t is introduced to measure the absolute time. For each production order (defined by its release date, due date and a recipe number), one job automaton and a clock Q are used.

operation

wait

exec

done

resource

idle ^ ^ ^ ^ b u s y

Figure 2. A simple operation and the corresponding resource modeled as timed automata.

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S. Panek et al dosing 1 Ci < 1058

dosing 2 Q < 411

filling d < 1322 r. > -^q^

r^-rO

^2

t < 5700 Q > 120

022 Q > 411 Q < 411

quality check 1

Q > 1322

a •= 0

'Cj

:=

a > 1322

Ci< 1322 quality check 2 dosing station 2

busy

—•I

busy

idle

Figure 3. One job automaton and two resource automata for both dosing stations. The recipe determines the general structure of the automaton, i.e. the locations, transitions and clock constraints. In Fig. 3 the leftmost transition is enabled bycomparing the clock t to the release date 600, and a deadline is established by enabling the rightmost transition only when the due date is not yet missed {t < 5700). For each resource, a TA is created by defining two locations idle and busy. For each operation that can use this resource, a pair of transitions is introduced: one from idle to busy and one in the opposite direction. For each allocation, the resource automaton must synchronize with the corresponding job automaton, where transitions allocating a resource are labeled by a, and transitions releasing a resource by (j). For each alternative resource on which a task can be executed, a separate transition is introduced (into the job automaton) which synchronizes with a transition of the corresponding resource TA. In Fig. 3, both dosing operations can be executed on two different dosing stations. Since the allocation of mixing vessels does not have fixed durations, the 5 vessels are modeled by a shared integer variable v which is decreased by 1 when a vessel is allocated and increased by 1 when it is released. The following limitation of the modeling scheme should be mentioned: continuous degrees of freedom (such as batch sizes) cannot be modeled directly but must be approximated by discrete decision alternatives. 5. C o m p u t a t i o n a l E x p e r i m e n t s a n d R e s u l t s To compare MILP-based and the TA-based solution approach for the case study, a test environment of a 2.4-GHz Pentium 4 machine with 1 GB memory and SuSE Linux 8.2 was chosen. The MILP model was solved with GAMS 21.7 and CPLEX 9.0.2. In the GAMS model, a sequential order-based formulation [9] was combined with an EDD heuristics. The computation time of G A M S / C P L E X was limited to 2 hours and a configuration of CPLEX with the non-standard setting dpriind = 1 and an optimality gap of 0 was chosen. For the comparison, the tool TAopt was used to derive schedules by cost-optimal reachability analysis [11]. It combines a reachability algorithm for TA with specific state-space reduction techniques, and enables to use different cost functions, including the minimization of the makespan, or of storage and delay costs. From the input information (recipes for the lacquers, resource data, and a table of pro-

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Table 1 Feasibility problem (FP) and Makespan optimization (MO): computation time in sec. (time), index of the node representing the solution (ind.), the total number of nodes (nds.), number of discrete variables (disc), number of continuous variables (cont.), number of equations (eq.), length of the path (len.), makespan (ms.). FP GAMS/CPLEX TAopt len. time ind. jobs time ind. disc. cont. eq. 10 0.1 125 0.6 0 833 1115 7877 128 0.2 15 2.7 0 1851 3232 17688 206 200 20 7.7 20 0.3 273 265 3219 3797 31141 19.2 25 40 0.5 418 335 4976 5702 48488 135.4 192 29 0.6 490 389 6677 7519 65148 GAMS/CPLEX MO TAopt ms. ms. jobs time nds. disc. cont. eq. time nds. len. 21904 10 6 28 125 833 1115 7877 21470 176 10^ 15 58 1851 3232 17688 29740 215 200 31046 300 10^ 241 20 1300 3219 3797 31141 39748 282 10^ 265 39748 7121 36600 4976 5702 48488 44111 309 25 335 47547 10^ 29 7200 18200 6677 7519 65148 59223 417 389 59223 10^ duction orders), TAopt automatically creates a modular TA model with one job automaton for each of the 29 orders. In addition, 14 automata were generated to model the pre-processing steps of the parallel branches contained in two of the three recipes. Finally, the model comprises 8 resource automata and one shared variable to represent the mixing vessels. The memory usage of TAopt was limited to 10^ visited nodes, where a node represents a combination of locations of the automata and clock valuations. The search algorithm of TAopt was configured to employ a combination of depth-first search with a cost minimization, and different techniques for state-space reduction (weak non-laziness, sleep-set method, EDD heuristics, and cutting of symmetric resource allocations - see [11] for details). For both approaches, the maximum tardiness of all jobs was fixed to zero and the EDD heuristics was applied to reduce the possible sequences of operations. In order to provide a fair comparison, two tests were defined: 1) finding a tardiness-free schedule (feasibility problem), and 2) minimizing the makespan of the tardinessfree solution. The results are shown in Tab. 1: Results of both approaches reveal that the feasibility problem is not very difficult and delay-free schedules can be computed quickly. Nevertheless, TAopt performs better, and for all instances schedules were computed in less than 1 second. The ratio between the number of nodes required to reach the solution and the length of the path shows that only a very moderate amount of backtracking was necessary. For the makespan minimization, CPLEX provided better or the same solutions as TAopt with respect to the makespan. While TAopt required longer computation times for problem instances with a small number of jobs, the situation seems to reverse for larger problems. The last row shows that CPLEX terminated due to the limitation of the computation time and thus the optimality of the solution cannot be guaranteed. The node limit of 10^ (required to account for memory restrictions) caused termination in all experiments with TAopt. Note that the

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computational effort of the reachability analysis is mainly limited by the available memory required to store the reachability graph, and the time limit of 2 hours was never reached by TAopt. 6. C o n c l u s i o n s The investigation of scheduling techniques based on timed automata was the theme of the research project AMETIST funded by the European Union ( h t t p : / / a i n e t i s t . c s . u t w e n t e . n l ) . TAopt was developed within this project, and the apphcation to the lacquer production problem shows that TA-based methods are competitive to more established techniques. From the example, it can be concluded that TA-based scheduling is advantageous compared to MILP optimization if feasibility (i.e. meeting deadlines) is the prime concern. In situations, where makespan minimization is crucial, state-of-the-art MILP solvers perform slightly better and have the additional advantage of providing information on the maximal gap to the optimum. We are currently extending the TA-based optimization algorithm by the computation of lower bounds from relaxed algebraic formulations in order to combine the strengths of both approaches [11]. REFERENCES 1. Y. Abdeddaim and O. Maler, Job-shop scheduling using timed automata, Comp- Aided Verification, LNCS 2102, Springer, 2001, pp. 478-492. 2. R. Alur and D.L. Dill, A theory of timed automata, Theor. Comp. Science 126 (1994), no. 2, 183-235. 3. G. Behrmann, A. Fehnker, T.S. Hune, P. Petterson, K. Larsen, and J. Romijn, Efficient guiding towards cost-optimality in UPPAAL, Proc. Tools and Algorithms for the Constr. of Systems, 2001, pp. 174-188. 4. G. Behrmann, K.G. Larsen, and J.I. Rasmussen, Optimal scheduling using priced timed automata, ACM Sigmetrics 32 (2005), no. 4, 34-40. 5. C.A. Floudas and X. Lin, Continuous-time versus discrete-time approaches for scheduling of chemical processes: a review, Comp. Chem. Eng. 28 (2004), 2109-2129. 6. M. lerapetritou and C. Floudas, Effective continuous-time formulation for short-term scheduling, Ind. Eng. Chem. Res. 3 7 (1998), 4341-4359. 7. E. Kondili, C.C. Pantelides, and R.W.H. Sargent, A general algorithm for short-term sched. of hatch operations - MILP formulation, Comp. Chem. Eng. 17 (1993), 211-227. 8. L. Mounier, S. Graf, and M. Bozga, If-2.0: A validation environment for component-based real-time systems, Proc. CAV'02, LNCS, vol. 2404, Springer, 2002, pp. 343-348. 9. S. Panek, S. Engell, and C. Lessner, Scheduling of a pipeless multi-product batch plant using mixed-integer programming combined with heuristics, Proc. Europ. Symp. on Comp. Aided Process Eng., 2005, pp. 1033-1038. 10. S. Panek, O. Stursberg, and S. Engell, Job shop scheduling by combining reachability analysis with linear programming, Proc. IFAC Workshop on Discrete Event Systems, 2004, pp. 199-204. 11. S. Panek, O. Stursberg, and S. Engell, Efficient synthesis of production schedules by optim. of timed automata, Accepted for: Contr. Eng. Practice (2006).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Novel continuous-time formulations for scheduling multi-stage multi-product batch plants with identical parallel units Yu Liu, I. A. Karimi* Department of Chemical & Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore, 117576 Abstract Scheduling production optimally in multi-stage multi-product plants is a very difficult problem that has received limited attention. While the case of non-identical parallel units has been addressed, the case of identical parallel units is equally worthy of attention, as many plants are or can be approximated as such. In this paper, we construct and compare several novel MILP formulations for the latter. In contrast to the existing work, we increase solution efficiency by considering each stage as a block of multiple identical units, thereby eliminating numerous binary variables for assigning batches to specific units. Interestingly, a novel formulation using an adjacent pair-wise sequencing approach proves superior to slot-based formulations. Keywords: MILP; multi-product; batch plant; scheduling; makespan. 1. Introduction Multi-stage multi-product batch plants with parallel units are quite common in the batch chemical industry. Scheduling of production operations is a routine activity in such plants. Due to the many alternate ways in which batches can be assigned to various units and produced in different sequences, the task of optimal scheduling is formidable. So far, the literature has focused mainly on plants with non-identical parallel units. However, there exist plants with identical parallel units in each stage or plants that can be approximated by that simplified model. For this difficult problem, this simplified configuration needs to be exploited, as it should be easier to solve than its more complex and general counterpart. This is precisely what we wish to focus on in this paper. Kuriyan & Reklaitis (1984) termed multi-stage, multi-product batch plants with identical parallel units as network flowshops and presented several heuristic algorithms for scheduling a set of batches for minimum makespan. They included setups in processing times, assumed no resource constraints, examined several variations of listscheduling and multi-fit strategies, and produced numerical results using many simulated test problems. After this work, a few continuous-time MILP formulations have appeared in the literature for this challenging short-term scheduling problem. Pinto & Grossmann (1995) presented a continuous-time formulation for sequenceindependent set-ups and no resource constraints. They used asynchronous time-slots on parallel time axes for units and tasks. To assign orders to these slots, they used tetraindex binary variables, i.e. (order in a slot of a unit at a stage). This obviously required a large number of binary variables. In addition, their formulation required one to prepostulate a number of slots for each stage, which, as we see later, is crucial in determining the number of binary variables and solution optimality. They used

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minimum earliness as the scheduling objective. Later, Hui, Gupta, & Meulen (2000) presented a formulation with tri-index (order, order, stage) instead of tetra-index binary variables. They used direct (adjacent) pair-wise sequencing of batches at each stage instead of slots. While they could deal with sequence-dependent set-up times, they could not solve problems with six or more orders to optimality. Recently, Gupta & Karimi (2003) developed several improved MILP formulations using the same sequencing approach. Compared to previous work, their formulations used fewer binary variables and constraints, and solved faster. In some cases, they even gave better objective values. All this work assumed unlimited intermediate storage and unlimited wait (UIS/UW) policy (Reklaitis, 1989). We have not included work on single stage process here due to lack of space, but readers can refer Lim & Karimi (2003). In this paper, we develop a novel continuous-time formulation for scheduling a set of batches in a multi-stage batch plant with identical units in each stage to maximize productivity or minimize makespan. We begin this paper with a detailed problem description and then develop an MILP formulation for this problem. Next, we compare our model with our two other slot-based models for this problem and the existing models for non-identical parallel to demonstrate the superiority of our model with three examples. 2. Problem Description Fig. 1 shows the schematic of a multi-stage, multi-product batch process with identical parallel units. The plant has ^S* batch stages (^ = 1, 2, ..., 5) with /w^ identical parallel batch units in stage s. In this paper, we assume the UIS/UW storage configuration and hope to address other configiirations in future communications.

Stage 1

Intermediate Storage

Stage 2

Stage S

Batches Fig. 1. Schematic of a multi-stage, multi-product batch process with identical parallel units The plant operates in a batch mode, i.e. produces individual batches of products rather than long campaigns of identical batches. Let / denote the number of batches {i= 1, 2, ...,/) that the plant must process in the upcoming scheduling horizon. Some of these batches may be identical. As is common for a multi-product plant, we assume that each batch follows the sequence 1,2, ...,Sof stages for processing. If a batch skips a specific stage, then its processing time is zero at that stage. Since the parallel units in each stage are identical, any unit can process any batch. In addition to the above process features, we assume the following.

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1. A unit cannot process more than one batch at a time. 2. Processing is non-preemptive. 3. Processing xmits do not fail and processed batches are always satisfactory. 4. Start of the current scheduUng period is zero time. 5. Neglect transition times. 6. The size of each batch is known a priori. 7. More than one unit cannot process a single batch. 8. Resources are unlimited. With these, we can state a short-term scheduling problem for the above process as follows. Given S, ms, and the processing times of batches at stages, assign batches to units and identify the times at which they should start/end their processing to optimize a scheduling criterion such as minimum makespan. 3. Mathematical Formulation Because the parallel units in each stage are identical, we can process a batch on any unit and the processing time will be the same. Thus, we define a unique processing time PTjs for batch / in stage s. Some recent formulations (Hui et al., 2002 and Gupta & Karimi, 2003) have successfully used pair-wise sequencing of batches for scheduling multi-stage batch plants with non-identical parallel units. In addition to sequencing pairs of batches on each stage, they had to assign batches to each unit separately. However, by doing this, they successfully reduced the number of binary variables compared to the earlier slotbased formulations in the literature. For plants with identical parallel units, we reduce the number of binary variables even further by eliminating the need to assign batches to units in each stage. The definition of batch sequence and sub-sequences in stages is shown in Fig 2.

^

2

r

1

r

2

^

2

r

1

r

2

1 ^2

^

o o o o o o

o o o o

^ I

Stage 1

2

o

I ms

Stage S

Fig.2 Continuous-time representation for sequence-based model To sequence pairs of batches on each stage, we define for /^/', i'elis'. if batch /' follows right after batch / on some unit in stage s

^»'^-lO cotherwise

r <=> <^

W2

Stage 2

1

o

000000

I

mil

V

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where, Ijs = {/' | batch /' can follow directly after batch z on stage s}. Since a batch / can have at most one successor and at most one predecessor, we write,

X^-^i

(la)

Because two batches / and /' cannot follow each other (subtour of two batches not allowed) on the same unit, we have, V.+^r/. ^1

(2)

Having sequenced the pairs of batches for stage ^ as a whole, we now need to ensure that the batches do not use more than rris parallel units at any time. We ensure this by breaking up the stage sequence into a subsequence for each unit. Then, we ensure that the number of such subsequences does not exceed w^. To this end, we define the following 0-1 continuous variables (Ujs and V/^): if batch / processes first on some unit in stage s otherwise if batch / processes last on some unit in stage s otherwise Since the number of first batches in stage s must equal the number of last batches and that number should not exceed the number of units in that stage, we have.

If a batch processes first (last) in a stage, then it cannot follow (precede) any other batch in that stage, and vice versa. In other words,

Y,Xus+^is=^

(4b)

The above two equations allow us to eliminate Ujs and Vjs. Then, substituting for them in eq. 3, we obtain, / . / . -^i'is ~ 2^ 2^ ^ii's

Novel Continuous-Time Formulations

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In fact, as long as / > m^, with no loss of generality, we can write,

i i'^Iis

Z

Z ^i'is=I-^s

(5b)

In the absence of pre-specified sequencing constraints, i.e. a job can follow any other job at all stages, eqs. 5a and 5b become identical and we ignore any of the two. However, if this condition does not hold, then both eqs. 5a and 5b must be used. In this paper, we ignore eq. 5b. If a batch i' follows /, then i' cannot begin processing, until / finishes. Therefore, tSi^,>ts,,^PT,^-H{\-x,,^,)

(6)

where, tSis is the start time of batch / on stage s as defined earlier. Note that we must write the above for all valid permutations of / and /'. Similarly, a batch / cannot begin processing on a stage s, until it has left the previous stage, thus, ^^/.^^^/(.-i)+^Vi)

s^i

(^)

Finally, we compute the makespan (MS) by using eq. 8. MS>ts,s+PT,s

(8)

Thus, our sequence-based formulation (Fl) comprises eqs. 1, 2, 5a, 6, 7, and 8. We have developed two different slot-based formulations (F2 and F3) also. For the lack of space, we do not present them in detail. However, we include them in the the following model evaluation section. 4. Model Evaluation We have three new formulations (Fl, F2, and F3) and two literature formulations (PG with /-w^+1 slots for stage s and GK). We evaluated these numerically with three test problems with varying numbers of stages and parallel units and progressively larger sizes. All batches pass through stages 1, 2, and so on, in that order. For the sake of brevity, we do not present the processing time data for the examples. We ensure that all stages in each plant are relatively well balanced in terms of using processing times and number of units in a stage. To this end, we have designed the average processing time at each stage to be proportional to the number of identical units in the stage for the entire evaluation, we used CPLEX 9.0 in GAMS 21.4 on a DELL workstation with a single 3.20 GHz Intel Pentium processor and 2 GB RAM running WINDOWS XP. Table 1 shows the performances of and statistics for various models. From Table 1, we see that F2 is unable to solve all these examples within 5000 s. F3 gives higher RMIP values for examples 1 and 3 and the same RMIP value for example 2, which indicates that F3 is tighter than all the other formulations. However, like PG, F3 is unable to solve examples 2 and 3 within 5000 s. Although GK is able to find all

1984

Y. Liu and LA. Karimi

the optimal solutions within 5000 s as Fl, the latter which involves fewest binary variables clearly is an order-of-magnitude faster than all the other four formulations. Table 1. Model and solution statistics for Examples 1-3 Example 1(5' = 2,mi = 2, W2 == 2,1=6) Model

Non-

Variables

RMIP

MIP

(tu)

(tu)

Nodes Relative CPU Gap (%) Time (s)

Binary

Conti.

Zeros

Constraints

Fl

60

73

504

158

51

74

0

0.50

1720

F2

72

325

1749

507

51

76

12.9

5000

2368233

F3

144

181

1474

423

63.5

74

0

16.2

17422

GK

84

121

1188

1382

51

74

0

1.71

2478

PG

140

205

1372

348

51

74

0

10.4

18566

Example 2 (5"= 3,Wi = 2, ^2 = 5, ^3 = 3,/=7)

Fl

126

148

1050

318

29

34

0

12.9

47003

F2

147

652

3543

1013

29

42

31.0

5000

953948

F3

294

358

2896

817

29

36

15.3

5000

1684525

GK

196

286

3444

836

29

34

0

132

178479

PG

336

463

3278

799

29

35

14.3

5000

6267340

Example 3(^=2, Wi = 2, W2 = 4,7=8)

Fl

112

129

928

274

53

76

0

12.0

44929

F2

128

561

3085

867

53

83

34.2

5000

837625

F3

256

305

2638

707

58.25

76

7.57

5000

2388991

GK

160

85

2784

678

53

76

0

3752

5835400

PG

360

407

2998

716

53

76

9.21

5000

5885695

5. Conclusion An MIL? formulation (Fl) using continuous-time sequence-based time representation was developed for scheduling multi-stage multi-product batch plants with identical parallel units. Evaluation on three examples shows that our formulation (Fl) is simpler and much faster than the best existing formulations and our other two slot-based formulations.

References 1. 2. 3. 4. 5. 6.

K. Kuriyan and G. V. Reklaitis, PSE Symposium Series, 92 (1984) 79. K. Kuriyan and G. V. Reklaitis, Comput. Che. Eng., 13 (1989) 187. J. M. Pinto and I. E. Grossmann, Ind. Eng. Chem. Res., 34 (1995) 3037. C. H. Hui, A. Guputa, and H. A. J. Meulen, Comput. Chem. Eng., 24 (2000) 2705. S. Gupta and I. A. Karimi, Ind. Eng. Chem. Res., 42 (2003) 2365. M. F. Lim and I. A. Karimi, Ind. Eng. Chem. Res., 42 (2003) 1914.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Routing and cargo allocation planning of a parcel tanker Kim-Hoe Neo^, Hong-Choon Oh^, LA. Karimi^* The Logistics Institute - Asia Pacific ^Department of Chemical & Biomolecular Engineering National University of Singapore, 4 Engineering Drive 4, Singapore 117576 Abstract A parcel tanker is an ocean carrier that has multiple compartments to hold multiple liquid chemical cargoes and prevent them from mixing. Management of parcel tankers entails several operational constraints. The types of cargoes that their compartments can hold depend on (1) construction materials and coatings of compartments and (2) contents of neighboring compartments. Furthermore, the weights of cargoes among the compartments must be properly distributed to ensure ship stability. Cargo loading and unloading, compartment cleaning requirements, and draft limitations complicate the parcel tanker operation and route planning even further. To the best of the authors' knowledge, none of the existing models that address routing and scheduling of parcel tankers considers the aforementioned operational constraints simultaneously. To address this research gap, we introduce a new routing and scheduling model that explicitly considers the unique operational limitations of parcel tankers. Essentially our new model involves deciding which ports the ship should visit and in which sequence, which cargoes it should pickup and unload, and to which compartments each cargo should be assigned and when over the entire trip so as to maximize the profit for the ship. Finally, we also apply our new model on a case study to illustrate the importance of including cargo compatibility and ship stability constraints in routing and scheduling of a parcel tanker. Keywords: parcel tankers, routing, scheduling. 1. Introduction Over the recent years, the global chemical industry has achieved consecutive annual positive growth in the value of world merchandise exports, which hit US$794 billion in 2003. As a result, the global demand for sea transport such as the versatile parcel tankers has remained strong. In fact, parcel tankers are currently built in record numbers (Shaw, 2003) by shipyards due to the increasing world demand. Unlike other bulk carriers, parcel tankers offer the convenience of transporting different liquid cargoes simultaneously. Most compartments of parcel tankers are equipped with separate pumps and lines so that multiple commodities can be handled at simultaneously. However, shipmasters of parcel tankers have to contend with two additional compatibility constraints that others can afford to ignore. The first constraint arises due to the fact that the compartments are usually coated with materials such as epoxy, phenolic resins, zinc silicate, polyurethane, or rubber and not all chemicals are compatible with all coatings. On the other hand, the second constraint arises due to the safety requirement stipulated

Corresponding author (E-mail: [email protected])

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K.-H. Neo et al.

by the International Maritime Dangerous Goods (IMDG) code that cargoes in neighboring compartments of parcel tankers must be non-reactive. Like other ocean carriers, a parcel tanker must also ensure proper weight distribution of cargoes among the compartments so that it does not trim or heel excessively. Trim is the difference between the draughts forward and aft. An excess draught aft is called trim by the stem (i.e. the rear end of a ship), while an excess draught forward is called trim by the bow (i.e. the forward end of a ship). To ensure ship stability during its voyage, moments that arise due to trims by the stem or bow must not exceed their respective limits which depend on the ship design. On the other hand, heel is basically a measure of the extent of left or right tilting of a ship. Similar to trim, moment that arises due to heel must not exceed its limits to ensure ship stability. Clearly, optimal assignment of cargoes in and schedules for a parcel tanker is a complex combinatorial problem. However, only one existing model (Jetlund and Karimi, 2004) in literature addresses parcel tanker routing and scheduling problem. But this model fails to accommodate the aforementioned unique operational limitations of parcel tankers. This paper aims to address this research gap by introducing a new model that (1) models these operational constraints for optimal routing and scheduling of parcel tankers and (2) serves as an effective basis for various possible extensions that are relevant to the industry. 2. Problem Description Essentially, our problem is similar to the 1-ship problem described in Jetlund and Karimi (2004). We make the same assumptions as in that work with the following information available at the start of the planning horizon: (1) current location of parcel tanker, (2) the set L (loaded) of cargoes on board the parcel tanker, (3) the set U (unloaded) of potential cargoes that can be served by the parcel tanker, and (4) pickup, discharge ports (/ = 1, 2 , . . . , P) and laycan (permissible period of pickup) of each cargo j (j e U). We use / (/ = 1, 2, ..., TC) to denote the compartments of the parcel tanker, where TC is the total number of compartments. The objective of this single parcel tanker routing and scheduling problem is to (1) select the cargoes that it should load and unload, (2) assign cargoes to compartments, and (3) devise an optimal route and schedule to maximize its profit. 3. Problem Formulation Our problem formulation is basically an extended version of that stated in Jetlund and Karimi (2004) for 1-ship routing and scheduling. All the equations and notation used by Jetlund and Karimi (2004) remain valid in this paper and readers may refer to that paper for details on these equations. We now describe the additional constraints that we impose in our parcel tanker routing and scheduling problem. Unless stated otherwise, all indices assume their ftill values. To represent if a cargo j is stored in compartment / at the end of leg A: (A: = 0, 1, 2, ..., K) of the tanker, we introduce a binary variable U^k where, [l if compartment / carries cargoy attiie endof 1^ A: ^^* loothawise In the model of Jetlund and Karimi (2004), binary variable Yjk defines whether or not the parcel tanker carries cargo y on board during leg k. Thus, Uijk and Yjk must be related by the following to ensure mathematical legitimacy.

Routing and Cargo Allocation Planning of a Parcel Tanker

1987

Z^'/'ijk^yjk

(Ai)

^Uy,
(A2)

/

For a similar reason, we have the following two constraints to relate Uijk with Yj, where the latter is a binary variable that equals one if parcel tanker serves cargo j during the planning horizon and zero otherwise.

ZZ^p^^y /

(A3)

k

YL^m^YjTC /

(A4)

k

Each compartment can either be empty or only hold chemical from at most one cargo during any sailing leg. If Um equals one when / is empty during leg k and zero otherwise, then we also have,

f/;«+Z^P = *

(A5>

J Each cargo j may be loaded into more than one compartment of the parcel tanker. We let Wijk be the weight of chemical from cargo 7 that is loaded into compartment / during leg k. Clearly, the volume of chemical in a compartment cannot exceed the storage capacity of that compartment, i.e., W,j,
(A6)

where, Vi and Dj denote the maximum storage volume of compartment / and density of cargo 7 respectively. In addition, the total weight of cargo 7 in the compartments of the parcel tanker must equal to the known weight of cargo7 {Wj) that is to be served. Thus, we also have, Yj^ijk=WjYj,

(A7)

The moment causing trim of a parcel tanker during sailing leg k is 2^ 2^ ^Ijkh where /

j

ij is the longitudinal distance of compartment / from the tanker's centre of flotation. The latter refers to the point in the water-plane (i.e. cross-sectional area of the hull which is at the water level of the tanker) with zero first moment. On the other hand, the moment causing heel of a parcel tanker during sailing leg k is / , / ,'^lfk^l , where KI is the / J lateral distance from compartment / to the tanker's centre of flotation. Assuming that the centre of flotation of the tanker does not vary significantly during its voyage, all the ii and Ki values are constants over the planning horizon. Note also that all the z/ and KI values are measured with respect to one reference longitudinal and lateral direction

1988

K.-K Neo et al.

respectively. Since the moments causing trim and heel of the tanker have to be less than their respective limits to ensure stability, we have,

~a
(A8)

J

-p
(A9)

j

where, a and ^ are respectively the maximum absolute permissible moments causing trim and heel of the tanker during its voyage. The draft of the ship cannot exceed a limit {Dynax) set by individual ports concerned. Given any load, the draft of a parcel tanker reaches its maximum, when its trim angle is at its maximum permissible value, OmaxThus, we write, KZZ^/^+^)/^>^]-(^^^^-ax/2)
(AlO)

J

where, ^, p, X, TU, denote the weight of empty tanker (no cargo), density of water, length and cross-sectional area of parcel tanker submerged in water respectively. We assume all ports share the same Dmax, and Omax, ^, p, ^ and TT values do not vary significantly during the parcel tanker's voyage over the planning horizon. Based on its coating, each compartment / of the tanker has a predetermined list (LQ) of chemicals that it cannot store. Thus, compartment / cannot hold cargoes (j) that are members of this list (i.e. Uyk = 0,j G U D LQ). Similarly, we define NQ as the set of neighbouring compartments (as defined in IMDG code) of /. For a cargo 7 (J e U KJ L) that is loaded into a compartment, there is a set of other cargoes that must not be stored in the neighbor compartments of 7 according to the safety regulations of the IMDG code. We use ICj to denote this set of incompatible cargoes. Thus, we have,

^ijk+ Z

Z ^'V*^l

(All)

The shipmaster of a parcel tanker may change the contents of its compartments to ensure cargo compatibility and ship stability. We use COyj to denote the fixed changeover cost incurred when the content of compartment / is changed from liquid chemical of cargo j to that of cargo j " where 7, j ' e {0, Z u U} with 0 representing empty clean compartment and COijj= 0. To compute the total changeover costs over the planning horizon, we define a transition variable, Mjjj'k (A: > 1) as: fl if compartment / holds cargoy during leg (k -1) andy' during k ^^^'^ ~ | o Otherwise Recall that A: = 0 in Jetlund and Karimi (2004) represents the tanker's current sailing leg at time zero. Clearly, Mijj'k =Uijk.iUij'k and we replace the nonlinear term by the following linear equivalents as in Jetlund and Karimi (2004). ^M,if'k=Uij,-i

k>l

(A12)

Routing and Cargo Allocation Planning of a Parcel Tanker

1989

'Yj^ijrk=u,r,

(A13)

k>\

J

These two constraints allow us to treat Mijj-k as 0-1 continuous variables. We use an objective function that is similar to eq 15 in Jetlund and Karimi (2004), except that we add compartment changeover costs for cargos (i.e. XiZ_i X i LL ^^W'k^ijj'k ) as an operating expense for the tanker over the k>l

I

;G{0,UUL}7'G{0,UUL}

planning horizon. Thus, we write, I

Profit = ^SRjYj

-^FCvTTk-^TCCI

ZZ^^^^'^-ZZ Z /•

k

y

vm

^

T^ + X DRj ~

Z ^^^^/^-^

(A14)

k>l I ye{0,UuL}7'e{0,UuL}

where, the summation terms on the right hand side are revenues from serviced cargoes, sailing fiiel cost, time-charter cost, all port charges, and compartment changeover costs respectively. This completes our formulation for the parcel tanker routing and scheduling problem in the presence of cargo compatibility and ship stability constraints. It is basically a mixed integer linear programming (MILP) model that comprises maximizing profit (eq A14) subject to eqs 1-14 of Jetlund and Karimi (2004) and eqs A1-A13. We now use a case study to illustrate the significant impact of cargo compatibility and ship stability constraints on routing and scheduling of a parcel tanker. 4. Case Study We apply our model to a parcel tanker with 10 compartments and 10 possible cargoes for pickup from 5 ports. The reader may obtain the frill data for our case study by contacting the corresponding author. We solve our model for two scenarios. In scenario 1, we include the cargo compatibility and ship stability constraints, and we ignore the same in Scenario 2. Key aspects of the solutions of these two scenarios are summarized in Table 1. The solutions were determined using CPLEX 9.0 in GAMS (Distribution 21.4) running on a Windows XP PC with a Pentium 4 (3.0 GHz) processor and with relative optimality gap limit (i.e. OPTCR) set to 0.00005. The model for scenario 1 involved 14,333 continuous variables, 1143 binary variables, 37,248 constraints, and 153,727 nonzeros, while that for scenario 2 involved 14,333 continuous variables, 1143 binary variables, 3948 constraints, and 48,211 nonzeros. CPLEX solved scenario 1 in 18,335 s and gave a maximum profit of $78K ($3.6K), while it solved scenario 2 in 72 s and gave a profit of $140K ($4.5K), where the numbers in the brackets are the total cleaning costs due to compartment changeovers in the two scenarios. Evidently, omitting cargo compatibility and ship stability constraints led to a routing and scheduling plan with a profit that is almost double that of scenario 1. Moreover, the plan in scenario 2 has several instances where the corresponding cargo pickup and allocation among compartments violate the cargo compatibility or ship stability constraints. This clearly illustrates the importance of accounting for cargo compatibility and ship stability constraints during the routing and scheduling of a parcel tanker.

1990

K.-H. Neo et al Table 1. Key Results of Solutions for the Two Scenarios

Scenario 1

Scenario 2

Leg ,,, ^^^

Port ^^ . , (Arrival TimeMir)

Loaded Cargo ,^ , (Compartment)

0

P,(1.62)

1

P3 (15.93)

2

P4 (20.40)

3

P5 (33.86)

(^^

, , . , (Arrival Time\hr)

Loaded Cargo ^^ (Compartment)

Jiids)

0"

Pi(1.62)

Jiidio)

Jl(]7.[8),J2(ll,l2,l3),

1

P4 (16.46)

j^(,3^,^)^j,(,^^i,^)

2"

P3 (19.43)

J,(l,,l4),J2(l5,l8,l9)

%'°

V nf, iQ\ t-scjo./y;

^'•C^' '5> '10), Wl?), J3(i,,i3,i„i,),j,„(i,)

4

P6 (38.84)

NIL

\ir\''Yif\a) J sill»A3, MJ k)^ J10U5J J6(l2),J8(l3,l4,l6,l9),

Leg

J 10(15)

4

P6 (35.36)

NIL

^The arrival time is with respect to the start of planning horizon. ^Ship stability constraints are violated. ^Cargo compatibility constraints are violated.

5. Conclusions This paper introduces tvs^o critical operational constraints pertinent to cargo compatibility that only the shipmasters of parcel tankers have to contend with. In addition, it also reports a new MIL? model for routing and scheduling of a parcel tanker that accounts for compartment changeover costs and other constraints that are common to all ocean carriers such as port draft and deadweight limitations. To the best of our knowledge, a parcel tanker routing and scheduling model that accounts for all the abovementioned features does not exist in the literature. However, our new model does have limitations that arise mainly due to the considerable model solution time as illustrated by the preceding case study. Clearly, this needs to be addressed before this model can be extended to account for more factors (variable cruising speed, centre of flotation, etc.) that will further enhance its industrial realism and application potential. References Jetlund, A. & Karimi, I. (2004). Improving the Logistics of Multi-Compartment Chemical Tankers. Computers and Chemical Engineering, 28, 1267-1283. Shav^, J. (2003). Forty-Five Years of Parcel Tanker History. Harbour & Shipping, November, 1617.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

An approximate framework for large multistage batch scheduling problems focusing on bottleneck resources Pablo A. Marchetti and Jaime Cerda"" INTEC (UNL - CONICET), Guemes 3450, 3000 Santa Fe, ARGENTINA ^E-mail: jcerda@intec. unl. edu. ar Abstract A rigorous representation of the batch scheduUng problem is often useless to even provide a good feasible schedule for many real-world manufacturing facilities. In order to derive simpler scheduling methodologies, some usual features of multistage processing structures should be exploited. A common observation in industry is the fact that most plants have very few bottleneck operations and the bottleneck resource controls the plant throughput. Consequently, the quality of the plant schedule heavily depends on the proper resource assigment and sequencing of bottleneck operations. Every other part of the processing sequence should be properly aligned so that the right amount of material required by the bottleneck timely arrives. This work introduces a bottleneck-based MILP scheduling model for multiproduct batch facilities that can also account for bottleneck resources other than equipment units and critical operations performed at different processing stages. Three large-scale examples have been solved at very low CPU time, despite near-optimal schedules are still encountered. Keywords: Scheduling, Multistage batch plants, Bottleneck, Optimization 1. Introduction The scheduling of large-scale multistage batch plants is an NP-complete combinatorial problem requiring a very high computer time even for finding a feasible solution. In order to derive simpler scheduling approaches, some usual features of multistage processing structures should be exploited. A common observation in industry is the fact that most plants have very few bottleneck operations. A bottleneck is any resource whose capacity is less than the demand placed on it. It may shift with the production mix. Since the bottleneck resource controls the plant throughput, it is working all the time and there usually is a buffer inventory in front of it to make sure that it never remains idle. On the contrary, a nonbottleneck resource has a capacity greater than the workload placed on it and does not work constantly. Idle time is a common feature of nonbottleneck resources and the main reason for the existence of alternative solutions with a similar performance. In turn, a capacity-constrained resource is one whose utilization is close to its capacity and may become a bottleneck if it is not carefiilly scheduled (Chase et al, 1998). By definition, the goal of the scheduling fiinction is to optimally allocate resources to processing tasks over time. However, not all resources but just those constraining the production flow through the processing sequence really deserve especial attention. Critical operations that should be carefiilly scheduled are those requiring the bottleneck resource. Every other part of the processing sequence should be properly aligned so that the right amount of material required by the bottleneck timely arrives.

1991

1992

P. A. Marchetti and J. Cerdd

The strategy of focusing the scheduhng effort on bottleneck resources was first explored by Goldratt (1990) through the so-called Optimized Production Technology (OPT) and implemented in different OPT-software packages commercially available. Though generic OPT-rules for the generation of production schedules are widely known, Goldratt's algorithm was never published. An OPT-package performs the following steps: (i) identify the bottleneck resource, (ii) schedule first just the critical operations for a most effective use of the bottleneck resource, (iii) schedule the upstream stages from the bottleneck so as to meet the scheduled starting times of bottleneck operations, and (iv) schedule the downstream stages assuming that the end times of bottleneck operations are given. As non-bottleneck resources have idle capacities, Goldratt's algorithm assumes that feasible schedules for upstream and downstream nonbottleneck resources can always be found. This work introduces a bottleneck-based scheduling model for multiproduct batch facilities. The proposed MILP framework can be regarded as a compact version of previous batch scheduling formulations (Mendez et al, 2001; Mendez and Cerda, 2002 & 2003) that still uses the general global precedence concept and handles limited discrete and continuous resources in a similar manner. As the best ordering of processing tasks at the bottleneck stage BS mostly determines the whole production schedule, the related sequencing decisions become the problem critical variables. Any pair of batches (z, i") allocated to the same unit at some non-bottleneck stage will be arranged as required by the bottleneck stage schedule. Consequently, a frequent sequencing pattern observed in optimal schedules is a common ordering of batches (/, /) at any stage whenever they share any resource item. Such a optimal pattern leads to a problem formulation with a unique sequencing variable Xw for any pair (/, i^ through the whole processing system. In this way, the number of sequencing variables and the computational cost are both diminished by orders of magnitude and near-optimal schedules can be efficiently encountered. The approach first assumes that the bottleneck resource is the set of equipment units running at the bottleneck stage and it is later extended to account for bottleneck resources other than equipment units and critical operations performed at different process stages. 2. Model Assumptions a) Setup times are sequence-dependent. b) Batch mixing and splitting operations are forbidden. c) The linear processing sequence includes a bottleneck resource and possibly a few other capacity-constrained resources. d) The ordering of each pair of batches at any processing stage is subordinated to the optimal ordering of them either at the bottleneck resource or at the next capacity-constrained resource.

3. Bottleneck-based Batch Scheduling Formulation Let us consider a multiproduct batch plant where the bottleneck resource is the set of equipment units running at the bottleneck stage BS. Then, the sequencing variables for the ^iS-stage are the critical problem variables also controlling the sequence of operations at non-bottleneck stages. The optimal sequencing pattern is a unique ordering of batches (/, 0 , similar to that followed at the shared bottleneck resource item, whenever they belong to the same queue. If not, such an optimal pattern still holds so that non-bottleneck operations end or start as required by either the bottleneck stage and/or the next capacity-constrained stage schedules. Therefore, the optimal sequencing

An Approximate Framework for Large Multistage Batch Scheduling Problems

1993

of any pair of batches (/, i^ through the processing system is fully specified by the critical variables Xu'. Since non-bottleneck stages always have idle capacities, there usually are several ways of arranging non-bottleneck operations so as to support the best batch ordering at the stages constraining the production throughput. In particular, the allocation of batches (z, i") to different equipment units at some non-bottleneck stages permits to automatically relax the unique ordering condition. 3.1. Unit allocation constraints: ^y;., = l

\fisI,seSi

(1)

3.2. Timing constraints: Sis = Cis - ^ ptisj Yisj Cis<Si,s + i

Vz el,se

S

yieI,seSi-{s/}

Sis > ^ [ruj + suisj) Yisj

Vz e I,se Si

(2) (3) (4)

jeJis

3.3. Sequencing constraints: Accounting for the model assumptions, whenever a pair of batches (z, z') is allocated to the same processing unity'G J/^rV/^ at some non-bottleneck stage s, the ordering of tasks (z, s) and (/', s) becomes determined by the bottleneck stage BS: Xjsj's = XIBSJ'BS = ^n'- If Xii' has no meaning for the bottleneck stage because tasks (i, BS) and (z', BS) are performed in two different units J'J'EJBS, then the rol of the bottleneck stage BS is taken by the next capacity-constrained stage (CCR) where batches (z, i^ share the same queue. Cis + Twj + surj < Si^ 5 + M(1 - Xii) + M{2 - Yisj - Y^ sj) Vz,z'€ /,5e Siv, jG [Jis n Jvs) : \i < V) Ci^ s + Tnj + suij <Sis-^M Xii' + Mil - Yisj -Yi'sj) ( \ ( \ \^-^) yij'e I,se Sii',je [Jis r\Jvs) : \i < V) 3.4. Objective Function: The problem goal is to minimize the overall batch tardiness. 3.5. Example 1: A large-scale multiproduct batch plant with a strong bottleneck stage To show the effectiveness of the proposed MILP approximate formulation, a large batch scheduling problem involving 5 processing stages and 12 equipment units has been tackled. Eight batches are to be processed over a time horizon of 100 hours. Problem data are given in Tables 1 and 2. Stage III performing the longest operations arises as the process bottleneck stage with almost no idle time. In order to compare their computational performance and the optimal solutions found. Example 1 was solved by using both the proposed approach and the previous formulation of Mendez et al. (2001). Results show that the bottleneck-based model provides the same optimal solution at much less computational effort (see Table 3 & Figure 1). The number of binary variables drops to a half of the previous value and the CPU time is 20 times less.

,.,.

1994

P. A. Marchetti and J. Cerdd Table 1. Data for Example 1

Stage I

Batch Ui

1 2 3 4 5 6 7 8

U2

8.5 10.2 9.5 8.4 9.7

9.8 7.4 8.1

9.6 10.8

9.7

0.5

0.8

Stage II U3

7.2 8.4 6.7

U4

Us

6.2 4.5 8.3

7.2 8.1 8.1 8.2 8.5

6.3 8.8 8.4 1.0

0.4

Stage III U6

Uy

18.2 15.2 17.1

12.4

12.7 12.5

15.5 16.6

Stage IV Ug

U9

9.1

Uio

11.2 10.5 10.2

10.5 11.8

16.4

17.8 15.4 16.7

10.1

0.4

0.3

1.0

0.4

Stage V

1.2

8.2 8.2

10.5 0.9

U12

8.2 8.2

9.8 9.5 11.7

Uii

Due Date 70 70 80 95 75 100 70 80

6.4 7.3 7.1 7.2

8.3

6.3 7.9

0.4

0.3

Table 2. Sequence-dependent setup times & steam/manpower requirements for Examples 2&3 Bi Bi B2

B2

B3

B4

B5

Bg

B7

Bg

1.4

0.7 1.9

1.1 0.9 1.5

1.4 1.0 1.4 0.8

0.7 2.0 1.0 1.1 0.8

1.1 1.6 1.9 1.1 1.0 1.5

0.6 1.6 2.0 0.6 1.7 1.4 1.3

0.7 0.7 1.6 1.4 1.5 1.7 3

1.9 0.7 1.8 1.6 1.1 2

0.6 2.0 0.8 0.7 1

0.8 0.7 1.1

Manpower

1.5 1.6 1.2 1.1 2.0 1.4 0.8 2

Steam (Stage I) (Stage IV)

9.0 7.0

7.0 9.0

9.0 12.0

15.0 9.0

Bs B4 B5 B6 B7 Bg

3

1.4 0.8 2

1.2 3

2

12.0 6.0

9.0 18.0

15.0 12.0

8.0 18.0

4. Extended approach to account for other types of bottleneck resources If the bottleneck is a resource other than equipment, then two different cases can be considered depending on whether the tasks requiring the bottleneck resource: (1) all belong to the same processing stage, or (2) they are performed at different stages. Case (1) looks quite similar to the one discussed in Section 3 and the notion of bottleneck stage or capacity constrained stage can be easily extended. Whenever a pair of batches belongs to the same queue the unique ordering condition, controlled by the critical variables sequencing batches at the bottleneck and/or at the next CCR stage, still holds. No additional binary variables are required and an important reduction in the problem size is further achieved. On the other hand. Case (2) involves the ordering of tasks competing for the same bottleneck resource at different processing stages. Tasks from different stages are not necessarily synchronized as the operations in the same stage and the unique ordering condition can no longer be applied to them. As a result, some additional sequencing variables and constraints are to be incorporated in the problem formulation but in a much lower amount than in previous approaches (Mendez and Cerda, 2002 & 2003). Let R be the set of resource types r other than equipment. Let us assume that the set of resource items of type r is given by Z^. Resource elements z ^ Zy for any discrete resource type r can be treated as unary items (Mendez and Cerda, 2002) or gathered into a small number of clusters or groups that can be assigned to tasks one at a time (Mendez and Cerda, 2003). Similarly, continuous resources can be partitioned into a low number

An Approximate Framework for Large Multistage Batch Scheduling Problems

1995

of sources whose capacities (t)^ are chosen by the model. Therefore: R = R^ Kj R^, where R^ includes the unary resources and R^ comprises discrete or continuous resources grouped on elements of unknown capacities. Proper allocation constraints are defined for each type of resource re R and for each resource item ze Zy. The allocation variable Wi^z denotes if the resource item z has been assigned to task (i, s\ 4.1. Allocation of resource items to tasks: Enough amount of each required resource r should be allocated to meet the requirement pisrj of any task (i, s). Y, pisrjYisj < X ^- ^ Y,(t>^ = Dr

V/ el.se

Si, r e R?,

\/rGR^

(6) (7.a)

zeZr

/iisz<(pz

jSJis

,Pisz
\/iGl,SGSi,rGR,^,,ZGZr

(7.b)

ZSZr

4.2. Resource sequencing constraints: Since a general notion of resource item is used, the following generalized sequencing constraints can be stated: Cis < Sv. + M(1 - Xiv)+ M{2 - Wisz - Wv sz) I \ ( \ V/,/'G /,5G Siv, rG \RisnRvs),zeZr : (/ < /'j Cvs <Sis + M Xiv+ M{2 - Wisz - Wi^sz) V/,fe /,5G Sii^,rG [Rs ni?/'s^),zeZr \\i< V)

(o.a)

The set of sequencing constraints for resource items other than equipment units required by tasks (i, s) and (/', ^-^ on different stages is derived from that of Mendez and Cerda (2003) by simply allowing s-^s'. 4.3. Example 2: Limited manpower Example 1 is revisited but this time there is, in addition, limited manpower to operate the units running at Stage I (Example 2a) or at Stage IV (Example 2b). The available manpower consists of 5 workers and the manpower requirement for each batch is given in Table 2. Results are shown in Table 3 and Figure 2. As the model of Mendez and Cerda (2002) does not end up with the optimal solution within the time limit of 3600 s, it cannot be ensured that the production schedules found by the bottleneck-based model in a few CPU seconds are optimal. 4.4. Example 3: Limited steam flow for simultaneous multistage requirements Example 1 is again revisited but the additional resource bottleneck is the available steam flow of 30 ton/h rather than the limited manpower. Table 2 shows the batch steam requirements in ton/h in Stages I and IV. The steam flow has been divided into 5 flow fractions of variable size. Computational results are depicted in Table 3. The same optimal value found in Example 1 has been encountered. Therefore, the steam resource is only a capacity-constrained resource and the solution found is consequently the optimal one. Despite the larger number of problem variables and constraints, the required CPU time remains low.

1996

P. A. Marchetti and J. Cerdd Table 3. Computational results for Examples 1-3 Binary vars, Continuous vars, Constraints 87, 88, 370 163, 88, 370

Example No. & Solution Approach 1. Proposed Formulation 1. Mendez etal. (2001) 2a. 2a. 2b. 2b.

127, 88, 658 203, 88, 658 127, 88, 658 217,88,658

Proposed Formulation Mendez and Cerda (2002) Proposed Formulation Mendez and Cerda (2002)

3. Proposed Formulation 3. Mendez and Cerda (2003)

Objective Function

CPU time (sec.)

Nodes

5.7 5.7

2.38 52.64

3923 95224

6.6

32.5*

33.92 3600 6.70 3600

28706 1768052 5339 1619341

5.7 45.4*

47.45 3600

12657 1053097

25.3*

5.9

223,173, 1667 313,173,1667

'• Best solution found within 1 hr. limit, on a Pentium IV PC (1.8 GHz) with ILOG OPL Studio 3.6

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.

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i

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i

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l

Fig. 2: Best schedule for Example 2a.

5. Conclusions An approximate bottleneck-based batch scheduling problem formulation has been developed. It assumes that the sequence of operations at non-bottleneck resources is aligned to that at the bottleneck and/or at the next capacity constrained resources. This allows a large saving in 0-1 variables and a significant reduction in the solution time, despite a near-optimal schedule is still found. A remarkable feature of the approach is that bottleneck resources must not be identified beforehand but through solving the proposed problem formulation.

References Chase, R.B., NJ. Aquilano, F.R. Jacobs, 1998, "Production and Operations Managemenf. Ed. Irwin/McGraw-Hill, 8* Edition. Goldratt, E., 1990, "What is this Thing called the Theory of Constraints and how should it be implemented". Ed. North River Press. Mendez, C.A., G.P. Henning, J. Cerda, 2001, Computers and Chemical Engineering 25, 701-711. Mendez, C.A., J. Cerda, 2002, Computer-Aided Chemical Engineering 10, 721-726. Mendez, C.A., J. Cerda, 2003, Computer-Aided Chemical Engineering 15, 984-989.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 Pubhshed by Elsevier B.V.

On the dynamic management of chemical engineering knowledge using an ontology-based approach A. Kokossis, E. Gao and A. Kourakis Department of Chemical & Process Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK Abstract The paper presents an ontology-based approach to monitor and assess changes and trends of technology intensive domains. The research, produced in collaboration with a variety of chemical engineering users of knowledge, promotes ontologies and semantically enriched models to enable knowledge sharing, classification, monitoring and reuse. Ontologies are deployed to search, retrieve and analyze technical structured and unstructured context from heterogeneous sources of information. The environment integrates models, language processing tools, knowledge engineering tools (crawlers, tokenizers), and agents. Besides the search and the management of information, the approach can be used to vaUdate the suitability of ontologies and their relevance to the particular application, suggesting modifications and alterations when appropriate. Keywords: knowledge management, ontology-based search, intelligent production 1. Introduction Chemical engineers are challenged to manage an increasing amount of technical knowledge critical and essential to process engineering and general business functions. The knowledge is diverse and accounts for innovations, new products and markets, emerging standards and environmental constraints, as well as important trends in technology and product design. The paper invariably refers to knowledge domains, knowledge models and applications. Domains of knowledge contain in-house repositories, external proprietory repositories or public domains. Domains represent resources available to search, scope and classify. Their information may be structured or semi-structured (models, property databases, software) but in the majority of cases it is unstructured (technical reports, documents, url sites). Knowledge models are used to describe a subject or a context. For the purposes of the paper models take the form of ontologies. Challenges in the development of ontologies have been addressed extensively in the literature. Challenges include general ontological formalisms (Vet

1997

1998

A. Kokossis et al

and Mars, 1998) to address biochemical and molecular problems of data management and business workflows. Schneider and Marquadt (2002) use ontologies in the context of a life cycle and with an emphasis on the design of chemical processes. They describe the integration of work process and product data, proposing a conceptual information model (CliP) as a comprehensive ontology to this purpose. Bayer and Marquadt (2004) extend and explain the structure of CliP and demonstrate its potential. For either structured and semi-structured resources (models, software, product data) associations between ontology concepts and resources is automatic (or semiautomatic). For unstructured resources, however, the association is implicit and needs to be discovered. Unlike previous work, the paper does not develop specific ontologies and reasons that, to a significant extent, ontologies should be developed according to the particular application, context or task. Moreover, ontologies are expected to also evolve in time (dynamic ontologies) as the underlying knowledge is not static. The paper explains the tight links between the knowledge models (ontologies) and the underlying domain that presents a number of illustrative applications in problems related to chemical engineering. The work explains the development of ontology-based search tools (supported by general-purpose search engines such as google, altavista or yahoo) in applications that discover relationships, new concepts or monitor trends. In reverse mode, the chemical engineering domains is finally used to validate knowledge models and correct their structure. 2. Problem description A generic version of the problem addressed with this paper could be stated as follows. Given is, (a) an ontology (OM) as a knowledge model that may account for the description of a project, system, task or assignment; the model accounts for the problem to tackle, and (b) a selected domain (D) that accounts for all the resources that contain potentially useful information related to this problem. The task would be then to (i) identify all subsets from (D) that relate to the problem, (ii) associate concepts and develop relationships between the components of OM and the information resources; and (Hi) assess (OM) in relation to (D) and propose posssible upgrades that better match the domain. For this purpose, the paper proposes a systematic and generic procedure that makes use of: (i) a 4. Dynamic analysis reference domain, (ii) a knowledge model 1. Modeling and information retrieval (ontology) of unknown On h-TechSighr and unspecified KMP associations with the domain (iii) agents to process ontologies and 2. Esrtraction and EvaluaJ assume tasks to collect.

On the Dynamic Management of Chemical Engineering Knowledge

1999

classify and cluster domain resources, (iv) Natural Language Processing tools that tokenize and annotate documents in preparation of stage analysis, (v) Analysis tools that process the results of stages (iii) and (iv). The procedure enables to map ontology concepts with the domains, review trends and relationships and present a list of possible modifications to the ontology. Developments are presented and integrated in the knowledge management platform (h-TechSight) that contains tools for editing ontologies. The figure above describes the general frame of developments and the applications presented.

4. System design and implementation The implementation has integrated technologies in ontology editors, agent systems, annotation tools, and databases. 4.1 Agent-enabled system Agents are assigned to extract resources and analyse. The former use Query Agents of the WebQL Enterprise studio. They use Perl-like scripts representing regular expressions for describe literals and meta-characters. Literals target information that matches exactly (literally). Meta-characters are employed through MASh (Aldea et al, 2003) and use ontologies. Agents make use web services of Internet Search Engines (google; yahoo, altavista) to collect url's. The agents process concepts and properties that include synonyms and abbreviatons. Additional properties are possible to build according the the needs of a particular application. 4.2 Annotation and tokenization Semantic annotation is implemented through GATE (Maynard et al, 2004), a development environment and middleware framework for creating, adapting and deploying Human Language Tools (HLT) components. GATE IE enables extraction from texts (url's, user Semantic AnDotstor files, text). The annotation uses grammar rules and JAPE (Java Annotations Pattern Language). JAPE implements a set of finite-state transducers, each containing rules based on pattern-matching. Rules include information about the class and the ontology. Instances can be visualised (Maynard et al, 2004) and populate databases that feature links with the ontology (DAML+OIL or RDF). Annotations are stored in the format: Token -Concept -ID-Time Stamp. The application supports MS

2000

A. Kokossis et al

Access, SQL Server 2000, Oracle 8i and Postgres. Connection with the RDBMS is accompHshed with an ODBC driver. The figure above outhnes the information flows of the stage. 4.3 Generic and targeted modes The methodology is applied in two separate modes: generic and targeted. The generic mode is responsible to search, retrieve and analyse resources, not specified a priori with respect to their content, location and structure. The targeted mode retrieves and analyses resources with approved content. An ontology editing environment is deployed to create, modify and browse ontologies stored in a domain repository. The search agent system launches the ontology-based search over matching concepts to further classify and cluster the results. Statistical components monitor metrics and statistics. 4.4 System Integration The methodology is implemented in the Knowledge Management Platform of hTechSight (Kokossis et al, 2005). The ontology enrichment tools consist of two web services: a text mining and a classification service. The former processes ontology inputs (and text) used to extract and identify relationships between concepts and resources. Results are produced in XML and inlcude an evaluation for each association. Recommendations are produced to update ontologies. A web content mining module performs searches and returns structured, domain-related information to the database. The information is used as input by modules of the application mode. The procedure is automated. Scheduling mechanisms can be invoked for periodic evaluations. 5. Illustrations Examples are highlighted to demonstrate the potential of the enviroment. They include internet empowered flowsheets, customized search engines, and monitoring tools to trace changes in selected areas (technology, research, and employment). 5.1 Internet-empoweredflowsheets translated

a knowledge model that comprises units, streams, materials, equipment, and technologies. Such a model would have access to resources in the form of property databases, suppliers, technology providers.

On the Dynamic Management of Chemical Engineering Knowledge

2001

experts (and expertise), patents and markets. In-house repositories would use such a model to integrate resources as a means to improve the management of a project, engineering workflows or assist general and targeted search queries. As in-house repositories are confidential, the demonstration had to rely on The Web, an alternative reference domain that is large(r) and, at least, equally diversified. An HDA flowchart has driven the search (figure above) with the agent-assisted environment of hTechSight that trasnforms the flowchart into a "clickable" interface that links with sites of suppliers, technologists, models and data. 5.2 Advanced Search Engines The search quality can be improved and organized with an ontology-based model. An example is presented around toluene diisocyanate (TDI) that is used in the production of polyurethanes for flexible foam applications (ranging from furniture, bedding, and carpet underlay, to transportation and packaging as well as in the manufacture of coatings, sealants, adhesives, and elastomers). A simple ontology for TDI is illustrated and includes concepts related to the production process (reactor, distillation) and the product qualities (anilin, DNT, TDA, nitrobenzene). The objective is an ontology-based information retrieval to analyse data and update the ontology (default maximum: 5 urls per concept). Retrieved urls and results are clustered. Even for such a small ontology most clusters are relevant. Exception is Cluster 7 and indicates that results could be improved as better and larger ontologies are employed. |TD1

F-TDI (5)

[pF-DNT (5)

t;

- D e v e l o p m e n t (0)

- P r o c e s s (0)

^ - R e a c t o r (0)

N i t r o b e n z e n e (5)

[ph-TDA (5)

L - D i s t i n a t i o n (5)

- a n i l i n (5)

CLUSTER: 8 URL: httpi//www.cdc,gov/niosh/90101_53.html CATEGORIES; Process. Reactor KEYWORDS; summary, niosh, publication, cdc

URL; http;//www.press.bayer.com/news/news,nsf/id/0C3D6AADC383470EC1256DF100333231 CATEGORIES; Process. Reactor KEYWORDS; news, new, gif, image, bayer, nsf

CLUSTER: 7 URLi http://www,tdi,state,tx,us/general/aspurch,htm CATEGORIES; Development, Process, TDI KEYWORDS; opportunities, bid, insurance, hub, texa

URL; http;//oem.bmjjournals,com/cgi/content/full/57/l/43 CATEGORIES; Process, T D I , Development KEYWORDS; t e x t , full, exposure, asthma, fev

CLUSTER: 6 URL; http;//www.cirec,net/summary-of-latest-issue-12-1999,shtml CATEGORIES; Process KEYWORDS; posted, i d , plant, July, tons, summary

53 Monitoring tools ^W§0n^^^M^^^f^^M

Applications included technology p^npmptM iM0imm<^9$0^E^^ .l:^Niaii;.' .:imm and employment watch. The former processed technical abstracts (AIChE ^^0^^M<:' meetings). The latter used an active •'f:/:;;^'s|'::5:;';..^ portal of the IChemE (member of the h-TechSight consortium) and ::.'''\-!:-'w,;;:;\/''^: monitors trends in employment. GATE is deployed for NLP-based mm¥:'i^::%^- ^'MMc^W:'' annotation, extraction and statistical |Jii||M||; ••,>lp:IE>i&:' :;•

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iai^«(«iE:^«i^if.; |«gf;;:::^v|o-; ^•^•W0:^::4f::\^^^^ '%^:::W^4$:;:-m:\: 4::S:::|-i(t'4:,.::;v'':'-^">. i ^ i i o d .,.,;:;;/.,::.:;••: • :;.;35C::;:,,,;,;:.;,,; ::.; ,:,K!v,4ii; ...is; \ •;;;: ••:;;•;,; \ i f i i r ; ; : ;.|;;'; ;;.,...:>S,,g|^^.:,;j..:.....g,.

^•^••^••^••••••:f«t'?;;'t-'-''-'

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2002

A. Kokossis et al

analysis. Sample ontologies included expertise, salary, company, location. Agents mined resources spanned over a period of 8 months (2.500 entries: January to July 2004) and findings are sorted with respect to the strongest frequencies, the emergence of new entries, and the development of the most significant trends. Technology watch agents processed 20,316 abstracts (Annual meeting: 1999-2004) and temporal analysis modules tested trends during the period. A screenshot from the portal is given above. 5. Remarks and Conclusions The work presents an ontology-based approach to manage and monitor chemical engineering knowledge. Conventional engineering concepts, such as flowsheets, are used to define and prototype ontologies used to search and manage information. The work takes up applications in general (advanced search engines) and targeted fields (technology watch, employment trends, monitoring tools). Preliminary results are highlighted in the paper. The concept of dynamic ontologies is new and is supported by search engines, and semantic web and reasoning tools available (WebQL, GATE IE, adhoc tokenizers, and crawlers) as developed in the course of the h-TechSight project. Acknowledgements The authors acknowledge generous financial support from the h-Techsight project (1ST 2001-33174). The examples in employment had been in collaboration with the Institute of Chemical Engineers (IChemE) in UK.

References Aldea A., R. Banares-Alcantara, J. Bocio, J. Gramajo, D. Isem, A. Kokossis, L. Jimenez, A. Moreno, D. Riano, An Ontology-Based Knowledge Management Platform", Z/C4/'03, Acapulco, pp77-182, 2003 Bayer B, Marquardt W, Towards integrated information models for data and documents, Computers & Chemical Engineering, 28 (8): 1249-1266, 15, 2004 Kokossis AC, R. Banares-Alcantara, L. Jimenezc, and P. Linke, h-Techsight: a knowledge management platform for technology intensive industries, ESCAPE-15, Vol 15, pp 1345-1350, 2005 Maynard D, H. Cunningham, A. Kourakis, and A. Kokossis, Ontology-Based Information Extraction in hTechSight, ESWS 2004, Proceedings in Lecture notes in Computer Science Notes S, Heraklion, Crete, Greece, May 10-12, 2004 Schneider R, Marquardt W, Information technology support in the chemical process design life cycle. Chemical Engineering Science, 57 (10): 1763-1792, 2002 van der Vet PE, Mars NJI, Bottom-up construction of ontologies, IEEE Transactions On Knowledge and Data Engineering, 10(4): 513-526, 199, 1998

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Lagrangean-based Techniques for the Supply Chain Management of Flexible Process Networks Peter Chen, Jose M. Pinto Polytechnic University, 6 Metrotech Center, Brooklyn, NY 11201, USA Abstract The supply chain optimization of continuous process networks is essential for most chemical companies. The dynamic nature of this problem leads to systems that involve types of chemicals as well as multiple time periods, and ultimately many complex and large combinatorial optimization models. These models become very difficult to solve, and sometimes not even solvable. Hence, such models require the use of decomposition methods, so that they could be solved efficiently and effectively. This work develops decomposition techniques to a continuous flexible process network MILP model. The techniques include Lagrangean Decomposition, Lagrangean Relaxation, and Lagrangean/surrogate Relaxation, coupled with subgradient and modified subgradient optimization. Several schemes derived from the techniques are proposed and applied to the process network model. The results from the full-scale solution and the proposed decomposition schemes are presented and compared. Keywords: Lagrangean relaxation, continuous process network, MILP, decomposition 1. Introduction Chemical process industries are composed of various production sites with many interconnected processes and flows of chemicals and their interconnection forms a chemical process network. It ensures the maximum flexibility in production by utilizing the various synergies between the processes. Processes are classified as either dedicated or flexible. The former are used for large volume chemicals and only operate with one production scheme at all time, whereas flexible processes are used for small volume chemicals and can operate with different production schemes at different times. Bok et al. (2000) developed a multiperiod MILP model for flexible process networks that incorporates inventory profiles, changeover costs, intermittent supplies, production shortfalls, and transportation costs for delivery when the network consists of processes located at different sites. The incorporation of various features in the model proposed by Bok et al. (2000) made it relatively difficult to solve as the horizon considered expands and as the number of chemicals and/or processes involved increases. It must rely on decomposition methods to solve the optimization problem faster and more effectively. Some of the famous techniques are the Lagrangean decomposition discussed in Guignard and Kim (1987), the Lagrangean relaxation and subgradient optimization from Marshall and Fisher (1985), the Lagrangean/surrogate relaxation in Narciso and Lorena (1999), and the modified subgradient optimization proposed by Fumero (2001). Thus, different decomposition methods exist and it is not straightforward to discriminate which is the most effective. A comparison of decomposition techniques applied to a MINLP long-range production planning model of petroleum refineries is discussed in Neiro and Pinto (2004). The results showed significant improvement in computational efficiency for all the techniques with respect to the full-scale solution.

2003

2004

P. Chen andJ.M. Pinto

but none of them stands out better as compared with the others. A more systematic comparison of various decomposition techniques for linear models is necessary since Hnear models present better convergence properties; in particular non-convexity issues are eliminated. The objective of this work is to develop and analyze decomposition techniques for large scale process networks. This work takes the continuous flexible process network (CFPN) model proposed in Bok et al. (2000) and applies various decomposition techniques to it. The techniques used include the Lagrangean decomposition, Lagrangean relaxation, and Lagrangean/surrogate relaxation. The results from the full-scale solution and the results from the various decomposition methods and relaxation strategies are then compared.

2. Problem Statement The CFPN model of Bok et al. (2000) is used as the basis for the present work. The CFPN model is a production/distribution network consisting of a set of processes that interconnect in a finite number of ways. The processes can be either dedicated or flexible based on their production schemes. While only one scheme means that the process is dedicated, two or more schemes mean the process is flexible. A set of chemicals is involved for each process with a specific scheme. The chemicals can be raw materials, intermediates, products, or byproducts. The raw materials are bought from a set of markets and the end products are sold in another set of markets. All the processes are located in a set of production sites. The objective is to maximize the operating profit of the network over a planning horizon. The planning horizon is expressed as a set of time periods of equal length during which prices and demands of chemicals and cost of operation and inventory can vary. For all processes and schemes, the material balances for raw materials and byproducts are assumed to be linear ratios to the main product of that scheme. All the processes are assumed to have fixed costs. The operating costs for each process and production scheme are assumed to be proportional to the amount of main product produced for that scheme. Last but not least, changeovers are assumed to only imply costs and the overall time spent in changeover is assumed negligible. 3. Mathematical Formulation and Solution Approach The techniques presented here are based on the decomposition and multiplier updating methods. The general structure of the algorithm is shown in Figure 1. The CFPN model is decomposed into |T| sub-problems through either Lagrangean or Lagrangean/surrogate relaxation methods, and the updating methods of multipliers are either subgradient or modified subgradient optimization. The linking constraints that contain variables at different time periods have these variables duplicated and substituted. These variables include the inventory level of chemical 7 for site c at time t (Vjct), shortfall of chemical J for market / at time t (SFjit), production changeover of process / with scheme k for site c at time t (YjkctX ^^^ intermittent delivery type d from market / to site c at time t {YPdict)- Each constraint that enforces the equality of the duplicated variables is divided into two inequality constraints. One of the two is relaxed into the objective frmction, while the other is added to the set of constraints. For instance, consider the following inventory level constraint for thefrill-scalemodel. Vjct - Vjc,t-i = Inputj^, - Outputj^,

jeJ,ceC,teT

(1)

Lagrangean-Based Techniques

2005

In equation (1), the left-hand side denotes the change of inventory level from time t-\ to time t, whereas the right-hand side contains all inputs and outputs at time t. The inputs consist of chemicals that are either purchased or produced at time t, and the outputs include the chemicals that are either sold or consumed. The following are steps for applying the decomposition approach. 1. Variables are duplicated, expressed as an equation, and converted to the equivalent inequality form. V > V and V < V ^ jet ' jet ^ jet — ^ jet " ' ^ ' ^ ^ jet — ^ jet 2. The variables are replaced by the duplicated variables. jet ^

'^ jet'

'^ jc,t-\

^

'^ je,t-\

3. One inequality is relaxed into the objective ftinction with multiplier // j&J c&C tGT

4. The remaining inequality is added as a constraint. ^ jct - ^

jeJ.csCjeT

jct

(2)

5. The model is decomposed into |T| sub-problems with equation 3 as the inventory level constraint.

Vjc,-Vl,_,=Inputj^,-Outputj„

j^J,c^C

(3)

The |T| decomposed sub-problems are solved either sequentially or independently. Setting bounds and determining Parameters Starting next iteration Update the multipliers using the following methods:

1) Solve |T| decomposed sub-problems (solution = Z) using the following methods;

•Subgradient optimization

• Lagrangean decomposition

• Modified subgradient optimization

• Lagrangean relaxation • Lagrangean/surrogate relaxation 2) Upper bound = min {Upper bound, 2 ^^ Y[ Fixing binaries 1)

Return lower bound and corresponding solution

Solve the full scale problem, (solution = Z)

2) Lower bound = max {Lower bound, 2}

Figure 1 - General algorithm for the CFPN model

4. Decomposition Results The example studied in this paper is as follows. There are four processes and six chemicals considered in this process network. The processes are categorized as II, 12, 13, and 14. The chemicals are Jl, J2, J3, J4, J5, and J6. II involves only one production scheme Kl. 12 and 14 both have two production schemes, Kl and K2. 13

2006

P. Chen andJ.M. Pinto

consists of four production schemes, Kl, K2, K3, and K4. There are two production sites CI and C2. All processes with their respective schemes are located in CI. C2 is very similar to CI, but it does not have process II and process 13 only have three schemes, Kl, K2, ad K3. Also, unlike CI, process 14 in C2 does not produce byproduct J6. The raw materials can be purchased from two marketplaces, LI and L2, and final products or intermediates can be sold in two markets, L3 and L4. Chemicals Jl and J2 can be purchased, J3 can be sold as a product or used as raw material in scheme Kl of process 14, J4 and J6 are either purchased or produced, and J5 is sold as a product. The diagram of the overall process network is shown in Figure 2.

Figure 2 - Continuous flow process network diagram (Bok et al., 2000) Four strategies (A, B, C, D) are proposed for solving the CFPN model. Strategy A solves the |T| decomposed sub-problems using Lagrangean relaxation, and then updates the multipliers using the subgradient method. Strategy B uses the same decomposed sub-problems, but updates the multipliers using modified subgradient instead. Strategies C and D use the corresponding subgradient optimization methods from A and B, but solve the Lagrangean/surrogate relaxed |T| sub-problems instead. Strategies A, B, C and D are applied to the solution of models ranging from 7 to 63 time periods, at every 7 time periods. The solution method terminates if the lower bound repeats for more than 10 times or if the difference between the upper and lower boxmd is less than 0.01. The percent difference from fiill-scale solution for strategies A, B, C and D are shown in Table 1. The computational results are presented in Figure 3. Table 1. Difference from the optimum |T|

Strategy A

Strategy B

Strategy C

Strategy D

7 14 21 28 35 42 49

0.000% 0.536% 1.195% 0.826% 0.824% 0.679% 0.765%

0.000% 0.554% 1.726% 0.829% 0.833% 1.317% 0.837%

0.000% 0.536% 1.723% 0.826% 0.824% 1.317% 0.837%

0.000% 0.554% 1.726% 0.829% 0.833% 1.317% 0.837%

Lagrangean-Based

2007

Techniques

(a) Computational time

(b) Solution Value

Figure 3 - Results for the proposed strategies The comparison between strategies showed that Lagrangean/surrogate relaxation uses less time than Lagrangean relaxation. Although the Lagrangean/surrogate relaxation strategy converges faster, the solutions generated from the Lagrangean relaxation are better. As for multiplier updating methods, results showed that the modified subgradient optimization is computationally more expensive than the traditional subgradient optimization and does not guarantee better results. However, the modified subgradient optimization might generate better solution than subgradient optimization at the expense of fine tuning the parameters of the algorithm. Sixteen cases are proposed for analyzing the solution quality of the subproblems. Each case fixed a different set of linking variables between consecutive subproblems. The case number and the respective set of fixed variables and free variables are listed in Table 2. Table 2. Cases and fixed variables Case

Vjct

^^,7.

^ikct

YPdict

Case

Vjct

SFju

J^ikct

YPdict

1 2 3 4 5 6 7 8

Fixed Fixed Fixed Fixed Fixed Fixed Fixed Fixed

Fixed Fixed Fixed Fixed Free Free Free Free

Fixed Fixed Free Free Fixed Fixed Free Free

Fixed Free Fixed Free Fixed Free Fixed Free

9 10 11 12 13 14 15 16

Free Free Free Free Free Free Free Free

Fixed Fixed Fixed Fixed Free Free Free Free

Fixed Fixed Free Free Fixed Fixed Free Free

Fixed Free Fixed Free Fixed Free Fixed Free

Table 3 summarizes the percent difference from fiiU-scale for all cases using Strategy B with 21 time periods. Table 3. Cases and their percentage difference from optimum Case

Identical Cases

Percentage difference from optimum

1 2 3 9 11 12

5 4,6,8 7 10, 13, 14 15 16

1.72% Infeasible 2.27% 92.56% 93.10% 93.86%

2008

P. Chen andJ.M.

The comparison between the relaxed solution and fiill-scale solution is shown in Figure 4a and the comparison between the primal problem and the full-scale solution is presented in Figure 4b. 21000 1

10000

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9000

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1000 0

5000 10

15

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^ '^"' ' '

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# of iterations

(a) Relaxed solution vs. Optimum

(b) Primal solution vs. Optimum

Figure 4 - Solution performance of the proposed cases Regarding the proposed cases, when all variables are fixed sequentially (Case 1), the upper bound crosses over the optimum and the lower bound. Since the primal problem only fixes the binaries, it is less constrained than the sub-problems that fix both continuous and binary variables in sequential iterations. The gap between lower and upper bounds widens as more variables are made independent between sequential sub-problems; moreover, infeasibility always occurs when Vjct is fixed and YPdict is free. 5. Conclusion The results showed that the proposed solution strategies are much more efficient than the full-scale method. The computational time is greatly reduced and the difference in the objective function value from the full-scale solution is less than 2%. The comparison between the cases for all strategies and numbers of time periods indicated that fixing values between consecutive sub-problems made the relaxed problem more constrained. However, by making the variables independent, the solution generated will be far off from the optimum and sometimes even infeasible for the primal problem. References Bok, J. K.; Grossmann, I. E.; Park, S., 2000, Supply chain optimization in continuous flexible process networks, Ind. Eng. Chem. Res., 39, 1279-1290. Fisher, M. L., 1985, An applications oriented guide to Lagrangian relaxation. Interfaces, 15,2, 10-21. Fumero, F., 2001, A modified subgradient algorithm for Lagrangean relaxation, Comp. and Oper. Res., 28, 33-52. Guignard, M.; Kim, S., 1987, Lagrangean decomposition: A model yielding stronger Lagrangean bounds. Mathematical Programming, 39, 215-228. Narciso, M. G.; Lorena, L. N., 1999, Lagrangean/surrogate relaxation for generalized assignment problems, EJOR, 114, 1, 165-177. Neiro, S; Pinto, J. M., 2004, A general modeling framework for the operational planning of petroleum supply chains, Comp. Chem. Engng., 28, 871-896.

Pinto

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

2009

Restructuring methodology in process engineering for sustainable development I. Koshijima^, A. Shindo^, Y. Hashimoto^ and T. Umeda'^ ^Chiba Institute of Technology, Narashino, Japan ^Nagoya Institute of Technology, Nagoya, Japan ^Aoyama Gakuin University Research Institute, Tokyo, Japan Sustainability of the chemical plant becomes a key issue to achieve corporate social responsibility. To meet this requirement, the authors discuss investment and technical transfer strategy by extending the concept of Michael Porter's value chain. After providing mathematical formulations to describe trade-off relationships among investment, profit and intangible assets, a development of process design methodology is taken as an illustrative example. Keyword: sustainability, value-chain, restructuring 1. INTRODUCTION Technological and managerial achievements should be simultaneously attained in design and engineering activities to contribute chemical plants to the sustainable development of the society. There is strong interdependency between the above two targets. In spite of this, a framework of corporate strategy connecting these targets has not been provided in the design and engineering activities, which may cause failure in accomplishing the sustainability as well as competition on the global market. According to G.E. Keller and P.P. Bryan[l], there are seven key themes for process-design improvements in the future. Those are 1) raw material-cost reduction, 2) Capital-investment reduction, 3) energy-use reduction, 4) increased process flexibility and inventory reduction, 5) ever greater emphasis on process safety, 6) increased attention to quality, and 7) better environmental performance. Sustainability of the chemical plant has to be a keyword to make answers to these questions. In a design for sustainable chemical plant, a value-based management framework should be prepared to solve two major managerial requirements. One requirement is to create the high value in the design and manufacture stages as much as possible. The other should be improved through the horizontal value chain in the primary activities and support activities, respectively to assure not only the sustainability of the plant itself but also the sustainability of the corporation. In this paper, the authors disclose two problems. The first problem in the research and development of process design methodologies is related to the value added in the support activities that has to be maximized by using the investment. The second problem in the above design stage is related to the value chain in the primary activities that has to be maximized by the support activities providing technical solutions. The authors propose a method for solving these

/. Koshijima et al

2010 f i r 1 Investmg

1 Horizontal value chain of the support activities 1

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Fig. 1. Simplified model of sustainability oriented value-based structure

two problems simultaneously by utilizing Pareto optimal solution. To show the validity of this proposed method, a typical chemical design methodology is used as an illustrative example. 2. PROCESS SYSTEMS ENGINEERING FOR SUSTAINABILITY The activities require more expanding scope of work than those in traditional productionoriented engineering. The scope should cover the whole product life cycle stages from raw material acquisition to disposal except use and service and retirement and also the stage of manufacturing and assembly is usually excluded from the traditional scope of process systems engineering. The process systems engineering for attaining sustainability can be rephrased by the design for the environment. Various efforts for achieving the objective have been made in the past decades and the results from these efforts are available to process synthesis, analysis and optimization. Based on these understanding, the ways of process systems engineering and design for the environment may be gradually established. The waste minimization, material recycles, as well as energy recovery for system design are examples of those fruits of efforts. The meaningful index for the present purpose of engineering design aims to minimize environmental impacts and it is useful to specify the so-called resources productivity as an index. The waste minimization and input energy minimization are to be accomplished by maximizing the resource productivity. It should be emphasized that the whole engineering and design activities in product development require continual search for sustainability and many elements of activities feed back to others. 2.1. PROBLEM STATEMENT Several frameworks connecting technological and managerial problems was introduced with focusing on the sustainability-oriented value chain.[2-4] Based these concept, each technical development stage in the process design methodology should be considered as a support activity and certain process design evolution corresponded with the said support activity is arranged as a primary activity as shown in Fig. 1. The value chain in the primary activities can be expressed as the input-output relationship

Restructuring

Methodology

in Process Engineering for Sustainable Development

2011

Table 1 Typical Values for Sustainability Oriented Value Chain Activity

Design

Manufacture & Assembly

Use or Service Treatment or Disposal

Value Chains Horizontal Vertical (caused by the primary activities) 1. Competitive, Energy saving and pollution 1. Design methodology protecting products 2. Expert Engineers 2. Design for assembly, disassembly and manufacturing 3. Latest computer systems and tools 3. Design for environment 4. New materials 1. Manufacturing quality 1. Educated labors 2. Energy conscious inbound and outbound logistics 2. High performance and energy conscious production facilities 3. Operation method for conserving resources, preventing pollution 1. Operation records 1. Preventive Maintenance 2. Maintenance records 2. User maintainability 3. Operability or usability 1. Reusability 1. Material information 2. Selectivity 2. Reuse information. 3. Disassembly

of Stage-by-stage design evolution. The primary value added activities are included in the each evolutional stage where specified design package must be physically created. Support activities, especially research and development (RD, here in after) activities of process design methodologies, are also heaped up technical advantages and linked to the primary activities by providing technology developed. The value added activities must be concerned with the decision-making processes whose stages possess the higher values in the entire value chain, or the significant activities that influence the final value of the design package. The first problem in the RD of design methodologies is that the value added activities are subject to the given research organization and maximizing the horizontal side of value chain in the support activities has to be accomplished with the investments of each stage. The second problem in the above mentioned design stage is that the value added activities are subject to the given chemical processing system and maximizing the horizontal value chain in the primary activities has to be accomplished with the support activities that provide technical solutions. These two problems have to be simultaneously solved as a Pareto optimal solution. Table 1 shows the value chain activities from design stage to disposal stage and typical values gained. 2.2. METHOD FOR SOLUTIONS Based on Fig.l, the Pareto-optimum solution among the horizontal value-chain of the research and development activities (HSN), the horizontal value-chain of the practical design activities (HPN) and the earned profits (EN) should be found by using a function Popt that gives the Pareto-solution: Hsi{Zsi,Dsi(Ii))^Esi^i{^Si(Zsi,Dsi{Ii))), Parato — Solution — Popt Hpi{Zpi,Dpi{Vi))+Fpi^i{^Pi{Zpi,Dpi{Vi))), Ei + Gpi+x{Evi{Zpi,Dpi{Vi)))

(1)

under the following constraint that shows a fixed budget IMOX/ . ^phase — -'Max phase

(2)

/. Koshijima et al

2012 Design with Implicitly Concerned about Operation (A) Conventional Simulation based Design

Design with Explicitly Concerned about Operation (B) Design for Operability

•IS (C) Advanced Process Control

Q S

(D-1) Design Margin

(D-2) Design with Auxiliaries

(D-3) Situation Awareness

(D-4) Autonomous System

with Fixed Structure

with Rexible Structure

Fig. 2. Classification of process design problems in seven areas where Z/ stands for a stage-to-stage transitional of state vector, D, is a decision vector for the stage structure and O/ is a transforming function from input values to output values at the stage i. Evi and Vvt are also functions to set a earned profit {Et) and a vertically transfered value (V/) at the stage i. (Suffix s and p shows the support and the primary activity, respectively.) 3. RESTRUCTURING OF PROCESS DESIGN METHODOLOGY Integrating the RD and design activities by taking the congruence with the corporate strategy, previous section proposed the conceptual framework for restructuring of process design methodology in the form of multiple objectives. Though there are several commutating methods to solve the above equations quantitatively, the conceptual framework described in the preceding sections will give an analytical insight into the logical structure of restructuring direction to have the deeper insights behind the solutions. It is assumed that each activity's input-output state variable consists respectively of three factors such as efficiency (measured by a comparison of production with cost as in energy, time, and money), effectiveness (ability to produce a decided, decisive, or desired effect) and efficacy (the power to produce an effect) for evaluating the values. In each support activity, the efficiency in the state variable is always a matter of great concern and expresses the effect of investments for the RD. The efficacy in the state variable is typical of value added distributed through vertical value chain and the the effectiveness may show the current status of the research to proceed to the next problem. The efficiency in the state variable of each primary activity directly affects to the creation of monetary values and explain the cost effect in each stage. In the primary activity, the efficacy in the state variable justifies the client benefit, because the effectiveness prepares evidences of engineering efforts. Though the change of design methodology from one stage to the next is innovative, as a whole set of stages, the changes become evolutional. Because the optimum system is constrainted by the current configuration, the environment changes cause mis-matches in the configuration. In order to compensate the changes caused by reduced efficiencies and create new values based on effectiveness, a better methodology that increases a certain efficiency has to be developed and transfered from the RD to the practical design and engineering as shown in Fig.l.

Restructuring Methodology in Process Engineering for Sustainable Development

2013

Research and Development Activities Zs,

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Fig. 3. Illustrative value chain in the process design and engineering 4. AN ILLUSTRATIVE EXAMPLE Each technical development stage in the process design methodology is considered as the support activities and process design evolutions corresponded with the said support activities are assumed as the primary activities. The technical development activities are assigned in 7 stages (A, B, C, D-1, D-2, D-3 and D-4) as shown in Fig.2 and the design activities are also adopted in the same number of stages. It is assumed that the corporate strategy requires a certain extent of flexibility in the continuously operating process system for adapting various markets' needs. A process under consideration consists of a pretreater, a reactor and a separator with a purifier, this structure is only intended to have some practical image. To illustrate the approach in the previous section, 4^^, 5^^ and 6^^ stages are selected as shown in Fig. 3. Restructuring context of each stage can be read from the state variables (as efficiency effectiveness and efficacy) transferred between stage to stage and the decision variables in each stage that are summarized in Table 2. The 4^^ Stage for D-1: Even thought heat and mass balances of the process system are strictly calculated by using numerical models, it is difficult to apply the calculated result to make and build physical process plants. Some range of flexibilities is implicitly emerged as operational margins caused by over-sizing of equipment. The 5^^ Stage for D-2: There are several alternatives to achieve the same result. Under the traditional process design methodology, a proper selection of unit operations is the most important task for process engineers, because the selection fixes engineering and construction cost. The e^ Stage for D-3: At this point, we can install advantageous mechanisms to our plant as situated actions. Economical advantage could be gained, if these actions are properly activated. Situation awareness, which makes appropriate decisions based on the perception, may install loss prevention capabilities to our plants, though reasoning about diverse types of quantitative and qualitative information in uncertain environments is a challenging problem.

/. Koshijima et al

2014 Table 2 State and decision variables in the value chains of the illustrative example Zpi,

DpA 2p5

Dp5 2P6

Dpe Zpi

Value Chain in Primary Activities Efficiency Effectiveness Efficacy Increased prodAutomated Dynamically uct quality under verified soluplants in control reduced labors tions with computer control system how to decide allowable range of over-sizing Increased operEconomically Robustness able range and convinced sizing against deviasolution reduced productions tion loss how to decide auxiliary units in each task Increased Economically Robustness adaptability convinced auxilagainst large by switchable iary solution deviations equipments how to collect plant operation data and abnormal data Increased agility Loss preventing Automated by intelligent solution plants in diagnomonitoring sis systems

Z54

DSA ZS5

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Dse Zsj

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Efficacy Dynamic lator

simu-

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Modulized process design and control problem

Diagnosis methods

5. CONCLUDING REMARKS In this paper, the authors propose a sustainability oriented value chain combining corporate activities with product lifecycles, and define two types of value chain; horizontal value chain and vertical value chain. The research activities to develop process design methodologies and the practical process design activities based on the developed methodologies are formalized as the horizontal value chains that increase the corporation's sustainability. Strategic investment and technology transfer to both horizontal value chains are also defined as the trade-off problem for determining the engineering structure, and increase the process system's sustainability. ACKNOWLEDGMENT This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B), No. 11680451 (2005), titled "Innovation for business strategy based on integrated PLM".

REFERENCES 1. 2. 3. 4.

G. E. Keller and P. R Bryan, Chem. Eng. Progr., 96 (2004) 41. S. Ghara and T. Umeda, Proc. of Int. Symp. on PSE94, Kyongju, Korea (1994) 829. T. Umeda, Ind. Eng. Chem. Res., 49 (2004) 3827. I. Koshijima and A. Shindo and T. Umeda, Proc. of Int. Engineering Management Conference 2004, Singapore, Singapore (2004) 104.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Development of a Multiobjective Scheduler for Semiconductor Manufacturing O. Baez Senties, C. Azzaro-Pantel, L. Pibouleau, S. Domenech Laboratoire de Genie Chimique, UMR 5503, ENSIACETINPT. 5, ruePaulin Talabot-BP 1301 -31106 Toulouse, Cedex 01 -France

Abstract Scheduling of semiconductor wafer fabrication system is identified as a complex problem, involving multiple and conflicting objectives (meeting due dates and minimizing waiting time for instance) to satisfy. In this study, we propose an effective approach based an artificial neural network technique embedded in a multiobjective optimization loop for multi-decision scheduling problems in a semiconductor wafer fabrication environment. Keywords: Discrete Event Simulation, Artificial Neural Networks, Multicriteria Genetic Algorithm, Semiconductor manufacturing. 1. Introduction Scheduling of semiconductor wafer fabrication system is identified as a difficult task, mainly because of the typical features of the process scheme, such as complex product flows (the so-called wafer fab is indeed a multipurpose plant), high uncertainties in operations, rapidly changing products and technologies (Ellis et al., 2004). It is thus a significant challenge to develop effective scheduling methods in wafer fabrication. Discrete-event simulation (DES) is one of the most widely used methods to study, analyze, design, and improve manufacturing systems. The combined used of a DES and an optimization procedure based on a genetic algorithm was an efficient solution to short-term job-shop scheduling problems and was adopted in our previous works (Charles et al., 2003). In spite of its acknowledged benefits, this kind of approach often reaches its limits in the industrial practice because of the highly combinatorial nature of the problem. In addition, the main emphasis of much of the work on scheduling has been on the development of predictive methodologies with a single objective. Actually, production managers have to cope with various objectives, which contributes to scheduling complexity: meeting due dates is an important goal in low-volume and high variety production circumstances within competitive market environments; another major objective in scheduling of semiconductor wafer fabrication is reducing waiting time for work-in-process (WIP) inventory to improve responsiveness to customers: in addition, the shorter the period that wafers are exposed to contaminants while waiting for process, the smaller the yield loss. Increasing the throughput is also an important stake, since the investment in fabrication equipment is capital intensive. In this study, we propose an approach based on an artificial neural network (ANN) technique coupled with a multiobjective genetic algorithm (MUGA) for multi-decision scheduling problems in semiconductor wafer fabrication. The paper is organized as follows. Section 2 is devoted to the general modelling framework. Then, the principles of the multiobjective optimization procedure are presented with some typical results. Then, we draw the conclusions on this work.

2015

2016

OB. Senties et al

2. General framework and plant modelling The main emphasis of much of the work on scheduling has been on static systems with a single objective. As noted above, wafer fabrication is complex, dynamic and highly stochastic. Satisfying the multiple objectives might be more important than only optimally meeting a single objective. In this study, we propose an effective approach based on an artificial neural network technique for multiobjective and multi-decision scheduling problems in semiconductor wafer fabrication. Figure 1 shows the role of the proposed methodology, which will be presented in more detail what follows. Artificial

Semiconductor manufacturing system

MUGA MUltiobjective Genetic Algorithm Decision maker

backfeed Discrete Event Simulation Model

Fig. 1. General methodology 2.7. Discrete-event simulation To model the wafer fab in a high degree of detail. Discrete Event Simulation (DBS) techniques were previously implemented, leading to the development of MELISSA software (see (Charles et al., 2003) for more detail). Typical events taken into account and managed in the simulation core have been widely presented in (Berard et al., 1999) and will not be recalled here. It was largely used and validated in previous works and reflects the behaviour of the plant. The different runs performed with MELISSA allow identifying the more sensitive variables on some performance criteria, such as makespan, cycle time, respect of due dates, limitation of WIP etc... The example which serves here to illustrate the methodology is presented in Figure 2 and imitates the fabrication plant of a typical semiconductor manufacturing process.

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Development

of a Multiobjective

Scheduler for Semiconductor

Manufacturing

2017

sequence, to reproduce the so-called re-entrant flow. Some units are in parallel, which confers more flexibility to the process. We consider the production of 5 different families of products with their associated recipe identified by a number (1, 2, 3, 4, 5). A batch is constituted by 50 wafers and 3 levels of productions are to be considered according to market demand, i.e. 16, 24 and 32 batches. A horizon time of 10000 min was chosen as a reference. 2.2. Decision variables The defmition of each decision variable is proposed in Table 1, where an example is developed. Each wafer batch is identified by a number corresponding to its family. A campaign is constituted by the release of several products belonging to various family sequences into the wafer fab. Table 1 Decision variables Decision variable Maximum number of recipes Maximum number of recipes treated simultaneously (NC) Identification of family sequence (FS) Total of batches (NP)

Number of campaigns (NC)

Time Between Batches (TBB) (identical from a batch to the following one, whenever the campaign)

Time between Campaigns (TBC)

Number and type of parameter 5 ex . 4 among the 5 available recipes 24 combinations, ex : 1,2,3,4 Batches are always released with this sequence order into production 3 discrete values (16, 24, 32) ex : 16 3 discrete values (2,3,4) ex : 2 This variable sets the number of batches within a campaign [ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1 [ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign2 3 discrete values (120, 750, 1250) min ex : ex: [ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1 TBB = 120 min 5 discrete values (500,1500,1900,2400,2800) [ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign 1 Separated by TBC = 500 min from [ 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] campaign2

2.3. Performance criteria There are multiple criteria that can be used in evaluating the system performance and system status of the semiconductor fabricator. The criteria are mainly based on completion times, due-dates, inventory level, or machine utilization. Two criteria were selected here, involving the computations of delay/advance of the due date (Ci) called ADV/DEL and of the average waiting time (AWT) of all products (n) at each processing unit (C2) (WTi refers to the waiting time of product /). (Q) = ADV/DEL = yfld^^nc^ + iDelaysf

(1)

Y^WT, (C2) = AWT--

(2)

2018

O.B. Senties et al

It must be emphasized that the expression selected for criterion Ci will penalize more the delays than the advances of products. These expressions will then been used at the optimization step with the objectives of Min (Cj) and Min (C2). 2.4. ANN modelling We then resort to an artificial neural network technique (Dreyfus et al, 2004) to model the semiconductor plant, since its efficiency has been successfully demonstrated in semiconductor processing by many researchers (Min and Yih, 2003) (Sun and Choung, 1999). The training phase was carried out by a classical backpropagation algorithm. For this purpose, the MATLAB software package dedicated to neural network modelling was used. The neurons of the input layer correspond to the wafer manufacture process parameters (decision variables). The output neurons correspond to the wafer manufacture performance parameters to be evaluated (criteria). The same network structure was adopted for each criterion. The network also includes one hidden layer, which helps the network in learning the non-linear mapping between the input and output layers. The training was carried out systematically with varying the number of hidden neurons (a number of 30 was finally selected). All these models were based on the multilayer perceptron architecture (Cheng and Billings, 1992). Tan-sigmoid transfer function was used as an activation function for hidden and output layers. The values of the test data were normalized within the range from -1 to 1. The root mean square error (RMSE) was chosen as a criterion for supervised training. The error is computed as the difference between the target and network output values. The total combination set of decision variables was generated to build the neural network, i.e. 5400 (training and test data, 2:1) at the preliminary stage of the study. Typical results obtained with MELISSA and ANN computations can be shown in Figure 3 and exhibit a good agreement between the two sets of values, thus showing the efficiency of ANN to model the batch plant.

Measured values for ADV/DEL

x 10'

Measured values for AWT

Fig. 3- Comparison between ANN and MELISSA computed values for ADV/DEL criterion and for AWT criterion [min]

3. Multiobjective optimization The neural networks have then been embedded in a multiobjective genetic algorithm (MOGA) to optimize the decision variables and to deal with the set of compromise solutions for the studied criteria, thus giving the optimal Pareto zone solutions. Lately, there has been a large development of different types of multiobjective genetic algorithms, which are reflected in the literature. The big advantage of genetic algorithms

Development

of a Multiobjective

Scheduler for Semiconductor

Manufacturing

2019

over other methods, particularly over other stochastic procedures such as Simulated Aimealing, is that a GA manipulates a population of individuals. It is therefore tempting to develop a strategy in which the population captures the whole Pareto front in one single optimization run. Following the guidelines proposed by (Dietz et al., 2005a, b), this approach was adopted in this work. The same encoding procedure was used since only integer variables are involved in the problem formulation (more detail concerning crossover and mutation can be found in (Dietz et al., 2005a, b). Initial Population Generation

I

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Fig 4. Combination between neural networks models and multiobjective genetic algorithm Figure 5 visualizes some typical results obtained for the two criteria, i.e. the compromise zone. The occurrence of the Pareto front underlines the conflict in minimizing both objectives and Pareto compromise solutions can be found. Two zones can be distinguished corresponding either to delays or advances relative to a time horizon of 10000 min. The decision maker can thus select one or several compromise solutions corresponding to a schedule (decision rules) for the best decision in semiconductor wafer fabrication. He can thus validate his decision by using the discrete-event simulator to determine precisely all the scheduling parameters. A major interest is that the approach may find wide applications in practice since the data base may be updated with the feedback from the real process, thus improving continuously the model accuracy in time, as seen in Figure 1.

4. Conclusion Semiconductor wafer fabrication involves an important number of decision problems. The objective of this paper was to propose an optimization strategy in order to assign appropriate decision variables. More precisely, a scheduler for selection of decision

2020

OB. Senties et ah

variables in order to obtain desired performance measures at the end of a certain production interval was developed. In the proposed methodology, a three-level strategy based on the combined use of a simulation technique, a neural network and a multiobjective genetic algorithm is suggested. The results indicate that this methodology is an effective method considering the complexity of semiconductor wafer fabrication systems, as a time-saving way to achieve a prompt response in a dynamically changing environment. The computational time for reaching the Pareto's solutions is about 2000 seconds, that is much faster than the process real-time (about 6 days). If the DES model is used instead of the ANN the time is multiplied by 100, and the computational time is in the same order of magnitude than the real time.

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±_ 2000

|Avance| or|Delay| [min] Figure 5-Pareto's optimal solutions for the pair of criteria AWT- [ADV/DEL]

References Charles A.S., Floru I. R., Azzaro-Pantel C , Pibouleau L., Domenech S., 2003, Optimisation of preventive maintenance strategies in a semiconductor batch plant, Computers and Chemical Engineering, 27, 449 - 467. A. Dietz, C. Azzaro-Pantel, L. Pibouleau and S. Domenech, 2005a, A Framework for Multiproduct Batch Plant Design with Environmental Consideration: Application To Protein Production, Industrial Engineering and Chemistry Research, 44, 2191-2206. A. Dietz, C. Azzaro-Pantel, L. Pibouleau and S. Domenech, 2005b, Multicriteria optimization for multiproduct batch plant design under economic and environmental considerations, Computers and Chemical Engineering, in press. G. Dreyfus, M. Samuelides, J. Martinez, M. Gordon, F. Badran, S. Thiria, L. Herauh, 2004, Reseaux de neurones - Methodologies et applications, EyroUes. K.P Ellis, Y. Lu, E.K. Bish, 2004, Scheduling of Wafers Test Processes in semiconductor Manufacturing, International Journal of Production Research, Vol. 42 No. 2, 215-242. MATLAB, 2001, Neural Network Toolbox, User' Guide, Inc. H.S. Min, Y. Yih, 2003, Development of real-time multiobjetive scheduler for a semiconductor fabrication system. International Journal of Production Research, Vol. 41 No. 10, 2345-2364. F. Berard, C. Azzaro-Pantel, L. Pibouleau, S. Domenech, D. Navarre, M. Pantel, 1999, Towards an incremental development of discrete-event simulators for batch plants : use of objectoriented concepts, Comm. Escape 9, Budapest, (Hongrie) 31 Mai - 2 Juin, Comp. And Chem. Eng. Supplements, S565-S568. C.S. Sung, Y.I. Choung, 1999, A neural approach for batching decisions in wafer fabrication. International Journal of Production Research, Vol. 437 No. 13, 3101-3114.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Ontology-Based Information Management in Design Processes Sebastian C. Brandf, Jan Morbach^, Michalis Miatidis^, Manfred TheiBen^, Matthias Jarke^^, Wolfgang Marquardt^ ^Informatik V(Inform. Systems), RWTHAachen University, 52056 Aachen, Germany ^Process Systems Engineering, RWTH Aachen University, 52056 Aachen, Germany "^Fraunhofer FIT, Schloss Birlinghoven, 53754 St. Augustin, Germany Abstract Engineering design processes are highly creative and knowledge-intensive tasks that involve extensive information exchange and communication among diverse developers. In such dynamic settings, traditional information management systems fail to provide adequate support due to their inflexible data structures and hard-wired usage procedures, as well as their restricted ability to integrate processes and product information. In this paper, we advocate the idea of Process Data Warehousing as a means to provide an information management and integration platform for such design processes. The key idea behind our approach is a flexible ontology-based schema with formally defined semantics that enables the capture and reuse of design knowledge, supported by advanced computer science methods. Keywords: Process Data Warehousing, Ontologies, Information Management 1. Introduction Knowledge about engineering design processes belongs to the most valuable assets of an enterprise. Typically, a vast amount of this design knowledge is manipulated by legacy tools and stored in highly heterogeneous sources, such as electronic documents and data bases. To fiilly exploit this intellectual capital, the knowledge must be made explicit and shared among designers and across the enterprise. Thus, the prominent concern of any successfiil approach is the integration of all these knowledge sources in a coherent framework that supports the mining of knowledge and its reuse on demand. In the literature, we can identify a plethora of contributions for the support of engineering knowledge management inside manufacturing enterprises. Document management systems are widely used in industrial praxis for the storage, maintenance, and distribution of documents. A step ftirther, Product Data Management (PDM) systems provide extended facilities for the handling of detailed product information. Regarding the process support, however, current PDM systems have largely focused on the workflow management level [14], while the fine-grained support of development activities (e.g. engineering best practices) has attracted less interest. The identified contributions adequately support information exchange, especially in the later phases of the engineering lifecycle, which are characterized by complete and well-known processes and product models. They lack essential knowledge management capabilities [5] and are less suited for the conceptual design stage [5; 14]. Conceptual engineering design processes are highly creative and dynamic processes, which are hardly predictable [12]. Any software solution has to cope with the continually changing requirements and the many degrees of freedom within these processes. Because of their

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hard-wired usage processes and restricted ability for interoperation, the integrated environments available today are usually unable to offer appropriate support. This paper presents a novel approach for supporting creative, non-deterministic design processes: The Process Data Warehouse (PDW) allows the capture and reuse of design knowledge - data, documents, work processes, and decision making procedures, as well as their interdependencies - through ontologies. In this context, the term "ontology" denotes a conceptual data schema that represents the relevant domain entities and their interrelations by concepts and relations. Within the PDW, ontologies are used to store design knowledge in a formal, machine-interpretable way. This enables the provisioning of advanced computer science methods for managing, enriching, and searching the knowledge within the PDW. Ontology-based knowledge repositories are currently being developed in other areas than process engineering (e.g., [10; 18] in electromechanical engineering). However, these repositories are limited to the storage of product data and documents and do not record the associated work processes and decision making procedures. 2. The Process Data Warehouse The PDW has been derived from the concept of Data Warehousing [7], where large amounts of fixedly structured data (e.g., from sales or accounting) are stored, aggregated, and then presented. To support design and other creative work processes, more flexible structures are needed, which allow the integration of complex and changing domain models [8]. This flexibility has been achieved by building the system on top of loosely connected ontology modules which are held together by a central Core Ontology [3]. The Core Ontology introduces top-level concepts and their relations, which arefiirtherrefined within the peripheral ontology modules.

OntoCAPE

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Figure 1. Simplified view of the Core Ontology and some peripheral ontology modules

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Fig. 1 displays a simplified view of the Core Ontology. Four prominent areas of conceptualization are arranged around the Object as the abstract central concept. Relations associate the concepts of one area to another. Product area. The product area (top) contains concepts for the description of the type and version history of electronic documents and other information resources, as well as their mutual dependencies and their structural decomposition. The Product concept denotes all kind of information elements, such as data items or decision representation objects (e.g. DRL [13]). Products can be aggregated into Document Versions. The different versions of a Product can be bundled by a Version Set. A specialization of Version Set is the (logical) Document, which bundles different Document Versions. Storage area. The storage area (right) describes at which Storage Place a particular Version Set is located, i.e. in which data base, document management system, or external tool it is stored. A Storage Place forms part of a Store, such as a document management system. This allows a tight integration with document management systems: When a user edits a document with the appropriate tool, the changes can automatically be correlated with their representation in the PDW. Descriptive area. The descriptive area (left) contains basic concepts for describing the content or the role of Product Objects on a high semantic level. This includes Content Descriptions and Categorizations, which are grouped into Categorization Schemes. Thus, the descriptive area provides the necessary vocabulary for the content-based retrieval of data and documents. Process area. The process area (bottom) contains the concepts needed to represent the Process Objects that create, use, or modify Product Objects. They comprise general process definitions (Actions) as well as Process Traces resulting fi-om concrete executions of the Actions by Users or Software Tools. The dependencies between elements of all four areas are explicitly modeled as an additional area orthogonal to the other four through the depends On relation. This allows the formulation of specialized relations between objects, independently of their concrete relationships and attributes. Around these fiindamental and domain-independent areas, extension points are placed that can be used to add ontology modules for specific application areas or other specializations. The most elaborate of these extensions is OntoCAPE, a large-scale ontology for the description of the process engineering domain [20], which covers fields like physicochemical properties, process equipment, and mathematical modeling. Here, it extends the descriptive area by refining the Content Description concept (refinement is indicated by dashed arrows in Fig. 1). Unlike conventional tools that store their data in fixed schemas, the PDW uses ontologies with explicitly defined semantics to structure the data domain. The concrete data is then stored as instances of the ontological concepts. This approach allows modifications and extensions of the data structures, even during project execution. This contributes to the flexibility necessary for supporting creative design processes, as mentioned in the beginning of this section. For reasons of interoperability, the Ontology Web Language (OWL) [2] standard fi-om the Semantic Web approach would be the first choice for the representation of ontologies. However, since current OWL-based ontology repositories do not offer an efficiently searchable storage in relational databases, the RDFS-based KAON system [15] is used instead. KAON enables semantic queries directly on the repository, at the

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cost of losing some of the expressiveness of OWL. A converter to translate from OWL (e.g., OntoCAPE) into KAON's RDFS format is currently under development. 3. Using the Process Data Warehouse In this section, we illustrate the utilization of the PDW in engineering design processes. The chosen application scenario is part of a large case study on the design of a polyamide 6 production facility which has been described in [4]: For the design of the polyamide 6 reactor, the engineer creates a reactor model and performs simulation experiments, from which he or she derives design parameters. Based on these parameters, the reactor equipment (apparatus type, heat transfer equipment, etc.) is specified within a process flow diagram (PFD). Model file and PFD are represented as instances of the Document Version concept, while their respective contents are annotated by concepts from the OntoCAPE ontology, which refine the Content Description concept: For example, the content of the PFD is described by instantiating the OntoCAPE concept "Pressure Vessel". For a more precise description. Pressure Vessel can be frirther characterized by supplementary concepts from OntoCAPE: For instance, it could be associated with the ontological concepts Blade Agitator and Heating Jacket. The activities that the engineer performs in the scenario are represented by instances of Process Trace, which are frirther characterized by instantiating specializations of Action (e.g.. Modeling Action to denote the creation of the reactor model). Appropriate instances of the concepts User and Tool Element describe the human and computer agents involved in the work process. As the model file and PFD mentioned above are produced during the engineering activities, the corresponding Document Versions are associated with the Process Traces. Based on the knowledge representation described above, the PDW can support the following tasks: Design documentation. The PDW offers a unified access structure for all types of design information. Such documentation is valuable during the design process itself, for example to facilitate communication among distributed development teams. In later stages of the plant lifecycle, this information can also be used for the support of tasks like change management, plant expansion, or claims management. Information can be accessed independently of the original formats and storage locations. Content-based retrieval of resources. Relying on the content annotations of the resources, content-specific queries can be submitted to the PDW, for instance ''Retrieve all documents that specify a reactor for the polymerization of polyamide 6'\ A graphical query editor supports the composition of such queries in a "query-by-example"-based fashion, using concepts, relations, and already known instances fi-om the ontology. Since the semantics of the query terms have been formally defined within the ontology, the computer "understands" the meaning of the query and is therefore able to retrieve the appropriate documents, even if they are represented by different (but semantically equivalent) ontological concepts within the PDW. Navigation between resources. Content annotation and structural analysis establish semantic connections between products or documents, which can be utilized to retrieve semantically or thematically related resources. In the above scenario, for instance, model file and PFD are connected via their Content Description, as they both represent (different aspects of) the same reactor. Exploring and navigating between the available resources is supported by custom tools like the PDW fi-ont-end.

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Documentation of organizational context. From the Process Objects associated with the respective resources, the user can gain information about the organizational context (i.e., the work processes and decision making procedures) the resource has been created, used, or modified in. This allows to answer questions such as ''What has this model file been utilized for?'' or ''Which decisions have been taken on the basis of this data?'\ Moreover, by analyzing the Process Traces of already accomplished projects, a user can find out who already solved a certain type of problem, and contact this expert directly. To enhance usability and user acceptance, information acquisition (i.e., the process of getting information into the PDW) must be simple and time-efficient. User interaction can be minimized by means of the following acquisition techniques and tools: Simple document metadata, as commonly stored in a document management system, can be easily derived via the storage area concepts and integrations. Annotating the content of highly-structured and formalized documents with well-known syntax and semantics (databases, xml files, model files, etc.) can be achieved by converters that extract information from the documents and map it to the ontologies of the PDW. For demonstration purposes, two prototypical converters are being developed that automatically transform and annotate model files of the process simulator Aspen Plus [1] and flowsheet objects of the CAE system Comos PT [6], respectively. Informal text documents need to be annotated by hand. Markup tools like the OntoMat-Annotizer [16] or SemanticWord [19] support the annotation of such text fragments with ontological concepts via simple drag and drop operations. The tool TRAMP [9] has been especially developed for structuring and annotating multimedia content as produced during threedimensional simulations in plastics engineering. The traces of coarse-grained work processes can be extracted from project management and other planning tools, or captured with the Workflow Modeling System WOMS [11]. For supplementary information like decisions taken and their arguments, specialized tools (e.g., a decision editor) are being developed. Complementary to project coordination, environments for fine-grained process support at the engineering workplaces can store traced experiences in the PDW and reuse them when demanded by providing experience-based methodical guidance for developers. To this end, the PRIME process-integrated modeling environment [17], developed by the authors' research group over the last years, closely interweaves with the PDW. PRIME exploits the process area of the Core Ontology for the definition of guidance and traceability models. Based on the storage area, PRIME'S process-integration mechanism offers the potential to couple the engineering tools in a coherent manner and provide high quality support for the user at the same time. 4. Summary and Outlook The Process Data Warehouse has been developed as an ontology-based repository for the support of design processes. Its key features are (1) flexible data structures and work processes for the support of creative and dynamic work processes, (2) improved retrieval mechanisms based on the use of Semantic Web technologies, and (3) integrated representation of resources, content descriptions, and organizational context to enable experience reuse. Possible application areas have been described, and the possibilities to capture different forms of design knowledge in an integrated manner have been specified. We feel that an evaluation of our approach in an industrial setting will greatly help us to improve our concepts according to real-world requirements and needs. In the near

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future, we are planning to apply our approach to a large-scale industrial project dealing with continuous production processes for rubber profiles. The research described herein has been funded by the German National Science Foundation (DFG) as part of the CRC (SFB) 476 "IMPROVE".

References [I] Aspen Technologies Inc. Aspen Plus, http://www.aspentech.com/, accessed Feb 2006, 2006. [2] S. Bechhofer, F. van Harmelen, J. Hendler, I. Horrocks, D. McGuiness, P. Patel-Schneider, and L. Stein. OWL Web Ontology Language Reference. http://www.w3.org/TR'owl-ref/, accessed Feb 2006, Feb. 2004. [3] S. C. Brandt, M. Schliiter, and M. Jarke. A Process Data Warehouse for Tracing and Reuse in Engineering Design Processes. \nProc. of the Second Internal Conf. on Innovations in Information Technology, Dubai, UAE, Sept. 2005. [4] M. Eggersmann, J. Hackenberg, W. Marquardt, and I. Cameron. Applications of modelling: A case study from process design. In B. Braunschweig and R. Gani, editors. Software Architecture and Tools for Computer Aided Process Engineering, pages 335-372. Elsevier Science, 2002. [5] J. X. Gao, H. Aziz, P. G. Maropoulos, and W. M. Cheung. Application of product data management technologies for enterprise integration. Intern. Journal of Computer Integrated Manufacturing, 16(7-8):491-500, 2003. [6] Innotec GmbH. Comos PT. http://www.innotec.de/en/, accessed Feb 2006, 2006. [7] M. Jarke, M. Lenzerini, Y. Vassiliou, and P. Vassiliadis. Fundamentals ofData Warehouses. Springer-Verlag, 2nd edition, 2003. [8] M. Jarke, T. List, and J. Koller. The Challenge of Process Data Warehousing. In Proceedings of the 26th International Conference on Very Large Databases - VLDB, Cairo, Egypt, 2000. [9] M. Jarke, M. Miatidis, M. Schliiter, and S. Brandt. Media-Assisted Product and Process Traceability in Supply Chain Engineering. In 37th Hawaii International Conference on System Sciences - HICSS, Big Island, HI, USA, January 2004. [10] J. Kopena, W.C. Regli. Functional Modeling of Engineering Designs for the Semantic Web. IEEE Data Eng Bull. 26(4): 55-61, 2003. [II] LPT, RWTH Aachen. Workflow Modeling System - WOMS. http://www.lpt.rwth-aachen.de/WOMS/, accessed Feb 2006, 2006. [12] W. Marquard and M. Nagl. Workflow and information centered support of design processes - the IMPROVE perspective. Computers and Chemical Engineering, 29:65-82, 2004. [13] J. Lee and K.-Y. Lai. What's in design rationale? Human-Computer Interaction, 6:251-280, 1991. [14] S. Mesihovic, J. Malmqvist, and P. Pikosz. Product data management system-based support for engineering project management. Journal of Engineering Design, 15(4): 389-403, 2004. [15] D. Oberle, R. Volz, B. Motik, and S. Staab. An extensible ontology software environment. In S. Staab and R. Studer, editors. Handbook on Ontologies, chapter III, pages 311-333. Springer, 2004. [16] OntoMat. Annotation Portal: OntoMat Homepage. http://annotation.semanticweb.org/ontomat/index.html, accessed Feb 2006, 2006. [17] K. Pohl, K. Weidenhaupt, R. Domges, P. Haumer, M. Jarke, and R. Klamma. PRIME: Towards Process-Integrated Environments. ACM Transactions on Software Engineering and Methodology, 8(4):343^10, 1999. [18] S. Szykman, R.D. Sriram, C. Bochenek, J.W. Racz, J. Senfaute. Design Repositories: Engineering Design's New Knowledge Base. IEEE Intelligent Systems, 15(3): 48-55, 2000. [19] Teknowledge Corporation. Semantic Word, http://mr.teknowledge.com/daml/ Semantic Word/Semantic Word.htm, accessed Feb 2006, 2006. [20] A. Yang and W. Marquardt. An ontology-based approach to conceptual process modeling. In European Symposium on Computer Aided Process Engineering - ESCAPE 14, pages 1159-1164,2004.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Workflow Support for Inter-organizational Design Processes Ri Haf, Markus Heller^, Wolfgang Marquardf, Manfred Nagl*', Rene Worzberger^ ^Process Systems Engineering, RWTHAachen University, 52056 Aachen, Germany ^Computer Science 3 (Software Engineering), RWTH Aachen University, 52074 Aachen, Germany Abstract Inter-organizational design processes are inefficiently supported by management tools w.r.t. to their dynamic nature and inherent complexity. In this paper, we present an innovative framework for the definition, analysis, improvement, and management of such design processes. Key features are to bridge the gap between modeling and execution of inter-organizational design processes and the seamless execution support for both dynamic and static parts of the overall process both by appropriate process management systems. A case study is given for the conceptual design process of a plant for polyamide 6 production by hydrolytic polymerization. Keywords: process design, workflow management, computer aided engineering, interorganizational design processes 1. Introduction Design processes in chemical engineering are often carried out by engineering teams distributed across different organizations. Each team of engineers works on a certain part of the overall design process and is lead by di process manager who plans and controls this process part. Management support of inter-organizational design processes for process managers and engineers should not only address the management of the process parts inside one organization but also the coordination of the distributed process parts across organizational boundaries. A design process part can be identified to be dynamic or static, depending on its scope and character. When desiging complex, unpredictable processes, many alternative and successive technical products are generated and evaluated. Such a design process (part) is not predictable in every detail and is referred to as "dynamic", e.g., the overall design process is dynamic and may contain dynamic design process parts. In contrast, some design process parts, e.g. the design of an apparatus, and can be specified in advance on a fine-grained level. Such design processes are called "static". Obviously, different types of design processes lead to different requirements of management. To coordinate the distributed design process parts, design tasks for each organization must be clearly defmed to avoid misunderstanding. Especially, the documents to be exchanged and the context of each design step, e.g. its dependence of other design steps must be considered. It is also important to note that one organization may want to protect its sensitive information. 1.1. Requirements for Tool Support of Inter-organizational Design Processes Management tools must be able to deal with both static and dynamic design processes [1]. For dynamic design process parts, the tools must support process

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managers and allow for interleaved planning and execution of the process parts overseen by them. For static process parts, the automatic execution by management tools must be supported. Within each organization, it should be possible to choose the management tools used for executing either dynamic or static process parts individually Additionally, the coordination of all design process parts across organizations requires adequate tool support. Firstly, the tools must provide a unified understanding of the entire design process as well as of the process parts for all involved organizations before the design process begins. Secondly, corresponding management tools must be coupled with each other to coordinate the work of all involved engineering teams. Not only the exchange of process status information and resulting documents among the organizations should be supported, but also protection of the sensitive information should be handled appropriately. 7.2. State of the Art Only a few existing systems such as the n-dim system [2] and KBDS [3] address dynamic design processes in chemical engineering directly. Conventional workflow management systems (WFMS) [4] can only be used to execute static (parts of) processes. Advanced or dynamic WFMS support interleaved planning and process execution. An example of such a system is AHEAD [5]. It has been developed in the context of the long-term research project IMPROVE [6] which is concerned with models and tools for design processes in chemical engineering. AHEAD offers a management environment to process managers to plan, control and monitor dynamic processes as well as a work environment for engineers to access and execute assigned design process activities. In the medium time range, such advanced systems are considered to be of high industrial relevance. A delegation-based approach for the management of dynamic inter-organizational design processes is described in [7], where a client and a contractor organization couple their AHEAD-instances for executing the overall design process. However, we assume that the overall process only includes dynamic processes parts. 2. A New Framework for Workflow Support for Design Processes Based on the above requirements, we now present a framework developed within the IMPROVE project that offers a suite of modeling languages, methodologies and tools to support inter-organizational process design with both static and dynamic process parts. The framework spans the entire lifecycle of a design process from its early definition to its execution within conventional and advanced dynamic WFMS (Fig.l). Four different phases are identified: (1) process definition and distribution, (2) process formalization, (3) process execution, (4) and process evaluation and improvement. In the following, the phases are described. 2.1 Process Definition and Distribution In this phase, the focus is put on the understanding of the entire inter-organizational design process. To clarify which organizations have to perform which process steps and deliver what results to whom, an initial draft of the overall process is created, including the planned public design processes of all organizations and their dependencies. This overall process is split up into process parts for each organization. A process part contains all the public activities to be carried out within an organization and it may depend on other process parts. Therefore, a process definition of a design process part should include both the public activities within one organization and the dependencies to other organizations' activities. These process definitions are passed to the corresponding organizations and can be extended inside one organization with private process details (not shown in Fig. 1).

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Figure 1. New framework for workflow support of inter-organizational design processes. In the definition phase, a semi-formal process modeling language is sufficient, since it allows process modelers and analysts of the participating organizations to create clear and simple process models and to gain a common understanding of the overall process. For this purpose, the semi-formal language C3 [8] was developed within IMPROVE and the workflow modeling tool WOMS [9] was developed based on C3 language to define, analyze, and improve design processes on process definition level. When a design process is defined in WOMS, it is possible to analyze the process with different aspects, such as which design tasks are to be undertaken by a certain role, or which document is the input and output for a certain design step. 2.2 Process Formalization The focus of this phase is on process execution within management tools of each organization. In the process definition phase and the process execution phase, instances of the same design process are on different abstraction levels. Accordingly, the modeling languages used in each phase differ in the modeling capabilities. A semantic gap exists between the two types of modeling languages. For example, a semi-formal modeling language targeting process understanding purposes (cf section 2.1) contains no information regarding the execution state of a process, but an execution-oriented modeling language addresses such information. Further, while the data type definition is an important part of the later modeling language, it is not necessary for the former modeling language. During semi-automatic transformation between both modeling languages the missing content can be added to obtain an executable process definition from the semiformal process definition. The transformation also realizes a mapping from a platform-independent notation to a platform-dependent notation. Therefore, the transformation depends on the type of WFMS, which is used in the process parts for execution. Within our framework, process modeling experts and process managers of each organization can choose different platforms for process execution, i.e. conventional WFMS to automate static process parts, advanced WFMS for dynamic process parts, or both systems in parallel. We use the aforementioned advanced WFMS AHEAD for the management of dynamic processes which models design processes as dynamic task nets with the DYNAMITE modeling language [5]. We use a conventional WFMS, SHARK

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from ObjectWeb Consortium, because it is able to process workflow definitions in the standardized workflow definition interchange format XPDL [10], so that other WFMS using this standard format can be used without further adaptation of the framework architecture. If there are significant interactions among processes parts, an AHEADAHEAD coupling should be used, because specifical modeling elements are introduced in AHEAD for the dynamic design processes, e.g., feedback relationships for iterations [7]. To deal with both WFMS, two different transformers have been realized via simple XSL transformations in our framework: one transformer for transforming the C3-nets into the dynamic task nets for execution within AHEAD, and the second for transforming the C3-nets into the XPDL-nets for execution within the SHARK. 2.3 Execution In the execution phase, either the conventional or the advanced WFMS of the organizations are coupled with the WFMS of other organizations. The coupling is bi-directional in each case to avoid the need for a centralized coordination system among all WFMS. Several instances of AHEAD can be coupled for delegation-based cooperation relationships [7], and the coupling of a conventional WFMS (SHARK) with an advanced WFMS (AHEAD) has been recently realized (cf. section 3). Tool integration of AHEAD and SHARK is done in an a posteriori manner, so that no significant modifications of the publicly available source code of SHARK were required. We only need to extended the SHARK system with an event-based coupling component to exchange workflow-related events between the coupled workflow systems. 2.4 Evaluation and Improvement The process evaluation phase focuses on the re-use of previously executed instances of design processes. Comparison and analysis of the to-be process and the as-is process indicate the deficiency in the process definition. Additionally, the accumulated process instance data can be used to determine the beneficial process details for the optimized and more efficient design processes in the future. In this framework, all the information gained can be recorded with WOMS manually to improve the design process step by step. As described in [11], it is also possible to capture the design process history into process traces automatically as a basis for design process improvement. Summary The proposed fi-amework meets all requirements of section 1.1 by providing adequate modeling languages and tool support for process modelers, process managers and the engineers within all phases of the design process lifecycle. Process managers can use different management tools specialized on the execution of dynamic or static processes to support process management. Within the framework architecture, the management tools can be coupled to form 2i federation of process management tools and exchange process information and data with each other. 3. Case Study In the following, the concepts and tools described so far are implemented through a case study on the design process of a plant for polyamide 6 (PA 6) productions, including the design of the reaction and separation system as well as the extruder configuration. More information about the PA6 design process can be found in [6]. Figure 2 shows a simplified overview on the inter-organizational PA 6 design process. In this scenario, a chemical engineering company and an extruder manufacturer are involved. The roles manager, reaction expert, and separation expert belong to the chemical engineering company and the plastic engineering belongs to the extruder manufacturer. We assume

Workflow Support for Inter-Organizational Design Processes Reaction Expert

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Figure 2. Design process for Polyamide 6 production. that the chemical engineering company is responsible for the entire process and the extruder manufacturer is a contractor for the design of the polymer processing section. This corresponds to the classical disjunction of the two domains of chemical and plastics engineering. In order to illustrate the cooperation between the two companies without blurring the overview, only the information, which is delivered by one company to another, is shown in Fig. 2. Now we briefly illustrate the application of the proposed framework by means of this scenario (Fig. 3). Firstly, WOMS is used for the modeling of the entire design process (e.g., by an expert in the chemical engineering company). An initial draft of the design process is created (cf. Fig. 2, without the highlighted blocks). This initial draft is improved by inserting an additional design step (cf. Fig. 2, the highlighted blocks) to account for the fact that the separation system can be reduced considerably in size, when part of the degassing is performed in the extruder. After that, individual design process parts are extracted from the overall process definition and distributed to both companies: the chemical engineering company as the client receives the overall process definition. The plastics engineering company as the contractor receives his polymer part and additionally all inter-organizational control and data flow, which depend on activities of the chemical engineering company. A similar scenario is given in [7], where coupled instances of AHEAD are used in both organizations. In this paper, we focus on the mixed case: the client uses AHEAD to manage and execute the overall dynamic design process while the plastics engineering company can use SHARK. The process fragment for the plastics engineering company is small, static and well-understood, e.g. it may be repetitively carried out and thus offered as a service to customers like the chemical engineering company. In the formalization phase, the client uses the C3-AHEAD-transformator to create a dynamic task net and imports it into AHEAD. The contractor uses the C3-XPDL-transformator and installs the resulting XPDL workflow description in the SHARK system. The client starts the process within AHEAD and calls for the execution of an instance of the installed workflow within SHARK after providing needed input parameters. Both systems are coupled with a CORBA-based communication server (providing message passing and message queuing) for the exchange of process state information and data transmission.

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%

Figure 3. Workflow support for polyamide 6 design process.

4. Conclusion In this paper, we have presented a framework of models, methodologies and tools firom the IMPROVE project to support inter-organizational design processes. It supports process modelers, process managers and the engineers within all phases of the design process lifecycle. Dynamic and static process parts are executed by coupled advanced or conventional workflow management systems. The application of the framework was demonstrated through a case study on the conceptual design process of a plant for polyamide 6 production. In the future, we aim at extending the existing transformers between modeling languages of different phases to address advanced problems, e.g. hierarchical design process structures, and we will work on improved user interfaces.

Acknowledgement Financial support of Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Center 476 IMPROVE is gratefully acknowledged.

Reference [1] B. Westfechtel. Models and Tools for Managing Development Processes, LNCS 1646, Springer, Germany, 1999. [2] A. W. Westerberg, E. Subrahmanian, Y. Reich, S. Konda, and the n-dim group. Designing the process design process. Comp. & Chem. Eng., 21(Suppl): S1-S19, 1997. [3] R. Banares-Alcantara and H. Labadibi. Design support systems for process engineering II KBDS: an experimental prototype. Comp.& Chem. Eng., 19(3):279-301, 1995. [4] P. Lawrence, editor. Workflow Handbook. John Wiley, Chichester, UK, 1997. [5] D. Jager, A. Schleicher, and B. Westfechtel. AHEAD: Graph-Based System for Modeling and Managing Development Processes. In M. Nagl, A. Schurr, and M. Miinch, editors, ACTIVE - Applications of Graph Transformations with Industrial Relevance, LNCS 1779, pp. 325-340, Castle Rolduc, The Netherlands, 1999. Springer. [6] W. Marquardt and M. Nagl. Workflow and information centered support of design processes — the IMPROVE perspective. Comp. & Chem. Eng., 29(l):65-82, 2004. [7] M. Heller, D. Jager, M. Schliiter, R. Schneider, and B. Westfechtel. A management system for dynamic and interorganizational design processes in chemical engineering. Comp. & Chem. Eng., 29:93-111, 2004. [8] C. Foltz, S. Killich, M. Wolf, L. Schmidt, and H. Luczak. Task and information modeling for cooperative work. In Smith and Salvendy. Proc. HCI Intl., vol. 2, pp. 172-176, 2001. [9] R. Schneider and S. Gerhards. WOMS - A Work Process Modeling Tool. In M. Nagl and B. Westfechtel, editors, Modelle, Werkzeuge und Infrastrukturen zur Unterstiitzung von Entwicklungsprozessen, pp. 375-376, 2003. Wiley VCH. [10] Workflow Management Coalition. Workflow Process Definition Interface - XML Process Definition Language. WFMC-TC-1025 Version 1.0, 25 October 1999. [11] M. Jarke, T. List, and K. Weidenhaupt. A Process-Integrated Conceptual Design Environment for Chemical Engineering, Proc. 18th Intl. Conf on Conceptual Modeling (ER '99), pp. 520-537, 1999.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

Rigorous scheduling resolution of complex multipurpose batch plants: S-Graph vs. MILP Sergio Ferrer-Nadal,^ Tibor Holczinger,^ Carlos A. Mendez,^ Ferenc Friedler,^ Luis Puigjaner^ ^Chemical Engineering Department -CEPIMA, Universitat Politecnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain ^Department of Computer Science, University of Veszprem, Egyetem u. 10, H-8200, Veszprem, Hungary Abstract Highly efficient optimization methods are needed to deal rigorously with complex short-term production scheduling problems. This work presents a comparative study of two alternative solution strategies based on a mathematical programming approach and a graph-oriented method. Different scenarios of increasing complexity are addressed in order to evaluate the current capabilities and limitations of each solution method. Keywords: Batch operations, MILP model, S-graph. 1. Introduction Batch plants are the best option to manufacture relatively small quantities of specialty chemicals, pharmaceuticals and agrochemicals high-value products with variable demand patterns. The most general case corresponds to the so-called multipurpose batch plants where a wide variety of products with different processing recipes can be produced by sharing flexible processing equipment capable of performing different batch operations. This work is focused on the short-term scheduling of multipurpose batch plants comprising (i) multiple batches for each product, (ii) different intermediate storage policies, (iii) sequence-dependent setup times, and (iv) non-zero transfer times between stages. A direct comparison between the performance of an enhanced S-graph scheduling algorithm introduced in Romero et al. (2004) and an extended version of the scheduling framework proposed by Mendez and Cerda (2003) is presented. The S-graph algorithm is developed as a PhD work, its components are free softwares, e.g. the embedded LP solver is also freely available nonprofessional software, whereas the MILP framework is implemented within the commercial modeling language GAMS (Brooke et al., 1998) and solved with CPLEX 7.5. 2. Alternative resolution strategies: MILP and S-Graph 2.1. MILP-based approach Different MILP-based scheduling models have been proposed to manage the inherent complexity of multipurpose batch processes. Mathematical formulations provide a rigorous and explicit representation of the main decisions and constraints involved in the scheduling problem. MILP models can be solved to global optimality through different general-purpose optimizers, commonly LP-based branch and bound methods. Mathematical programming decouples the modeling issue from the solution algorithm.

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S. Ferrer-Nadal et al

Therefore, MILP-based methods can take advantage not only of better modeUng techniques but also of significant improvements in the optimization codes. Following this direction, Mendez and Cerda (2003) developed a continuous-time MILP formulation that relies on the notion of general precedence. This generalized precedence notion extends the immediate predecessor concept to consider all batches belonging to the same processing sequence. This model defines a couple of sets of binary variables in order to sequence and allocate processing tasks as well as related storage operations, without compromising the global optimality of the solution. The model contains the following general constraints: • Processing unit allocation constraints state that a single processing unit must be assigned to every required processing task. • Duration of a task constraints define that the duration of a task must consider the overall time required to perform: (i) the loading of material from the previous stage; (ii) the batch processing operation itself; (iii) a possible holding time in the processing unit and; (iv) the unloading of the material to either next stage or to a suitable intermediate storage tank. • Task precedence constraints are defined for every pair of consecutive tasks that must be sequentially performed for a particular product. The task can never begin before the material from the preceding task starts being transferred to the unit assigned. Moreover, non-zero transfer times enforce that unloading and loading operations from/to units involving consecutive tasks must be synchronized, unless the material is previously stored in an intermediate storage tank. • Task sequencing at every processing unit constraints define the order in which a pair of tasks is performed whenever both are allocated to the same processing unit. • Storage task sequencing constraints define the order of storage tasks assigned to the same tank. Because of the general precedence notion, the same sequencing variables used for a pair of processing tasks can be utilized for their related storage tasks. 2.2. S-Graph algorithm, a graph-oriented method Besides mathematical programming models, other rigorous alternative methods have emerged over the last years. This is the case of the S-graph framework proposed by Sanmarti et al. (2002), a sophisticated graph representation which exploits specific knowledge of the problem to develop efficient algorithms for solving the production scheduling of batch processes. In contrast to MILP-based methods, the S-graph embeds the modeling aspects into the solution algorithm. Recipes can be conveniently represented as a directed graph, where the nodes represent the production tasks and the arcs the precedence relationships among them. There are two specific S-graphs, the recipe-graph and the schedule-graph. The recipe-graph is based on the recipe and describes the inputs of the scheduling problem. The schedule-graph describes a single solution of the scheduling problem. Because of its combinatorial characteristics, a branch-and-bound procedure may generate the optimal schedule of a scheduling problem, i.e. the schedule-graph that corresponds to the minimal makespan. The recipegraph with no equipment unit assignment serves as the root of the enumeration tree of the B&B procedure. At any partial problem, one equipment unit is selected and then all child partial problems are generated through the possible assignments of this equipment unit to unscheduled nodes. The bounding procedure tests the feasibility of a partial problem by a cycle detection algorithm. If this test is positive, it determines the lower bound for the makespan of all solutions that can be derived from this partial problem simply by using the well-known longest path algorithm. The search space can be reduced when there are multiple batches for the products (Holczinger et al., 2002). In

Rigorous Scheduling Resolution of Complex Multipurpose Batch Plants

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the branching step the infeasible solutions (cyclic S-graph) can be detected before their appearance with a simple algorithm, so called "look ahead" algorithm (Holczinger et al., 2004). Using the longest path algorithm in the bounding step is not always the most effective method because the lower bound can be sharpened by a LP model (Sanmarti et al, 2002). 2.2.1. Handling transfer times within the S-graph algorithm Neither additional nodes nor any modification of S-graph is needed to handle transfer times. The increase of the weight of the recipe-arc (or arcs) starting from the second task node by the transfer time means that the equipment unit of the second task node can start to perform the task with transfer time later than the previous equipment unit finishes its work. The increase of the weight of the schedule-arc starting from the second task node by the transfer time means that the changeover (cleaning) of the equipment unit of the first task can begin with transfer time later than it finishes its work (see Figure 1). If transfer times are considered for the first or the last task of the batch, the specific transfer time can be contained within the processing time.

Figure 1. Handling non-zero transfer time by modifying the weight of the arcs. 3. Comparison study and computational results Three case studies have been analyzed in order to compare the main strengths of these two approaches. All runs were performed in a AMD Athlon XP 2600 MHz. 3.1. Case Study 1 This case study comprises fifteen batch processing units and 33 batches of eight different products. Problem data is reported in the Example 4 in Romero et al. (2004). Different instances of this case study have been considered to account a non intermediate storage policy (NIS) as well as unlimited intermediate storage (UIS). In some instances, transfer times are assumed to be non-zero with a duration equal to 10% of the corresponding processing task. Main computational results are summarized in Table 1. The analysis of the results reveals that both methods were able to generate the optimal solutions in all the cases with modest CPU time. The MILP-based approach was more efficient than S-graph algorithm, although the difference in terms of time could be neglected in an industrial environment.

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S. Ferrer-Nadal et al Table 1. Results for case study 1 Makespan, min.

Instance - Description

CPU Time, s

MILP

S-graph

MILP

S-graph

11

NIS Zero TT

2000

2000

0.093

1.313

12

UIS Zero TT

1830

1830

0.078

3.828

13

NIS10%TT

2311

2311

0.109

1.54

14

UIS10%TT

2081

2081

0.093

4.34

3.2. Case Study 2

Problem information for case study 2 is described in the algorithm evaluation section in Romero et al., (2004). This problem comprises eight products and six units in parallel. A single batch of each product is considered and transfer times are assumed, in some instances, to be 5% of the corresponding processing times. The performance of the methods was tested for different storage policies and the results obtained are reported in Table 2. Table 2. Results for case study 2 Instance Description

Makespan, h

CPU Time^, s

MILP

S-graph

MILP

S-graph

11

NIS Zero TT

77 (cyclic schedule)

97

0.406

18.718

12

UIS Zero TT

64

64

0.468

2.75

13

NIS 5% TT

104.15

104.15

0.265

35.515

14

UIS 5% TT

70.3

70.3

0.531

11.125

For instance 1, the MILP-based approach found a solution with a shorter makespan than the S-Graph algorithm. At first sight, it seems an advantage of the MILP formulation with respect to the S-Graph. However, a detailed analysis of the schedules shown in Figures 2 and 3 reveals the occurrence of cycles in the schedule generated through the MILP-approach. A cycle appears if: (i) a set of processing or storage units are simultaneously receiving and transferring their intermediates and (ii) no intermediate storage is available. Cycles may appear in job-shop or multipurpose processes (bidirectional flows) whenever zero transfer times are considered in a non-UIS transfer policy. Although this situation can be easily identified during the search procedure embedded in the S-graph algorithm, it can be an important drawback for most of the MILP models, which can have problems to eliminate these infeasible solutions. Zero transfer times seems to be quite realistic if we only look at the small percentage that they suppose in most chemical processes compared to other operations in the plant. However, the chemical industry is commonly characterized by shared tanks as well as zero-wait, non-intermediate or finite storage policies. In these cases, the transfer of intermediates from unit to unit need the proper synchronization of the units involved during the transfer operation. In cyclic schedules this synchronization is impossible to perform. Therefore, neglecting these aspects may lead to these infeasible situations. However, it is worth mentioning that zero transfer times represent a problem

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simplification and cycle generation can be avoided if actual transfer times are included in the problem formulation.

Figure 2. Infeasible solution including cycles for instance 1 of case study 2

___

___

•72h"

Figure 3. Optimal S-graph solution for instance 1 of case study 2 33. Case Study 3 Case study 3 deals with a large-scale scheduling problem introduced in Sanmarti et al. (2002). This challenging problem comprises nineteen pieces of equipment to manufacture ten different products. Unit-dependent changeover times and transfer times are also taken into account. Table 3. Results for case study 3 Instance - Description 11

NIS Zero TT

Makespan, min

CPU Time, s

MILP

S-graph

MILP

S-graph

7740

7740

725.11

1.00

12

NIS 5% TT

8157

8163

1937.8

1.92*

13

UIS Zero TT

7740

7740

221.75

1.06

14

UIS 5% TT

8157

8163

506.36

2.48*

* Optimality could not be proved in 1 hour CPU time. Table 3 summarizes the computational results for the several instances addressed of case study 3. In turn. Table 4 reports the detailed solution search corresponding to instance 2 of case study 3. As can be seen, S-graph was able to find a very good solution in only 1.9 seconds while MILP generated the first feasible solution after 45 seconds. The initial solution of S-graph was improved by the MILP model after 1639 seconds. On the other hand, S-graph was not able to prove the global optimality of the solution after Ih. CPU time. This situation reveals that the better guided search

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implemented in the S-graph algorithm may allow finding very good schedules with low computational effort, which is a desirable feature in any industrial environment. Table 4. Evolution of the search for instance 2 of case study 3 Time, s

MILP current best solution

S-graph current best solution

1.91

-

8163.0

45

10359.0

8163.0

85

10155.0

8163.0

1937.8

8157.0 (optimum)

8163.0

4. Conclusions Two alternative ways of representing and solving the scheduling problem of a multipurpose batch plant have been presented and compared. As the MILP formulation, S-graph can easily manage non-zero transfer times without greater computational effort. For multipurpose batch plants with zero transfer time, it was observed that MILP models may generate infeasible solutions because of cyclic schedules. In contrast, Sgraph has implemented an efficient pre-processing algorithm for cycle identification and for some specific problems, was able to find very good solutions in a short time. Finally, it is worth to point out that a nonprofessionally realized students' software has been compared with a highly professional modeling environment (GAMS) and optimizer (CPLEX). On the one hand, GAMS provides an user interface where the model can be easily adapted to new process features. On the other hand, S-Graph can be a bit rigid in that sense but the use of this framework does not require any expensive software, which can be a significant advantage in an industrial environment. Acknowledgements Financial support received from the European Community projects (MRTN-CT-2004512233; RFC-CR-04006; INCO-CT-2005-013359) and the Generalitat de Catalunya with the European Social Fund (FI grant) is fully appreciated. References Holczinger, T., Romero, J., Puigjaner, L. & Friedler, F. (2002). Scheduling of Multipurpose Batch Processes with Multiple Batches of the Products, Hungarian J. Ind. Chem., 30, 305. Holczinger T., Biros G., Friedler F., Sarkozi N. (2004). Acceleration tools for batch process scheduling, presented at the CHISA 2004 (16th Intemational Congress of Chemical and Process Engineering), Prague, Czech Republic, August 22-26. Mendez, C.A. & Cerda, J. (2003). An MILP continuous-time framework for short-term scheduling of multipurpose batch procecess under different operation strategies. Optimization & Engineering, 4, 7 - 22. Romero, J., Holczinger, T., Friedler, F. & Puigjaner, L. (2004). Scheduling intermediate storage multipurpose batch plants using the S-Graph. AIChE Journal, 50, No. 2, 403 - 417. Sanmarti, E., Holczinger, T., Friedler, F., & Puigjaner, L. (2002). Combinatorialframeworkfor effective scheduling of multipurpose batch plants. AIChE Journal, 48, No. 11, 2557 - 2570

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Integrative optimization of refining and petrochemical plants Chufu Li^, Xiaorong He^, Bingzhen Chen^, Bo Chen^, Zhenzhi Gong^,Liu Quan'' ^Department of Chemical Engineering, Tsinghua University, Beijing 100084, China ^ Lanzhou Petrochemical Company, PetroChina, Lanzhou 730060, China

Abstract Integrative optimization of refining and petrochemical plants are proposed based on GIOPIMS (Graphic I/O Petro-chemical Industry Modeling System) in this paper. The optimization of raw materials to ethylene cracking unit is a key step for integrative optimization of refining and petrochemical plants. Naphtha is the primary raw material to ethylene cracking unit, and is also the raw material for continuous reformer to produce benzene, toluene and xylene. It is an approach for profit increase to optimize amount of naphtha which feeds into ethylene cracking unit and continuous reformer. Application case study shows that profit has an about $1.0 million increase per month than the case without optimization. It can be concluded that integrative optimization of refining and petrochemical plants is the developing trend and should attract more concern in the future. Keywords: integrative optimization; production planning; refining and petrochemical; GIOPIMS 1 Introduction Integration of refining and petrochemical plants as an operation strategy has been adopted by many petrochemical companies such as Exxon^ MobiK ShelK BP^ Fina. They have studied the integration strategy for many years and procured a great profit. For example, BASF and Fina jointly built 900,000 t/a ethylene and refining integrated plant, producing $60 million coordinated benefit a year (Li Xuejing, 1999). In petrochemical supply chain management, integrative optimization of refining and petrochemical plants has been concerned widely in the word. It is very important for profit increase to optimize raw materials' supply each other between refining and petrochemical plants, seeing Fig 1. The optimization of raw materials of ethylene cracking unit is a key step especially for integrative optimization of refining and petrochemical (Joly.M, Moro.L.F.L, Pinto.J.M, et al, 2002, Wu Xiaoyun, 2003, Zhang Deyi, 2003). ^1 Uvi Inln'i; ] [ prndui t s i

Crude o11

\rjphthri, 'fupped rii ] Refi ru'ri^

IJght du^st'l

oil ical

Crude o i l

i Petrncljenjic
Itsicnit'tJi riles

Fig 1 Integrative optimization of refining and petrochemical plants The raw materials to ethylene cracking unit involves naphtha, topped oil and light diesel oil. At the same time.

•Corresponding author. Tel.: +86 1062 784572; fax: +86 1062 770304 E-mail address: [email protected]

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naphtha is also the raw material for continuous reformer to produce benzene, toluene and xylene. In refining process, one kind of crude oil could produces one type of naphtha, and different naphtha has a different production rate of ethylene, propylene and butadiene in ethylene cracking unit (Table 1). Therefore, the amount of naphtha fed to the ethylene cracking unit or continuous reformer should be optimized. In this paper, these studies are proposed based on GIOPIMS (Graphic I/O Petro-chemical Industry Modeling System) developed by Department of Chemical Engineering of Tsinghua University. 2 LP model for integrative optimization of refining and petrochemical plants In the LP (Linear Programming) model for integrative optimization of refining and petrochemical plants (Li, W.K 2004& Li, P 2004), the objective is to maximize the profit. max Z = 2 ] ;?.x. - ^ p .x. - ^ ^ q^x', -C

(1)

Constraints include: 1). Material balance constraints of units k

(1-<J')X4=I4 (2) k

x[<2Xx[

h

2). Production rate constraints of units

*

(5)

5). Raw materials supplies and products demand constraints

(3)

(6)

k

6). Inventory constraints

3). Capacity constrains of units

(7) k

*

h

7). Product quality constraints

(4)

Y,aix[>4x[

k

(8)

4). Feedstock flowrate proportion constraints of units Nomenclature

K2[

the maximum production rate of the hXh out

p. price of product/; stream of unit/; Pj price of raw material/; V\

the minimum capacity of unit /;

X. output of product/; V2^ the maximum capacity of unit/; Xj consumption of raw materialy; ^1^ the minimum proportion of the ^ h feedstock x[ the Ath feedstock flowrate of unit /; of unit/; x^ the Mh out stream flowrate of unit/; ^2/

the maximum proportion

of the Ath

qj operating cost per unit of unit /; C fixed cost; 5 K\[

loss rate of unit/; the minimum production rate of the Mh out

stream of unit/;

feedstock of unit/; Z)l.. s the minimum supply of raw material / or demand of product/; iD2.... the maximum supply of raw material / or

Integrative Optimization of Refining and Petrochemical Plants

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demand of producty

cdj^ product quality of the Ath feedstock of unit /;

Vl initial inventory of tank /;

OJ^ product quality of the h\h out stream of unit /.

V^ final inventory of tank /; 3 GIOPIMS (Graphic I/O Petrochemical Industry Modeling System) GIOPIMS is a graphical system for optimal production planning in petrochemical company (Fig l)(Li Chufu, 2005 ). In GIOPIMS, users only need to draw a flowsheet and input related data, the system will generate the LP model (Eq.l- Eq.8) for production planning in refining and petrochemical process and solve it. The system can freely integrate refining model and petrochemical model for production plarming into one to solve.

IWiilfflMlfflMlllllillllHiillllMiliyMi ^1ili(l)

Editd)

View (2)

P»g«(J)

InpatCt)

Solve (£)

Output®)

Fornd)

Winao«(|)

HelpQC)

Yaha naphtha (1 25)

Changijing naphtha (1 96j

Tuha naphthaCl 78) (0 00).

J

', Qinhai naphtha©. 88)

' 'Ksnjiang naphtha© 07)

Fig .1 A graphical model for integrative optimization of refining and petrochemical plantsin GIOPIMS GIOPIMS mainly covers following functions: 1) Complicated flowsheet's analysis and sub-flowsheets' integration; 2) Graphic edits such as addition, deletion, copy, paste same as Microsoft Office; 3) Automatically generating mathematical model for production planning according to flowsheet and related data; 4) Quickly solving model and showing optimal results in graphic interface; 5) Sensitivity analysis and stability analysis of optimal solution; 6) Strong data errors detection and infeasible model diagnosis and so on. 4 An application to integrative optimization of refining and petrochemical plants In a petrochemical company, it can process 8,500,000 t crude oil and produce 240,000 t ethylene per year. Its primary products are gasoline, kerosene, diesel oil, lube, LDPE, HDPE, SAN, ABS, polypropylene and etc. The LP model of production planning in the company involves about one thousand constraints and two thousand

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C Li et al.

variables. 4.1 Optimal distribution of naphtha In the petrochemical company, there are 6 kinds of crude oil in petroleum process, so there are 6 kinds of naphtha fed to the ethylene cracking unit or continuous reformer. Production rate of ethylene, propylene and butadiene in cracking unit of all naphtha are shown in Table 1. Therefore, amount of naphtha should be optimized to feed into ethylene cracking unit or continuous reformer. Table 1 Yield of cracking product when different naphtha from various crude oils i n the same cracking conditions Cracking product/wt% Torch gas Methane S D Hydrogen Ethylene Propylene C4

Gasoline Heavy diesel oil Loss Total

Beijiang 3.20 12.51 0.97 31.32 16.15 11.37 20.00 3.88 0.60 100.00

Nanjiang

3.10 9.38 0.76 31.48 15.96 12.01 22.71 4.00 0.60 100.00

Naphtha Qinhai T\iha Yaha

3.01 9.00 0.81 29.26 14.06 11.01 27.00 5.25 0.60 100.00

3.22 9.20 0.81 30.49 15.53 11.26 22.00 6.89 0.60 100.00

Changqing 3.50 9.38 0.84 28.73 14.21 10.33 25.00 7.41 0.60 100.00

3.10 8.88 1.00 27.26 11.35 8.99 30.00 8.82 0.60 100.00

Integrative optimization of refining and petrochemical plants in GIOPIMS gives the optimal distribution of different naphtha fed to continuous reformer and ethylene cracking unit (Table 2). The optimal results show that Nanjiang naphtha is most suitable for ethylene cracking, and other 5 naphtha is fit for continuous reformer. It is in accordance with experiment data (Table 1) because Nanjiang naphtha has the highest production rate of ethylene, propylene and butadiene. Table 2 Optimal distribution for different naphtha fed to continuous reformer and ethylene cracking unit Item

Total Continuous reformer Proportion(%) Ethylene cracking unit Proportion(%) Sales

Beijiang Yaha

Naphtha (10^ t) Nanjiang Qinhai Tuha

Changqing

0.3108

3.6488

0.6772

1.7848

2.747

1.2453

0.3108

0.0

0.6772

1.7848

1.0783

1.2453

100.0

0.0

100.0

100.0

39.3

100.0

0.0

3.6488

0.0

0.0

0.0

0.0

0.0

100.0 0.0

0.0 0.0

0.0

0.0

0.0

0.0 1.6687

0.0 0.0

0.0

0.0

0.0

0.0

60.7

0.0

Proportion(%

)

Integrative Optimization of Refining and Petrochemical Plants

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4.2 Contrast on optimal results Table 3~4 give the contrasts in raw materials purchased and product output after integrative optimization of refining and petrochemical plants. A is a basic scheme. In scheme A, all naphtha are simply mixed nor optimized into ethylene cracking unit, and no light diesel oil replace naphtha to feed into ethylene cracking unit. In scheme B, some diesel oil replace naphtha to feed into ethylene cracking unit and naphtha are optimized into ethylene cracking unit and continuous reformer. After integrative optimization of refining and petrochemical plants, raw materials purchased and products' output has some change (Table 3-4). The profit of scheme B has an about $1.0 million increase than scheme Ao Table 3 Change of raw materials purchased in a month after integrative optimization of refining and petrochemical plants Raw materials Benzene Butadiene Methanol Other Total cost ($)

Purchase amount (10'* t) £,. A C U r» « A Scheme A Scheme B B-A 0.1096 0.03 -0.0796 0.1921 0.1701 -0.022 0.0518 0.0559 0.0041 0.0041 0.0036 -0.0005 -1,080,358

Table 4 Change of product output in a month after integrative optimization of refining and petrochemical plants Products Naphtha Light diesel oil Benzene Toluene Xylene 90# gasoline 0# diesel oil Dry gas Fuel gas Styrene Propylene MTBE C4

1-butylene Extracted oil Total value ($)

Output (lO'^t) Scheme A

Scheme B

B-A

0.00

1.6687

1.6687

4.3296

2.173

-2.1566

0.1822 0.6329 0.7279 7.7046 23.6766 1.0157 1.0284 0.124 0.1328 0.1616 0.0719 0.0252 0.2055

0.2221 0.7537 0.8795 7.7008 23.8151 1.0663 0.9782 0.0634 0.1039 0.1744 0.0776 0.0306 0.2271

0.0399 0.1208 0.1516 -0.0038 0.1385 0.0506 -0.0502 -0.0606 -0.0289 0.0128 0.0057 0.0054 0.0216 22,619

4.3 Analysis on economical increase 1). Light diesel oil is a raw material to ethylene cracking unit, but it is not suitable for continuous reformer. When some light diesel oil replaces naphtha to feed into ethylene cracking unit, more

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C.Lietal

naphtha is fed into ethylene cracking unit to produce valuable benzene, toluene and xylene. Thus, the profit has an increase o 2). After integrative optimization of refining and petrochemical plants, purchased amount of raw materials (benzene and butadiene) have a great decrease because the output of C4 and crack gasoline increase to 7242 t and 14506 t from 6789 t and 13424 t respectively. C4 is raw materials to produce butadiene, and crack oil is used to produce benzene. Therefore the output of benzene produced in chemical process increases to 3642 t from 3337 t, and butadiene increases to3527 t from 3306 t. The purchased amount of benzene and butadiene decrease to 300 t and 170It from 1096 t and 1921 t respectively, so the total cost of raw materials has a great decrease.

5 Conclusions The optimization of raw materials to ethylene cracking unit is a key step especially for integrative optimization of refining and petrochemical plants 1) Naphtha should be optimized to feed into ethylene cracking unit or continuous reformer according to different production rate of ethylene, propylene and butadiene in cracking unit. 2) Naphtha replaced with some light diesel oil is fed into ethylene cracking unit, thus more naphtha can be fed into continuous reformer to produce valuable benzene, toluene and xylene. It is approach to produce profit in petrochemical company. It can be concluded that integrative optimization of refining and petrochemical plants is the developing trend and should attract more concern in the ftiture.

References Joly.M, Morol.L.F.L, Pinto.J.M, 2002, Planning and scheduling for petroleum refineries using mathematical programming, Brazilian Journal of Chemical Engineering, 19, 2, 207-228. Li Xuejing, Yang Deen,1999, Comment upon refining / chemical integration. Petrochemical technology & application ,17,3, 131-133. Wu Xiaoyun, 2003, Recovering Ethylene and Light Olefms from Refinery Dry Gas to Exploit Refining and Chemical Integration Advantages to the Full, Petroleum & petrochemical today, 11,6,18-27. Zhang Deyi, 2003, Outlook of oil refining industry in the early 21^^ century. Petroleum refinery engineering, 33, 1, 1-6. Li, W.K., Hui, C.W., Li, P, Li, A.X , 2004, Refmery planning under uncertainty, Ind.Eng.Chem.Res , 43,21, 6742-6755. Li, P., Moritz, W., Gvnter, W, 2004, Optimal production planning for chemical processes under uncertain market conditions, CET, 27, 6, 641-651. Li Chufii, He Xiaorong, Zhang Qiuyi, etc, 2005, Development of a graphic modeling system for optimal production planning in petrochemical industry and its application. Petroleum processing and petrochemicals, 36, 10, 45-48.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

2045

Scheduling of identical and parallel on/off production units under uncertainty in cost and demand prediction Petteri Pulkkinen and Risto Ritala Tampere University of Technology, PB 692, FI-33101

Abstract In this paper a production system of identical and parallel production units is studied. The emphasis is on including stochasticity in the optimization. Methods for using stochastic constraints and objectives are presented an experimented with. An application of the results is scheduling of thermo-mechanical pulping (TMP) at paper mills, which, due to high electricity consumption, has a significant the cost reduction potential with optimal scheduling. It is found that the scheduling problem with uncertainty can be solved for practical cases, but its formulation requires specifying preferences that process operating personnel is rather unaccustomed to give. Keywords: optimization, uncertainty, scheduling 1. Introduction When operating industrial processes the uncertainty in decision making is rather often neglected or dealt with implicitly and intuitively. In the optimization problem presented in this paper the effect of uncertainty in production cost and demand on operative decisions is studied. Cost uncertainty is modeled with AR(N) or random walk process, and that of demand with a two state Markov process. The two formulations are compared of the objective: either the decision maker's attitude towards risk is modeled with exponential utility ftinction or the expected cost is minimized subject to an additional constraint for probability of exceeding user specified maximum cost allowed. Transforming the constraints of the corresponding deterministic problem into probabilistic constraints of the stochastic problem is also discussed. In all cases discussed in this paper the probability distributions of the stochastic variables are calculated directly rather than obtained via Monte Carlo simulations. With the exception of stochastic demand the stochastic problems are analytically formulated as deterministic optimization tasks. The paper presents the solutions of these robust optimization problems with genetic algorithms and simulated annealing. The role of constraint violation probabilities and how their dependence on time is selected are discussed. Scheduling of thermo-mechanical pulping (TMP) at paper mills is an application of the results. Production costs vary with electricity market price. Demand varies according to paper grades produced Due to high electricity consumption in TMP production, the cost savings of optimal scheduling are up to millions of €/a at one production site.

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2. Problem formulation A production system of A^ identical and parallel production units is studied. The production units are operated either at full speed or not at all. The production flows to a storage of volume Vmax- The cost of operation per unit process is predicted to vary in time as hi = h{iAt), / = 1.. .M To start up (shut down) a production unit has a cost hstart {hstop)- The demand from the storage is predicted to vary as dj-d(iAf), i=\...M. Scheduling the production so that the demand is always satisfied and that overall production costs are minimized constitutes an MfNLP problem with search space {J?/}/=i^, Rt € {Q,...,Rmax}Uncertainty is an essential element in decision making and optimization. In this optimization problem the effect of uncertainty in production cost and demand is studied. The models for cost uncertainty are AR(iV) and random walk, and that of demand a two state Markov process. 2.1. Deterministic case Let/be the production of one unit, and Ri the number of production untis on, hi the average price, 4 the average demand and Vi the end volume of the storage at time interval [/-I, /]. nshutiR) and nstop(R) are the number of startups and shutdowns in schedule R. Then the scheduling optimization problem reads as

nnnh^R + n,,^,, {R)K^rt + ^shut WKhut. Vi=Vo-

/ / Y.dj ^fY.Rj. 7=1

subject to

. . V^^
(1)

7=1

2.2. Stochastic components

The the price and the demand predictions in (1) are stochastic. The price is described with a time dependent known component and a stochastic fluctuation: hi=h,,,^+3ii,

(2)

The stochasticity is modeled as a random walk (RW) or as an autoregressive process (ARiN)): AR(N):

Shi=Y,cCjShi_j+ei

£,-7V(0,o-^)

7=1

RW\

dhi=Shi_^+£i

(3)

£i-^N(0,ap)

Scheduling of Identical and Parallel on/off Production Units Under Uncertainty

2047

The demand is a two state Markov process with transition probabiHties/> and q: Pidj^,=Q\dj=df)

=p (4)

2.3. Stochastic objectives The uncertain scheduUng problem is formulated in two alternative ways. In the first formulation the decision maker's attitude towards risk is modeled through exponential utility function U = I - exp(-a*(r -f)), where r is the revenue, a n d / i s the cost of production. The revenue is assumed deterministic and independent of schedule. The parameter a is user specified. The optimal schedule is the one maximizing the expected utility. In the second formulation, the optimal schedule is defined to be the one that minimizes expected cost subject to that the probability of exceeding a (user-specified) cost Hfnax has to be less than (user-specified) PQ. The deterministic constraints of not exceeding storage volume of V^nax and not falling below a storage volume Vmin= 0 at any time instant, are replaced by conditions that the probability of violating these constraints at time /, must be less than q/"^"^^ and q/'"'''\ respectively. Following [1-4] the stochastic problems are transformed into deterministic robust versions. Simply calculating the expected utility with (3) the deterministic objective with respect to schedule Rj i=l...Mis replaced as

iir{h^R + n,,^,,{R)h,,^,, + n^top{R)K R

n[/2^i^ + ^,,,,,(^)^..... + ^..op(^)^..op + ^ * ^^C7?/2]

The second formulation has the same objective ftinction as the deterministic case with expectation values of price and demand, but adds a probabilistic cost constraint:

^max -h^R-n,,,,,{R)h,,^,,

-nstop{R)Ktop >F-\\-P,y[R'CR]'^

(6)

with F as the cumulative of A^(0,1) -distribution. Both in (5) and in (6) C is the covariance matrix of the cost prediction errors, analytically calculated from (3).

3. Results The methods described in the previous chapter are applied on the production plant with five production units producing raw material for the manufacture line. The units are operated either at full speed or not at all. The production flows to an intermediate storage tank. The cost of operation per unit process is predicted to vary in time. The produc-

P. Pulkkinen and R. Ritala

2048

tion unit has also start up and shut down costs. The demand from the storage is defined by the schedule of the manufacture line. The optimization is done with a moving horizon. Optimization over 12 decision interval horizon is repeated at each decision time. The performance is monitored over 48 hours. After each optimization the first action is executed. Figure 1 presents the results of the deterministic and stochastic cases. The stochastic components used are random walk for the price and Markov chain for the demand, the objective is formulated with the utility fimction. The variance of the RW is 0.04 and the transition matrix of the Markov chain is [0.95 0.05; 0.5 0.5], meaning that the probability for a break is 0.05 and the probability of recovering from the break is 0.5. The optimization is done with simulated annealing [5]. Genetic algorithm [6] gives similar result. The optimization with the stochastic components requires 3-4% more CPU time compared to the deterministic case. With stochastic, soft constraints ql"^"^^ and ^Z'"'"'^, the operation cost is lower, but the number of start ups and shut downs increases Demand of the manufacture line

Demand of the manufacture line • Predicted Real

0

10

20

30

40

0

10

20

30

40

Raw material production

Raw material production

L-n_n.y\n 10

20

30

0

40

10

20

30

40

Electricity price

Electricity Price

rVVV^J 10

20

30

40

"0

10

20

30

40

Volume

Volume

• Predicted

300

Real

200 10 20 30 40 Deteministic model

10 20 30 40 Stochastic model

Figure 1 Left side presents the results of the deterministic case and right side stochastic case. The manufacture line is assumed to have 5 breaks. Stochastic model can predict the effect of breaks better than the deterministic.

Scheduling of Identical and Parallel on/off Production Units Under Uncertainty

Optimal schedule

2049

Volume

ll;

llj

M

1 1 1 1 1 1

lii 10

20

300 41 41 jl —a-i

LLlii 11 ' • I ' 1 1 I 1

• I 1 • 1 1 1 1,

30

:

•0.1 • 0.5 •0.9 200

40

Figure 2 The impact of different probabilistic constraints can be seen especially between time steps 20 and 30. Looser constraints allow exceeding of the limits. Figure 2 illustrates how the optimal schedule changes as a function of constraints. Loosening the limits allows more radical control actions and results in lower cost. In this example the costs are 100, 97.3 and 95.5 for q = [0.1 0.5 0.9]. The example values for q are rather large, which can be seen in the figure as exceeding of the limits. Here the constraints were constant in time. If the constraints are time dependant the problem becomes more flexible. The constraints can change from tight to loose, keeping the process in strict control in the begiiming and allowing more violations at the end of the time horizon. The violations at the end can be tolerated since the optimization is with rolling horizon and repeated at each decision time. Using the second formulation of the objective function, the one with stochastic cost constraint, increases the CPU time by half. Added constraint decreases the amount of feasible solutions and makes the problem harder for both of the stochastic algorithms. Proposed schedules have higher cost than the ones given by the utility function based optimization. The formulation introduces the parameters: Hmax and PQ. Hmax defines the maximum cost allowed subject to probability PQ. The optimization is extremely sensitive to the selection ofHmax' Lowering the Hmax forces the optimization algorithms to make more radical changes to the production schedule to satisfy the request of the constraint. The robustness with respect to PQ depends on the selection of Hmax and also on the behavior of the process. If P^ and H^ax are selected wrong, the optimization algorithms have difficulties in finding feasible and optimal solutions.

Presented example case can be directly applied in the paper mill. The raw material for the paper machine is produced with refiners by disintegrating fibers through a mechanical action on wood chips in a high temperature and pressure environment generated by steam addition. The process requires a large amount of energy, 2.0-3.5 MWh/ton pulp and a TMP plant produces between 500-1500 tons a day. Electricity consumption comprises a large amount of the variable production costs and offers an area of optimization and cost savings. In order to minimize the operating costs, the TMP production schedule is optimized around the market energy price and constraints of the mill. The deterministic TMP optimization problem has been presented in [7, 8].

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4. Conclusions Methods presented can be applied in the decision support systems; they are computationally on acceptable level. In the paper mill the online optimization is in most cases sufficient, if it provides a result every half an hour. Clearly decision support systems should deal with stochastic aspects. The mathematical framework to support decisions in the presence of stochastic elements is well established The question is: "How to determine the decision makers' attitude towards risk?" The uncertainty-related parameters for the objective and the constraints are for the decision makers to set. However, they are expected through concepts that is rather alien to decision makers' context. Stochasticity as a phenomenon appears confusing to process operators and engineers, and the mathematical formulation does not provide any help. Therefore, educating the end users will be an integral part of setting up support for decisions under uncertainty.

References [1] Sahinidis, N. V., 2004, Optimization under Uncertainty: State-of-the-art and Opportunities, Comp. Chem. Engng 28, 971. [2] Lin, X., S.L. Janak and C.A. Floudas, 2004, A New Robust Optimization Approach for Scheduling under Uncertainty: I Bounded Uncertainty, Comp. Chem. Engng 28, 1069. [3] Janak, S.L., X. Lin and C.A. Floudas, 2005, A New Robust Optimization Approach for Scheduling under Uncertainty: II Uncertainty with Known Distribution submitted for publication. [4] Janak, S. L., C. A. Floudas, 2005, Advances in Robust Optimization Approaches for Scheduling under Uncertainty, Proceedings of ESCAPE-15, 29 May - 1 June, 2005, Barcelona, Spain, eds. L. Puigjaner, A. Espuna, Elsevier. [5] Otten, R. H. J. M, van Ginneken, L. P. P. P. (1989), The Annealing Algorithm, Kluwer Academic Publishers. [6] Goldberg, D. E(1989). Genetic Algorithms, Addison-Wesley. [7] Pulkkinen P., R. Ritala, A. Mosher and M. Tienari 2004. Designing Operator Decision Support System for TMP Production Based on Dynamic Simulation and Optimization. Proceedings - Preprints. Control Systems 2004 Conferences. 14-18 June, 2004, Quebec City Quebec, Canada. Publications Administrator; PAPTAC, Montreal, QC, Canada 2004. ISBN 1-897023-01-2. p. 215-218. [8] Mosher, A., M. Tienari and R. Ritala, 2005, Dynamic Optimization of the Electricity Cost at an Integrated TMP Plant, submitted for publication.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

2051

A Flexible Design of Logistic Network against Uncertain Demands through Hybrid Meta-heuristic Method Yoshiaki Shimizu^, Shigeharu Matsuda^ and Takeshi Wada^ '^Production Systems Engineering, Toyohashi University of Technology 1-1 Hibarigaoka,Tenpaku-cho, Toyohashi, Aichi 441-8580 Japan

Abstract This paper is concerned with a flexible logistics optimization problem under uncertain demand forecasting. Formulating the problem based on flexibility analysis, we apply a meta-heuristic method termed hybrid tabu search to solve the problem. We also give a procedure amenable for the trade-off analysis between cost and flexibility. Finally, the validity of the proposed method is examined through numerical experiments. Keywords: Logistics, Tabu search. Flexibility analysis. Uncertain demand 1. Introduction Under the influence of globalization and the introduction of advanced transportation systems, industrial markets are acknowledging the importance of flexible logistic systems favoring just-in-time and agile manufacturing. Focusing on the logistic systems associated with supply-chain management (SCM), we developed a method termed hybrid tabu search [1] to solve the problem under deterministic customer demand. It is a two-level method whose upper level problem decides the location of hub facilities using tabu search while the route selection problem will be solved under the upper level decision based on the graph algorithm at the lower level. It can cope with large systems in an efficient numerical way compared with the solver based on mixed-integer programming approach. In reality, however, a precise forecast of demand is quite difficult to obtain, which results in either insufficient production when forecast goes below the actual demand or undue expenditure due to large inventory. One solution is to formulate the problem by taking uncertainty in the demand into account. In fact, assuming certain stochastic deviation, two-stage formulations using stochastic programming have been published in recent studies [2], [3]. However, these approaches seem to be ineffective for designing a flexible logistic networkfi*omthe following two reasons. First, customer satisfaction is evaluated by the demand basis but it is left unrelated to the other important factors like cost, flexibility, etc. Second, they are unconscious of taking a property of decision variables into account whether they are soft (control) or hard (design). In comparison, this study will formulate the problem by employing the idea of flexibility analysis [4], [5], which is a procedure on how to find the extent of parameter deviations by managing control within their admissible region under the prescribed design. Eventually, the approach introduced in this paper aims at

Y. Shimizu et al

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deriving a flexible logistic system that is highly desired in real world applications since it can cope with any deviation while keeping performance as high as possible at the nominal state. 2. Flexible Logistic Network Design 2.1. Prelim inary statement Let consider a hierarchical logistic network as shown in Fig.l, and define index sets I, J and K for customer (CT), distribution center (DC) and plant (PL), respectively. It is assumed customer z^I has an uncertain demand Di under normal distribution. To work out this problem, first, we will define a fill rate of demand termed service level as follows. fOCd

s{ao)=

I N(pQ,G)dp

(1)

(a: natural)

where N( •) stands for the normal distribution with average po and standard deviation a. It is the probability that the network can deliver products to the customers whatever deviation might occur within the prescribed extent at the demand. After

I

Open / C b s e \ 0 pen

:distribution center (DC),

icustomer (CT)

Fig.l A hierarchical logistic network problem concerned here For example, the network designed for the average demand can present 50% service level, and 84.13% for the demand corresponding topo-^G. Then, we are interested in minimizing total transportation cost with respect to the location of DC and selection of a route between the facilities while satisfying the service level. In the following development, we also assume that: (1) Every customer is supplied via a route only in order such like PL-DC-CT. (2) To avoid a separate delivery, each connection is limited to only one linkage. Now, the problem without taking the demand deviation into account is given by the following mixed 0-1 programming problem. (PN) MimmizeJ^^fijEij i

J

(2)

+ Y^H^J^^J^ J

k

subject to (3) J

(4)

A Flexible Design of Logistic Network Against Uncertain Demands Y.f,<XjUj

2053

yjeJ

(5)

yjejykeK

(7)

i

k

gj,
Z^,.=I/, vyey

(8)

z

k

^gj,<S,

ykeK

(9)

J

J

f,g:

integer, x, y, ze {0,1}

where Xj denotes a binary variable that takes 1 when DC opens at they-th candidate and 0 otherwise. The binary variables y^ and zjk represent the status of connection between CT and DC, and DC and PL, respectively. These two binary variables (ytj and Zjk) become 1 when connected and 0 otherwise. Quantities fij and gjk are shipping amounts from DC to CT, and from PL to DC, respectively. The objective function stands for the total transportation cost, and each constraint denotes the conditions regarding demand satisfaction (Eq.(4)), the single linkage (Eqs.(3), (6)), capacity constraints (Eqs.(5), (9)), flow balances (Eqs.(7), (8)), and the required opening number of DC (Eq.(lO)). 2.2. Problem formulation To consider a flexible decision problem, we present a formulation based on the flexibility analysis. For this purpose, first, we classify the decision variables into hard or soft depending on their generic natures. Hard variables are not allowed to change once they are determined (for example the DC location). On the other hand, soft variables can change according to the demand deviation (for example the distribution route). Then we derive a two level problem formulation as follows. (PD)

Min

CT(X,U,W\PQ)

(11)

x,u,w

subject to (x,u,w)e

F(x,u,w

Min Cj^(x,v,w'\ p^)

\PQ) subject to (x,v,w')e

(12) F{x,v,w'\

p^)

(13)

v,w'

||ii-v||<2f

(14)

jc, II, VG {0,1}, w, w' : integer where x denotes the location of DC, u the route at the nominal (average) demands, and V the route at the deviated demands corresponding to the service level. By letting II*II denote the Hamming distance, ^ refers to the allowable number of route exchange. Moreover, w and w' represent the other variables in (PN) at the nominal and the deviated states respectively.

Y. Shimizu et al.

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Also, Cj {'\PQ) andF{-\PQ) express symbolically the objective function (Eq.(2)) and the feasible region (from Eq.(3) through Eq.(lO)) at the nominal, respectively. Similarly Eq.(13) stands for the optimization at the deviated state. Due to the linearity of the constraints regarding demand satisfaction, we can decide easily the permanently feasible region [4] that will guarantee the feasibility even in the worst case of parameter deviations regardless of the adopted design and control. Accordingly, the demand Dt appearing in Eq.(4) must be replaced with the value corresponding to the prescribed service level in F{'\p^). Eventually, the lower level problem tries to search the optimal route while satisfying the feasibility against every deviation under the DC location decided at the upper level problem. 2.3. Resolution procedure Even in the case where uncertainties are not considered, the formulated problem belongs to the class of NP-hard problems. It becomes especially difficult to obtain mathematically a rigid optimal solution as the problem size expands. Using the hybrid tabu search as a core method, therefore, we derive a procedure to solve (PD) while being aware of a parametric study regarding ^, which will be amenable for the trade-off analysis on flexible logistics decision at the next stage. As shown in Fig.2, it begins with setting the service level and f. After deriving randomly the initial DC location x, it decides the optimal route at the nominal demands u. Then define the permanently feasible region, and derive the optimal route that will satisfy the condition on ^, i.e. Eq.(14) by using a penalty function method. This route optimization in the lower level should be repeated until a certain convergence criteria will be satisfied while updating the DC location at the upper level. Since from all of these, we can solve (PD) for the prescribed f, we utilize its result as the initial location of the next decrement problem of ^. That will enhance the efficiency of the parametric study regarding ^. C start

)

Set service level & i

Initial location, x (Foregoing result)

Hybrid Tabu Search

~ . \ ~ I J PK,:Optimal route Revise location H H ^^ nominal, u

X IJOut) Set permanently [feasible region (PFR)| Adjust I PniOptimal route p e n a l t y l ^ -at " " the ' - - deviated, -•- -=-^--' -v\

u = y®z, v = y©z'

Fig.2 Flow chart of resolution procedure 3. Numerical Experiments To examine the effectiveness of the proposed approach, we solved a variety of problems where the number of customers rangesfi*om50 through 150. Moreover, number of plant |K|, potential DC |J|, designated open DCp and customer |I| are set at the ratio like 5:15:7:50, and these facilities are located randomly on the plane.

A Flexible Design of Logistic Network Against Uncertain Demands

2055

Then unit transportation costs Ey, Gjk are given to be proportional to the EucUd distance between them. We prepared three benchmark problems (GRl) to examine the properties of the flexible solution through comparison. On the other hand, problems in set GR2 are composed of problems having particular properties as presented in Table 1. In what follows, we will show numerical results and discuss about the requirements for flexible logistics strategies. Table 1 Property of benchmark problem, GR2 Problem ID S-A/B C-A/B P-A/B

Property Small/Large standard deviation Small/Large capacity of DC Small/Large number of potential DC

|K|

|J|

p

|I|

15 15 12/40

7 7 7

50 60 60

3.1. Comparison among the different strategies In Table 2, we show the results for GRl obtained from three strategies, i.e. the proposed flexible decision (F-opt.), nominal one (N-opt.), and conservative one (Wopt.). There, the objective values are compared with each other both at the nominal (po) and the worst (po+3a) states when ^=5. The values in the parenthesis express the rates to the respective optimal values. In every case, N-opt. is known not to cope with the deviated state at all. On the other hand, though W-opt. has an advantage at the worst state, its performance degrades outstandingly at the nominal state. In contrast, F-opt. can present better performance in the nominal state while keeping nearly optimal value in the worst case. Results obtained from the problems in GR2 reveal that the more hardly the decision environment and the more seriously the deviated situation become, the more especially the flexible design takes the advantage. All these problems are solved at most within a few minutes using PC with CPU as Intel Pentium 4 (3.0GHz). Table 2 Comparison of result for benchmark problem, GRl Problem ID Strategy at Nominal at Worst V-opt, 45846 (1.25) 77938 (1.04) D-5-15(7)-50 N-opt.*^ 36775(1.00) NA (D-|K|-|J|(p)-|I|) W-opt.*^ 58377 (1.59) 74850 (1.00) V-opt. 38127 (1.03) 47661(1.04) D-10-30(14)-100 N-opt. 36918(1.00) NA W-opt. 39321 (1.06) 45854 (1.00)

D-15-45(21)-150

V-opt, N-opt. W-opt.

40886 (1.07) 38212(1.00) 45834 (1.19)

48244 (1.05) NA 45899 (1.00)

* Min Cjix^ w, w I /7o) s.t (jc, w, w) ^ F ( x , w, w |/?o) *^ Min Ciix, V, w' Ipr) s.t. (x, v, >vO^F(jc, v, w' \p^) 3.2. Trade-off analysis toward flexible decision To make a final decision under the flexibility consideration, we need to examine the dependence of adjusting margin ^ on the system performance or total cost. Since the appropriate amount of margin will reduce the degradation of performance effectively, we can derive a rational decision by compromising the attainability of

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Y. Shimizu et al

84.13% (1 a ) 97.72% (2 a ) 99.87% (3 g )

-*-84.13% (1 a ) - ^ 9 7 . 7 2 % (2 a ) -^99.87% (3(T)

100 0

1 2

3

4

Adjisting m ai:gii f 0 - 5 - 1 5 (7)-50)

Adjisting m ai^ii 5 (t)-15-45ei)-150)

Fig.3 Relation between total cost and adjusting margin ^ these factors. From such aspect, we engage in the trade-off analysis for various problems belonging to GRl and GR2. In Fig.3, results obtained for the problems in GRl are shown for examples. There, we can observe the trade-off between total cost and ^, which increases along with the amount of deviation. In the next step, therefore, we need to discuss about the sufficient service level and/or the allowable adjusting margin together with the cost factor. 4. Conclusion In this paper, we have applied the hybrid tabu search to solve a logistics optimization problem under uncertain demands. For this purpose, we have formulated the problem based on flexibility analysis. Then we have given a procedure amenable for carrying out parametric study for trade-off analysis. Through numerical experiments, we show the proposed method can derive a flexible logistic network against the deviation of demands, and reveal some insights useful for the trade-off analysis at the next stage. References 1. Shimizu, Y. and T. Wada, 2004, Hybrid Tabu Search Approach for Hierarchical Logistics Optimization, Trans. ISCIE, 17, 6, 241. (In Japanese) 2. Jung, J Y., G. Blau, J F. Pekny, G. V. Reklaitis and D. Eversdyk, 2004, A Simulation Based Optimization Approach to Supply Chain Management under Demand Uncertainty, Computers & Chemical Engineer in, 28, 2087. 3. Guillen, G, F. D. Mele, M. J. Bagajewicz, A. Espuna andL. Puigjaner, 2005, Multiobjective Supply Chain Design under Uncertainty, Chemical Engineering Science, 60, 1535. 4. Shimizu, Y. and T.Takamatsu, 1987, A Design Methodfor Process Systems with Flexibility Consideration, Kagaku Kogaku Ronbunshu, 13, 574. (In Japanese) 5. Swaney, R. E. and I E. Grossmann, 1985, An Index for Operational Flexibility in Chemical Process Design Part 1: Formulation and Theory, AIChE J., 31, 621.

Acknowledgements This study was financially supported in The 21^* Century COE Program "Intelligent Human Sensing," from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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A Flexible Framework for Optimal Biorefinery Product Allocation Norman Sammons^, Mario Eden^, Harry Cullinan'^, Lori Ferine'', Eric Connor^ ""Department of Chemical Engineering, Auburn University, AL 36849, USA ^Alabama Center for Paper and Bioresource Engineering, Auburn, AL 36849, USA ^Agenda 2020 Technology Alliance, American Forest and Paper Association, USA "^ThermoChem Recovery International, Baltimore, MD 21227, USA Abstract The integrated biorefinery has the opportunity to provide a strong, self-dependent, sustainable alternative for the production of bulk and fine chemicals, e.g. polymers, fiber composites and pharmaceuticals as well as energy, liquid fuels and hydrogen. Although most of the fundamental processing steps involved in biorefining are wellknown, there is a need for a methodology capable of evaluating the integrated processes in order to identify the optimal set of products and the best route for producing them. The complexity of the product allocation problem for such processing facilities demands a process systems engineering approach utilizing process integration and mathematical optimization techniques to ensure a targeted approach and serve as an interface between simulation work and experimental efforts. The objective of this work is to assist the bioprocessing industries in evaluating the profitability of different possible production routes and product portfolios. To meet these ends, a mathematical optimization based framework is being developed, which enables the inclusion of profitability measures and other techno-economic metrics along with process insights obtained from experimental as well as modeling and simulation studies. Keywords: Integrated biorefineries, sustainability, optimization 1. Introduction Current chemical and energy industries are heavily reliant upon fossil fuels, and these fuels are unsustainable and contribute to economic and political vulnerability [1]. Biomass, a renewable resource, has incredible potential to fulfill the energy and chemical needs of society while minimizing environmental impact and increasing sustainability [2]. The process of separating biomass constituents and converting them to high value products is known as biorefining, and the integrated biorefinery provides a unique opportunity for reinvigorating an entire manufacturing sector by creating new product streams [2]. Economic and environmental sustainability are achieved through the optimal use of renewable feedstocks, and a need exists for a process systems engineering (PSE) approach to ensure maximum economic and societal benefit through minimizing the usage of raw material and energy resources. The bioprocessing industries are slowly becoming aware of the benefits of infusing PSE methods to this emerging field. To maximize the applicability of such systematic methods and to integrate experimental and modeling work, a unique partnership has been established consisting of researchers in academia and industry along with government entities, equipment vendors and industry stakeholders to procure the wide range of information necessary such as data needed for process simulation models, information on capacity constraints, financial data, and nonlinear optimization techniques. This information is

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obtained from a variety of collaborations to be formed and strengthened involving industrial partners, internal academic partners in both chemical engineering and business, and external academic sources. This ensures that the data used in the decision making process is realistic and that the research addresses problems of industrial and regulatory interest. 2. Scope and Complexity of the Biorefinery Production Problem A plethora of combinations of possible products and process configurations exists for the conversion of biomass into chemicals and fuels. Figure 1 provides an illustration of some of the many processing steps and possible products available in a biorefmery, but it should be noted that Figure 1 does not include all possibilities and serves primarily to illustrate the complexity of the product allocation problem. The diamonds represent products that can either be sold or further processed to other products, while the boxes denote conversion processes that may be comprised of several processing steps.

Figure 1. Schematic of biomass conversion and biorefmery production routes. It is apparent that such a large number of possible process configurations and products results in a highly complex problem that can not be solved using simple heuristics or rules of thumb. Business decision as well as policy makers must be able to strategically plan for and react to changes in market prices and environmental regulations by identifying the optimal product distribution and process configuration. Thus, it is necessary to develop a fi-amework which includes environmental metrics, profitability

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measures, and other techno-economic metrics. Such a framework should enable policy and business decision makers to answer a number of important questions like: • For a given set of product prices, what should the process configuration be, i.e. what products should be produced in what amounts? • For a given product portfolio, how can process integration methods be utilized to optimize the production routes leading to the lowest environmental impact? • What are the discrete product prices that result in switching between different production schemes, i.e. what market developments or legislative strategies are required to make a certain product attractive? In the following sections, the developed framework for answering these questions is presented along with a discussion of some preliminary results. 3. Methodology for Integrating Modeling and Experimental Efforts The introduction of PSE methods into biorefining research provides a systematic framework capable of seamlessly interfacing results generated in simulation studies as well as experimental work. Such a framework is imperative when attempting to combine knowledge and information from a variety of research areas and disciplines. The objective of this approach is to create a library of rigorous simulation models for the processing routes along with a database of corresponding performance metrics. Wherever possible experimental data is used to validate the performance of the simulation models, and for processes that commercial software packages are incapable of describing adequately, the performance metrics are initially based on experimental results until a satisfactory model has been developed. Published Data

Semi-empirical Data

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Superstructure of Processing Routes Tree Striicture Incorporating All O p t f m i ^ Models

Figure 2. Strategy for identification of performance metrics.

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Figure 2 shows a schematic representation of the strategy employed for identification of characteristic performance metrics of the individual subprocesses. The simulation models for each process are developed by extracting knowledge on yield, conversion, and energy usage from empirical as well as experimental data. If a given process requires the use of a solvent, molecular design techniques like CAMD and property clustering techniques are employed to identify alternative solvents that minimize environmental and safety concerns [3,4]. Process integration techniques are then used to optimize the simulation models. This is an integral step in the model development as it ensures optimal utilization of biomass and energy resources. Finally, the optimized models are used to generate data for the economic as well as environmental performance metrics. The estimation of environmental performance is achieved through the use of the US-EPA Waste Reduction (WAR) algorithm [5]. It should be noted, that only the economic and environmental performance metrics are incorporated in the solution framework described below, thus decoupling the complex models from the decision making process. This approach allows for continuously updating the models as new data becomes available without having to change the selection methodology. Similarly, if new processes are to be included for evaluation, an additional set of metrics are simply added to the solution framework, thus making it robust and flexible. 4. Methodology for Biorefinery Optimization The optimization framework, which combines the library of processing routes and corresponding economic performance metrics with a numerical solver, is given in Figure 3 below. It should be noted, that the environmental performance is not included in the optimization step, thus avoiding identification of the zero impact facility as a solution, corresponding to no biomass being processed at all. PROCESS OPTIMIZATION FRAMEWORK Process Design Objectives Quantify desired performance

Processing Superstructure Optimized process models

Constraints Technical, economic, structural

Perfonnance Metrics Database Economic potential

Numericai Soiver Routines Handling real and integer variables (MILP.IMINLP)

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Figure 3. Methodology for identification of optimal biorefinery structure.

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The objective of the optimization step is to identify candidate solutions that maximize economic performance and then the candidates are ranked according to environmental performance. If a candidate satisfies the environmental objectives, then the optimal production scheme has been identified. If none of the candidates satisfy the environmental impact constraints, then the desired economic performance requirements are relaxed until a solution with acceptable environmental performance has been identified. It should be emphasized that by decoupling the complex models from the optimization and decision making framework, the methodology is more robust and also provides added flexibility by only having to update the performance metrics for a given process as new information, e.g. a new catalyst with higher conversion, is identified. This approach is analogous to the reverse problem formulation fi-amework used for decoupling the complex constitutive equations from the balance and constraint equations of an individual process model [6]. The design targets linking the two reverse problems are constitutive or property variables, which in this framework are represented by performance metrics. 5. Generalized Biorefinery Model A generalized biorefinery model, which has been used to develop the structure of the optimization framework, is given in Figure 4. The model structure was formulated to include a variety of basic complexities encountered in the decision making process, e.g. whether a certain product should be sold or processed further, or which processing route to pursue if multiple production pathways exist for a given product. The objective function maximizing the overall profit of the biorefinery is given below: m11

m14 m 13

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profit=x i^„.c;-xz^.,c:,-crz^,

m\j

(1)

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In Eq. (1), Rmk denotes the production rate of product k from bioresource m\ C/ is the sales price of product k\ Rmij is the processing rate of route //; C ^ / is the cost of processing bioresource m through route ij; and finally Cm^^ is the purchase price of bioresource m. Using this nomenclature, the first term in Eq. (1) represents the sales revenue from the products made from each bioresource m. The second term represents the total processing cost incurred by the pathways pursued in production, and finally the third term represents the total cost of the biomass resource m. This generalized model, where the objective function and constraints are linear, is easily solved using commercially available software. It should be noted here that while earlier work such as the proposed solution by Sahinidis and Grossman [7] incorporate process models into the optimization problem, the proposed framework separates the wide range of

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biorefining models from the optimization portion, thus reducing the complexity of the problem for the solver while maintaining the robustness achieved with proven optimization techniques. Without including any constraints on capacity of the processing steps, the solution is a single-product configuration in which all available biomass is converted into the most profitable product. However, if constraints are imposed on the most profitable route, the framework identifies the additional products and processing routes required to maximize the overall profit, thus leading to a polygeneration facility [7]. Approximate capacity constraints are based on a variety of sources, e.g. existing equipment, vendor data and qualitative process information provided by academic and industrial collaborators. In order to effectively address the strategic planning objectives of business decision makers, it is necessary to incorporate the total capital investment as a constraint in the formulation. The capital investment for a given unit or process can be approximated as a fiinction of its capacity or processing rate, and both linear and nonlinear expressions have been successfiilly implemented in the framework. Inclusion of capital cost constraints is crucial for practical application of the results, i.e. enabling evaluation of the potential benefits to be obtained for a given maximum investment by retrofitting an existing facility or constructing new plants. 6. Discussion and Future Work A general systematic framework for optimizing product portfolio and process configuration in integrated biorefineries has been presented. Decoupling the process models from the decision-making framework reduces problem complexity and increases robustness. The next phase of this work involves development of additional process models for the generation of performance metrics. Qualitative predictive models for capital investment as a fiinction of processing capacities will also be developed. Finally, it is desired to include concepts of optimization under uncertainty to quantify the effects on process configuration resulting from minute changes in product prices [8]. This will lead to increased robustness of the methodology and thus better recommendations. 7. Acknowledgements The authors greatly appreciate the financial support for this research provided by the Auburn University Competitive Research Program, the Auburn University Presidential Graduate Opportunity Program, and the Alabama Agricultural Experiment Station. In addition, the authors are thankfiil for access to information and process expertise through ongoing partnerships with the US Forest Service Forest Products Laboratory, the Auburn Pulp and Paper Foundation, and PureVision Technologies. References [1] [2] [3] [4] [5] [6] [7] [8]

US Department of Energy, World Energy Report, 2003 A. V. Bridgwater, Chemical Engineering Journal, 91, 2003 P. M. Harper and R. Gani, Computers and Chemical Engineering, 24, 2000 M. R. Eden, S. B. Jorgensen, R. Gani and M. M. El-Halwagi, Computer Aided Chemical Engineering 15B, 2003 D. M. Young and H. Cabezas, Computers and Chemical Engineering, 23, 1999 M. R. Eden, S. B. Jorgensen, R Gani and M. M. El-Halwagi, Chemical Engineering and Processing, 43, 2004 N. V. Sahinidis, I. E. Grossmann, R. E. Fomari and M. Chathrathi, Computers and Chemical Engineering, 13, 1989 I. Banerjee and M. G. lerapetritou. Computers and Chemical Engineering, 27, 2003

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Systems for Decisions Support in Industrial Use Katrin Coboken, Georg Mogk, Thomas Mrziglod , Ulrich Telle Bayer Technology Services GmbH, D-51368 Leverkusen, Germany Abstract In this paper we present some systems for decisions support used in industry for different subjects. In all cases it is important to consider most parameters as uncertain. Therefore these variables are supposed to satisfy a given probability distribution. To ensure authorized users reliable access to consistent information at any time all over the world these tools are realized as web based systems with a central database. Keywords: decisions support systems, uncertain parameters, web based systems 1. Introduction In this paper we present some systems for decisions support used in industry for different subjects. These are systems for • risk management - to control the costs and risks of business sales, • project management - to control single projects, • project portfolio management - to support the decision process for a well balanced project mix. In all cases it is important to consider most parameters (e.g. sales price, costs of raw materials, quantity of sales, project success) as uncertain. Therefore these variables are supposed to satisfy a given probability distribution. To ensure authorized users reliable access to consistent information at any time all over the world these tools are realized as web based systems with a central database. This allows to collect all relevant data fi^om different sites in a single consistent database and to make the results conveniently accessible. 2. Risk Management The control of costs and risks of business sales of a specific manufacturing product is based on the costs of the main ingredients, additional variable and fixed costs as well as transportation costs. In addition idle time costs are taken into account. If the production takes place at different sites over the world it is essential that decisions are based on actual and consistent data. Therefore a web based system with central database is used to carry out the marginal costing analysis. Besides a reporting part (based on historical data) to compare costs of raw materials and energy from different suppliers and costs of production processes at different sites the system has means to estimate margin-risks caused by uncertainties of time dependent cost driver (raw materials and energy). Therefore marginal costing is based on the average over a given period of historical cost data or to support the pricing process on individual user-specified data. In the case where the main costs of a product depend on the heavy varying costs of energy or raw materials the risk of an intended business can be estimated with the help of a tornado diagram. In a tornado diagram the bars show the change in profit contribution due to a

K. Coboken et al.

2064 Profit Contribution 408

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Figure 1. The risk of uncertain parameters can be estimated with the help of a tornado diagram. change in costs of raw materials or conversion costs holding all other cost and revenue constant. Thereby the costs vary in a range given by e.g. the 10th and 90th percentile points of the data distribution of a selected time period of historical data. The profit contribution is very sensitive with respect to parameters corresponding to wide bars. If the actual costs (marked with a dot in Figure 1) are in the future uncertain an intended business is risky if the dot lies in the lower third of a wide bar. 3. Project Management Usually uncertainties (e.g. in prices, costs and sales) are not considered during Net Present Value (NPV) calculations for projects. In an approved tool the density probability function of the NPV is interactively estimated depending on the probability distribution of all input variables.

8

10

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NPV [MIo Euro]

Figure 2: NPV of a nearly riskfi-eeproject.

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In figure 2 and 3 examples of NPV distributions of different projects are shown. Example 1 (Figure 2) represents a nearly risk free project, since the NPV is almost surely positive whereas the project illustrated in Figure 3 is more critical. The NPV expectation is positive but there is large probability (nearly 50%) that the project generates a clearly negative NPV. 0.09

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Figure 3: NPV of a critical Project. The tool enables a project manager to identify the really critical parameter, which control the expectation value and variance of the NPV distribution. On the parameters which have high influence on the width of the distribution (large width means high risk) the project leader should initiate activities for risk reduction e.g. hedging by options, out or in sourcing, additional consulting, etc. Usually risk reduction results in increasing costs. But it is possible that the total expected project NPV remains constant or even increases in spite of higher costs by decreasing the total risk. Hence the project leader is in a position to find an optimal balance between risk, cost and NPV. 4. Project Portfolio Management For a sustainable business development it is important to have the right profitability-risk structure of the total project portfolio, especially for invest or R+D projects. For an invest decision not only the profitability of a project counts, but also how a new project fits into the portfolio of the current projects from a risk oriented point of view. An industrial used software solution supports to hedge risky projects by some solid value creators and to identify the no goes. Error! Reference source not found, represents an example of a risk profitability portfolio. In addition a feature to carry out project tracking during its live cycle helps to push or cancel projects at the right time. The path of a project across the portfoHo during its duration is shown in Error! Reference source not found.. With its help a portfolio manager can react fast, if a project develops in the wrong direction.

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Figure 5: The path of a project across the portfolio during its life cycle.

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Systems for Decisions Support in Industrial Use

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The reasons for the changes of the complete project portfoHo from one year to the next can be analyzed easily by means of a bridge diagram (Error! Reference source not found.).

Years Figure 6: Bridge diagram

5. Method The calculation of the probability distribution of the NPV is based on Monte Carlo simulation of a deterministic NPV computation. Therefore some of the input parameters are regarded as uncertain. These input parameters are for example volume and price of marketing, different types of costs per unit volume and different investment kinds. Usually for those input parameter a triangle probability distribution is assumed. For each uncertain input parameter the boundaries and the position of the maximum value are individually adjustable. In each calculation the actual value of the parameter is randomly chosen with respect to the distribution. The result is then an NPV distribution. In an alternative method the NPV distribution is evaluated by convolution of the underlying density functions. During the NPV calculation the technical risk can be taken into account. The success probability is calculated for each subproject by a risk matrix. For several project properties, which are important for the project success, the user fixes an importance for decision and a realization probability. If the realization probability is high, the technical risk is low. The importance for decision defines the weight of the property.

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In each Monte Carlo calculation it is evaluated on the basis of the technical risk if the subproject is successful or not. If it is successful the NPV calculation as described above is accomplished, if not the NPV calculation is stopped after the development phase. In this case only the investment and research costs are taken into account. Further key data e.g. the expectation value of the NPV or the 10% and 90% quantiles are calculated from the resulting NPV distribution. 6. Software structure and user roles The software system is web based using a Java Application Server on the server side. Only a modem web browser is required on the client side. The system was tested with Tomcat 5.0 and MS Internet Explorer 5.5, but also works with other modem web browsers and any Java Application Server conforming to Servlet 2.4 API and JSP 2.0 API. Communication between client and server may be secured using SSL. The user interface is implemented with JavaServer Pages, while the mathematical kemel, the database layer and the charting component consist of Java classes. Data management is based on a SQL relational database using JDBC drivers. Currently supported databases are Oracle 9i and MS SQLServer 2000. Access to the data is controlled by a simple user rights management based on four different user roles: A project leader may create projects in all business areas, but is allowed to edit and view only his own projects and to carry out simulations, version comparisons and portfolio analysis based on his own projects. A portfolio manager for a business area is restricted to create projects belonging to his own business areas, but may create, edit and view simulations, version comparisons and portfolio analysis based on all projects in his business areas. Additionally he may view all public analysis in his business areas. A portfolio manager has for all business areas the same rights as a portfolio manager for a business area. An administrator is responsible for the user and database administration. He authorizes new users, assigns user roles and maintains the master data like business areas, country dependent tax rates or weight factors for risk calculations. Optionally arbitrary documents may be attached to each project. These documents are stored at a configurable location of the server file system. The database contains references to these documents only.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Medium term planning of biopharmaceutical manufacture under uncertainty K Lakhdar^ SS Farid^ J Savery", NJ Titchener-Hooker^ & LG Papageorgiou*' a Department of Biochemical Engineering, The Advanced Centre for Biochemical Engineering, University College London, Torrington Place, London JVCIE 7JE, UK b Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WCIE 7JE, UK c BioPharm Services UK, Lancer House, East Street, Chesham, Bucks HP5 IDG, UK Abstract Manufacturers in the biopharmaceutical industry face greater scheduling and planning challenges as the trend of employing multiproduct manufacturing facilities continues to grow. These challenges are complicated by the randomness inherent in the biopharmaceutical manufacturing environment. This work focuses on capturing the effect of uncertainties in fermentation titres when optimising planning of biopharmaceutical manufacturing campaigns In this paper we extend our previous deterministic medium term planning formulation to include uncertain production rates resulting in a two stage, multi-scenario, mixedinteger linear programming (MILP) model. When tested on industrial-sized problems, the resulting MILP problem proved intractable. An iterative solution algorithm is proposed for solving the resulting large scale MILP planning problem. The applicability of the algorithm is demonstrated through three illustrative examples. The computational results indicate that the proposed solution algorithm offers a significant reduction in the computational requirements whilst maintaining solution quality. The proposed optimisation-based framework presents an opportunity for biomanufacturers to make better medium term planning decisions, particularly under uncertain manufacturing conditions. Keywords: Biopharmaceutical manufacture, Planning, Uncertainty, MILP 1. Introduction In the biopharmaceutical industry, production planning and scheduling methods have traditionally been spreadsheet-based and hence static. However, these methods are difficult to maintain and provide limited decision-support. More recently, discrete event simulation techniques have gained popularity for modelling the logistics of operations. However, with the increasing use of multiproduct facilities in the biopharmaceutical industry, more systematic methods for optimising planning and scheduling are needed so as to meet pressures to cut costs and maximise production throughput [1]. Strategic planning of biopharmaceutical manufacture is further complicated by inherent technical uncertainties that can impact cost and delivery time. These include fluctuations in fermentation titres, purification yields, campaign lengths and contamination rates [2]. Previous work [2, 3] exploring the impact of such uncertainties on biopharmaceutical manufacture, through the use of Monte-Carlo (MC) simulations, found the fermentation titre to be the most critical driver affecting both the cost of goods and the throughput. Given that variable fermentation titres (grams of product per litre of broth) directly determine the number of batches required to satisfy demand, schedules that do not

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account for this uncertainty may be rendered inadequate due to poor quality performance or even infeasibility. Hence in this paper, the issue of how to adequately plan in the medium term given uncertainties in fermentation titres is addressed. In the traditional process industries, there have been many relevant articles dealing with scheduling and planning under uncertainty. Typically problems are represented via twostage stochastic programming formulations whereby strategic decisions are made in the first stage (here and now) while operational decisions are made in the second stage (wait and see). The characteristic problem of such two-stage programming problems is the inevitable explosion in the number of scenarios with increasing products and/or outcomes. Sahinidis [4] presents a recent review of the literature on optimisation under uncertainty; topics covered include two-stage programming, probabilistic programming, fuzzy programming and dynamic stochastic programming. A number of recent works addressing these problems include the work of Maravelias and Grossmann [5]. They introduce linking constraints to link two or more independent sub-models. The authors duplicate the linking constraint variables and build a Lagrangean decomposition scheme, resulting in two independent sub-models. Gupta and Maranas [6] formulated a two-stage stochastic program. The two-stage model is composed of a here-and-now production model which constitutes the first stage and a wait-and-see inventory and distribution model which constitutes the second-stage problem. The uncertain demand parameter is included in the inventory and distribution model. The advantage of their approach lies in the fact that they derived a closed-form solution for the second-stage problem, such that the expectation of the second stage can be evaluated directly via analytical integration. Levis and Papageorgiou [7] developed a hierarchical algorithm in which the first step uses an aggregated version of the model, with a reduced variable space. This problem is solved initially and first stage (strategic) decision variables are calculated. In the second step, a detailed model is solved subject to the decision variables estimated in the previous step. The algorithm proposed in this paper is based on a similar concept to that introduced in the work of Werner and Winkler [8]. They present a heuristic algorithm with two parts, a constructive and an iterative part. The algorithm uses heuristic insertion rules in order to investigate the neighbourhood graph for the best solution path based on the combinatorial path structure of feasible solutions. We present a construction/ improvement algorithm for the efficient solution of the proposed large scale two-stage, multi-scenario, MILP model. This is tested on three illustrative examples and the computational results presented. 2. Problem Description In our earlier work [1] we presented a deterministic MILP formulation for the medium term planning of multiproduct biopharmaceutical manufacturing facilities. The production planning problem is to meet all demands whilst maximising profit and hence minimising operating, changeover, storage and late delivery costs. Here, we extend the deterministic medium term planning model to allow for variable fermentation titres. Production of product/) is approximated by a rate, Vp. This is assumed to be uncertain in order to account for the variability in fermentation titres. This is accepted as an approximation due to the discretisation required at this timescale.

Medium Term Planning of Biopharmaceutical Manufacture Under Uncertainty

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Probability of Outcome 25 %

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rp

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Fig 1: Typical probability distribution and realisations of r^ Industrialists report that within a typical commercial manufacturing enviroimient 2 - 8 products are run over a 1-year production horizon. Assuming 3 possible outcomes for uncertain parameter r^, this results in between 8 and 6561 possible scenarios meaning that for larger examples solving the full space multiscenario model would be computationally intractable. Fig. 1 shows the typical probability distribution for the uncertain parameter, r^. 3. Problem Formulation The formulation presented in this work is a simplification of that presented earlier [1] as the focus is shifted to tackling the impact of uncertainty. Manufacture is assumed to occur over one stage as opposed to being divided between upstream and downstream manufacture as previously. The production constraint shown in Eq. 1 shows how an index k (for scenario) is added to all second stage variables as shown in the number of batches produced, Bptk and production time, Tptk. All parameters are scenario independent as in a p, while r^ becomes rpk to allow for the multiscenario representation required to capture different outcome realisations. Bptk = Zpt + ^pk {Tptk - oCpZpt) "^pj.k (1) The binary variable Zpt allows for the start of a new production campaign for a given product/? at time t and Ypt allows for the start of production for a given product/? at time t. Both binary variables are scenario independent as they are 1^^ stage decisions. This leads to a single schedule with different production timings and production variable realisations for each scenario. Other constraints used within the model include: production timing, minimum and maximum campaign durations, inventory, shelf-life, late delivery penalties and an objective fiinction which is shown in Eq. 2. P ^ ^ f i t " ^ ^ ! ) Pk^I^ptk-Oli>^ptk-C^pt-Stplptk-Sp^pt]^ (2) p t k The objective fiinction is to maximise Profit and is weighted by the probability of each scenario/?yfc. Profit is represented as the difference between total sales (VpSptk) and costs including operating costs (OppBptk), changeover costs (ChpZpt), storage costs (Stplptk) and late delivery penalties {dj4ptk} The resulting mathematical formulation is a two stage, multiscenario MILP model.

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4. Solution Methodology Solving realistic size problems can result in large scale mutliscenario MILP problems that are intractable. More efficient solution procedures are needed. Here, we present a construction/ improvement algorithm for the efficient solution of large scale planning problems under uncertainty. By way of consecutive insertion, individual products are expanded to include multiple scenarios for the selected uncertain parameter and then optimised one by one. Products assume their mean value when they are not being expanded. Production schedules are optimised under uncertainty one by one thereby eliminating the inevitable scenario explosion associated with large scale two-stage programming. In the second stage of the algorithm one or more products are released iteratively in a bid to improve the solution quality achieved at the end of the first "construction" stage. The impact of uncertainty is often quantified via Monte-Carlo (MC) simulation for a more accurate realisation of uncertainty and in order to demonstrate the need for better decision-making [2]. Products can be selected for construction and improvement either randomly or by heuristic rules. The order of product insertion and the number of products released have an impact on the time taken to reach convergence but not the solution quality. The steps of the proposed algorithm (CON/IMP) are shown below: step 1 - Construction (CON): (i) Decide order of insertion of products, either randomly or by heuristic rules. (ii) Select product for expansion using three scenarios for different production rates; all other products use mean value. (iii) Solve 3-scenario model. Fix binary variables/decisions of selected product and then change its production rate back to a single scenario using the mean value. (iv) Repeat (ii) and (iii) until all remaining products are inserted.

Step 2 - Improvement (IMP): (i) Decide which p r o d u c t s a r e t o be r e l e a s e d , e i t h e r randomly by h e u r i s t i c r u l e s . ( i i ) S e l e c t p r o d u c t ( s ) for expansion u s i n g t h r e e s c e n a r i o s d i f f e r e n t p r o d u c t i o n r a t e s per r e l e a s e d product thus r e s u l t i n g 3^"" s c e n a r i o s (where p r i s t h e number of p r o d u c t s r e l e a s e d ) ; o t h e r p r o d u c t s use mean value and remain f i x e d . ( i i i ) Repeat ( i i ) u n t i l convergence i s reached. Step 3 - E v a l u a t i o n : (i) Evaluate performance of proposed s c h e d u l e ( s ) through simulation.

or for in all

MC

5. Illustrative Examples Three examples are solved to illustrate the applicability of the approach/algorithm: two small examples with three products and a large example with seven products. Each example is solved using three alternative approaches; a deterministic model based on mean values, a full space multiscenario problem and the proposed CON/IMP algorithm. In all three examples, MC simulation is used to quantify the impact of variability in fermentation titre on the deterministic schedule as well as for the validation of all solution schedules. Since the fluctuations in time have a direct impact on the number of batches required, they are captured as fluctuations in the production rate Vp. This is carried out using the equivalent normal distribution to the discrete probability data used in the proposed problems, using mean = Vp and standard deviation s = 0.071* r^ for 10% variability and s = 0.1414* r^ for 20% variability. All problems were implemented

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in GAMS [9] using the CPLEX solver with a 1% optimality margin. All runs were performed on an IBM RS/6000 workstation. The results are shown in Table 1. Table 1: Computational results for Examples 1, 2 and 3; OF: Objective function, CPU: Solution time in seconds, MC: Monte-Carlo simulation objective function.

OF

(CPU)

CON

Full space

Deterministic

Example

MC

OF

(CPU)

MC

OF

IMP

(CPU)

MC

OF

(CPU)

MC

1

108.5 (0.2)

99.5

105.8 (78.2)

105.7

105.4 (1.4)

105.3

105.8 (169.7) 105.7

2

108.5 (0.2)

96.4

105.8 (85.8)

102.8

105.5 (1.4)

102.7

105.8 (153.6) 102.8

126.9 (1.14)

108.8

119.2 (17.1)

116.9

125.8 (80.6)

3

Unsolvable within 20 h

123.8

In example 1, a typical biomanufacturer wishes to schedule 3 products over 6 time periods assuming a 10% variability in production rate. Three possible outcomes for rp are assumed as shown in Fig. 1. This results in 27 possible scenarios for the full space problem. The MC simulation result provides an indication of the quality and robustness of the deterministic schedule. In this particular case it highlights the significant impact that slight variations in the titre can have on the profitability of a schedule. IMP makes a marginal 0.3% improvement on CON which converges to the same profit as that of the full space model. IMP makes a valuable 6.2% improvement on the deterministic solution. Fig. 2 shows a graphical representation of the production schedules for example 1 using the deterministic and CON/IMP models. Production schedule for Example 1: (Determinstic)

1

P1

1

1

P2 P3

•™"

1

2

3

4

5

"™™™™" 6

7

8

9

Time Periods

Production schedule for Example 1: (CON/IMP) P1

I

I

I

I

I

P2

1 ^

P3 1

2

3

4

5

6

7

8

9

Time Periods

Fig 2: Deterministic and CON/IMP model solution schedules for Example 1 The figure shows that in the CON/IMP solution schedule time period 6 is used to manufacture product 1 instead of product 3. In example 2, the same problem as that presented in example 1 is solved assuming a 20% variability in production rate. IMP makes a marginal 0.1%) improvement on CON which converges to the same profit as that of the full space model. IMP makes a

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valuable 6.6% improvement on the deterministic solution. The higher variability in this example results in a higher impact on the profit and a marginally higher relative improvement in solution quality by the proposed algorithm. In example 3, a biomanufacturer wishes to schedule 7 products over 12 time periods assuming a 10% variability in production rate. Again three possible outcomes for Vp are assumed. This results in 2187 scenarios for the full space problem. IMP makes a 5.9%) improvement on CON. IMP makes a very considerable 13.8%) improvement on the deterministic solution. The full space model proves intractable due to the large number of scenarios generated. One final schedule is generated for each of the deterministic, full space, CON/IMP models. The relevant scenario related variables (production times, inventories and late penalties) are calculated for individual scenarios. 6. Conclusions A mathematical optimisation-based framework for medium term biopharmaceutical manufacturing planning under uncertainty has been presented together with a three-step algorithm for the efficient solution of large-scale MILP problems. The results from the three illustrative examples indicate that while solving the full space two-stage programming model may be sufficient for problems with a smaller or more modest number of scenarios, this is however not sufficient for problems with a large number of scenarios. The impact of uncertainty on the solution schedules was quantified for all three examples via MC simulation. The construction algorithm generates good quality solutions which are improved by varying degrees by the improvement algorithm in reasonable solution times. The full space model proves intractable for example 3, while CON/IMP returns a considerably improved solution on the deterministic solution. This is a valuable framework for industrialists wishing to incorporate and address the impact of uncertain parameters within their existing scheduling systems. References [1] S.S. Farid, J. Washbrook, and N.J. Titchener-Hooker, (2005), Combining multiple quantitative and qualitative goals when assessing biomanufacturing strategies under uncertainty. Biotech. Prog. 21, 486-497. [2] N. V. Sahinidis, (2004), Optimization under uncertainty: State-of-the-art and opportunities. Comp. Chem. Eng. 28, 971-983. [3] C.T. Maravelias and I.E. Grossmann, (2001), Simultaneous planning for new product development and batch manufacturing facilities. Ind. Eng. Chem. Res. 40, 6147-6164. [4] A. Gupta, and CD. Maranas, (2000), A two-stage modeling and solution framework for multisite midterm planning under demand uncertainty. Ind. Eng. Chem. Res. 39, 3799-3813. [5] A.A. Levis AND L.G. Papageorgiou, (2004), A hierarchical solution approach for multi-site capacity planning under uncertainty in the pharmaceutical industry. Comput. Chem. Eng. 28, 707-725. [6] F. Werner and A. Winkler, (1995), Insertion techniques for the heuristic solution of the job shop problem. Discrete Appl. Math. 58, 191-21. [7] K. Lakhdar, Y.H. Zhou, J. Savery, N.J. Titchener-Hooker and L.G. Papageorgiou, (2005), Medium term planning of biopharmaceutical manufacture using mathematical programming. Biotech. Prog. 21, 1478-1489. [8] A. Brooke , D. Kendrick , A. Meeraus and R. Raman, (1998), GAMS: A user's guide, GAMS development Corporation, Washington.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

2075

A Multi-criteria Optimization Model for Planning of A Supply Chain Network under Demand Uncertainty C.L.Chen,

T.Y. Yuan, C.Y.Chang,

W.C.Lee,

Y.C. Ciou

Department of Chemical Engineering, National Taiwan University, Taipei 10617 TAIWAN ROC Abstract We consider the planning of a multi-product, multi-period, and multi-echelon supply chain network that consists of several existing plants at fixed places, some warehouses and distribution centers at undetermined locations, and a number of given customer zones. Unsure market demands are taken into account and modeled as a number of discrete scenarios with known probabilities. The supply chain planning model is constructed as a mixed-integer linear programming (MILP) problem to satisfy several conflict objectives, such as minimizing the total cost, raising the decision robustness to different product demand scenarios, lifting the local incentives, and reducing the total transport time. For purpose that a compensatory solution among all participants of the supply chain can be achieved, a two-phase flizzy decisionmaking method is presented and, by means of application of it to a numerical example, proved effective in providing a compromised solution in an uncertain multi-echelon supply chain network. Keywords: Supply Chain, Uncertainty, Multiple Objectives, Mixed-integer Linear Program 1. Introduction The location of manufacturing and warehousing facilities has received considerable attention over the past four decades. Location models have been developed to answer questions such as, how many facilities to establish, where to locate them, and how to distribute the products to the customers in order to satisfy demand and minimize total cost (Melachrinoudis et al, 2000). In this work, we will deal with warehouses and distribution centers location-allocation problem. When making location decisions, in addition to the total cost, we also consider the influence of local incentives and transport time. The whole supply chain plarming model would turn into a mixed-integer linear program (MILP). The compromised solution for simultaneously ensuring minimizing the total cost, raising the decision robustness to uncertain product demands, lifting the local incentives, and reducing the total transport time will be determined by applying the fuzzy multi-objective optimization method. 2. Problem Description We consider a typical multi-products, multi-echelon, and multi-periods supply chain network originally studied by Tsiakis et al. (2001). The reviewed supply chain network consists of several existing multi-product plants at fixed places, some candidate warehouses and distribution centers at specific but undermined locations, and a number of known customer zones.In this mid-term supply chain planning problem, each customer zone place demands for one or more products. The candidate warehouses and distribution centers are described by upper and lower bounds on their handling capacity. The establishment of warehouses and distribution centers will result in a fixed infrastructure cost. Operational costs include those associated with production, handling of material at warehouses and distribution centers, and transportation. The numbers and the locations of selected warehouses and distribution centers are left to be determined for establishment of a cost-effective supply chain network.

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3. Supply Chain Modeling with Demand Uncertainty We extend the integrated multi-echelon supply chain model of Tsiakis et al. (2001) for optimal decisions. The scenario-based representation for uncertain product demands is considered in the modeling. Several conflict objectives can be considered simultaneously for the supply chain network design, as stated in the following. Objective 1: minimizing the total cost The total costs are summation of the total establishment costs, the total production costs, the total handling costs, and the total transportation costs, such as, TCO =

TEC -f ^

^

PPD^ (TPCt3 + THCt^ + TTC^,)

(1)

vreT¥3€5

Objective 2: maximizing the robustness of total cost to demand uncertainties We use the upper partial mean as the measure of robustness, where only costs above the expectation are penalized and are weighted by probabilities of related scenarios. UPM = YL P P ^ ^ niax{0, Js - J)

(2)

As the upper partial mean decreases, the robustness of total cost will increase, thus we define the robustness index as below. HI = - U P M

(3)

Objective 3: maximizing the local incentives A meaningful local incentive can be defined by firstly identifying all important factors which causing great affects of location-allocation problem, and secondly giving weight coefficient to each factor according to its importance, and subjectively scoring factors of each candidate location. The weighted average of those scores can be defined as the local incentive of each candidate location. Although our target is to maximize the average local incentive of all chosen locations, it will cause nonlinear term. Thus we will apply the following linear model, Eq.(4), to simplify the solution procedure. TLI

=

min{LI« + £/(l - Y^.) I Vw € W } + mm{LId + t/( 1 - Yd) I Vd e V)

(4)

Objective 4: minimizing the total transport time We c£in set the total transport time as the objective to be minimized, as shown below.

OTT = 5]] Yl TTp.Xp^ + I ] I I TTwdX^ + IZ H ^dc^c 'ip€P'iw€W

Vwewvdei?

(5)

vrfcPVcec

In summary, the supply chain planning model can be constructed as a multi-objective mixed-integer linear program (MILP). The multiple objectives for maximizing, Jm(x),meM, the decision vector, x, and the feasible searching space, CI, are shown below. max(^(^),... ^ J M H ) =

( - T C O , M , TLI, - OTT)

(6)

A Multi-Criteria

2077

Optimization Model for Planning of a Supply Chain Network

\

^Sti>

Vi el, p€V.

S

w €W,d£t>.

c€C, keK,

Yu-; Xu-rf < Y^.; Xud < Yd; X ^ < Y^;

^

teT.

X^.^ = Y^;

Vw€W

(7)

seS

Nemofk J Z " ^ ^ " " ^ l structure J constraints

VdtP

TCLi-^Zi',, < TJq%, < TCLjzi^,,; J ^ ^i^,, < U Y. ^^ts = E T"^^'^Transport constraints

Qr«^x. < Y, Qi.« 1- Qfrx.: Q:,, > O; (* € {pw^wd^dc} ^^pts = Z^

Qputts'i PQipe ~^ 0; 2^

Qpu-tB = 2^

Qwdts

Material balance constraints Production resource constraints

Q=Iw SQS'^Y^. < SQ^. < SQ«?^^Y^; SQj^^'^Yd < SQ^ < SQ^^^-^^Y^

SQ-> E E E E TEC = 5 3 UEC„Y„+ Yi L-EQY,; TPC., = £ VtueW

Vd€P

^

\

Capacitvconstramts

^^ts

UPC=PQ=.,

'V'i^I'^pEV

THC,. = E E UHCLi £ Qi„,.) + 5 : 5 : UHCi! J^ Q;,„) TTC*.= E

E "CU+ E

E TTCt^+ E ETTCi,,

> Costs

TTC.t. - E (F^*^*2^,, + UTCiTTQj,J; .> € {pit'.u.'d,dc]) Vi el, pe'P, weW, deV, ceC. keKl.teT,

s€S (8)

4. Numerical Examples Considering a typical supply chain consists of 2 plants, 4 candidate warehouses, 7 candidate distribution centers, 8 customer zones, and 5 products. Two plants manufacture 5 different types of products and are located in two different location. Each plant produces several products using a number of shared production resources. There are 4 candidate locations of warehouses and 7 candidate locations of distribution centers. Each candidate warehouse and distribution center has its own establishing cost, capacity, and local incentive. The whole planning horizon is 3 periods. The product demand scenarios are given elsewhere (Lee, 2004) and the assigned probabilities are PPDs=i = 0.4, PPDs=2 = 0.3 and PPDs=3 = 0.3, for the case study. And in order to simplify the problem, we neglect thefluctuatingrate for cost parameters. Other indices and sets are [A^ = 4 and [A^ = 6. Values of all fix transport cost parameters, unit transportation cost and transportation time, resource coefficients, and other parameters can be found in Lee (2004). The problem includes 11, 478 3 equations, 7, 622 continuous variables, and 3, 415 binary variables. To solve this mixed-integer linear programming problem for the supply chain model, the Generalized Algebraic Modeling System (GAMS, Brooke et al., 2003), a well-known high-level modeling system for mathematical programming problems, is used as the solution environment. The MILP solver used is CPLEX 7.5. We can firstly apply the single objective programming method to minimize the total cost, the most common method in the traditional supply chain planning. Then we project the result, caused by single objective programming, to the membership functions. Obviously, the satisfaction levels are extremely unbalanced, since the objective function is only taking the total cost into consideration. So we should

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consider all objectives simultaneously, and use multi-objective programming methods to elevate satisfaction level of individual objectives. According to the problem description, mathematical formulation, and parameter design mentioned previously, we solve the multi-objective mixed-integer linear programming problem by using the typical fuzzy procedure. Form the results obtained by directly selecting minimum as t-norm, we can get a more balanced satisfaction among all objectives where the degrees of satisfaction are all around 0.55. By using average operator to guarantee a unique solution, however, the results are unbalanced with lower degree of satisfaction for local incentives. On the other hand, the high robustness measure are given very high emphasis. Obviously this is not desirable for obtaining a compromise solution. Overcoming the drawbacks of the single phase method, the proposed two-phase method can incorporate advantages of these two tnorms. The minimum operator is used in phase /to find the maximal satisfaction for worst situation, and the average operator is applied in phase // to maximize the overall satisfaction with guaranteed minimal fiilfillment for all fuzzy objectives. The resulting optimal network structure is shown in Fig.l, where triangle means plant, square means warehouse, hexagon means distribution center, and circle means customer zone. The values in the network structures are the total transport quantities through the whole planning horizon. 5. Conclusion This paper investigates the simultaneous optimization of multiple conflict objectives problem in a typical supply chain network with market demand uncertainties. The demand uncertainty is modeled as discrete scenarios with given probabilities for different expected outcomes. In addition to the total cost, we consider the influence of local incentives and transport time to location decision. The problem is formulated as a mixed-integer linear programming (MILP) model to achieve minimum total cost, maximum robustness to demand uncertainties, maximum local incentives, and minimum total transport time. To find the degree of satisfaction of the multiple objectives, the linear increasing membershipfimctionis used; the the final decision is acquired by fuzzy aggregation of the fiizzy goals, and the best compromised solution can be derived by maximizing the overall degree of satisfaction for the decision. The implementation of the proposed fuzzy decision-making method, as one can see in the case study, demonstrates that the method can provide a compensatory solution for the multiple conflict objectives problem in a supply chain network with demand uncertainties.

A Multi-Criteria

Optimization Model for Planning of a Supply Chain Network

^ ^

3

AlAMWyp^/

2079

-^

Figure 1: The optimal network structure by using proposed two-phase optimization method Notations Index/se ceC dsD iel keK me M ne N peP seS teT weW Parameters FCD:„

FTC* LI. PPD,

pg; y^max V*5 ^nts

SQ:

Dimension [C] = C [D] = D [I] = I [K] = K [M] = M [N] = N [p] = P [S] = S [T] = T [W] = W

Physical meaning customer zones distribution centers products transport capacity level all objectives resources plants scenarios periods warehouses *G Physical meaning forecasting customer demand of / for customer c {c} {pw, wd, dc) Ath level fix transport cost, pXow,w to d, dXo c {M^,d} local incentive of w, d probability of product scenario s {max, min} maximum, minimum manufacturing quantity of product i {pw, wd, dc) maximum transport quantity ofp to w, wXo d, dXoc {pw, wd, dc} minimum transport quantity ofp to w, wXo d, dXo c total resource « at;? {p) {w,d) maximum, minimum capacity of w, J, +E {max, min}

2080

C.L. Chen et al {pw, wd, dc) {pw, wd, dc) {w,d) {w,d} {p) {pw ,wd, dc} {w} {d} {P}

Ath transport capacity level, plow, w to d, dto c transport time ofp to w, wto d, dtoc unit establishing cost of w, d unit handling cost of product / for w, d unit production cost of product / for plant/? Ath level unit transport cost,p to w,w to d, dto c coefficient relating the capacity of d to flow of product / handled coefficient relating the capacity of w to flow of product / handled coefficient for resource m used in plant/? for product /

X. Y. Z*

*e {pw wd, dc} {w,d} {pw,wd, dc}

Meaning when having value of 1 a link between/? and w, w and d, d and c exists warehouse w or distribution center d is to be established Ath transport capacity level,p to w,w to d, dto c

Real var.

*G

TCU TT. UEC. UHCl

upc: UTC'

al

P\ P*nts

Binary var.

J.

Pel

Ql« SQ, "^TQl Fuzzy var. FD m

Physical meaning {m} objectives manufacture quantity of / {p} {pw, wd, dc} total transport quantity,p to w,w to d, dto c capacity of w, d {w,d} {pw, wd, dc} Ath level transport quantity,p to w,w to d, dto c Physical meaning fuzzy set for final decision fuzzy set for objective m

References [1] Brooke, A., Kendrick, D., Meeraus, A., Raman, R., 2003. GMAS: A User's Guide, GAMS Development Corporation, USA, Washington, DC. [2] Lee,W.C., 2004. Multi-echelon supply chain network optimization for chemical processes. Ph.D Thesis, Department of Chemical Engineering, National Taiwan University. [3] Melachrinoudis, E., Min, H., Messac, A., 2000. The relocation of a manufacturing/distribution facility from supply chain perspectives: a physical programming approach, Multi-Criteria Applications 10, 15-39. [4] Tsiakis, P., Shah, N., Pantelides, C.C., 2001. Design of multi-echelon supply chain networks under demand uncertainty. Industrial & Engineering Chemistry Research 40 (16) 3585-3604.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 Published by Elsevier B.V.

2081

A view-based information model for enterprise integration in process industries Ping Li*," Ming L. Lu,'' Yuan S. Peng," Ben Hua" ^ Key Lab of Enhanced Heat Transfer and Energy Conservation, Ministry of Education, South China University of Technology, Guangzhou, Guangdong 510641, P.R.China ^ Aspen Technology Inc., Ten Canal Park, Cambridge, MA 02141, U.S.A. Abstract This paper presents a view-based information model for the enterprise integration in process industries, which divides the domain space into 16 views organized as 4 subspaces: entity, behavior, task and performer, and 4 types of models: data, behavior, activity, and performer. Then an analysis conducted on using the models in enterprise integration explores an important insight-the vertical relationship between different levels of model decomposition is the key to enterprise integration. Specifications of 4 types of models are then given and used in a Traditional Chinese Medicine(TCM) enterprise integration project to demonstrate the useftilness of the models. Keywords: Information model; Domain modeling; Enterprise integration; Process enterprises. 1. Introduction Data and knowledge sharing and communication in enterprises in process industries are critical to the improvement of their operation competitiveness. However, existing systems were mostly developed separately and have their own data models and interfaces, which makes the communication and information sharing seriously impaired. In recent years, a lot of work has been done in both process systems engineering and enterprise integration and some efforts towards development of standards are actively made in supply chain management and other areas. A multi-dimensional integrated platform was proposed to support chemical process design(Batres, Lu, & Wang, 2003). A general model architecture called multi-dimensional object-oriented model(MDOOM) as a design framework was developed and implemented in an engineering design environment to support concurrent process engineering(Lu, Naka, & Wang, 2003; Lu, Batres, Li, & Naka, 1997; Batres, Naka, & Lu, 1999; Lu, Yang, Li, & Wada, 2000). Bayer and Marquardt gave an overview of different data models related to the chemical engineering and made good efforts in developing the integrated information models for data and documents(Bayer, Schneider, & Marquardt, 2000; Bayer, Eggersmann, Gani, & Schneider, 2002; Bayer & Marquardt, 2003; Bayer & Marquardt, 2004; Bayer, Krobb, & Marquardt, 2001). The basic concepts of the domain are covered in a framework and described in the sense of an ontology by Uschold & Gruninger(1996). The ontology-based approach is presented to support enterprise process integration(Ninger, Atefi, & Fox, 2000). Some other efforts focused on integrating the existing tools based on the integrated data models (Becker, Haase, * Corresponding author: (current address) Clarkson University, Potsdam, NY 13676, U.S A. Email: [email protected]

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Westfechtel, & Wilhelms, 2002; Gruner, Nagl, & Schurr, 1998). The MDOOM model has been extended to support both plant operation and supply chain management(Lu, 2003). A hierarchical model architecture was proposed to support enterprise integration in process industries(Hua, Zhou, Yang, & Cheng, 2000). McKay & de Pennington attempted to develop a conceptual model that serves as a framework for the integration of existing tools and the development of new data models(2001). In addition, there are a lot of work towards standard development (ISA S95; ISA SP88; SCOR; CIDX Chem eStandards; ISO 10303 STEP; CAPE-OPEN; OPC, etc.), and the models developed in all of these standards are categorized into two types: fixed and generic models, the former is efficient while the latter is more flexible... Despite of the tremendous efforts made in the integration of existing models and tools and the development of new data models, the extended and general information models for enterprise integration are still missing as they are needed for the efficient support of enterprise processes and for the development of integrated application systems, and in the industrial practice, legacy systems still hinder the integration work, while many efforts focus on developing mapping techniques. There is a need to bring the existing work together to provide a consistent model framework for enterprise wide information integration. Towards such goals, this contribution attempts to develop a view-based information model and its implementation. The rest of the paper will first identify the nature of enterprise integration problem and then present a view-based information model. Then, 4 types of models are specified, and used in a case study on a Traditional Chinese Medicine(TCM) enterprise integration project. 2. A view-based information model 2.1. Enterprise integration Enterprise integration is to align business processes and information systems across organizations and business parties to support fast decision-making and efficient execution. Essentially, this means a consistent set of terminology, models, data, and business application systems that share all of the above to support efficient process execution. This requires business process modeling and domain modeling to align them both vertically and horizontally. Based on the SCOR model, a business process modeling effort has been conducted for a TCM enterprise, which has identified more efficient ways of operating the enterprise by sharing latest information and promotion of concurrency(Li, Lu, & Hua, 2005). Domain modeling is to bring all needed data together and develop a unified, consistent and accurate representation of domain system to support data sharing among the enterprise processes. Whereas the domain problem is huge, this paper focuses on a key part of the consistent models towards enterprise integration. 2.2, The 16 views of the domain space An analysis on the nature of the data integration problem has resulted in such guidelines that data can be classified according to the frequency of change, history and future predictions, location and topology, process behavior, business and manufacturing processes, tasks, users and systems in the enterprise. Based on the extended MDOOM model(Lu, 2003), a view-based information model is presented below, which has 16 views divided into 4 subspaces and 4 types of models. As shown in Fig. 1, the 4 subspaces are: Entity, Behavior, Task and Performer, and the 4 models are: Data, Behavior, Activity, and Performer.

A View-Based Information Model for Enterprise Integration in Process Industries

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Fig. 1 A view-based conceptual model for process enterprise integration The separation of the domain space into 4 subspaces simplifies the complexity of the domain. The physical entity subspace is concerned with the tangible enterprise entities such as enterprise, plant, site, product line, utility, equipment, etc. The behavior subspace addresses various behaviors such as market trend, process phenomena, project plan and schedule, logistics and organization dynamics. The task subspace considers the enterprise business and manufacturing activities, such as market forecasting, decision making, planning & scheduling, operation optimization, simulation & design, monitoring, control, fault diagnosis, etc. The performer subspace considers people and/or tools that carry out manufacturing tasks and perform business activities. The 4 subspaces interact with each other as shown in the right hand side of Fig. 1 to support enterprise operation. Within each of these 4 subspaces, the 4 types of models are used to capture all aspects of the domain. The data model defines data structures to hold data. The behavior model describes various system behaviors such as market trend, process phenomenon, etc. using mathematical equations, neural networks, etc. The activity model captures all the activities and business processes and their relationships, while the performer model specifies the performer's configuration and management, such as their ability, availability and responsibility for conducting activities. In other words, the 4 models specify all the answers to domain problems. 2.3. Vertical relationship management-Key to integration of enterprise applications Each of the 16 views of the information model presented above contains many domain objects which can be organized both vertically and horizontally. Vertically, domain objects are organized in levels. The higher level domain objects can be decomposed into multiple detailed domain objects at the lower level. Horizontally, domain objects are interconnected as seen in a process flowsheet or a business process model. Examination of the current applications reveals that most of the enterprise application systems only work on one level of the above information model and they usually cover all needed domain objects in the same level together with their interconnections. For example, a steady state process simulation system covers all information in the process flowsheet level, while a planning system works at a higher level.

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Currently, most existing systems do not have consistent models and many kinds of conversion or mapping were done to allow application systems at the same level to communicate with each other. Even this is not value added activity but it does work for the application systems at the same level. However it does not work for the application systems at different levels. Current practice is to hard code some so called adaptors to do the tricks. This can cause many issues such as system upgrades and system configuration changes. It has been found that the vertical relationship within the above model is the key to enterprise integration since they represent the relationship between application systems at different levels. The integration among these applications relies on such relationships. The subspaces are provided to cover the domain objects at all levels of decomposition. Therefore any changes to objects at any level can all be captured in this model. Integration mechanism is then built upon this model. 3. Model specification for the four subspaces This section addresses some aspects in specifying the 4 types of models within all the 4 subspaces. First of all, the performer model is specified using UML use case diagrams in which each actor represents either human user role or software tool. These use cases describe the way these performers cooperate to solve domain problems rather than requirements. UML sequence diagram can also be used to model performer collaboration. Behavior models are represented as mathematical models. There are several representations used by solution systems such as gPROMS, Aspen Custom Modeler. Standards such as Modelica, MathML and OpenMath also provide behavior model representation in XML. Our attention is directed to the integration of these models with the data model. Relevant results will be presented separately. UML use case and activity diagrams are used to represent activity model that describes activities/tasks and their relationships. A UML sequence diagram can be used to combine performer and task together to show who does what. Data models are represented in UML class diagrams. 4. Case study The above models are converted into a consistent set of entity relation diagrams (ERD) which are used to develop an enterprise database and a set of message schemas for managing data and sharing data across a TCM enterprise. A software system has been prototyped in an intelligent integrated system with the following components: • A master data store that persists asset data, raw materials, products, business process and tasks, users and their role, and locations • A planning and scheduling system for production and logistic planning and scheduling(Li, 2004) • A knowledge-based enterprise performance monitoring systems that observes current status of the operation(Li, Lu, & Hua, 2005) • A task oriented and role-based web user interface(Li, Lu, & Hua 2005) Fig. 2 is a screendamp of the prototype system. Various model views are shown on the left pane of the window of the model view browser. The main pane displays the selected model views. This current view displays the prices of some raw materials change over time. The system allows different organizations to share the latest information for faster

A View-Based Information Model for Enterprise Integration in Process Industries

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and more accurate decision-making and efficient execution.

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5. Conclusions Significant progress has been made in different areas of enterprise integration in process industries, however consistent models are still missing and adaptor mapping are still the main method used to solve the problem. There are many limitations and disadvantages such as flexibility and extendibility to meet future changes and upgrades, not to mention the development and maintenance cost for the hard coded integration adaptors. To tackle this problem, based on a brief review of existing work, a view-based information model is presented which divides the domain space into 16 views, and organizes them into 4 subspaces: entity, behavior, task and performer, and 4 types of models: data, behavior, activity, performer. An analysis on using the models in enterprise integration has explored an important insight - the vertical relationship between different levels of model decomposition is the key to integration. Specifications of 4 types of models are then given and used in a case study on a TCM enterprise integration project. This demonstrates a consistent set of information models is critical to enterprise integration.

Acknowledgements The authors gratefully acknowledge the financial support from the Major State Basic Research Development Program (G2000026307) and the Natural Scientific Foundation of China (79931000).

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References Batres, R., Naka, Y., & Lu, M. L. 1999. A multidimensional design framework and its implementation in an engineering design environment. Concurrent Engineering: Research and Applications, 7(l):43-54. Batres, Rafael, Ming L. Lu, Xue Z. Wang. 2003. Concurrent Process Engineering and Integrated Decision Making. Proceedings of the Process Systems Engineering Conference, pp. 160-165, P. R. China. Bayer, B., Schneider, R., & Marquardt, W. 2000. Integration of data models for process design— first steps and experiences. Computers&Chemical Engineering, 24(2-7):599-605. Bayer, B., Eggersmann, M., Gani, R., & Schneider, R. 2002. Case studies in conceptual process design. In B. Braunschweig, R. Gani (Eds.), Software architectures and tools for computer aided process engineering. Amsterdam: Elsevier, in press. Bayer, B., & Marquardt, W 2003. A Comparison of Data Models in Chemical Engineering.Concurrent Engineering Research and Applications, 11(2): 129-138. Bayer, B., & Marquardt ,W. 2004. Towards Integrated Information Models for Data and Documents. Computers and Chemical Engineering, 28:1249-1266. Bayer, B., Krobb, C , & Marquardt, W. 2001. A data model for design data in chemical engineering—information models. Technical report LPT-2001-15, Lehrstuhl fur Prozesstechnik, RWTH Aachen. Becker, S., Haase, T., Westfechtel, B., & Wilhelms, J. 2002. Integration tools supporting cooperative development processes in chemical engineering. In H. Ehrig, B. J. Kramer, & A. Ertas (Eds.), Proceedings of the sixth biennial world conference on integrated design and process technology (IDPT-2002) [CD-ROM], Pasadena: Society for Design and Process Science. Ben Hua, Z. Zhou, S. Yang, S. Cheng. 2000. Integration of technology and management for process industry in 21st century. Proceedings of the fifth world conference on Integrated Design and Process Technology, June 4-8, Dallas: ppl2. Gruner, S., Nagl, M., & Schtirr, A. 1998. Integration tools supporting development processes. In M. Broy, B. Rumpe (Eds.), Requirements targeting software and systems engineering. Lecture notes in computer science, 1526:235-256. Berlin: Springer. Ming L. Lu, Rafael Batres, Hua Sheng Li and Yuji Naka. 1997. A G2 Based MDOOM Testbed For Concurrent Process Engineering. Computers and Chem. Eng., 21:11-16. Ming L. Lu, Aidong Yang, Huasheng Li and Tetsuya Wada. 2000. An application driven approach to the development of a data model standard for process plant operation. Computers & Chem. Eng., 24(2-7): 463-470. Ming L. Lu. 2003. A Model Architecture for CPE, Proceedings of 8th International Symposium on Process System Engineering 1322-1327. Ed. B.Z. Chen and A.W. Westerberg. Pubhsher Elsevier. Micheal Gru Ninger, Katy Atefi, Mark S. Fox. 2000. Ontologies to Support Process Integration in Enterprise Engineering. Computational & Mathematical Organization Theory, 6:381-394. McKay, A., & de Pennington, A. 2001. Towards an integrated description of product, process and supply chain. International Journal of Technology Management, 21(3-4): 203-220. Ping Li, Ming L. Lu, Ben Hua. 2005. A Business Process Model for Traditional Chinese Medicine Manufacturing Enterprises Based on Concurrent Engineering. Chinese Journal of Systems Engineering Theory & AppUcation, 26(1): 48-52. Ping Li, Ming L Lu, Ben Hua. 2005. A QFD-based Dynamic KPI Selection Method for Enterprise Performance Monitoring. Invited Presentation in Session 4 by the 8th World Conference on Integrated Design & Process Technology(IDPT),June 12-16, Beijing. Ping Li, Ming L Lu, Ben Hua. 2005. An Intelligent Method for Real-time Enterprise Performance Management, Chinese Journal of Computers & Applied Chemistry (English), 22(6):411-420. Qian Li. 2004. Integrated modeling and optimization for production system of Traditional Chinese Medicine (TCM) enterprises, Ph.D dissertaion. South China University of Technology. Uschold, M., & Gruninger, M. 1996. Ontologies: Principles, methods and applications. The BCnowledge Engineering Review, 11 (2):93-136.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Strategic planning and design using MILP: an industrial application from the tissue manufacturing industry Joakim Westerlund^, Pedro Castro^ and Sebastian Forsseir

"Abo Akademi University, Biskopsgatan 8, Turku FIN-20500, Finland ^DMS/INETl 1649-038 Lisboa, Portugal ^Metsd Tissue Corporation Abstract Due to rising costs of energy, transportation and raw materials, the tissue market situation in Western Europe is increasingly challenging. In the quest for a competitive advantage in this tough environment, efficient production planning and design tools play a key role. While the production may be more or less optimised from everyday experience, a more systematic method must be used to help corporate level strategic decision-making, more specifically in deciding on how to achieve an optimised production scheme. This paper presents a strategic design and operation study performed at one of Metsa Tissues Mills in Europe. Keywords: Scheduling, production planning, MILP, Resource Task Network, discretetime, continuous-time 1. Introduction The efficient usage of available production resources has a vital importance to the manufacturing industry (Roslof et al. 2002). This work is concerned with simultaneous design (of intermediate storages and equipment items) and scheduling of a tissue mill in Europe. Different production designs and strategies were investigated to form a supportive platform for strategic investment decision making. Two conceptually different Mixed Integer Linear Programming (MILP) formulations were additionally used and compared in the study. On the one hand, a stable periodic mode of operation was assumed to achieve the yearly demands, and a Resource Task Network continuoustime formulation (Castro et al. 2005) used to model the problem. Novel features of the approach include the ability to handle variable recipes in order to incorporate broke recycle, which cannot exceed given limits. On the other hand, a conventional discretetime MILP formulation, where the scheduling horizon was divided into a number of time intervals of uniform duration, was used.

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2. Background and motivation The Metsa Tissue Mill in question is a comparatively large tissue-producing mill with a yearly production of around 70,000 tons of tissue paper. Tissue paper is produced continuously using recycled fibre and virgin fibre at 2 stock preparation (SP) lines, 2 tissue machines (TM) and several converting lines. The main products are: bathroom tissue, household towels, hand towels, and industrial wipes. Due to increased pressure on profitability in the European tissue market, it is essential to strive for continuous improvements concerning quality and productivity. The objective of the study presented in this paper was to analyse different production arrangements and given the production capacities and demands, allocate production runs on different production units. The results should serve as strategic decision-making support for future investment decisions and production optimisation. A number of different scenarios, considering various production design alternatives were analysed. In this paper, one of the scenarios is presented. 3. The process model The model includes existing process unit arrangements and connections as well as potential future process units, to enable strategic simulations for investment decisionmaking. The production arrangement at the mill is, in this paper, simplified to involve 6 raw material and 5 base paper product categories. These categories are principally grouped according to raw material and product brightness. Raw materials are processed on the SP lines and thereafter on the TMs. Some end-products are mixed qualities made of two or more different raw materials. Limited intermediate storages are available for two raw materials, increasing flexibility and capacity of the plant. A part of the raw material, fed into the SP lines and the TMs, is lost as sludge and reject. The percentage raw material lost is given by a raw material specific fibre factor. A flowchart of the production scheme is presented in figure 1. Even though the model balance boundary excludes the converting lines, which is the following step in the production flow after the TMs, the amount of broke generated on the converting lines is taken into account. The broke generated at the converting lines is mainly trim losses and is recycled back into the production cycle at the SP lines, thereby saving expensive raw material costs. Changeovers play an important role in the production planning process on most production facilities. In this case, changeovers are penalized with a specific change-over cost, depending on the specific product qualities involved. It is evident that a change-over firom the darkest tissue quality to the brightest quality is penalized more harshly than vice versa since no cleaning of the machinery is needed at "bright to dark" changeovers, while "dark to bright" changeovers tend to require cleaning. Since this particular planning and design project was made as a non order-driven strategic tool, considering monthly demands and resources, no job-specific due or release-dates were taken into account.

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Ash • • Product A

Figure 1. Flowsheet of the tissue mill, used as a basis for the MILP models Discrete decisions, such as, equipment assignment and product allocation over time makes the problem in question inherently combinatorial, and hence very challenging from the computational complexity point of view (Pekny & Reklaitis, 1998). In the following sections, two conceptually different solution approaches are presented and evaluated. 3.1. The continuous-time

model

A continuous-time scheduling formulation is used to determine the yearly profit of the plant assuming a stable periodic operation with a cycle time of one week. It is based on the formulation of Castro et al. (2005) although no design variables are required since the production rates of the two SP lines and the two TMs are known as a function of the material being processed. There is however one important conceptual difference. To model the continuous consumption of broke by the TMs and its possible simultaneous consumption by the SPs the original Resource Task Network (RTN) model must be modified. This, because RTN models assume that the proportion of materials being produced/consumed by a task (/^^,), like those of the equipment resources {pirj and //^ ^), is fixed and known a priori. Now, however, the amount of broke being recycled back to the SP lines is unknown and its proportion of the total feed may vary up to a prespecified limit (mrecf). In more technical terms, previous RTN models only consider a set of continuous extent variables per task $,^ (the same variables can be used for tasks with fixed proportions), which can be related through model parameters to the consumption/production of all resources r involved in task /. For the new model, tasks with variable proportions (set f^) require another set of variables ^^n, where parameter TJ^J equals 1 if resource r is consumed by task /. The other model variables are the following: A^^^ is a binary variable indicating the execution of task / at event point t; Rrt are the excess resource variables (positive continuous); A^ represents the net production/consumption of resource r over the cycle; Tt the time corresponding to event point t and H is the time horizon.

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The most relevant constraints are shown below. Other entities used are: R^^, the set of equipment resources; R®^ the set of broke material resources; p^^^ the maximum processing rate of task /; and H, the wrap-around operator.

Y^if^rj^i,, +/^../^,-,a(M) +^.,;^/.a(M) yn-r^^" i,t "^/^r,/^^/,Q(M) '^ ^^r,ihi,a{t-\)

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(3)

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The mathematical formulation uses a single time grid where the total number of event points |T| is set a priori. The solution of the MILP is highly dependent on |T| and global optimality is only guaranteed for a sufficient number of event points. However, due to the complex nature of the model, we must often stop at a suboptimal solution. 3.2. The discrete-time

model

In order to get a good prediction of the maximum achievable throughput, a discrete-time formulation was used, based on a short term mode of operation where the time horizon is divided into a number of uniform time sequences. The time horizon used in this case was 1 month, consisting of 35 time intervals of approximately 20h each. The discretetime representation also uses a single time grid, which is now uniform (all time intervals have the same length), to keep track of all events taking place. This is particularly useftil in problems featuring operations competing for shared resources (Floudas & Lin 2004). When compared to continuous-time models, the discrete-time model has the advantage of generating simpler models, enabling easy incorporation of intermediate storages or setup-costs, etc. It is, however, clear that the discrete-time model requires significantly more CPU time for solution. When using a discrete time-representation, there is always a tradeoff between solution efficiency and quality. In this study, the chosen time interval of 20h may be considered fairly coarse. The size of the discrete-time model using a discretisation grid of 20h, and a simulation horizon of 29 days was 4068 variables, of which around 770 binaries. 5125 linear constraints were used in the process model.

Strategic Planning and Design Using MILP

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The combinatorial complexity increases dramatically as the time-discretisation grid becomes finer. Better quality solutions may, thus, be obtained, but at the expense of the solution efficiency. There is also a purely operational aspect in favour of longer time intervals. Shorter time intervals allow more frequent changeovers between qualities on SP lines as well as TMs, which may result in lower runability and endproduct quality. As this case study was intended purely for strategic purposes and not for operational use, the discretisation grid used was considered appropriate. The objective function used in the formulation is concerned with maximising the profit, taking into account endproduct flow and price, raw material flow and price and changeover costs. The main constraints used in the discrete time formulation are the following: material balances, unit to task, capacity, storage and change-over constraints. A total material balance and partial material balances are introduced to cover all flows, mixes, process units and storages. The unit to task constraints are used to manage competing operations on shared resources and are shown in equation (6) where ytjj is a binary variable equal to 1 if task i is processed at unit j at time t; 0 otherwise. The capacity constraints are used to define the momentary maximum production capacity of each unit and are presented in equation (7). The intermediate storages are managed with the storage constraints, presented in equations 8a-8d. Change-over constraints (equation 9) are used to identify the number of changeovers at each unit, enabling changeover penalties in the objective function. Wu'j^t is a binary variable equal to 1 if a changeover from task / to /' takes place at time ttot+1.

Y,y,j,,
^jeJ,

teT

(6)

ielj

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— '^i.slor,max'

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— ^i,SP\,t

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Although a discrete-time model adds a significant amount of binary variables to the problem formulation, good quality solutions to problems of very large scale may be reached using reasonable computational effort. The discrete-time MILP also allows us to easily incorporate changeover costs in the objective function to reduce the total number of changeovers.

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4. Computational results The continuous-time formulation reached a solution with an annual production of 82,102 tonnes, using 7 event points on the time grid, in just 200 CPUs (optimality gap==7.3%). This was indeed a very good solution to the problem since all products but two are produced at their minimum demands, one of these two being the most valuable product. Furthermore, one TM is always at its maximum processing rate and the other is working with a very high efficiency. The model also finds out that the option of recycling the broke is made very efficiently for all product qualities but one. For that broke quality, an alternative recycling strategy must be considered. The first feasible solution using the discrete-time model is reached at 520 CPUs, at an annual production of 79,821 tonnes (optimality gap=18.97%). The annual profit reached at the first feasible solution is, however, only 1.7% lower than that of the continuous-time model. If the discrete-time model is terminated at 2500 CPUs, the optimality gap is still rather high (15.95%) and the annual production capacity is at a level of 81,387 tonnes with a annual profit very close to the one reached with the continuous-time model in 200 CPUs (the discrete model reached a 0.5% higher profit than that of the continuous model). 5. Conclusions This study presents an industrial application of strategic production planning and design. Furthermore, it presents two conceptually different solution approaches to a large scale industrial problem. On the one hand a continuous-time model, and on the other hand a discrete-time model. The study shows that the continuous-time model is more efficient in maximising the production rate while the discrete-time model is able to reach similar profitability levels using slightly less production capacity. Regarding the usage of computational resources, the continuous-time model is significantly faster than the discrete-time model, reaching similar profit levels orders of magnitude faster. Overall, we were able to tackle a challenging industrial problem and reach very good quality solutions in reasonable computational time and, thus, come up with a useful tool for the decision-making process. The results from a large number of production scenarios were used as an important strategic platform for investment decision-making as well as for strategic production planning at the considered tissue mill. Acknowledgements The first author gratefully acknowledge the supportfi"omthe Academy of Finland. References Castro P. M., Barbosa-Povoa A. P. & Novais A. Q. (2005). Ind. Eng. Chem. Res. 44, 343-357. Floudas C.A. & Lin X. (2004). Comp. Chem. Engng. 28, 2109. Pekny J. F. & Reklaitis G.V. (1998). In Proc. FOCAPO-98. Snowbird, Utah, USA, pp. 91-111. Roslof J. Harjunkoski I. Westerlund T. & Isaksson J. (2002). EJOR., 138, 29.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Multiple Time Grid Continuous-Time Formulation for the Short Term Scheduling of Multiproduct Batch Plants Pedro Castro^ and Ignacio Grossmann^ ""DMS/INETl 1649-038 Lisboa, Portugal ^Dep. Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Abstract This paper presents a new multiple time grid continuous time MILP model for the shortterm scheduling of single or multistage, multiproduct plants. It can handle both release and due dates and different objective fimctions efficiently, such as the minimization of total cost, total earliness or makespan. This formulation is compared to other existing MILP approaches with the results showing that the proposed formulation is much more efficient than its uniform time grid counterpart. It is also better than models that use global precedence sequencing variables for single stage problems, while performing similarly for multistage problems. Keywords: Mixed-integer programming; Resource task network; Parallel machine scheduling.

1. Introduction Scheduling approaches can be classified according to different criteria. The type of network structure under consideration is one possibility and can range from the simplest single machine single stage problem, to those with several machines and multiple stages. Another classification criterion is the type of method used. Well known examples are mathematical programming, generalized disjunctive programming (Raman & Grossmann, 1994) and constraint programming (Hentenryck, 1989). The type of time representation is yet another possibility and one can use discrete or continuous-time representations. The later can feature one (Castro et al., 2004) or multiple time grids (Janak et al., 2004) with several event points that are defined explicitly a priori or are implicit on the model constraints through the use of sequencing variables (Jain & Grossmann, 2001, and Harjunkoski & Grossmann, 2002). Some scheduling approaches are only suitable for a specific type of problem, while others are more general in the sense that they can handle different network structures, storage policies and even changeover requirements. General mathematical programming approaches rely on unified networks for process representation that include information on both the network structure and recipe information. Well known examples use the State Task Network of Kondili et al. (1993), and the Resource Task Network of PanteHdes (1994). This paper focuses on single and multistage multiproduct problems and compares the performance of different continuous-time mathematical programming formulations. These consist of the general uniform time grid formulation of Castro et al. (2004), the formulations of Jain & Grossmann (2001) and Harjunkoski & Grossmann (2002), for singe stage and multistage, respectively, and a new multiple time grid formulation.

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2. Problem definition A set / of product orders must be processed on one or more stages, belonging to set K, to reach the condition of final products. It is assumed that all orders go through all stages and that there is a unique sequence of stages for all orders. A total of M equipment units are available, where machine m can process all orders belonging to set Im. Each unit belongs to a single stage, with set Mk including all units belonging to stage t Given are the processing times, /?,>,, release and due dates, r,, df. We assume unlimited intermediate storage for the multistage case and absence of changeovers. Three alternative objectives (to be minimized) are considered: i) total cost, where the processing cost of order / on unit m is given by c,^; ii) total earliness; iii) makespan. 3. New multiple time grid formulation In cases where common resources are not an issue, machines belonging to the same processing stage become independent and the use of multiple time grids is potentially attractive. In the proposed formulation, we consider |M| time grids featuring the same number of time points, |T|. The execution of order / on machine m is identified through the binary variable 7V,> ^ whereas Tf^rn are continuous variables that represent the time corresponding to time point / on machine m. The availability of machine m at time point t is given by the binary variable Rrrjj. Fig. 1 illustrates how the continuous-time formulation works for single stage problems. Any task starting at t, will end somewhere between points t and t+l, more specifically, at time Tt^m'^pi^m. This follows the basic idea of previous work (Castro et al. 2001) that eliminated the need of allocating time points to the end of non-limiting tasks. Naturally, the starting time of any order must be later than its release date, while the finishing times must be earlier than the orders due dates. In Fig. 1, orders 13 and 15 end exactly at the subsequent time point while orders 11-12 do not end at specific time points. This ensures that the number of time points required on time grid m is equal to the number of orders assigned to that machine plus one. For multiple machines, an iterative procedure is required to find the number of event points that ensures global optimality. Starting with a number that can lead to a feasible solution, each iteration features a single increase in the cardinality of set T, until no further improvement in the objective fimction is observed. 14

Mi i I—^^—>^—h M2

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-73 3

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Figure 1. Possible solutionfrommultiple time grid formulation, single stage (|I|=5, |M|=3, |T|=3) For the multistage case, a new set of variables is required to link consecutive stages and ensure that any process order can only start to be processed on stage ^+1 after stage k has been completed. The continuous variable TDfk represents the transfer time of order / in stage k and can take any value between the ending time of the order in stage k and the

Multiple Time Grid Continuous-Time Formulation

2095

starting time of the order in stage k+\. This can be seen in Fig. 2, which assumes one machine per stage. While the transfer time of order II in stage 2 (TDjj) can take any value between Tjj+pu and T2,3 (shown as a grey filled rectangle), the transfer time of order 12 in the same stage is exactly equal to both the ending time of the order in that stage T2,2^P2,2 and to the starting time of the order in the following stage, 7; 5. Notice also that order sequencing can vary from one stage to the other as seen for M2 and M3. M1

h r..>r,

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13

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^ dx

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Figure 2. Possible solution from proposed formulation, multistage case (|I|=3, |M|=3, |T|=4, |K|=3)

The model constraints are shown next. Eq 1 is a multiperiod balance on the unit availability. Any order starting on unit m at time point t decreases the resource availability when compared to M, while any order starting on the same machine at time point M increases the resource availability at /. Eq 2 ensures that all orders are processed exactly once on each stage. Eq 3 states that if order / is processed on unit m at time t, then the difference in time between points /+1 and / on unit m must be later than its processing time. Eq 4 ensures that if a given order i is executed on unit m at time t, then the time corresponding to that time point must not be earlier than the order's release date if the unit belongs to the first stage. For units belonging to subsequent stages, appropriate lower bounds are given by adding the minimum processing times over the previous stages. Eq 5 is the equivalent due date constraint. For multistage problems, eq 6 states that the transfer time of order i in stage kA is lower than the time corresponding to time point t, if the order starts to be processed on unit m (belonging to stage k) at that time point or at a previous one. Eq 7 ensures that the transfer time of order / on stage k is later than the ending time of the order on that stage. Eqs 8 and 9 define lower and upper bounds on the timing variables, while eq 10 defines variable MS (makespan) as the maximum ending time of the time grids corresponding to last stage machines. R.•m,t

+

••(%••

T.^i,n,,,-i^meM,teT

(1)

'£i„

Z Z ^ ' > . ' =1^''*; Vm meMiAmeMi, teT

-T,,n, ^ Y.^i,n,,,Pi,n. '^Tn,t*\T\

Tt,m^T^i,rnAn^Y.

p,,^OykeK,meM,,teT,t^\T\

. .3^^^.

(2-3)

i^In,

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r,,, < ^ N,^^^, {d, - p,^^ - ^ min p,^^,) + //(I - X ^i,m,t )yk,meM,,t^\T\ i^In,

m'eMi k'>k^.

i^Irr,

(5)

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t
Tf^> minr,. ;Tf^
= max J,- Vm,^; MS > T^^ rn^me M^^i

(^-^0)

iel

'

The three alternative objective flmctions considered in this paper are the minimization of total cost (eq 11), makespan (eq 12) and total earliness (eq 13). To better understand the objective of total earliness minimization, note that since every task only lasts one time interval, we know a priori how many time points (of machines belonging to the last stage) will end being equal to their upper bound, H. The last term on eq 13 subtracts the starting times, which are accounted for in variables Tt^^, of such time points. ^^^Z Z

H^i,m,t^i,m ;minMS

mmX^,-Z

Z

(T,,n, + JlN^,n,,,P,,n.)-m\I\-\M^^,^\(\T\-l)•\

(11-12)

(13)

4. Computational results The MILP models were solved to optimality (lE-6 relative tolerance), unless otherwise stated, by the commercial solver GAMS/CPLEX 9.0. The computer used was a Pentium 4 2.8 GHz processor running Windows XP Professional. The results for single stage problems (Table 1) showed that the proposed formulation is the best approach. The solution found always had the same or better value for the objective ftmction and the computational time required to prove optimality was usually lower. The proposed formulation is a significant step forward from its uniform time grid counterpart, mainly due to two factors: i) fewer time points are required to achieve global optimal solutions (in terms of |T| since the total number of time points is significantly greater due to the consideration of multiple time grids); ii) all tasks last only one time interval, whereas in the imiform time grid formulation the number can go up to |T|-1. Both factors lead to the generation of smaller problems, while the second also leads to a significant reduction of the integrality gap. For total earliness minimization, the form of the objective fimction leads to a significant reduction of the integrality gap, when compared to the typical 100% relative gap of other MILPs. Because of this, we were able to find the global optimal solution for problem PI2, for the first time, and also a better solution for problem PI3 than those reported in the literature so far. For multistage problems, the results given in Table 2 are somewhat different. The performance of the proposed formulation (FP) is now very competitive compared to that of the MILP with sequencing variables (FS). Nevertheless, we can say that FP is clearly better for the objective of makespan minimization, while FS is slightly better when minimizing total cost or total earliness. The general uniform time grid formulation (FG) continues to be a poor performer, although it can still find the optimal solution for a few problems. Overall, we can say that when going from single stage to multistage, the performance of the proposed formulation decreases while those of the general and

Multiple Time Grid Continuous-Time Formulation

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sequencing variables formulations increase. The reasons for this behavior will be explained in the next section. Table 1. Computational performance (CPU s) for single stage problems Objective function

Total cost

Total earliness

Problem/Formulation

FP

FS

FP

FS

PI (|Ihl2, |Mh3)

6.18

15.2

0.37

1663

P2 (|Ihl2, |M|=3)

0.04

0.05

8.66

95.0

P3(|Ihl5, |M|=4)

3.82

33.3

7.30

3600^'*

P4(|I|=15, |Mh4)

0.14

5.32

3600^

3226

P5 (|Ih20, \M\=S)

443

421

1232

-

P6 (|Ih20, |Mh5)

0.74

3600"^'*

3600^

-

P7 (|Ih25, |Mh5)

220

3600"'''*

-

-

P8 (|Ih25, \M\=S)

29.1

3600^'*

-

-

P9 (|Ih30, |Mh5)

13.6

107.8

-

-

PIO (|I|=30, |M|=5)

3600'''

3600^'*

-

-

P11(|I|=12, |M|=4)

-

-

0.38

0.30

P12 (|I|=29, |M|=4)

-

-

209

3600^'*

P13 (|I|=40, |M|=4)

-

-

3600^'^

-

^Maximum resource limit; *Suboptimal solution returned; "^Best solution found, maybe the optimal Table 2. Computational performance (CPU s) for multistage problems Objective function

Total cost

Makespan

Total earliness

Problem/Fomiulation

FP

FG

FS

FP

FG

FS

FP

FG

FS

PI (|I|=8, |M|=6, |Kh2)

0.19

0.07

0.07

4.23

2E5^'*

4.38

0.82

6214

0.51

P2 (111=10, |Mh6, |K|=3)

2.58

-

0.07

29.9

-

0.27

14.1

-

1417

P3 (|Ihl2, |Mh8, |Kh3)

104

-

3803

24.4

-

1.29

22.0

-

2E5^

P4 (111=15, |Mh6,|K|=2)

2E5^'*

2E5^'*

21.8

7.52

2E5^'*

2E5^'*

2E5^

2E5^'*

2E5^'*

P5 (|I|=10, |Mh8, |Kh4)

2262

-

4070

2E5^'^

-

2E5^

2E5"^'*

-

8742^'*

^Maximum resource limit; *Suboptimal solution; •'"No solution found; ^Solver ran out of memory

5. Discussion Each formulation is conceptually different from the others and that makes them more appropriate for a certain class of scheduling problems. For instance, the uniform time grid formulation was built having multipurpose plants in mind. Since these plants can have very complex network structures with the possibility of some equipments being used for tasks belonging to different stages of production, it is convenient to keep track of everything that is happening on a single time grid. This option makes it easier to

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write the resource balances for both equipments and materials but also makes it more difficult to locate the ending time point of every task, which leads to a large number of variables and also increases the integrality gap, both effects leading to a decrease in performance when at least one task needs to last several time intervals. In single stage plants without resources being shared by the several machines, these are totally independent and the use of one time grid per machine allows us to limit the length of all tasks to one time interval and hence overcome that difficulty. However, when going to multistage problems we need additional variables and constraints to link consecutive stages of production with the latter being complex big-M constraints, which are known to decrease the performance of MILP models. Finally, the option to use sequencing variables instead of an explicit time grid to order the several tasks is a totally different one. If we look at the models of Jain & Grossmann (2001) for the single stage and Harjunkoski & Grossmann (2002) for multistage problems, we see that the constraints that relate the completion times of orders in consecutive stages are rather simple, whereas those that relate the completion time of different orders in the same stage (and same machine) are big-M constraints. Thus, it is no surprise that MILP scheduling models with sequencing variables have strong performance in multistage problems, multiple time grid formulations in single stage problems, and general uniform time grid formulations in multipurpose problems, where the other two cannot be applied without compromising part of the feasible region.

6. Conclusions This paper has presented a new continuous-time formulation for the short-term scheduling of single or multistage plants where product orders are subject to both release and due dates. It relies on the use of multiple time grids, one for each equipment resource. The proposed formulation was shown to be very efficient for the three objective functions considered: minimization of total cost, total earliness and makespan. The performance of the formulation has been compared to other conceptually different formulations that can solve these specific types of problems. The results have shown that the use of multiple time grids is undoubtedly the best approach for single stage problems, whereas the advantages, when compared to MILP models with sequence variables, are only apparent for the objective of makespan minimization. Nevertheless, it is certainly a best approach than using a single time grid.

References Castro P., Barbosa-Povoa, A. & Matos, H. (2001). Ind. Eng. Chem. Res., 40, 2059. Castro, P., Barbosa-Povoa, A., Matos, H. & Novais, A. (2004). Ind. Eng. Chem. Res., 43, 105. Harjunkoski, I. & Grossmann, I.E. (2002). Comp. Chem. Eng., 26, 1533. Hentenryck, P.V. (1989). Constraint satisfaction in logic programming. Cambridge, Ma: MIT Press. Jain, V. & Grossmann, I.E. (2001). INFORMS Joumal Comp., 13, No. 4, 258. Janak, S.L., Lin, X. & Floudas, C.A (2004). Ind. Eng. Chem. Res., 43, 2516. KondiH, E., Pantelides, C.C. & Sargent, R. (1993). Comp. Chem. Eng., 17, 211. Pantelides, C.C. (1994). In Proc. FOCAPO-94; Cache Publications: NY, 253. Raman, R. & Grossmann, I.E. (1994). Comp. Chem. Eng., 18, 563.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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An Inventory Control Scheme for Simultaneous Production Planning and Scheduling under Demand Uncertainty T. Nishi^, H. Tominaga^ and M . Konishi^ ^The Graduate School of Natural Science and Technology, Okayama University 3-1-1 Tsushima-naka, Okayama 700-8530, Japan We propose an inventory control scheme for simultaneous production planning and scheduling for uncertain demand situations. The stochastic approach accompanied with feedback information from the results of production scheduling is used to determine appropriate inventory amount to maximize the total profit. Numerical results show the effectiveness of the proposed approach. 1. Introduction Simultaneous optimization of production planning and scheduling is widely required for multiproduct batch plants under demand uncertainty to reduce inventories satisfying various customer needs. Conventional production planning system and scheduling system has a hierarchical structure where production planning system derives the amount of production and inventories to maximize the total profit, and production scheduling is determined separately to satisfy the production goal set by production planning system[l]. However, such a hierarchical system has difficulties to accommodate various uncertain changes. Thus, it is necessary to integrate production planning and scheduling for uncertain situation changes. In this paper, a dynamic demand situation is developed to represent a practical demand uncertainty in the environment where the value of demand variation at each time period can gradually be available as time period proceeds. We firstly apply an MILP model incorporating a stochastic approach proposed by Petkov and Maranas[2] for supply chain planning for single stage production system to determine the quantity of material delivered to the production site and safety stock level for final products to accommodate the change of production demand on normal distribution. The optimal condition for trade-off between inventory holding costs and penalty costs for product shortage can be determined by setting a safety parameter. Production scheduling problem is solved at each time period to satisfy the production goal derived by solving the supply chain planning problem. Due date uncertainties are represented by Erlang's distribution that can represent both of regular and random arrivals. The main results for this paper are the following two points. • An optimal parameter to determine probability to exceed dynamic demands on normal distribution can be approximately calculated by solving a static inventory control problem under demand uncertainty according to inventory control theory. • The determination method of targeted safety stock level for final products for simultaneous planning and scheduling has been investigated. An inventory control scheme with feedback loop from scheduling results to planning with future information for demand uncertainty

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T.Nishietal has shown that it can effectively adjust the inventory level to minimize total supply chain costs for simultaneous planning and production scheduling under demand uncertainty.

The paper is organized into the following sections. Supply chain planning model under demand uncertainty is formulated in Section 2. The inventory control method for production planning is explained in Section 3. In Section 4, an inventory control method for simultaneous production planning and scheduling is proposed and numerical experiments are shown in Section 4.2. Section 5 concludes our research. 2. Supply chain planning model under demand uncertainty 2.1. Demand situation The supply chain planning model under demand uncertainty is explained in this section. The rolling planning at time period t is executed for dynamic situation. The total planning horizon is K time periods at the time period t. It means that the planning is generated from time period t, t + 1, •••, t-\-K — 1. The entire demand information is not always available. The due date of jobs are given only at the first period i, and the average and variation of demand at the second and third time period t + 1, t + 2 are given as ^t+i, ^t+2 and crt+i, 0-^+2 • Fig- 1 illustrates the timing of planning for demand situation. When the time proceeds, the demand is determined by a random number generated by normal distribution of average demand ^t+i and variation of demand (Jt+i. The lead time for raw material procurement is one time period. A practical demand situation can be precisely simulated by setting the demand variation as crt+i < cFt+2' Planning horizon Scheduling horizon "^^ {&2.CT,) -

t

a

'r+l

- ^ i +

, i&t,^K) ,

a

! Current time

— (gj.cTa)

0i

^ ' t

...

!

(grti.gr.,)

1 i i JJ

!

i

I CXirrent time

Figure 1. Situation for planning and scheduling under uncertain demand 2.2. Formulation of production planning problem under demand uncertainty The production planning problem under demand uncertainty at time period t is formulated as the following mixed integer linear programming (MILP) problem. Maximize Jt t+K-l

(1)

Np

s.t. A , t ' = ^ Pi,f - rii^t' ' Vi

J2^i^t'Pi
t+K-1 Nr

{t' = t),

Di^t' = 0^

(Vi, W)

it' = t + lr",t^K-l)

(2) (3)

(VO

(4)

i

Ii,t'=Ii,t'-i+Pi^'-Si,t,

{Vz,VO

(5)

An Inventory Control Scheme for Simultaneous Production Planning and Scheduling

2101

Si,v + S~t, = Di,v + S-^,_^ {'ii,\/t')

(6)

Cr,t' = Cr,t'-1 + Mr,t' - E

(7)

fi
ieQr

z-,,>\h^tt-ij\

(yi,W)

0 < ii,t' < ir'"'"^ 0 < Cr,t' < cr'''' (vi,viO

(8)

(9)

where the decision variables and constants are as follows. Cr,t' inventory amount for raw material t in time period t, Ii^t'- inventory amount for final product i in time period t, lj\ targeted safety stock level for product z, Mr^t'- material procurement amount for raw material r in time period t, ni^t'- number of jobs for final product i in time period t, Pi^t'- production amount of final product i in time period t, Si^t- delivery amount of final product i to customer, S~^\ shortage of delivery amount of final product i from customer demand A^t? ^i^t'- penalty for deviation from targeted safety stock level ij for product i in time period t. Np-. number of final product, Nr'. number of raw material, fi^r- unit amount of raw material r to produce unit amount of final product i, hi^t- unit inventory holding cost for product i per time period, Ht: length of hours in time period t, pf. processing time of product i, Qr^t' unit cost of raw material r in time period t, V^: amount of product i per a job, ji^t- penalty cost for deviation from targeted safety stock for product i in time period t, fii^t' revenue of product i in time period t, ui^t'- unit production cost for product i in time period i, pr^t'- material inventory holding cost for material t in time period t, C^i^t- penalty for shortage of delivery for product i in time period t, Qr'. set of products produced by material r. Eq. (1) represents the planning problem to maximize total profit at time period t. Eq. (2) specifies demand constraints under uncertainty. Oi^t denotes a random number generated on normal distribution on average Oi^t and variation (Ji^f (3) is concerned with relationship between production amount and number of jobs. (4) restricts the production capacity. (5) states the inventory balancing constraints. (6) is related to shortage of delivery to customer indicating that production shortage at previous time period is added to demand from customer at current time period. (7) indicates material balancing constraints. (8) expresses safety stock constraints. (9) denotes maximum inventory constraints. 3. Inventory control method for production planning The targeted safety stock is determined by the stochastic approach proposed by Petkov and Maranas[2]. A pseudo value of demand 0*j^i is determined so that the pseudo demand exceeds demand on normal distribution by probability A^ for product i. Pr[eit+x > Bi^t+i] = \i

(10)

^it+1 ^^^ b^ represented as the following inverse function of accumulative normal distribution function $~^. Equation (10) is converted into the following equation. ^M+l=^W + ^M+i-^"'M

(11)

The inventory amount of raw material is determined to satisfy the pseudo value of demand ^*^+i calculated by Eq. (11) using Xijeed-

Ii,t + Ptt+i>Olt+i+Sr^ + lT

(12)

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T. Nishi et al.

The safety stock target ij for final products are determined by the following equation. i^ = cri,t+iH\,prod) (14) Equations (11)-(14) are embedded into planning model in Section 2.2 for demand uncertainty. 3.1. Effects of Xijeed and Xi^prod t o t o t a l profit Three types of demand situations (Case 1: (Ti^t+i = 5%, cTi,t+2 = 10%, Case 2: (Ti^t+i = 15%, f^i,t+2 = 25%, Case 3: CTj^t+i = 30%, cri,t+2 = 40%) are simulated to investigate the effects of safety parameter Xijeed and Xi^^rod- In this example, we set the parameters K = 3 and Ht=720. The effects of Xijeed and Xi^prod to total profit are shown in Fig. 2. The total profit has a maximum point where Xijeed is approximately 0.7 for both cases. The results demonstrate that there exists an optimal operating point for Xijeed and Xi^prod to maximize the total profit even though demand variation changes. The optimal operating point may change when the cost information such as unit inventory holding cost or penalty of shortage is changed. The total profit can be maximized for planning problem under uncertain demand by setting appropriate value of safety parameter Xijeed and Xi^prod40000

40000

•

30000 ^

20000

*^:!1::^—*—^—^—^ ^

=^

o j

-•-Casel 1

/

-*- Case2 \ -*- Case3 :

10000 0

0.4

M-i

30000

*,*••*,i;;fe±zi

20000

-Casel - Case2 - Case3

10000 0 0.5

0.6

0.7

0.6

0.8

^ prodt"]

Figure 2. Effects of probability Xijeed and Xi^prod to total profit 3.2. E s t i m a t i o n m e t h o d of optimal safety p a r a m e t e r A The inventory control theory for static uncertain demand is applied to calculate approximate optimal A. Here, f{z) denotes a probability density function of amount of uncertain demand represented by a stochastic variable 2;. a; is material procurement amount and y is production amount. C: unit price of raw material, P: unit revenue price of product, Ct^iunit inventory holding cost for raw material, Cy-. unit inventory cost for product, Cso' penalty cost for delivery shortage, D: unit production cost. By using these variables, the total sales of products are Pmm{y,z}, penalty costs for delivery shortage are CsoToasix{z - y,0}, inventory holding costs for products are Cvinax{y — z,0}, and inventory holding costs for raw material can be given by C^j max{a: — z, 0}. It is assumed that production amount y is equal to material procurement amount x because lead time for material procurement is negligible for the static model. When the probability density function is considered for stochastic variable z, the total sales, penalty costs for delivery shortage, inventory holding costs for products, and inventory holding costs for raw material can be given by J^ Pzf{z)dz-\-J^ Pyf{z)dz, Cso J^{z-y)f{z)dz, Cy J^{y-z)f{z)dz, Cw ^^{y — z)f{z)dz, Cy^ Dy. The expected total profit Ej^ is calculated as ry

/"oo

E„ = / Pzf(z)dz Jo

+ / ,

Jy

/•oo

Pyfiz)dz

- Co /

py

(2 - y)f{z)dz

Jy

- C„

{y -

z)f{z)dz

Jo

-C^

['{y - z)f{z)dz Jo

-Cy-Dy

(15)

An Inventory Control Scheme for Simultaneous Production Planning and Scheduling

2103

Prom Eq. (15), when ^^L — Q, we can obtain the following equation. F[y )=

f[z)dz = 1 Jo p + Co + a F{y*) satisfies the following equation. Pr[y* > 2] = 0( y-

(16)

+a

(17)

= F{y*)

O'Lt

Here, F{y*) is probability that demand z is less than y*. Therefore, F{y*) is approximately equal to A and it can maximize the total profit.

0.8

I 0.6 "•• Dynamic model • Static model

0.2

0.4

0.6

40.4

••• Dynamic model • Static model

0.2

0.8

Raw material cost coefficient [-]

Penalty cost coefficient for product shortage [-]

Figure 3. Comparison of parameter A* for the dynamic model and the static model Fig. 3 shows the effects of unit raw material cost and unit penalty cost for product shortage to the optimal value of A for the dynamic model derived by simulating various condition for planning problem and for the static model calculated by Eq. (16). The results demonstrate that the optimal value of A can approximately be estimated by the static model. 4. Inventory control method for simultaneous production planning and scheduling Due date of jobs may change according to urgent production requests or cancellation from customers in near future periods. In this section, an inventory control method for simultaneous production planning and scheduling is investigated taking into account for frequency of due date is represented by stochastic variables. The following conditions are assumed in the simulation. 1) Due date of jobs are given only at the first time period. The frequency of due date depends on Erlang's distribution. The amount of order is generated by normal distribution. If the total delivery of products cannot satisfy the total demand at the time period, the rest of order is added to the following time period. 2) Production scheduling is executed only for the first time period. Production schedule is optimized by simulated annealing method. If the amount of order is more than production amount at each time period, safety stocks are allocated to product delivery. 4.1. Inventory control using feedback information from scheduling When the safety stock level is increased, product shortage from demand may decrease, but inventory holding costs for final products is increased. Therefore, the safety stock level for production planning is determined so that the ratio r of due date penalties and inventory holding costs derived by production scheduling is almost constant. The following feedback rule for simultaneous optimization of production planning and scheduling is developed. The ratio r=(tardiness penalties)/(earliness penalties) is calculated. U r > TH then, H is added to targeted safety stock level. Otherwise ii r < rn then, -L is added to targeted safety stock level. rn = 0.60 and r^ = 0.05, H = 1, L = 1 are tuning parameters.

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4.2. Numerical examples Numerical experiments are carried out to evaluate the effectiveness of the proposed inventory control methods. A single stage batch plant consisting of single machine is taken as an example. Four types of products are produced by two types of raw materials. The plant is operated 24 hours. A situation of demand changes is assumed for this simulation. The trajectory of demand is shown in Fig. 4. Three types of inventory control methods are compared to investigate the effects of integration of planning and scheduling. Method A: The targeted safety stock level is determined by incorporating stochastic variable in Section 3. Method B: The targeted safety stock level is determined by feedback information from scheduling results explained in Section 4.1. Method C: The targeted safety stock level is determined by the sum of the planning by incorporating stochastic variable in Section 3 and feedback information from scheduling results explained in Section 4.1. The trajectory of inventories are also shown in Fig. 4 for 180 periods of simulation for each case. The average results of 30 times of calculation are shown in Table 1. PLN, EAR, TAD, CHC, TPF respectively indicate total planning costs, earliness penalties, tardiness penalties, changeover costs and total profits. The results demonstrate that Method C can efficiently change the targeted safety stock level for demand changes without sacrificing total profit for three cases. Table 1 Results of total profit for each method Demand -o~ Method A -tr- Method B -*- Method C

^ ^ ^ y ^ Time step |-|

Case 1 PLN EAR TAD CHC TPF

Case 2 PLN EAR TAD CHC TPF

Case 3

Figure 4. Trajectories of quantity of inventory

PLN EAR TAD CHC TPF

Method A

Method B

Method C

974 6

964 149 264 164 387

961 128 249 158 426

Method B

Method C

905 159 368 171 207

897 117 300 167 313

Method B

Method C

823 165 560 181 -83

788 107 554 188 -61

1,194 342

-568 Method A 908 24 903 298

-317 Method A 791 62 805 230

-306

5. Conclusion Three types of inventory control methods are compared for simultaneous optimization of planning and scheduling under demand uncertainty. It has been demonstrated that the stochastic approach incorporating feedback information from production scheduling can effectively adjust the targeted safety stock level. REFERENCES 1. R.B. Bitran and A.C. Hax, On the Design on Hierarchical Production Planning Systems, Decision Sciences, Vol. 8, pp. 29-55 (1977) 2. S.B. Petkov and CD. Maranas, Multiperiod Planning and Scheduling of Multiproduct Batch Plants under Demand Uncertainty, Ind. Eng. Chem. Res., Vol. 36, pp 4864-4881 (1997)

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Implementation of an Integrated Platform of Process System Operations for Education and Research Xiuxi Li, Yu Qian*, and Yanbin Jiang School of Chemical & Energy Engineering, South China University of Technology Guangzhou, 510640, P. R. China Abstract Process system engineering is a cross discipline with characteristic of industrial practice. New advances and application of computer and information techniques, including process simulation, virtual reality, interaction courseware, remote network education, help to cultivate critical and creative thinking for students and professionals. In this paper, an experimental platform is implemented for the purposes of PSE education and research. It integrates the main components in chemical process operation to help students to adopt a 'systems approach' to engineering problem solving. Keywords: education and training, process system operation, experimental platform 1. Introduction Process System Engineering (PSE) is concerned with the development of techniques and tools to address the generic manufacturing problems in design, operation and control for the process industries. In the recent years, PSE education has been concerned and discussed widely. The most of chemical engineering departments set curricula of PSE, such as chemical process design, process control, process modeling and optimization etc, for undergraduates and graduates. The aim of these curricula is to help students to adopt a 'systems approach' to engineering problem solving and encourage students to adopt a systematic and practical approach to their entire professional career (John, 2002). However, many existing teaching materials and tools are very difficult for students understanding when there is not an advanced interactive Corresponding author, Tel: +86(20)87113046, E-mail address: [email protected]

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experimental platform. In this paper, an experimental platform is proposed and implemented for the purposes of education and research in the area of process operation integration. 2. The platform structure The tasks in PSE, such as process design, modeling, optimization, control, scheduling, are correlative rather than alone. It should be considered in systemic approach. The proposed integrated experimental platform in this paper integrates the main components in chemical process operation. The core of the experimental platform consists of five parts: processes equipments and simulators, control system, data acquisition and rectification, database, and advanced application, as shown in Figure 1. Process equipments and simulator are used as experimental objects. The control systems used are CENTUM-CSIOOO and SIEMENS profibus. The data acquisition and rectification and the database work together to realize data exchange among the basic control systems and advanced appHcations. Opening Database Connection (ODBC) and Common Object Request Broker Agent (CORBA) are followed to realize communication among these software systems in the platform. The Standard for exchange of product data (STEP) is used for information integration. Advanced applications include process monitoring, fault diagnose, production planning and scheduling, safety evaluation, online optimization, and steady state simulation of process etc. Fault Detection and Diagnosis G2

Scheduling and Optimization GAMS

f Process Simulation 1 Aspen plus, Pro/II

Safety Evaluation

Process Monitoring

^

Data Acquisition and Rectification, DataCON — •

CZL

i

Z!I Database ^MS SQL server^

Process Control, CS-1000 Process Simulator Aspen Dynamic, Simulink

Process Equipments miniplant

Figure 1 Structure of the experimental platform for process operation system

To imitate the real industrial production process, the miniplant was installed in the products and processes development laboratory, while the advanced application and teaching experiment are implemented in the system integration

Implementation of an Integrated Platform ofProcess System Operations

2107

laboratory in another building. The distance between the two laboratories is about 600 meters. Signals of data and video are transmitted by special light fiber as well as campus network. The physical structure of the platform is shown in Figure 2. The equipment Lab

The system integration Lab 52' plasma display panel

video camera

^^^^F^ s s s ^ s ^ ^ ^

l^^n •f^!

B

Fiber Switch

Campus Network

•r. '"

\

•r:":" ' i |

igiBgpjy

Workstation Control PC

Database server

Figure 2 The platform physical cormecting structure

3. Processes simulators and equipments Process simulation has been widely used in education. Process simulator is designed to represent the characteristic of the real plant productions, with necessary functions of the operation. Tennessee Eastman process simulator was developed by using C++Builder in the platform. Tennessee Eastman process is a typical chemical process found in the industry (Downs, 1993). There are 41 measurements, including 22 continuous process measurements and 19 composition measurements, and 12 manipulated variables. One of the interface of TE process simulator is shown in Figure 3. Besides the process simulator, a miniplant is used and connected to the platform as experimental object. The miniplant includes two batch reactor units, dosing, centrifuge, heating and cooling system, heat exchanger, vacuum pump using for negative pressure operation, and industrial control system operator station, as shown in Figure 4. In this miniplant, the raw material is pumped into the alternative reactor from the supply vessels. Flow rate of the feeding is controlled with dosing valves. The reactor unit consists of a 2 litre and a 10 litre jacked stirred reactors. They are used for batch and semi-batch chemical processes, such as reactions and crystallizations. When finishing reaction, the fluid product goes through the partial condenser and total condenser with heat exchange, and then flows into the product vessels. Meanwhile, the solid product is transported into the centrifiige from the reactor bottom valve. Crystal

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separation is operated in the centrifuge. The vacuum pump ensures negative pressure operation. There are 16 measurement variables and 7 control variables.

Figure 3 The interface of TE process

Figure 4. Structure of the miniplant

4. Software tools It is essential for students and engineers to be familiar with the concepts of applying software design, formal specification methods, programming languages and different software tools. Currently hundreds of software products are available for PSE. In the platform, we select some of the most popular software development tools for PSE education and researching, as listed in Table 1. Table 1. Software tools used in the platform

Software tool G2 Matlab/Simulink LabWindows/CVI Aspen Plus Pro/II SuperPro DataCON GAMS WinCC MSDN Control Station InTouch Rational XDE gPROMS SCA

Usage Real-time expert system development tool Numeric computation and visualization tool Virtual instrumentation programming environment Process simulator Process simulator Process simulator Data rectification MINLP solver Process monitoring Programming and database tools Process control Process control Data modeling Process modeling and optimization Statistical analysis to time series

Implementation of an Integrated Platform ofProcess System Operations

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5. Applications This platform is currently used in process system engineering teaching activities in South China University of Technology. Up to now, 5 years of graduate students courses of 'simulation and optimization of chemical processes', 'computer integrated process operations', 'chemical processes design', and 'process control' are delivered based on this platform, listed in Table 2. Table 2 courses based on the platform Course name Process control Analysis and synthesis of chemical process Simulation and optimization of chemical processes Design of chemical processes Computer integrated operation

Student

Hour

Undergraduate Undergraduate Graduate Graduate Graduate

40h 40h 60h 40h 40h

A series of research and development projects have been conducted based on this platform (Qian, 2000, 2003). One of the research is to integrate multi-task for chemical process operation using multi-agent technique (Cheng, 2003). The multi-agent system with CORBA as a middle ware of the information exchange of agents is an open architecture, which is essential to the requirements of the process operation system integration. Another research based on this platform is process monitoring and fault diagnosis. For miniplant, operating data are collected from WINCC using OPC technique. Operation monitoring module for miniplant was developed using VB, which 16 measurement variables and 7 control variables of miniplant are monitored, shown as Figure 5. Several control variables, such as temperature, stirring speed, are monitored on loop remotely. Many methods of process fault detection and diagnosis were studied on the platform, such as ART2 (Li, 2003a), PCA (Li, 2003b, 2004), ICA (Yao, 2004)

Figure 5 Operation monitoring for miniplant

Figure 6 TE process fault detection and diagnosis based on PCA

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etc. Shown in Figure 6 is the interface of TE process fault monitoring and diagnosis based on PCA. 6. Conclusion The proposed and implemented integrated experimental platform provides an easily used, systemic educational learning environment for students to solve engineering problems. The platform plays an important role in transferring new advanced technology in research to industry. A number of popular software development tools has been integrated in the platform. It has been used for education and research for several years. Furthermore, the WWW technology will be introduced in the platform to make the platform a virtual experimental environment on the campus so that more students and researchers from different disciplines make maximum use of the platform. Acknowledgments Financial support from the National Natural Science Foundation of China (No. 20225620, 20376025, 20536020), the international cooperation research project from Guangdong Provincial Sci. & Tech. Bureau are gratefiilly acknowledged. References Downs, J.J. and E.F. Vogel, 1993, A plant-wide industrial process control problem. Computers and Chemical Engineering, 17, 245-25 5. John P., 2002, Education in process system engineering: past, present and future, Computers and Chemical Engineering, 26, 283-293. Cheng, H. N., Y. Qian, X. X. Li, and H. H. Li, 2003, Agent-oriented modeling and integration of process operation systems. European Symposium on Computer Aided Process Engineering-13, 599-604. Li, X X, Y. Qian, Q.M. Huang, 2003a, Multi-scale ART2 for state Identification of process operation systems, 8th International Symposium on Process Systems Engineering, 523-529. Li, X.X, Y. Qian, and J. F. Wang, 2003b, Process monitoring based on wavelet packet principal component analysis, European Symposium on Computer Aided Process Engineering-!3, Lappeenranta, Finland, 455-460. Li, X.X.,Y.Qian, 2004, Process Monitoring based on nonlinear wavelet packet principal component analysis, European Symposium on Computer Aided Process Engineering-14, 685-690. Qian, Y., Q.M. Huang, 2000, An object/agent based environment for computer integrated process operation system. Computers and Chemical Engineering, 24, 457-462. Qian, Y., X.X. Li, Y.Q. Wen, 2003, An exper system for real-time fault diagnosis of complex chemical processes. Expert Systems with Applications, 24(4), 425-432. Yao, Z.X, 2003, Statistics Modeling of the state space and monitoring of chemical process systems, PhD Dissertation, South China University of Technology.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. PanteUdes (Editors) © 2006 Published by Elsevier B.V.

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Integration of multi-scale planning and scheduling problems Hlynur Stefansson^ Pall Jensson^, Nilay Shah^ ^Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, London SW72AZ. United Kingdom. Faculty of Engineering, University of Iceland, 105 Reykjavik, Iceland. Abstract In this paper we propose a multi-scale approach together with integration strategies for a continuous and dynamic planning and scheduling decision problem from the process industry. The decisions have to be made before all data are available in contrast to most sophisticated planning and scheduling approaches for the process industry that consider a fixed time horizon and assume that all data is given at the time of application. The approach is based on a hierarchically structured moving horizon algorithm. On each level we propose optimisation models to provide support for the relevant decisions. The levels are integrated with versatile integration strategies that transfer and implement the decisions at the adjacent levels. The algorithm based on the integration strategies restricts the solution space to eliminate infeasible solutions and uses hard constraints, bounds, shaping methods and penalty functions as guidelines for obtaining near-optimal solutions. Feasible solutions can still be obtained when the guidelines are violated although they become less optimal. Solution procedures have been developed and the integrated multi-scale approach has been validated and tested with data from the real world problem. Keywords: Planning, scheduling, multi-scale, integration, MILP. 1. Introduction A supply chain may be defined as an integrated process wherein various entities work together in an effort to meet the objectives of each entity as well as the common objectives of the overall supply chain. It is theoretically possible and preferable to build mathematical models for entire supply chains including all interacting strategic and operational decisions throughout the supply chain. Such monolithic models will not be consistent with the nature of the managerial decision process or practical due to computational complexity of models, data and solution techniques. Mathematical programming is most commonly used to formulate planning and scheduling problems within the process industry. The problems are combinatorial in nature which makes them very difficult to solve and it is vital to develop efficient modelling strategies, mathematical formulations and solutions methods. One of the major difficulties in building mathematical programming models is to keep the size within reasonable limits without sacrificing accuracy. To solve full-scale real-world planning and scheduling problems efficiently, simplification, approximation or aggregation strategies are most often necessary (Grunow et al., 2002, Engell et al., 2001). It is widely recognized that the complex problem of what to produce and where and how to produce it is best considered through an integrated, hierarchical approach which also acknowledges typical corporate structures and business processes (Shah, 1999). Production planning and scheduling in a typical enterprise involves managers at various

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echelons within the organization and the decisions that need to be made differ by scope and time horizon and the underlying input information differs by its degree of certainty and aggregation. The decisions also need to be made with different timing and frequency and according to the correct sequence which even further makes the case for an integrated hierarchical approach. The literature often describes problems solved individually but less often the integration of different problems or the integration of different detail levels of the same problems. An example of an integrated strategic and operational planning problem is described by Kallrath (2002) and an investigation on the integration of long-term, mid-term and short-term planning operations through a common data model is reported by Das et al. (2000). Some typical economical benefits of integrated decision making are listed by Shobrys and White (2002) who conclude that the major challenges in integrating planning, scheduling and control systems are involved in issues like changing human and organizational behaviour rather than technical issues. The general conclusion made in the literature is that the integration of decisions with synchronized models is desirable but at the same time it is very difficult to solve such models efficiently. 2. Problem Description Our study is based on a real world problem from a pharmaceutical enterprise and we focus on single plant production planning and scheduling for a secondary production facility with order-driven multistage flowshop production. The plant consists of a large number of multi-purpose production equipment items at each production stage, operated in batch mode. The plant uses campaign production and each product has a number of different feasible production routes through the plant and as the number of product families is over 40 and product variations more than 1000, the process of planning and scheduling the production in an optimal way is extremely complicated. The overall goal in the problem is to determine a campaign plan and to schedule customer orders within the campaigns. The customers request certain delivery dates for their orders and the plant attempts to meet those requests. The general objective is to meet the quantity and delivery date of customer orders and minimize the unproductive production time while respecting constraints. A further description of the problem environment is given by Stefansson and Shah (2005). 3. The Integrated Multi-scale Algorithm (IMA) 3.1. General approach Order driven production planning and scheduling in the pharmaceutical industry is a critical example of a continuous and online decision problem. The production is a dynamic and ongoing process that is affected by several uncertain inputs and the most important one is the demand from customers. To cope with the problem we propose an integrated multi-scale hierarchically structured algorithm. At each level we propose optimisation models to provide support for the relevant decisions, wherein the scope and availability of information at the time of solution differs. 3.2. Three hierarchical levels of decisions At the top level we propose a model to optimise a campaign plan for long term planning purposes. The campaign structure has to be decided earlier than when the orders become available and demand predictions are used instead of the actual orders. The purchasing of raw materials is based on the campaign plan and needs to be performed before the orders become available as the suppliers' lead-times are often greater than the promised

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lead-time of the production. Also maintenance, shifts and other events must often be planned beft)re orders become available. At the middle level we propose a model to reoptimise the campaign plan simultaneously with allocating orders within the campaigns. The plant receives new customer orders each week with a requested delivery date and the orders need to be scheduled and confirmed. When the orders are scheduled within the campaigns the decision maker does not have inft)rmation about other orders that will be added to the campaigns but have not been received yet. The current orders can change and other unanticipated orders may appear later with earlier due dates, larger quantities or higher priorities which can make it necessary to change the current schedule. However the confirmed delivery dates that have already been promised must be respected if at all possible. At the lowest level we propose an optimisation model for the detailed scheduling of all production tasks. The optimisation is based on the confirmed customers' orders together with the newest possible real-time information each time it is used. Level 1 - Campaign planning with 12 months horizon

Expected campaign and demand structure, resource availabilities

Level 2 - Campaign planning and order scheduling with 4 months horizon

T L

Demand information, confirmed deliveries, resource availabilities

Level 3 - Detailed scheduling with 1 month horizon

J.

Final production schedule with campaign structure and sequence of production tasks

Figure 1: Flow of information in the IMA. The proper integration of the different levels in the algorithm is fiindamental for its success. If the campaign structure based on the expected demand at the top level is not transferred to the lower levels it is likely that the schedules will not be able to respond to orders that have not yet been received. Some decisions are also made at the upper levels that must without any exception be respected and the lower level plans will be useless in practice if these constraints are violated. The levels in the algorithm are integrated with bi-directional flow of information. The decisions created by the upper level models are implemented at the lower levels either as guidelines or/and as strict constraints that must be respected. By selecting careftxlly which decisions are implemented as guidelines and which as strict constraints it is ensured that feasible solutions are obtained. 4. Integration Strategies The levels in the algorithm are integrated with integration strategies that provide the relevant flow of information between them. It is of great importance to ensure that the results transferred from higher levels provide feasible input for lower levels and in this

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research both mono- and bi-directional integration strategies have been developed and implemented to ensure feasibility. The algorithm based on the integration strategies restricts the solution space to eliminate infeasible solutions and uses hard constraints, bounds, shaping methods and penalty functions as guidelines for obtaining near-optimal solutions. We use hard constraints to implement decisions that must be respected without any exceptions. If hard constraints are used to implement upper level decisions at lower levels where less aggregation is used, it must be ensured that feasible solutions can be obtained and therefore can not all degrees of freedom be eliminated with hard constraints. As an example of decisions we implement with hard constraints is the availability of raw materials. Raw material inventories are often expensive to keep and it is therefore important to minimize quantities as well as ensure that the appropriate raw material is available when needed in production. The material requirement planning and purchasing of raw material with longer lead times than the promised lead time of customer orders is based on the long-term campaign plan created at the top level and these decisions need to be transferred and respected at the lower levels. The constraints we use are based on the stock of each raw material, the delivery schedule of raw material orders that have already been placed as well as delivery time of new orders. Hard constraints are also used for other limited resources such as information about the available manpower that is transformed from the middle level to the lowest level according the shift schedules made from the middle level plan. We use a "shaping" method to transfer demand information from the top level to the second level. The middle level is not as proactive as the top level and the decisions taken at the middle level are based on temporally incomplete datasets since all the orders are not yet available. The solution space for the second level is reduced by creating subsets of the sets of time related indices, based on the top level results. The subsets define where the production of each product is allowed to take place at each stage, i.e. specifies the machines and the time periods allowed, and thereby transfers information about expected demand as well as decreases the number of variables and constraints that are needed in the models. Machines - level 2

Machines - level 1

rrrr

I I I ga M

I ' m

I I I I I I I I I I I Weeks

^

I I I v/iwmis I I I wmwA •

Weeks

Figure 2: On the lefl side we have the allocation at the top level of certain product. This information is transferred to the middle level and flexibility added by allowing the production of that product to start either one week earlier or later and also to be performed on the other possible machines according to the products' recipe. Another example of the use of a shaping method is a constraint we use to shape the length and frequency of campaigns. The method analyses the maximum length of campaigns of each product family and the average frequency, at the top level and adds a set of constraints to implement it as guidelines at the second level. This is used to avoid too long or too many campaigns with low utilisation at the second level when we still have few orders available in the input dataset for the period under consideration at the second level and as a result little competition for resources.

Integration of Multi-Scale Planning and Scheduling Problems Machines

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Machines

\m^A \ \ w Tg^n I I mm

I mm

ii Weel<s

B^^ I E^ I I

m-r-m Weeks

y

gure 3: Examples of different campaign shapes. The maximum length of campaigns for the product family on the left is 3 weeks and in total 6 weeks can be used over the 12 week period, while the maximum length is 2 and 4 weeks can be used over a 12 week period for the family on the right. The number of v^eeks spent on production of each product family over each couple of months at the top level is transformed to the second level to account for demand that has not yet arrived. A set of constraints is added for each product family at each production stage and the constraints define upper and lover bounds for the number of constraints that should be used for production according to the expected demand. This W\\\ in some cases result in campaigns w^ith low^ utilisation or even empty campaigns but it is highly likely that the campaigns will be filled later on and by doing this v^e have reserved some production space for orders that w^ill most likely arrive later on. Bounds for the number of setups, i.e. the number of campaigns of each product family, are used in a very similar manner as the bounds for the number of production wrecks. Penalty functions are used to implement decisions that can be violated as much as needed however if they are violated they will affect the value of the objective function in the optimisation. As an example we use penalty functions at level 3 to implement the confirmed delivery dates that have been decided at the middle level. It is very important to respect the delivery dates but if it is not possible at all, e.g. because of equipment breakdowns, then the delivery dates can be violated and a penalty cost will be added to the objective value (when minimizing). The time horizons for the different levels in the IMA intersect temporally but the decisions made at the lower levels are based on more accurate and current information compared to the decisions made at the upper levels. The lower level decisions do therefore receive higher priority and are transformed with feedback loops to the upper levels where they are used to fix the overlapping period of the time horizon. 5. Results The Integrated Multi-scale Algorithm (IMA) has been tested with several full-scale datasets created from actual data from the production facility under consideration. The testing is at an early stage as some more variations of the datasets need to be created and used to test the characteristics and fixnctionality of the integration strategies under a great variety of conditions. Initial results strongly suggest that the integration strategies do create feasible problems when information is transferred between levels. Initial results also suggest that the integration strategies are quite effective in delivering the demand information between the levels although excess and imperfectly controlled capacity is sometimes reserved for production of overestimated demand. Some of the integration strategies did reduce the solution efficiency with the addition of complicated constraints, while others improved it by reducing the feasible solution space, i.e. the area searched by the algorithm. For the test cases under consideration the integration strategies in fact slightly improved the solution time at all the levels of the IMA, compared to the time required when solving the models in isolation.

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Table 1: Number of integer decision variables, constraints and computational time of the IMA (sub-problems are solved with CPLEX 9.0 and algorithms programmed in GAMS running on an Unix based machine with 2GB of RAM and 2.4 GHz Pentium IV CPU) Level

Integer variables

Constraints

Comp. time [CPU seconds]

2 3

41844 31726 8817

52478 46986 25344

31716 21060 1062

1

6. Conclusions We believe that proper integration of decisions of different levels and timescales has been lacking, both in practice and in approaches found in the literature. However due to the increased computer power available and the improved methods to manage and integrate complex data structures it has recently become reasonable to develop modeling approaches based on tightly integrated and interacting modules that provide monolith decision support. We have developed a multi-scale approach based on a hierarchically structured algorithm together with effective integration strategies for a continuous and dynamic plaiming and scheduling problem. On each level we propose optimization models to provide support for the relevant decisions together with efficient solution procedures. The integrated approach has been tested with full-scale real world datasets and the results obtained so far indicate that the feasibility is maintained when decisions are transferred between levels through the integration strategies and the solutions obtained are practical both in terms of computational time and usability. The results give some general directions for integrating multi-scale plaiming and scheduling problems. Work still remains on improving and testing the approach and it is of great interest to incorporate demand uncertainty into the top level model which should make the plaiming algorithm significantly more robust and improve the efficiency and accuracy of the integration strategies. References Das, B. P., Rickard, J. G., Shah, N. and Macchietto, S. (2000) An investigation on integration of aggregate production plaiming, master production scheduling and short-term production scheudling of batch process operations through a common data model, Computers & Chemical Engineering, 24, 1625-1631. Engell, S., Markert, A., Sand, G., Schultz, R. and Schutz, C. (2001) In Online Optimization of Large Scale Systems(Eds, Grotschel, M., Krumke, S. O. and Rambau, J.) SPRINGERVERLAG BERLIN, Berlin, pp. 649-676. Grunow, M., Gunther, H. O. and Lehmann, M. (2002) Campaign planning for multi-stage batch processes in the chemical industry. Or Spectrum, 24, 281-314. Kallrath, J. (2002) Combined strategic and operational planning - an MILP success story in chemical industry. Or Spectrum, 24, 315-341. Shah, N. (1999) Single- and multisite planning and scheduling: Current status and future challenges, AIChE Symposium Series, 320, 75-90. Shobrys, D. E. and White, D. C. (2002) Planning, scheduling and control systems: why cannot they work together. Computers & Chemical Engineering, 26, 149-160. Stefansson, H. and Shah, N. 2005, Multi-scale Planning and Schedulin in The Pharmaceutical Industry. European symposium on Computer Aided Process Engineering - 15, 20B, Puigjaner, L. and Espuna, A., Elsevier, Barcelona, Spain, 1003-1008.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Plant-wide planning and marginal value analysis for a refinery complex Wenkai Li," Xige Liang,^ Chi-Wai Hui"'* ^Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, PR China ^Daqing Refining & Chemical Company, PetroChina Company Limited, PR China Abstract Recently, besides the production of petroleum products and fiiels, modem refinery complexes are being reformed to become "energy plants" that are able to export electricity and steam. The success of this reformation depends heavily on effective integration of the production and the energy system. In view of this, this paper proposes a plant-wide multi-period planning model for a real world refinery complex. All main plants, such as the refining plant, the lubricant oil plant and the utility plant etc., are integrated into a site model, enabling the model to optimize operations among the plants at different periods and scenarios. Investment strategies and retrofitting suggestions are obtained which can contribute to an increase of the total profit by 6% in case studies. It has become common practice that electricity price is charged differently at different time shifts of a day in some countries. This paper proposes a multi-period model which examines how this pricing policy affects the operation of a refinery. Case studies show that adjusting the use of electricity to suit the price variance can significantly reduce operation costs in a refinery. Marginal Value Analysis (MVA), an analytical method that provides additional economic information for refinery complex, is also proposed in this paper. From the marginal value of final or intermediate products (e.g. electricity), important insights can be gained into the identification of production bottlenecks, decision-making and intermediate products pricing. Keywords: Planning, Marginal Value Analysis, Refinery, Site model 1. Introduction and literature review A typical refinery complex (e.g., the refinery complex shown in Figure 1) consists of several closely interacted plants, such as the refinery plant, the polymerization plant, the lubricant oil and the utility plant, etc. Products of the refinery plant such as propylene, fuel oil and fuel gas etc., are used in other plants as raw materials. The utility plant in turn produces steam and electricity which are used in all plants. Process units in the refinery plant rely on consistent steam and electricity supply that requires high capital and operating costs in utility systems. The interactions among these plants are rather complicated. In fact, by taking into consideration the interactions among the utility plant and other plants, energy can be transformed into materials and vice versa. As a result, the total cost can be minimized. The bottlenecks of the system not only exist in the refining plant or the lubricant oil plant but also exist in the utility plant or the streams among these plants in some scenarios. Some works relating to plan-wide planning have been reported. Hui et al.^ presented a multi-period MIP model for an industrial case. They considered seasonal variations. ' Author to whom correspondence should be addressed. E-mail: [email protected].

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day/night fluctuations of electricity demands as well as adjusting operational conditions to minimize the total cost. Zhang et al.^ integrated the hydrogen network and the utility system with the production system to optimize the overall profit. They found that the plant-wide integration model can increase the total profit by 1.0% or 1.9 m$/year compared with the sequential method. Some of the unit models in their paper are too simple, however, which may limit the accuracy of the planning strategies. Li et aP studied the material and energy integration in a petroleum refinery complex. The model in Li's work is not accurate enough because the demands for electricity and steam from the refinery and fertilizer plants are fixed into two scenarios (winter and summer). In a real world refinery complex, these demands vary with the operation conditions of the refinery and the fertilizer plants. Crude Oil

Propylene Fuel Gas, Fuel Oil

Refinery

1^ Steam _ • — ^

Polymerization Plant

Steam

j^

polymer

lectricity

Utility Plant Plant

T

Gasoline Diesel Oil Figure 1 The overall configuration of the "D" Refinery Complex 2. Problem description and the site model In this paper, a site model for a real world refinery complex (Figure 1) located in P. R. China (referred to as the "D" Refinery Complex later) is developed. If the amount of fuel oil and fuel gas from the refinery plant and the lubricant oil plant varies, the amount of steam and electricity generated in the utility plant changes subsequently. The change in electricity and steam will further influence the operation conditions of processing units in other three plants. Currently, plans in the "D" Refinery Complex are made manually which are not satisfactory in that: 1). Only the availability of the utilities is considered most of the time. The cost of utilities is also an important factor that needs to be taken into account; 2). The manual procedure consists of several steps that use different tools. Inconsistencies may occur among these tools. A systematic model is needed to integrate these steps into one; 3). Several planning schemes are compared to find a better plan, which requires heavy workloads; 4). An optimal plan for the refinery complex cannot be guaranteed by considering several schemes. Significant profitmaking potential may be lost. The above difficulties call for an integrated site model to figure out optimal planning scheme with efficiency. In this paper, a site model for monthly/yearly planning is developed as follows. In Eq (obj), the plant profit equals plant revenue minus the raw material cost Crw, the operating cost Cop, the inventory cost Cin and the investment cost Civ (new equipment installation or existing equipment retrofitting divided by their lifetime). Eq (1) is the turbine and boiler model to compute the EL(electricity) generated. Eq (2) is the EL balance (EL produced = EL consumed). Eq (3) is the steam (1.0 Mpa and 3.5 Mpa)

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balance and Eq (4) is the fuel (fuel gas, fuel oil and naphtha) balance. In Eq (5), product inventory InVp equals the initial inventory, InvOp, plus the product produced by unit /, Pdip, minus the the market demand for product p. Dp. In Eq (6), unit models which include CDU (Crude Distillation Unit), ARGG (Atmospheric Residuum maximum Gas & Gasoline), product blending, lubricant oil plant etc., are used for material balances of products (gasoline, diesel etc.) and intermediate streams.

max Profit=Revenue S.t.

EL,

-C^^-

cop

-C.in - CIV -other costs, (obj)

=f(ST,)

(1)

J:EL,=ZEL^

(2) (3)

^Fuel.

=^Fuel.

(4)

Inv„ = InvO „ + y Pd. - D^ p

p

V

L^

(5)

p

Unit models

(6) 2.1. Yearly planning using the site model The planning of the "D" Refinery Complex has two steps: the yearly planning and the monthly planning. The total annual amount of crude oil to be processed is decided by the authorities according to the national plan and the capacity of the "D" Refinery Complex. Yearly planning is to decompose the annual amount of crude oil to be processed into 12 months by considering the market demands, inventory, maintenance and other factors of 12 months comprehensively. A multi-period site model is developed. Note that another subscript, /, should be added for each term in (obj) and Eqs (1) to (6) in the multi-period site model. Figure 2 shows the monthly loads of CDU and ARGG from the yearly planning results of the site model. The annual amount of crude oil is 3430 KTON. Significant profit potential can be realized from the optimal yearly plan of the site model compared with the manual plan in "D" refinery complex.

Yearly Planning Results (Base Case) - CDU I + CDU 1 -ARGG

1

3

5

6 7 Month

10

11

12

Figure 2 loads of CDUs and ARGG from yearly planning "D" refinery complex is built in northern China where temperature variation is huge in a year. The demands for low pour point diesel oils (0# and 5#) are high in winter while low in summer. In winter, the refinery complex has to process more crude oil to meet the demands for low pour point diesels. Furthermore, at the beginning of the year, no initial inventory of low pour point diesel is assumed. To sell more profitable low pour point diesel, the site model allocates higher CDU loads in the first three months of the year (Figure 2). From October to December, the demands for low pour point diesel oils

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can be satisfied from inventory which is produced from April to October when demands ft)r low pour point diesels are low. The CDU load from October to December is thus lowered. 2.2. Monthly planning using the site model Monthly planning is to decide the refinery operation condition for one month by considering factors such as the allocated amount of crude oil from yearly planning, current market condition (crude oil available from suppliers, product demands from customers) and the plant condition (inventory, utilities, maintenance), etc. Figure 3 shows the monthly planning results of 12 months from the site model. From Figure 3, it can be seen that the profit increases as the average ambient temperature increases. The reason is that, with other conditions being the same (the crude processing amount, market demand and other conditions of each month are assumed to be the same as those of July), the increase in the ambient temperature will decrease the consumption of 1.0 Mpa steam by each unit. The consumption of fiiel gas and friel oil in the utility plant decreases subsequently. Thus, the reduced energy cost transformed to profitable products and higher profit in "D" refinery complex.

Total profit vs. month

10

11

12

Figure 3. Total profit (million Yuan) vs. month for monthly planning 2.3. Investment decision and debottlenecking from the monthly planning Site model can not only be used as an efficient planning tool for a refinery complex, but also help in investment decision and retrofitting. From the results of the monthly planning, we observe that the utility plant only produced minimal amount of 3.5 Mpa and 1.0 Mpa steam to meet the requirements of other plants. For instance, in monthly planning of February (case 1, the first row of Tables 1 and 2), the boilers in the utility plant produce 47.8 ton/hr of 3.5 Mpa steam. Two turbines (Tl#, T2#) are shut down because there is not enough 3.5 Mpa steam for them. Only 9% of the boilers total capacity (525 ton/hr) is utilized. The reason is that it is more profitable for fiiel gas and fiiel oil to be processed in the refinery plant and sell as final products than to be burned to produce steam and generate electricity. Table 1. 3.5 Mpa steam produced (ton/hr, monthly planning for February)

steam produced

^^^^ ^ ^^86 2

Bl#

B2#

B3#

B4#

B5#

B6#

ARGG boiler

Total

^ 44.8

^^-^ 0

^^-^ 0.0

0

0

0

35

82.8

103.7

130

130

35

443.5

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Table 2. 3.5 Mpa steam consumed (ton/hr, monthly planning for February)

easel steam consumed case 2

Tl#

T2#

DV1#,2#,3#

HT1#,2# ALF-DV LOSS

0

0

0

68.5

9.5

120

120

0.7

68.5

9.5

T5#

Total

4.8

N/A

82.8

4.8

120.0 443.5

We propose to retrofit several boilers so that they can process a cheaper raw material: coal. The investment needed to retrofit one boiler is 50 million Yuan. The life time of the retrofitting is 8 years. The results after the retrofitting are shown in the second row of Tables 1 and 2 (case 2) after taking the retrofitting depreciation cost into account. The site model suggests that we should retrofit three boilers (B4#, B5# and B6#) so that they can process coal and produce more steam. With enough steam available, a backpressure turbine (T5#) and a condensate turbine (T4#) are installed (whose investment cost is 40 million Yuan each). It can be seen that the utility plant produces more 3.5 Mpa steam by burning coal. The "D" Refinery Complex need not import electricity fi-om the market as more electricity is generated by turbines. The total profit increases from 187.43 to 198.61 million Yuan (6%) as a result. After analyzing case 2, we find that the bottleneck exists in the capacity of the condensate turbines (T3#, T4#). There are not enough consumers for LP. If we further increase the capacity of two condensate turbines, then the bottleneck will move to the capacity of the retrofitted boilers. 2.4. Marginal Value Analysis in site model Marginal Value Analysis'^ is used to get information for debottlenecking, retrofitting, pricing and investment evaluation by detailed analyzing of marginal values of all streams. Before the retrofitting, the MCp of electricity, which represents the production cost of electricity, is 0.68 Yuan/KW.hr (obtained by setting an appropriate upper bound to the amount of imported electricity). Since the production cost is higher than the market price of electricity (0.56 Yuan/KW.hr), the refinery complex prefers importing electricity from the market to generating it by its own turbines. After the retrofitting, the MCp of electricity drops to 0.28 Yuan/ton. Thus the production cost of electricity is lower than the market price and the utility plant prefers to generate electricity by itself The "D" refinery complex can even sell electricity to the market provided that the market price of electricity is higher than the production cost: 0.28 Yuan/ton after the retrofitting. 2.5. The site model when electricity price fluctuates The electricity price at peak hours may be several times higher than at valley hours as is the case in Japan and other countries. This fluctuating electricity pricing policy has caused further complications for large electricity users. It can be expected that this policy will be applied in P. R. China in the near future. Considering the optimal planning strategy for a big electricity user like "D" Refinery Complex in the fluctuating electricity price environment is still a new subject in China. Adjusting the usage of electricity to price change may bring huge profit potential to a company by shifting part of its electricity consumption away from peak hours. In this paper, a site model for fluctuating electricity price environment is developed. Besides t, another subscript, s, should be added for each term in (obj) and Eqs (1) to (6) to account for the shift and period variability of EL, steam and products. There are three electricity prices in three shifts: 0.56 Yuan/KW.hr for D (daytime) shift (7am to 3pm); 0.72 Yuan/KW.hr for P (peak hour) shift (3pm to 11pm) and 0.39 Yuan/KW.hr for M (midnight) shift (11pm to 7am). Note that the average price of the three shifts is still

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0.56 Yuan/KW.hr which is used in sections 2.1 and 2.2. The results of the model show that, by efficient adjustment of unit operation conditions and electricity usage, the total profit under fluctuating electricity price (1844.09 million Yuan) is higher than that when the electricity price is fixed (1842.7 million Yuan). To reduce the importation cost or increase the exportation revenue, the site model generates maximal amount of electricity when the electricity price is the highest. Some interesting results are obtained from this case. We find that the loads of CDUs are higher at shift "P". The results seem unreasonable because the CDUs consume more electricity when the electricity price is the highest. In fact, the results arise from appropriate tradeoff between the electricity consumption and the process bottleneck at different shifts. Firstly, the amount of fiiel gas is not enough and its amount becomes the bottleneck. Secondly, to reduce the consumption of the total electricity at shift "P", the load of lubricant oil plant, whose unit electricity consumption (amount of electricity consumed to process one ton of feed) is much higher than that of CDU, is decreased significantly. As a result, the loads of CDUs are increased by using the excess fiiel gas saved from lubricant oil plant to produce more profitable products. 2.6. Investment decision from multi-period site model The monthly planning site model (section 2.2) considers the conditions of one month and making investment decisions for that month. However, an optimal retrofitting for one month may not be optimal for other months. For example, in section 2.3, the monthly planning of February (inputting the market demands, ambient temperature, etc. of February into the site model) suggests that three boilers should process coal. However, after performing the monthly planning of July, the site model suggests 2 boilers to process coal. In fact, the monthly planning suggests a 3-boiler retrofitting for winter season while a 2-boiler retrofitting for summer season. In yearly planning, we can obtain the optimal investment decision by considering the conditions of 12 months as well as the depreciation costs comprehensively. Different from monthly planning, yearly planning suggests a 2-boiler retrofitting. The total profit after the retrofitting increases by 5%. In the fiuctuating electricity price environment (section 2.5), the retrofitting suggestions are similar to the suggestions from the yearly planning when electricity price is fixed. The total profit is 1945.26 million Yuan, which is higher than the profit when the electricity price is fixed (1942.1 million Yuan). Acknowledgments The authors would like to acknowledge financial support from the Research Grant Council of Hong Kong (Grant No. 614005) and the Major State Basic Research Development Program (G2000026308) of P.R.China. References 1. Hui, C.W. and Y. Natori, An Industrial Application using Mixed-Integer Programming Technique: a Multi-period Utility System Model, Comput & Chem. Eng., 1996, 20, SI577-81582 2. Zhang, J., Zhu, X.X. & Towler, G.P. A Simultaneous Optimization Strategy for Overall Integration in Refinery Planning, Ind Eng. Chem. Res. 2001, 40(12), 2640-2653 3. Li WK, Hui CW, Hua B, et al. Material and energy integration in a petroleum refinery complex, 8th Intemational Symposium on Process Systems Engineering, PSE2003, PTS A AND B : 934-939, Kunming, P.RChina, Jan 5-10, 2004 4. Hui, C.W., Determining marginal values of intermediate materials and utilities using a site model, Comput. & Chem. Eng, 2000, 24(2-7), 1023-102

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Refinery Planning under Correlated and Truncated Price and Demand Uncertainties Wenkai Li, I.A. Karimi, * R. Srinivasan Department of chemical and Biomolecular Engineering, National University of Singapore, 4 Engineering Drive 4, Singapore 117576 Abstract Because of the difficulty in computing the bivariate integral originated from the correlated demand and price, most research work on uncertainty assumes that the demand and price are independent. This can cause significant discrepancies in revenue calculation and hence yield sub-optimal planning strategies. This paper presents a novel approach to handle correlated and truncated demand and price uncertainties. A bivariate normal distribution is used to describe demand and price. The double integral for revenue calculation is reduced to several single integrals after detailed derivation. The unintegrable standard normal cumulative distribution fimction in the single integrals is approximated by polynomial ftinctions. Case studies show that, assuming independent price and demand may underestimate the revenue by up to 20%. Since the real world demands or prices vary in limited ranges, integrating over the whole range of a normal distribution, which some research has done, may give incorrect results. This paper uses a bivariate double-truncated normal distribution to describe demand and price. The influence of different degrees of truncation on plant revenue is studied. Keywords: Refinery, Planning, Uncertainty, Correlation, Truncation 1. Revenue calculation in the literature Since Dantzig's seminal work on uncertainty appeared^ research on uncertainty has been attracting the attention of numerous researchers. Most research work^'^ on uncertainty assumes that the demand and price are independent because of the difficulty in computing the bivariate integral originated from the correlated demand and price. However, by regressing real world demand and price data from EIA"^, the correlation coefficient between gasoline (New York Harbor Gasoline Regular) price and its demand is 0.44 for the year 2003 to 2004. For world crude oil in 2003 and 2004, the correlation coefficient is 0.30. These data show that the demand and price are far from independent. The computation of the expectation of plant revenue generates the main difficulty for a planning problem under uncertainty^. As pointed out by Petkov et aP and Li et aP, if the market demand is less than the product amount, only part of the product can be sold; otherwise if the market demand is higher than the product amount, then only part of the demand can be satisfied. The revenue should then be calculated by: Revenue = ^ [ X Z ^ * m i n ( P , x ) ] (1) cX Where c is the product price, P is the production rate of the product and x is the random demand. Different formulae have been developed in the literature^'^ from (1) to compute plant revenue by assuming that the demand and price are independent. ' Author to whom correspondence should be addressed. E-mail: [email protected]

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2. Revenue calculation for correlated and non-truncated price and demand A two-dimensional normal distribution is used to describe the relationship between correlated demand and price. Its pdf (probability density function), (p{c,x) , is represented by: ^1 (p(c,x) =

.

e ^

Ac-c)2 _2p(c-c)(x-e)

^ ^

(x-0)2.

c

^

(2)

27rac(Txyll-p^ Where, x represents random demand and c represents random price, c and GC are the mean and standard deviation of price, respectively; 0 and c^ are the mean and standard deviation of demand, respectively, p is the correlation coefficient. Combining eqs (1) and (2), the revenue is -j-oo

x*c*(p(c,x)dxdc

if x < P and

TOO

P * c * (p(c, x)dxdc if X > P. Then, eq (1) becomes C=-oo x=P

Revenue =

J

j

xc(p{c,x)dxdc-\-

C=-oo X = - o o

\

\

C=-oo

X=P

Pc(p{c,x)dxdc

After detailed derivations, the two double integrals in the above equation are reduced to several single integrals as following: Revenue = Al + A2 -{- A3 -\- A4 -\- A5 -\- B - CI - C2 (3) Where, the expressions of the single integrals in (3) are listed in Table 1. The term in Table 1, 0(.), represents the standard normal cumulative fianction. 3. Revenue calculation for correlated and truncated price and demand Underlying the formulae for revenue calculation in the literature^'^, the range of demand is assumed to be (-«>, +oo), which is not the case in the real world. Based on the data obtained from EIA"^, the world total oil demand and the crude oil price (Venezuelan Tia Juana Light) in 1970 to 1999 locate in the ranges (|i-2.2*a, |i+2.0*a) and (|i-1.3*a, 11+2.1 *a), respectively (|x and o respectively represent corresponding mean and standard deviation). A planning strategy based on a non-truncated distribution can be far fi-om optimal. This issue is addressed in this paper. Suppose that the range of demand is [XL, XU], where -oo < XL < Xy < +oo and the range of price is [CL, CU], where -00 < CL < Cu < +00. Then the pdf of bivariate bi-truncated normal distribution is: \g>(c,x) x^<x<x^ and c^
\ \(p{c^x)dxdc

Refinery Planning Under Correlated and Truncated Price and Demand Uncertainties Table 1. The expressions of single integrals in (3) (B) = Pc

-yjl-p^cT^a,

Al

^ e ^ dm

2K

A2 =

-V-

p^

P (^X^

e

2n

^e

^ dm

f\ e ^ Q>(U)clm ^{U)dn

C1 = -T=^

+«>

PCTr

C2--

'27r n

A4 =

GcO + pca^

-

°°

A5 = - ^ 427t

m=

[ me ^ ^{U)dn

e~^0(U)dn

^ (U)dm

c-c

p-e

m

{ J

I

w"

me

- pm

U-

^Table 2. The expressions of single integrals in (5) XL-0

27tFjrr

J

me ^ [e ^ -e

^ ]dm

L =-

- pm

V ^ Xjj

-

2nFrjr

Aj^3=

\ e '[e

'

-e

^]dm

~^

- pm

Ujj=-

^

^^'

f m^e ^ [^(U) -

7c

0(L)]difm

Cj7 =

Pc 27UFr,

ffj2

<7c

f„2

yJ27rF,

AT

J —

J

7 =

e~~[^{U)-^{L)\dm

42^F,,

Combing eqs (1) and (4), the plant revenue is then %

P

cu % JBBTN v^>

C=Ci X=Xi

C=Ci X=P

xjaxac

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W.Lietal

After detailed derivations, the revenue for truncated and correlated case is: Revenue = A^l + AT2 + A^S + Aj'4 + A^S + Q 7 + Q 2

(5)

Where, the expressions of the single integrals in (5) are listed in Table 2. 4. Approximation of the standard normal cumulative function In the single integrals listed in Tables 1 and 2, there is an unintegrable term, 0(.). There exist some accurate approximations to (.) in the literature^. However, those approximations are still too complicated to integrate the single integrals. In this paper, a polynomial fimction is used to approximate 0(.): 0 ( x ) = a + Z?x + cx^+Jx^+6x^ (6) When X is in the range of [-3, 3], the coefficients in eq (6) are: a, 0.5; b, 0.3942473009; c, -0.058125270; d, 0.0056884266 and e, -0.000228133. Replacing 0(.). with eq (6) in the single integrals, the revenue can be calculated. 5. Derivation of truncated loss function Loss fimction, defined as LF(P) = J (x - P)p(x)dx (for non-truncated and continuous p

distribution), represents the amount of unmet demand (the backorder level) of a plant facing uncertain demand^. It is essential to compute value of loss fimction to apply the Type II service level (fill rate) in a model. For non-truncated and normally distributed demand, loss fimction has been effectively applied and approximated to compute the actual fill rate in the literature^. In this paper, we extend Li et al.'s^ work to suit for truncated normally distributed demand. Assume that the range of demand is [XL, XU], the bi-truncated loss fimction, the bi-truncated density fimction of demand x is:


0(fz^)

PBTN W —

o-j«i>(^^^^)-
Where <j^^ =cr^[0(-^^^

—G

)-^{—

X —0

^^^

)] and (|)(x) is the standard normal density

fimction. Then, the bi-truncated loss fimction, LF^j^^ (P) , is derived: ^BTN

Where,Z;^ =—^

^BTN

.^;tp=

•

For non-truncated normal distribution, that is, X^ -^ +oo and X^ ^ -©o , eq (7) is reduced to the formulae used in the literature^'^: LF{P) = <7M2xp)-2xp[l-^i2^)\} (8)

Refinery Planning Under Correlated and Truncated Price and Demand Uncertainties 2127 6. Case studies Case studies are used to illustrate the influences of correlation and truncation of price and demand on plant revenue. The configuration of the case studies is taken from the case 1 of Li et al.^ Cases differ from each other on the standard deviations of 90# and 93# gasoline demand (Table 3). In Table 3, CV (Coefficient of Variation) is defined as CV = — . The means of 90# and 93# gasoline prices are 3215 and 3387, respectively. /^ The standard deviations of 90# and 93# gasoline prices are 600 and 620, respectively. Table 3. Means and standard deviations of product demands Products

Mean

Standard Deviation CV=0.2

CV=0.3

CV-0.5

90# gasoline

50

10

15

25

93# gasoline

70

10

20

35

6.1. Effect of correlation The revenues at different correlation coefficients (p) and CVs are listed in Table 4. It can be seen that, when CV is 0.5, the revenue at p of 0.4 (near the real world data) is 21.1% higher than the revenue when demand and price is independent (p= 0.0). That means, for a large enough CV, assuming independent price and demand may underestimate the revenue by up to 20%. It is found that the revenue difference between the independent and correlated cases increases as the increase of CV. In the above case study, when CV is 0.2 and p is 0.4, the revenue difference is 1.8%). This difference is 4.4%) for CV of 0.3 and p of 0.4. Table 4. Revenues at different correlation coefficients and CVs CV==0.2

CV==0.3

p 0

Revenue 38685.9

Gap*, %

0.1

38851.2

0.4

0.2

39021.0

0.3 0.4

Gap*, %

Revenue 10097.5

Gap*, %

28266.1

1.0

10589.8

4.9

0.9

28562.1

2.1

11096.9

9.9

39202.3

1.3

28878.3

3.2

11638.7

15.3

39398.5

1.8

29220.8

4.4

12225.6

21.1

Revenue at /? >0 - Revenue at

PI—*100%.

* Gan=

Revenue

CV==0.5

27978.6

Revenue at /7=0

6.2. Effect of truncation Integrating over the whole range of a normal distribution may give incorrect results to revenue calculation. The formulae derived for bivariate double-truncated normal distribution are applied in this paper. Some results are shown in Tables 5 and 6 (the product production rates of truncated cases are fixed to those of the non-truncated case for comparison). In Table 5 (CV=0.2), it can be seen that if the degree of truncation is |i±2o, integrating over the whole range will underestimate the revenue by 2 to 3%. If the degree of truncation is |a±a, integrating over the whole range will underestimate the revenue by about 12%). The revenue difference between truncated and non-truncated becomes much more significant for large CV. In Table 6 (CV=0.5), it can be seen that

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the revenue increases by more than 40% for degree of truncation |i±2a. The revenue even increases by more than 100% for degree of truncation a±|X. It is clear that the revenue difference between truncated and non-truncated cases can be very huge in some cases. In Tables 5 and 6, the difference is defined as: Revenue of truncated case - Revenue of non-truncated case Difference^ 100%. Revenue of non-truncated case Table 5. Revenues of truncated and non-truncated cases (CV=0.2) Non-truncated

Truncated, |i±2a

Truncated, a±|i

(-oOj +oo)

Revenue

Difference, %

Revenue

Difference, %

0

38685.9

39847.6

3.0

43513.6

12.5

0.1

38851.2

39948.7

2.8

43772.7

12.7

0.2

39021.0

40107.0

2.8

43958.2

12.7

0.3

39202.3

40418.5

3.1

44166.1

12.7

0.4

39398.5

40792.3

3.5

44369.3

12.6

p

Table 6. Revenues of truncated and non-truncated cases (CV=0.5) Non-truncated

Truncated, \\±1(5

Truncated, a±|>i

(-00, +00)

Revenue

Difference, %

Revenue

Difference, %

0

10097.5

14986.6

48.4

23935.1

137.0

0.1

10589.8

15234.4

43.9

24545.5

131.8

0.2

11096.9

15556.8

40.2

25030.0

125.6

0.3

11638.7

16067.2

38.0

25457.7

118.7

0.4

12225.6

16693.6

36.5

25907.6

111.9

p

7. Conclusion In this paper, the influences of correlation and truncation of product price and demand on plant revenue are studied. Detailed theoretical derivations are performed. Case studies are developed to show the significant revenue difference between the traditional method and the new formulae developed in this paper.

References 1. Dantzig, G. B. Linear programming under uncertainty. Management Science, 1955, 1:197206 2. Petkov, S. B.; Maranas, C D . Multiperiod planning and scheduling of multiproduct batch plants under demand uncertainty, Ind. Eng. Chem. Res., 1997, 36 4864 3. Li Wenkai, Chi-Wai Hui, Pu Li, An-Xue Li, Refinery planning under uncertainty, Ind. Eng. Chem. Res. 43, 6742-6755, 2004 4. The U.S. Energy Information Administration, http://www.eia.doe.gov 5. Kim, K. J.; Diwekar, U. M. Efficient Combinatorial Optimization under Uncertainty. 1. Algorithm Development, Ind. Eng. Chem. Res., 2002,41 1276 6. Abramowitz & Stegun, "Handbook of Mathematical Functions", Dover Publications, 1965 7. Hopp, W. J.; Spearman, M. L. Factory Physics: foundations of manufacturing management, Irwin/McGraw-Hill, 2000

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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An Integrated Model for the Design and Planning of Supply Chains with product return Maria Isabel Gomes Salema,^ Ana Paula Barbosa-Povoa,^ Augusto Q. Novais'' ""CMA, FCT-UNL, Monte da Caparica, 2829-516 Caparica, Portugal ^CEG-IST, Av. RoviscoPais, 1049-00ILisboa, Portugal 'DMS-INETI, Az. dos Lameiros, 1649-038 Lisboa, Portugal

Abstract The growth of waste generation has become an environmental concern. In order to prevent it, some companies are now rethinking their supply chains, where product return flows are being incorporated. In this work, we propose a model for the design and planning of supply chains with product return flows that integrate strategic and tactical decisions, within a single formulation. The model applicability is corroborated by an example, based on a published case-study. 1. Introduction Modem society generates waste in all its activities and all consumed materials will eventually become waste. In 2000, the Europe of the Fifteen (EU-15) generated about 3.8 tonnes of waste per capita, [1]. The EU-15 has established a set of targets in order to minimize environmental impacts, which include a combination of waste prevention, material recycling, energy recovery and disposal options. Within this fi-amework, companies have to redesign some of their processes in order to allow European countries to meet the said targets. In particular, companies have to take a close look into their established supply chain where it is urgent to introduce the reverse flow. This leads to several open questions such as: where to place collection centres? How to plan the collection? How to plan production when used materials exist along with new ones? In what ways may the new flow affect the prevailing forward flow?

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In the past decade, the research community has been proposing different models to help answering those questions. However, most published work considers independently the design and planning or is too case dependent. Fleischmann et al., 2001 [3] integrate the forward and reverse flows of a given product within the context of the supply chain design; Jayaraman et al., 2003 [4] discuss several issues related with reverse distribution and propose a formulation for a version of this problem; Fandel and Stammen, 2004 [2] propose a general model for extended strategic supply chain management, based on a two-step time structure. In this work, we go a step further and propose a model that integrates the design and planning aspects within a single formulation. This is achieved through a multi-period formulation, where a two-scale time formulation is used. The resulting MILP is an extension of the work presented by Salema et al., 2005a [5] where three different levels of decisions are considered: facilities sites location (time horizon), satisfied demand (macro time scale) and, finally production, distribution and storage planning (micro time scale). Additionally, in the present work the main types of products along the closed supply chain (forward, returned and sub-assemblies) are accounted for, together with the modelling of their bill-of-materials. However, when forward fiows are considered, no differentiation is considered between new and used components. The paper is structured as follows. The problem is first described and the model is briefly presented. Then an example, based on a previously published case-study, is applied. Finally, some concluding remarks are drawn. 2. Problem description Supply chain can be represented as a network where facilities act as nodes and links are related with direct flows between facilities. The network may have several levels (factories, warehouses...). Each facility in each level can be connected to another facility in a different (not necessarily consecutive) level. In this work, an extended supply chain is considered, with the introduction of reverse flows. Both the forward and the reverse chains have a two-echelon structure. It is assumed that some customers may not be supplied, meaning that they may not be selected to integrate the supply chain. The transformation of raw materials into final products with a view to meet customers' demand has, traditionally, been the main target of production planning models. However, in this paper context, products once used are collected, dismantled in disassembly centres and the assessed sub-assemblies sent to factories and/or disposal. Three different groups of products actually flow in the supply chain: forward productsfi*omfactories to customers through warehouses, reverse products from customers to disassembly centres and sub-assemblies fi-om disassembly centres to factories. Inventories are allowed in all facilities and are limited to a maximum level. Maximum and minimum limits are also imposed on production levels and distribution flows.

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2131

In this framework, two interconnected time scales allow for three levels of decisions: the time horizon, where the sites to locate facilities are chosen, a "macro" time where customers' demands and returns are satisfied and a "micro" time where production, storage and distribution are planned. These time scales can be years/months, years/trimester, month/days or whichever combination suits the problem. Two other features are considered in this work: the travel time which is modelled between network levels, and the product usage time. The former is defined as the number of "micro" time units needed for a product to flow from its origin to its destination. The latter is the minimum number of micro time units that a product remains in the customer before entering the chain as a return product. In short, the proposed model can be stated as: Given the investment costs, the amount of returned product that will be added to the new products, the relation between forward and reverse products, the travel time between each pair of network agents, the minimum disposal fraction and the minimum usage time for each return product, and for each macro period and product: customers' demand and unit penalty costs for non-satisfied demand /return, and for each micro period: the unit transportation cost between each pair of network facilities, the maximum and minimum flow capacities, the factory production unit costs, the facilities unit storage costs, the maximum and minimum production capacities, the maximum storage capacities, the initial stock levels, the transfer prices between facilities, customers' purchase and selling prices. Determine, the network structure, the production and storage levels, the flow amounts and the non-satisfied demand and return volumes. So as to maximize the global supply chain profit. The resulting model is a Mixed Integer Linear Program (MILP) which involves 15 types of variables (one production variable, four flow variables, four stock variables and two variables for the non-satisfied demand and return, all with a continuous domain, and four binary variables related with the location of the four different kinds of entities - customers included) and 24 types of constraints (four material balance equations, a demand and a return satisfaction constraint, one for the disposal option, three for the maximum storage, one for the minimum and another for the maximum production limits and, finally, a group of twelve constraints to assure minimum and maximum flow capacities). 3. Example This example (adapted form a previous published case - Salema et al., 2005b) was created based on a European company that plans the network design for a

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M.L G. Salema et al

supply chain. Several products are involved: three forward products (Fi, F2 and F3), two return products (Ri and R2) and four sub-assembly components (Ci, C2, C3 and C4). Five European cities are considered as possible sites to locate all the facilities: Amsterdam, Birmingham, Brussels, Frankfurt and Lyon. Clusters of customers (from now on designated simply as customers) are located in these 5 and other 21 cities: Berlin, Cologne, Copenhagen, Diisseldorf, Essen, Hamburg, Leeds, Lille, Liverpool, London, Manchester, Marseille, Milan, Munich, Newcastle, Nuremberg, Paris, Rome, Rotterdam, Turin and Vienna. Concerning time, a macro period is defined over a five years horizon and a micro period over four months: macro period = "year" and micro period = "four months". Over the horizon of five years, some data has to be estimated: demand volumes as well as variations in costs over the years. Some assumptions were made for these estimations: 1) transportations costs are proportional to the distance between each city; 2) after the first year an actualization rate of 3% (or some other convenient value) is applied to all costs; 3) in the first year, customers' demand is made equal to a fraction of the city inhabitants (a value between 0.08 and 0.105) while for the remaining years, this value is modified by a variation factor (ranging from 0.58 to 0.55), allowing for an increase or decrease in the demand volumes. After use, products Fi and F2 are returned as Ri and product F3 as R2. In terms of return fractions, all forward products have a return fraction of 80%; zero initial stock levels are assumed and the disposal fraction is set to 0.3, which means that at least 30% of returns are assumed not proper for remanufacture and have to undergo recycling or disposal. Minimum and maximum capacities are defined for production (10^ and 3*10^, respectively); maximum, but no minimum, limits are imposed on factories inbound and outbound flows; all flows from and to customers have maximum and minimum limits; travel time is set to nil, which seems a reasonable assumption given the chosen time scale and the particular geographical area under study. The usage time was set equal to all products and it is of four months (one micro time unit). Results The resulting MILP model was solved by GAMS/CPLEX (built 21.4), in a Pentium 4, 3.40 GHz. The model is characterized by 16 107 variables (866 binary) and 10 991 constraints, and took about 2700 CPU seconds to solve (0.1% optimality gap). The optimal value found is 23*10^ currency units. The optimal network has three factories located in Birmingham, Frankfiirt and Lyon. These three sites are chosen to locate warehouses and disassembly centres. Brussels is chosen as a fourth location to open a warehouse and a disassembly centre; both facilities are connected to the factory located in Frankfurt. In Figure la and lb, respectively, are represented the optimal forward and reverse networks. Apart from a few "special" customers, the remaining is served by the warehouse and disassembly centre with the same location. For the

An Integrated Model for the Design and Planning of Supply Chains

2133

"special" customers a more detailed analysis follows. All 26 customers have their demand and return fully satisfied.

Figure 1 a) Forward Network

b) Reverse Network

Three different analyses are carried out for the tactical decisions: production, storage and distribution. As the model produces a large volume of information, only some data will be presented. All four sub-assemblies are produced in all three factories. In Figure 2, the Frankfurt production plan is depicted over the time horizon. Note that after the third year, the production is reduced to its 4000000 3500000

DC4 nc3

• C2 @G1

1000000 500000 -

1

1 1 ' ' ' .! 1 1 2

3

2

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minimum level. This is caused by the decrease in customers demand. A zero stock policy is foreseen as far as inventories are concerned. Lastly, some examples will be given in order to illustrate the distribution results. Some uncommon customers are shown as they provide interesting insight into the analysis. Essen customer (Figure 3) is the only one served by three warehouses. Note that product F3 is firstly supplied by Frankfurt warehouse during the first three years and the last two years the supply is assured by Brussels. Product Fi is almost entirely suppHed by Birmingham. The main reason behind these is the fact that transportation costs are distance based.

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The example of the collecting plan for the Amsterdam customer is presented as an example of the flow between customers and disassembly centres. One interesting aspect of this collecting plan is that both Inn 1 Illlll llll 1 In products are handled by different .I3 . M = 1 . 1 = centres: product Ri is entirely collected by Brussels centre, while product R2 is collected either by Birmingham centre or by Lyon. The remained customers follow a Figure 4: Amsterdam collecting plan. more conventional plan where all three forward products are supplied by a single warehouse and the two reverse products are collected by one centre, usually located in the same city as the warehouse. n n n n

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4. Final Remarks and Future work In this work, a model for the design and planning of green supply chains is proposed. By incorporating a strategic and a tactical decision levels, we are not only able to find the best locations to install a set of facilities, but also to estimate the associated optimal production, storage and distribution plans. A European example, previously published, is modified in order to test the model applicability and adequacy. The results obtained show that the proposed model deals satisfactorily with problems with a considerable degree of detail and complexity. Thus, the proposed model appears as a promising decision support tool to help the decision-making process in the strategic and tactical planning of multi-product capacitated supply chains. As future work, we intend to investigate the application of decomposition methods, either generic or specially developed for this model, in order to speed up the solving process. Acknowledgment The authors gratefully acknowledge the support of the Portuguese National Science Foundation through the project POCTI/AMB/57566/2004. References 1. 2. 3. 4. 5. 6.

EEA (2005), EEA Report no 4. http://reports.eea.eu.int/eea report 2005 4 Fandel and Stammen (2004) IJPE, 89(3), 293-308. Fleischmann et al. (2001), POM, 10(2), 156-173. Jayaraman et al. (2003) EJOR, 150(1), 128-149. Salema et al. (2005a) CAGE, 19, 1075-1080. Salema et al. (2005b) EJOR. (submitted)

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Pipeline Scheduling and Distribution Centre Management - A Real-world Scenario at CLC Susana Relvas^'', Ana Paula F.D. Barbosa-Povoa^*, Henrique A. Matos^, Joao Fialho^, Antonio S. Pinheiro^ ""IST DEQ, Av. Rovisco Pais 1049 001 Lisboa, Portugal ^ CEG IS T, Av. Rovisco Pais 1049 001 Lisboa, Portugal ' CLQ EM 366, km 18, 2050-163 Aveiras de Cima, Portugal Abstract Competion is growing every day around petroleum industry, and more specifically in the management of the respective supply chains. Technology has been one of the main tools where development and benefits have been obtained. This work develops the basis of a decision support tool that relies on the development of a MIL? model to answer the real world scenario of CLC - Companhia Logistica de Combustiveis, a company that receives petroleum derivatives from a refinery through a pipeline and distributes them in the central area of Portugal. The model combines the pipeline operation with the end-ofpipe distribution centre management. The formulation relies on continuous time and pipeline volume representations. Solutions are obtained recurring to standard B&B techniques. Keywords: Multiproduct pipeline, scheduling, MILP, continuous-time. 1. Introduction The scheduling procedure of batches in pipelines through MILP techniques is a very complex task and involves several modeling issues, such has the choice of time and volume scale type, and has to manage a large set of restrictions that define the problem. The mathematical modelling of this operation has been studied increasingly over the last years where different problems within the supply chain of the petroleum industry have been considered. Neiro and Pinto (2004) developed a framework to describe the petroleum supply chain as a whole. A large scale MESfLP was obtained that represents a complex topology, using general representations of processes. A decomposing technique that reduces the computational effort was proposed. Additionally, pipeline usage has been studied recently by some authors where planning and scheduling are addressed. Pipelines are described as one of the most reliable and cost-effective entities in the petroleum supply chain. Magatao, Arruda and Neves (2004) and Rejowski and Pinto (2004) employed discrete representation approaches to model the pipelines. The former authors built an optimisation structure, which makes use of decomposition techniques and origins four different components. The proposed framework predicts the pipeline operation, minimising operation costs. The same scenario was solved for rough and detailed discretisation approaches and the computational effort was significantly higher for the last case. The latter authors studied different scenarios for product demand on a real world problem. However, the computational effort observed was high.

* Corresponding author: [email protected]

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Magalhaes (2004) proposed a continuous time formulation to address the same problem. The continuous time scale kept track of four different events, making the framework rather complex and with no feasible solution in a reasonable amount of time. Finally, Cafaro and Cerda (2004) proposed a novel continuous MILP formulation that explores a sequential spatial transportation along the pipeline. This framework proved to solve the real world scenario of the work of Rejowski and Pinto (2004) in much less computational time. The main characteristic is that this model reduces the amount of variables (continuous and binary) with no lost of information. The same authors Cafaro and Cerda (2005) presented a new version of their work where the time horizon was extended by considering a multiperiod approach and making use of rolling horizons. These works enforced the pros and cons of the different possible approaches. However, one of the issues not addressed in the previous works was that in a real world distribution centre there are usually daily client's demands. Moreover, most of the works focuses on the pipeline schedule itself or in the refinery operation as a whole, but no emphasis is found on the other end of the pipeline - the distribution centres. These have a complex type of operation and require detailed models to preview all the operations involved, such has settling periods and daily inventory management. In this paper we address some of these points. The proposed model is based on the work of Cafaro and Cerda (2004) which was extended to account for storage all along the time horizon, quality approving tasks and satisfaction of the daily's clients' demands. The current model is applied to a real case study that describes the planning and scheduling of the pipehne connecting the Refinery at Sines with the Portuguese distribution centre of refinery products (CLC - Companhia Logistica de Combustiveis, S.A.). Once the batches reach the end of the pipeline, they are allocated to storage tanks and after the required approving tasks they are distributed to CLC's clients. 2. Problem Description The system comprises a pipeline that pumps petroleum derivatives from a refinery to a single distribution centre. In this centre there are tanks, whose product service is fixed. Therefore, the arrived product is conducted to one (or more) of its pre-allocated tanks. Each new lot arriving to the distribution centre must settle at least for a certain period, due to quality control tasks. The clients provide their demands in a daily basis, which is needed to update the inventory level during the time horizon. Given: 1. the multiproduct pipeline data; 2. the available storage capacity for each product; 3. the matrix of possible transportation sequences; 4. pumping rate; 5. the initial inventory levels; 6. the initial lots inside the pipeline; 7. the daily products' demand; 8. the minimum settling period; 9. the time horizon extent; It is necessary to determine the pipeline schedule, including products' sequence, lot volume and continuous time scale, and the storage levels for each product over the given time horizon. In order to guarantee that the products' demands are satisfied over the time horizon and the quality control tasks are performed for each new lot, the objective is set to maximise the inventory level at the end of the time horizon, the amount of products pumped and total the pumping extent, all of them in normalised terms (using the respective upper

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bound). This objective represents one of the possible operational objectives maintained by CLC's schedulers. Other functions can be used such as costs' minimisation or profits maximisation. 3. Mathematical Formulation 3.1. Time and Volume Scales The main basis of the model is the continuous representation of both time and pipeline volume. The continuous time scale is controlled by the time when each lot / finishes to be pumped to the pipeline (event point driven). At each time interval, there can be two distinct situations: either the pipeline is working over the complete time interval or there is a stopping time allocated at the beginning of the interval (Figure 1). On the other hand, the continuous pipeline volume scale is controlled by the upper volumetric coordinate (related to the refinery as the origin) of each lot / inside the pipeline when a later lot / has finished to be pumped to the pipeline. Chronologically, pumping lot / is situated farther from the pipeline origin than lot /+7 (figure 1). (t = C5+At,J

Endofinterval5(t = C5)

^1

6

••i

iitii

^w^

imm^m

^ t hpipeline beginning

pipeline beginning

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(t = C5+5t)

6

s 1 ^ 1 pipeline beginning

pipeline beginning

6 78

10 11

12

intervals (z) days (k)

Figure 1 - Model representations: continuous time and pipeUne volume and lots sequencing

3.2. Mathematical Model The mathematical model is composed by a number of sets of constraints that control different aspects of the operation. These sets are revised in the following topics. • Pumping lots sequencing - the ideas described for figure 1 are implemented in this set of constraints to guarantee the time scale and that the time horizon is met; • Pumping volumes and product allocation - the pumping volumes of each lot are dependent of the associated product as well as of the volume of the allocated tanks; • Relation between pumping lot volume and extension - once the volume of each lot is set, the time scale is obtained relating pumping extension with the volume of the lot. In this set it also granted that if a certain lot is fictitious than it will be placed at the end of the schedule and it will have no volume; • Forbidden sequences - due to product transportation and certification, certain sequences are forbidden inside the pipeline, which is prevented by the model; • Upper and lower volumetric coordinates of pumping lot i - the lots (or fractions of them) inside the pipeline at each moment are given through their upper volumetric coordinate. The volume of/ still inside the pipeline is used to obtain the lower volumetric coordinate; • Discharge of pumping lot ifrom the pipeline to Distribution Centre (DC) - this set of constraints verifies when each lot can be unloaded to D C s tanks and to determine how much volume of lot i is unloaded at each interval. The discharge is only

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performed when the upper volumetric coordinate of pumping lot / reaches D C s volumetric coordinate; • Global mass balance - maintenance of volume balance between the pipeline ends; • Inventory control - when a lot arrives, it is necessary to identify which product it contains and redirect it to the respective tank. Each tank's inventory is tracked and maintained feasible through all the time horizon; • Clients allocation - the clients' demands are known in a daily basis. Therefore, at the end of each day the client's satisfaction is guaranteed. In a straight forward manner this would imply a detailed treatment of time leading to a cumbersome model. This was, however, overcome by considering clients requirements within a single interval of the continuous time scale built in the model (e.g. clients requirements of day 1 are allocated in interval 3, Figure 1); • Settling period - Each new lot that arrives to the distribution centre must settle a given settling period; • Initial conditions - information on the products inside the pipeline at the beginning. Apart from the model main constraints some auxiliary constraints are also introduced. These are the so-called speed-up constraints. These are essentially sequencing constraints and integer cuts that help to speed up the solver's performance. These constraints translate physical conditions and often reduce the feasible region. 33. Product s Sequence Usually, the pipeline schedule is built based on a pre-fixed sequence of products that results from the repetition of a certain number of times of a given base unit, which is the case of CLC (figure 2a). However, market evolution has led to a new situation where the products' sequence should be more flexible. At CLC the flexibility relies often on the choice of two single products for the same position (figure 2b).

P1

• P 3

Figure 2a - Base unit of products' sequence

m

P6 p% ^M

Figure 2b - Evolution of base unit Furthermore, the products' sequence is the core to obtain a feasible solution. Therefore, the model may be applied to several scenarios: (1) free sequence - where the sequence was left as a model decision; (2) fixed sequence - the sequence is made equal to the one predefined by the schedulers and (3) mixed sequence - where only some decisions are to be taken, namely the choice between the two products that fit in the open position. 4. Results The proposed MILP model will be used to solve a scheduling problem over a time horizon of one month. The initial inventory and clients' demands are set to the same as the ones of CLC (August 2005). The products are named P1-P6. The minimum settling period is 24 h. The pipeline is initially filled with product PI. The flowrate is defined at the average weighted value previewed during that period.

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The model was solved for two situations: "Scenario 1" where the sequence of products is set to the fixed sequence; "Scenario 2" that results from a mixed sequence. The results obtained are compared with the planned data at CLC. Table 1 represents the final inventories verified for each scenario. Both scenarios were solved using GAMS 21.5/CPLEX 9.0 in a Pentium IV 2.8 GHz with 512 MB RAM. The stopping criterion was either a resource time of 7200 CPU seconds (reasonable amount of time considering CLC operation) or a solution within a tolerance of 1%. For the free sequence case, no solution was obtained in the predefined CPU time limit. Table 1. Final inventories verified for each scenario and model performance Inventory (m^) PI P2 P3 P4 P5 P6 Total y© Pipeline Usage Objective Function Final GAP (%)

Planned at CLC

A Scenario 1

A Scenario 2

41554 24765 11927 15244 5965 6054 105509

+4500 0 +8000 +4200 +2588 +3980 +23268

+1400 0 +8000 +4200 +2588 +3980 +20168

91.4 -

97.5 2.52 2.05

96.7 2.51 2.72

For both scenarios, the total number of equations is 43871, involving 24725 continuous variables. In terms of binary variables Scenario 1 has 4672 and Scenario 2 has 4680. The difference on these values relies on the open positions of the product's sequence. The inventory profiles for all the products are shown in Figure 3. Generally, the model fits CLC's operation. The model was also capable of making more use of the pipeline, reaching end-of-period higher inventories, guaranteeing clients' demands. In this way, demand variability can be easily overcome. 5. Conclusions and Further Work In this paper a general model to the schedule of a real world multiproduct pipeline was developed. The main features are the continuous representation of time and pipeline volume as well as the daily representation of clients and distribution centre quality control tasks. Previous works usually account for clients only at the end of the period, which is quite apart from reality in a distribution centre under a dynamic market, as well as neglect the operations inherent to the centre. The model proved to obey to several decision types, however it was noticed that with tighter restrictions, the computational effort rises significantly. There are several possible reasons, such as the difficulty to report in a precise manner a number of different operations in a time scale that reports only one event. The option is to use bigM constraints that deteriorate the performance of the algorithm. Further work is being prepared in order to produce a tighter and more robust model, including flowrate as a variable and tanks' management. Furthermore, a high number of scenarios should be tested, to analyse model sensibility, formulation, solver options, among other aspects.

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Figure 3 - Inventory levels during the time horizon for all products Acknowledgment The authors gratefully acknov^ledge financial support from Companhia Logistica de Combustiveis and Funda9ao de Ciencia e Tecnologia, grant SFRH/BDE/15523/2004.

References D.C. Cafaro, J. Cerda, 2004, Optimal scheduling of multiproduct pipeline systems using a nondiscrete MILP formulation, Comp. & Chem. Eng., 28, Issue 10, 2053-2068 D.C. Cafaro, J. Cerda, 2005, Multiperiod Planning of Multiproduct Pipelines, ESCAPE15, Computer Aided Chemical Engineering, 20B, Puigjaner & Espuna (Editors), 1453-1458 M.V.O. Magalhaes, 2004, Refinery Scheduling, PhD Thesis, Imperial College, University of London L. Magatao, L.V.R. Arruda, F. Neves Jr., 2004, A mixed integer programming approach for scheduling commodities in a pipeline, Comp. & Chem. Eng., 28, Issues 1-2, 171-185 S.M.S. Neiro, J.M. Pinto, 2004, A general modelling framework for the operational planning of petroleum supply chains, Comp. & Chem. Eng., 28, Issues 6-7, 871-896 R. Rejowski, J.M. Pinto, 2004, Efficient MILP formulations and valid cuts for multiproduct pipeline scheduling, Comp. & Chem. Eng., 28, Issue 8, 1511-1528

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Scheduling under demand uncertainty using a new multiparametric programming approach Zhenya Jia, Marianthi G. lerapetritou Rutgers University, 98 Brett Road, Piscataway, NJ 08854, US Abstract In this paper, a novel framework is developed to deal with multiple uncertain parameters on the right-hand-side (RHS) that can vary independently. The issue is also addressed using parametric mixed integer linear programming (pMILP) analysis where uncertain parameters are present on the right hand side (RHS) of the constraints. For the case of multiple uncertain parameters, a new algorithm of multiparametric linear programming (mpLP) is proposed that does not require the construction of the LP tableaus but relies on the comparison between solutions at leaf nodes. Given the range of uncertain parameters, the output of this proposed framework is a set of optimal integer solutions and their corresponding critical regions and optimal functions Keywords: scheduling, uncertainty, multiparametric MILP. 1. Introduction A number of problems from the area of process design and operations are commonly formulated as mixed integer linear programming (MILP) problems. One way to incorporate uncertainty into these problems is using MILP sensitivity analysis and parametric programming methods. The main limitation of most existing methods is that they can only be applied to problems with a single uncertain parameter or several uncertain parameters varying in a single direction. In this work, we focus on the parametric MILP problems with RHS uncertain parameters which are allowed to vary independently. The proposed solution procedure starts with the B&B tree of the MILP problem at the nominal values of the uncertain parameters and requires two iterative steps: LP/mpLP sensitivity analysis and updating the B&B tree. Rather than checking all the neighbor bases of the associated LP tableau, a novel algorithm is developed for mpLP problems that can determine the optimal functions and their corresponding critical regions without constructing the LP tableaus. 2. Proposed multiparametric MILP approach For the general mixed integer problem (PI), minz = cx assuming a perturbation of problem RHS (PI) parameter values such that: Ax>d-\-AO, the s.t. Ax > (9 aim of is to investigate the effect of A9 on the x > 0 , Xj integer, j = l,. .,k optimal solution x and objective value z. 2.1 Single Uncertain Parameter For the case of single uncertain parameter, the proposed approach follows the basic ideas of the interactive reference point approach proposed by Alves and Climaco (2000) presented for multiple objective MILP problems. The proposed framework is shown in Figure 1.

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First the problem is solved at [ Solve the original problem 1 the nominal values of the using a branch and bound tree uncertain parameters using a branch and bound solution Find A6 that leaves the structure \ Collect zf AP at approach, and the dual each leaf node p of the tree unchanged information A,^, z^ is collected at each leaf node. Assuming that the optimal solution is For Ae= AG +e found at node 0, the LP I update the branch and bound tree sensitivity analysis is then performed at node 0 to Figure 1. Flow chart of proposed approach for single determine the range A6^^^^^ uncertain parameter case within which the current optimal basis does not change. We need to find the perturbation A9"^^^ beyond which the structure of the branch and bound may not remain the same. A9"'^'' can be found through the following equation:

n

ibasis

A<9"'^=min{A6''"'"%min{

}}

where z^ and X^ are the objective value and dual multiplier at the optimal node 0, p

respectively. Note that only the positive

0

need to be considered, because the

negative one means that node p can never provide a better solution than node 0 at a certain point. 2.2 Multiple Uncertain Parameters This subsection presents the detailed steps (Figure 2) of the proposed approach to deal with the case of multiple uncertain parameters. Assuming for simplicity in the presentation that we want to investigate two parameters, 6a and 9b, changing in the range of [a^, a^ + Aa] and

Solve the original problem using a branch and bound tree

mpLP at the leaf nodes

Compare the critical regions with the current upper bounds

mpLP algorithm

[bo,bo+Ab]. The MILP update the branch and bound tree problem is first solved at Figure 2. Flow chart of proposed approach for (a^, bo) using branch and bound multiple uncertain parameters case algorithm and the optimal solution is found at node 1 (Figure 3). Other leaf nodes of the B&B tree are denoted as node 2, node 3, ..., node n. Note that only the information at the leaf nodes is required. Then the multiparametric linear programming is solved at each of the leaf nodes including node 1, so as to identify the optimal value functions and their corresponding critical regions in the region of [a^, a^ + Aa] and [b^, b^ + Ab]. In this work, a new algorithm is proposed for the solution of mpLP. When the mpLP procedure is completed, the output will be a

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set of optimal functions z = z^ +X^d^ + yffj^ ^i,, k = 1,..., K , where K is the number of critical regions. For any point {6^ ^0^) in the range of [a^, a^ + Aa] and [b^, b^ + Ab], the objective value ex of the relaxed LP problem of that node can be expressed by m?o^{z^+X^e^+p^e^,'k = \,...,K]. If the procedure is not complete, then there must exist a that such point

io^A)^

maxlz*" +^^0^ + P^6^} is less than ex*. Thus

Figure 3. Branch and bound tree for the original problem a bilevel programming problem is formulated as shown in problem (P2). It is proved that linear bilevel programming problems (BLPP) are max{{imiicx | Ax > ^ } - z } strongly NP-hard. In order to avoid solving a S.t.Z>Z<''*+/l<''*^,+;5<'^'^„k == 1,...K BLPP, we propose to first convert the relaxed a „ < ^ ^ < a o + Aa (P2) LP problems (inner problem in (P2)) at the leaf nodes to its dual form, so that the bo<^,
the objective A^yz^'^+A^'^0,+j3^'^0,,k = l,..X ( P 3 ) constraints bQ0 bo0 constraints and the current optimal functions z = z^ -\-A^0^ -\-P^6^ , hence all the constraints are linear. If the objective value of the (P3) model is nonzero, it means that there exists at least one point {6^ ,6^) at which its real objective value cannot be represented by any of the current objective value functions. Therefore, the objective value function at that point is z — z-\-Xd^ +>^^^b ^^^ should be included in the next iterations. This procedure terminates when the objective value for problem (P3) is 0, which means that the entire uncertain parameter range is covered by the existing objective value functions. Since (P3) is a nonconvex problem, a global optimization algorithm should be utilized such as GAMS/BARON (Sahinidis, 1996), which relies on branch-andreduce algorithm. Therefore, by performing mpLP at each leaf node p, a number of critical regions (k), CRp,CRp,...,CRp are identified and in each CRp,k = 1,...,K, the max 6y subject to

max 0y - z subject to A'^y
in

n^

optimal value z^ is expressed as z^ = Zp + X^O^ + 0^0^ . The next step is to update the

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2144 B&B tree. The main procedure involves to compare the critical regions of the leaf nodes with the current upper bounds and finally identify a set of new critical regions, and their corresponding objective function values and optimal integer solutions. At the beginning, the upper bounds C R ^ are set to be the critical regions of the current optimal node (node 1), which are CR|^\CRp\...,CRS^^. Assuming that

min^ s.t.

z;'^>+f

=zr+r^a + A"'^. (P4) zf =: z f > + ^<^>^. +A*X b„^ ^ , < b „ + A b

o.A G C R *

we want to compare critical regions C R [ ^ and C R ^ , which have intersection CR"^*, the following constraint is defined: zf^ > z^ and a redundancy test for this constraint is solved in CR"^^ as shown in problem (P4) (Acevedo and Pistikopoulos 1997). The solution of this problem provides the optimal functional form in CR"^^ .At each iteration, the new leaf nodes in the updated B&B tree will be compared to the current upper bounds, so as to determine the new optimal functions in their intersected region. This procedure stops when no further branching is required and the uncertainty analysis of the entire uncertain space can be presented by a number of critical regions that contain their corresponding optimal functions and integer solutions. Comparing to the existing approach (Acevedo and Pistikopoulos 1997), the proposed method solves the mpLP at only the leaf nodes in the B&B tree instead of every node during the branch and bound procedure, and consequently reduces the computational efforts significantly as will be shown in the preliminary results in the next section. Moreover, the new mpLP approach can efficiently determine the optimal function with respect to the uncertain parameters and the critical regions without having to retrieve the optimal tableaus and investigate the neighboring bases. 3. Case Study This problem contains two continuous variables minz == -3xj - 8 x 2 + 4 y i + 2 y 2 Xi, X2 and two binary variables yi, yi- Oiand 02 in s.t. Xj •fX2 < 1 3 + (9i the RHS of constraints are uncertain parameters 5Xj - 4 x 2 <20 varying in the range [0, 10]. According to the approach proposed in section 2.2, the following -8X1+22X2 <121 + (92 steps are considered. 4x +X2 > 8 Step 1: Solve the original problem The problem is first solved with B&B algorithm ^x--lOy, < 0 for e^ = (0, 0). X,- -15y3 < 0 As shown in x > 0,yG { 0 , l } , 0 < ^ i , ^ 2 ^ 1 0 Figure 4, both nodes 1 and 2 provide integer solutions, which are (yu Yi) = (1,1) and (yi, yi) = (1,0), respectively, and the optimal values at these two nodes are -70.5 and 8, respectively. Therefore, node 1 gives the optimal solution. Node 3 is found to be an infeasible node Figure 4. Branch and bound and thus fathomed. tree at Step 1 Step 2: mpLP on the leaf nodes Multi-parametric linear programming is performed

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on these two leaf nodes, so as to determine how the objective value z changes with respect to 61 and 62 and the critical regions in which the optimal functions are valid. Node 1: The dual multipliers X-i and Pi of the relaxed LP at node 1 are -4.3333 and 0.1667, thus, the optimal function at 0^ is zj = -70.5-4.3333(9i -0.16676^2 • ^^ order to find the critical region for this function, problem (P4) is formulated and solved as follows: max -13(9i6fi -20(^2 +(-121-"(92) -70.5 - 4.3333<9i - OA66IO2 0<(9i,6>2 <10 The solution of this problem identifies if there exists a point (Oi, O2), at which the objective value z* of the relaxed LP problem cannot be expressed by the current optimal functions. The solution is found to be (0i, 62) = (10, 0) and the objective value is 16.74. The problem is solved using GAMS/BARON within one iteration in 0.02 CPU seconds. The same solution is obtained using GAMS/CONOPT in 6 iterations, while only a local optimal solution is found with an objective function of 0 using GAMS/MINOS. At (9i, ©2) = (10, 0), the optimal function of the LP problem at node 1 is z^ =-97.0909-0.36366^2. Then the constraint z > -97.0909-0.363602 is included in the above problem and the new objective value becomes 0. Therefore, the entire uncertain space can be covered by these two optimal functions. The critical regions of the two optimal functions can be derived from the inequality zJ > Zi ^ -70.5-4.3333(9i -0.16676>2 > -97.0909-0.3636<92 ^ 0.07333<9j -0.00333<92 < 0.45 Thus, we have: zf =-70.5-4.3333(9i-0.1667(92 CRI = {0.07333(9 -0.00333<9 < 0.45 , ^2^10} z^ = -97.0909-0.3636(92

CR^ = { 0.07333(9^ -0.00333(9^ > 0.45 , 0,,0, < 10 }

Node 2: The current optimal function is z = -8, which means the objective value doesn't vary with respect to 0^ and 0^. Similarly to node 1, problem (P4) is formulated and solved and the objective value is found to be 0. Therefore, the mpLP procedure stops and the result at node 2 is: z^ = - 8 CR2 = {(9i,(92 < 10 }. Step 3: Compare critical regions and determine optimal functions CRI

and CR2 : Since CR2 contains the entire uncertainty space, CRlnCR2

=CRI.

Then z| and Z2 are compared in CRI , which can be achieved by solving the following redundancy test formulation: min e

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Z Jia and M. G. lerapetritou -70.5-4.3333(9i -0.1667(92 + ^ = - 8 0.07333(9j -0.003336>2 < 0.45 6>2 <10

It is found that £* > 0 , therefore z j < Z2. CR^ and C7?2 • ^^ is trivial to see that CRj fl CR2 = CRj. The following redundancy test shows thatf > 0 , hence constraint z^ < z^ is redundant in CRI . min £ -97.0909 - 0.3636<92 + ^ = - 8 0.07333(9j-0.00333<92 > 0.45 Thus the final optimal solution for this problem is:

/=(U) Zj =-70.5-4.33336>i-0.16676^2

^^\ = {0.07333<9j-0.00333i92 < 0.45 , ^2 ^ 10}

Z2 =-97.0909-0.3636^2 ^^1 = { 0.07333(9^ - 0.00333/92 > 0.45 , ^i,(92<10} However, the result from the literature (Acevedo and Pistikopoulos, 1997) contains only z^andCR^ 4. Summary and Future Work The issue of uncertainty analysis for MILP problems is addressed in this paper. An integrated framework is developed that allows the parameters in the RHS of the MILP formulation to vary independently. It mainly consists of two steps: LP/mpLP sensitivity analysis and updating the B&B tree. For the case of mpLP, a novel algorithm is proposed which solves a set of NLP problems iteratively using the commercially available global optimization solver BARON. Work is under progress to further develop the proposed approach to enable the analysis of uncertainty in the constraints coefficients. Additional work is devoted in the application of the proposed method to scheduling problems where the main focus is to automate the information extraction at the branch and bound tree and investigate efficient computation schemes such as parallelization that will enable the application of this approach to larger problems. References Acevedo, J. and Pistikopoulos, E.N. 1997. A multiparametric programming approach for linear process engineering problems under uncertainty. Ind. Eng, Chem. Res., 36, 717. Alves, M.J. and CUmaco, J. 2000. An interactive reference point approach for multiobjective mixed-integer programming using branch-and-bound. Eur. J. Oper. Res., 124, 478. Dua, V. and Pistikopoulos, E.N. 2000. An algorithm for the solution of multiparametric mixed integer linear programming problems. Ann. Oper. Res.,99, 123. lerapetritou, M.G. and Floudas, C.A. 1998. Effective continuous-time formulation for short-term scheduling. 1. Multipurpose batch processes. Ind. Eng. Chem. Res., 37, 4341. Samsatli, N.J., Papageorgiou, L.G. and Shah, N. 1998. Robustness metrics for dynamic optimization models under parameter uncertainty. AICHE J, 44, 1993.

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Information Modeling for Pharmaceutical Product Development Chunhua Zhao ^, Leaelaf Hailemariam ^, Ankur Jain ^, Girish Joglekar ^, Venkat Venkatasubramanian ^, Kenneth Morris ^ and Gintaras Reklaitis ^ "^ Laboratory for Intelligent Process Systems, School of Chemical Engineering, Purdue University, West Lafayette, IN 47907 USA ^ Department of Industrial and Physical Pharmacy, Purdue University, West Lafayette, IN 47907 USA Abstract Development of a pharmaceutical product involves several inter-related steps with multiple decisions requiring iterative improvements. Large amounts of information, including the properties of a drug substance, interactions of materials, unit operations, equipment etc, have to be gathered and used for decision making. A systematic model of the associated information is thus needed to streamline the product development process and provide a common foundation to support the information. Following the information-centric infrastructure proposed in our earlier work, ontology has been used to model the information. The information modeled as ontology provides information in a way that can be easily used by humans and processed by machines. The information modeling process and developed ontology are discussed in detail. The benefits are demonstrated by using a case study for managing information generated from the preformulation stage of pharmaceutical product development. Keywords: Information Modeling, Pharmaceutical Product Development, Ontology, Information Management 1. Introduction The development of a product goes through several stages during its lifecycle with emphasis on shortening development time, cutting development costs and improving the process design to ensure higher flexibility (Schneider and Marquardt, 2002). This is particularly true in the area of pharmaceuticals and specialty chemicals. Commercial scale product and process development typically goes through the following stages after the viability of a newly discovered molecule is established: laboratory scale, pilot plant scale and commercial scale manufacturing. Laboratory scale experiments are used to determine various synthetic routes and characterize key steps in each route, as well as obtain process parameter values. Pilot plant studies provide a detailed understanding of processing steps in the selected route and data needed for scale-up to commercial manufacturing, at which stage information related to manufacturing is applied in debottlenecking and productivity improvement. The three stages are closely related through the information they exchange. Information generated at the lab scale can be used to improve manufacturing. Problems identified at the manufacturing stage are communicated to the lab scale to identify their root causes, and provide ideas on how to avoid similar problems in ftiture process development. The stage-wise development of a process for a drug substance and its dosage form(s) involves a substantial amount of information sharing by tools in each stage; and

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between stages, information and knowledge have to be exchanged at appropriate times. However, most of the available tools have reached maturity and exist in islands of automation with little interaction among them. Each tool uses its own format and interpretation of information, causing inefficient information and knowledge transfer. In the development and manufacturing of pharmaceutical products, use of new process analytical technologies (PAT) has enabled scientists to get a better understanding of the underlying physical and chemical phenomena. The knowledge created from the learning process can be in different forms: reports on paper or electronic format and experience gained by scientists. With more and more information and knowledge becoming available, however, it is clear that we need more intelligent software systems to effectively manage and access them for efficient decision making. Pharmaceutical product development involves the integration of process modeling tools, effective handling of laboratory generated information and knowledge as well as development of technical specifications and information base to satisfy regulatory requirements. We argue that in order to better support the activities and decisions in pharmaceutical product development, formal and explicit models of related information need to be developed. These information models should be easily accessible by human and tools, and should provide a common understanding for information sharing. The remainder of the paper is organized as follows: Section 2 discusses the ontology based approach to model the information, using the material property ontology for pharmaceutical product development as an example to discuss the steps used and lessons acquired in building ontologies. Section 3 illustrates the use of information modeling to support the information management during the preformulation stage. Examples are used to compare the proposed approach with existing solutions. 2. Information Modeling 2.1. Ontology Several approaches exist to model the information. However, such information models are usually in closed form and only provide a limited view of the information. An information-centric approach has been proposed (Zhao et al., 2005) in which information is modeled using ontology. An ontology is a formal and explicit specification of a shared abstract model of a phenomenon through identification of its relevant concepts (Gruber, 1993). This may be seen during the measurement of the flow rate of an active pharmaceutical ingredient (API) powder through an orifice, which conceptualizes the API, the flow rate value, the experiment and its context (Figure 1). API: Flow Rate. Ontology captures relations like has-value-of ^ ^ \ Relative Humidity: 78.0 ^ and was-done-on. In addition, ontology makes it possible to relate different concepts mujyiujjmmijif \ through their being instances of the same 1.0 g/s: [0.8 , 1.2]* class (e.g. flow rate of material A related to that of material B). Such relations are usable Figure 1: Relations between concepts by both human and computer tools. Compared to a database schema which targets physical data independence, and an XML schema which targets document structure, an ontology targets agreed-upon and explicit semantics of the information, and directly describes the concepts and their relations. Web Ontology Language (OWL, 2004) was used in this work to encode ontology.

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2.2. Ontology Building for Pharmaceutical Product Development Ontology building is an evolutionary design process consisting of proposing, implementing and refining classes and properties that comprise an ontology (Noy and McGuinness, 2001). The steps involved include: determining the domain and scope of the ontology, reusing existing ontologies, enumerating important terms in the ontology and defining the classes and class hierarchy. In pharmaceutical product development, one undergoes selection of a dosage form, selection of a processing route, selection of excipient (inert material added to impart desired properties on the final product) roles and selection of specific excipients and their composition. The best way to systematically capture the above sequence of steps as applied to the final product is through a construct called a development state, which includes information about the dosage form, processing route, excipient roles, excipients and their compositions. In addition, it should contain the description of the API, of the excipients and the dose amount. In turn, the description of the API should include the description of its properties and the experiments done on it. A central concept in the abstraction above is the material, which represents substances and mixtures (which are characterized by pure substances and their compositions). A material has several properties (e.g. specific heat capacity), can play several roles (e.g. API, flow aid) and can be involved in several experiments (e.g. Hosokawa Tests). The list of material properties may be classified into engineering properties, compound properties, particle properties and powder properties. Engineering properties include those properties which are used in engineering calculations, like heat transfer properties. Compound properties include the molecular properties like molecular mass, and a description of the chemical reactions the material undergoes. Particle properties include the crystalline properties and a description of the stability of the physical form (if crystalline, the crystal system). Powder properties describe the behaviour of a large number of particles of the material, like flow and deformation of the powder into tablets. Each property is represented by a class with its own set of attributes. A material property value can be measured experimentally, calculated mathematically or retrieved from literature. If measured experimentally, the conditions under which an experiment is performed defines its context, for example temperature, pH, relative humidity and so on. The description of an experiment would include the materials involved, the experimenter, the location of experiment, the date and time of experiment, the equipment used, the procedure followed and the experimental data. The relations between the experiments, the materials they were done on and the properties that were measured are explicitly described. Modeling of the domain information, i.e. creating the domain ontology, requires understanding the ontology building techniques as well as the domain one tries to model. In this project, we work very closely with collaborators in Industrial Pharmacy. The current ontology is the result of several iterations of propose-discuss-revise. The visualization tools provided by the ontology editor. Protege (see URL) and the plugins in the editor including the view of class hierarchy, the graph view and the automated generated form for information entry are very convenient in the collaboration. The developed ontology has been used as the foundation for the information repository and provides information to various tools including an engine to execute guidelines and another engine to utilize mathematical knowledge (Zhao et al., 2006) following the methodology developed in our previous work (Zhao et al., 2005). In this paper, we discuss how the ontology could support management of information gathered in product development.

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3. Information Management Voluminous information is generated during product development, such as raw data generated from analytical instruments, pictures from SEMs, pictures of experiment setups, experiment notes and reports, mass and energy balance results from simulation tools etc. The information could also be in different formats, including plain text files, WORD documents. Excel worksheets, JPEG files, MPEG movies, PDFs etc. How to effectively gather information from different resources and organize it for its end use are key information management tasks. A few solutions have been developed to manage the information, most important of which are laboratory information management systems, e-LabNotebooks and content management systems. The key frinctionalities and problems of these systems are discussed in the next section. 3.1. Current Information Management Solutions Laboratory Information Management Systems (LIMS) LIMS are database applications that are used to store and manage information associated with a laboratory (Paszko and Pugsley, 2000). Typical LIMS frinctionalities include sample tracking, data entry, sample scheduling, quality analysis/quality control (allowing users to generate control charts and trend analysis graphs), automatic electronic data transfer (from analytical equipment to the LIMS), chemical and reagent inventory, personnel and equipment management and maintenance of the database. A LIMS stores information in relational databases such as Oracle, DB2 or MS SQL (Grauer, 2003). Most LIMS have interfaces that give to access to the database for information retrieval or storage. As discussed earlier, such database schemas provide only limited semantics. Relational database structures also limit the capability of describing complex relations between information. E-lab Notebooks An ELN can provide frinctionalities like browsing online libraries, databases, other electronic devices, and remote sources such as the Web, writing documents and data sets, managing data, publishing and sharing information, and creating records (Zall, 2001). With an electronic notebook, the records will be published electronically and shared with collaborators and reviewers. The utility of an ELN to provide a collaborative environment has proven inadequate for industry, especially when quality assurance/control is expected to be a major factor (Pavlis, 2005). Quality assurance demands experiments to be performed following standard operational procedures. This may be accomplished by an automated interface for data entry, which is currently not available for ELNs. Integration with information management systems is also lacking. Content Management Systems (CMS) A content management system supports the process of publishing, maintaining and dissemination of documents (Noga and Kruper, 2002). The major components of a CMS are the data repository, user interface, workflow scheme, editorial tools, and output utilities. They allow writers to create or update content, track changes, and publish contents to make them available to all users in a variety of configurations. 3.2. Ontology Driven Information Management Without specifying the semantics of the information, it is very difficult for these tools to provide functionalities beyond sharing the information among users and keyword-based search on the information. From our experience of implementing information management systems in our project, we found two major problems with the current systems: (1) organization of related information; and (2) lack of an open and systematic

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way to manage meta-data. These problems are directly related to the lack of the semantics of the information. We argue that the semantics of the information should be provided by the user who creates the information. Since ontology defines the semantics of the information, information can be captured by the individuals based on concepts defined in ontology. Given the semantic richness of the information defined in ontology, the information entry form can be generated automatically (as demonstrated by Protege, the ontology editor used in this project). To manage the information associated with an experiment, the experiment individual is created which is linked to the raw files generated from experiment. Similarly, as shown in Figure 2 the individuals of material properties can also be created by the user directly in which the link to experiment individuals which have been carried out for this property is specified. The system can automatically locate the concepts and relations and provide an integrated view of the information. For example, for a specific material, all the experiments done on it and its properties are accessed. The information infrastructure described above makes possible the effective management of experimental files, which may exist in different forms (spreadsheets, movies etc.) with non-descriptive names and folder ^^^H^HBH|M|^ locations. This system is developed on top of an F>r ImiUvkUiM • API: Flow Rate (insta... existing content management system to utilize i,,,^,,^,^,,,^n,m. Pn available functionalities such as user management, p-^----I workflow, security etc. Instead of the user creating folder structures in an arbitrary way, the folder isei^ni* lc^f.iit^^ria! ^ %^ structure is created based on the concept hierarchy ^^''' defined in the ontology. The system allows surfing un^^M^ 4 %^ between related instances and search by keyword as #109/3: [0 8,i2] well as on the hierarchy. A web-based interface to the ^ 4. #^ information repository can be created for users to ^^^^^^^^^^ *1 ^'n ^ ' r ' • A P I : Flow Rate Measurement access, and modify the mformation. 4 %^ ih:»C.0 life 5
Figure 2: Interjace jor input oj

potentially affected by therelative humidity. In the material properties current case study, the micromeritics (solid surface properties) of 39 materials (including mixtures) were studied, each with 18 micromeritic properties. There were on average 5 experiments for each property for every material and each experiment had an average of 3 files associated with it. In semantic search, identification of relative humidity as an instance of context would lead to all instances of properties in which that context appeared. Through the relations between property and material, the instances are further filtered based on the specified drug substance. The instances of these properties which are subclasses of micromeritic properties are found given the expHcit definition of class-subclass relationships in the ontology. The individual experiments that are done on these properties are identified through the partwhole relationship with the property. The experiment files which are hnked to these experiments are presented as the search results. The semantic search engine found 8 experiments done to determine the flow rate of a powder through an orifice for the particular drug substance as the micromeritic experiments affected by relative humidity. In contrast, without the ontology to provide the semantics, a keyword based search would not be able to navigate using the relationships. Search using keywords 'relative humidity micromeritics' did not identify any of the experiments, while a similar search

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using 'relative humidity' identified many documents, most of which had very little to do with experiments. While it is acknowledged that such results are not indicative of all experimental work, they are indicative of the challenges faced by humans and machines alike in processing large amounts of information with little or no semantics. Conversely, they serve to illustrate the utility of semantic search made possible through the development of material, property and experiment ontologies. 4. Summary In this paper, we discuss the importance of modeling information with explicit and formal semantics. Only with the semantics which makes the information machine processable could tools better utilize the information and provide better functionalities. Ontology was used to model the information. In this work, we concentrate on the information related to pharmaceutical product development. The ontology building process and the final ontology were discussed. We also demonstrate an ontology-driven information management approach based on the developed ontology. This approach provides an easy way for the user to create semantics as well as relations on the information generated during process development. This approach is very general and could easily be applied to manage information in domains like chemical process development. To develop ontology to model the domain information could be a difficult task which requires information modeling techniques as well as understanding of the domain to be modeled. Nevertheless, the ontology provides a solid foundation to better utilize the information by supporting tool development, information sharing between tools as well as information management. References Grauer Z. (2003), Laboratory Information Management Systems and Traceability of Quality Systems, American Laboratory, 9, 15. Gruber, T. R. (1993), A Translation Approach to Portable Ontology Specification, Knowledge Acquisition, 5, 2, 199. Noga, M., Kruper, F. (2002), Optimizing Content Management System Pipelines, Lecture Notes in Computer Science, 2487, 252. Noy, N.F., McGuinness, D.L. (2001), Ontology Development 101: A Guide to Creating Your First Ontology, Stanford Knowledge Systems Laboratory Technical Report KSL-01-05. OWL (2004), Web Ontology Language Overview, W3C Recommendation, http ://www. w3 .org/TR/owl-features/ Paszko, C , Pugsley, C. (2000), Considerations in Selecting a Laboratory Information Management System (LIMS), American Laboratory, 9, 38. Pavlis, R., (2005), Scientific Computing, 9, 31. Protege (Version 3.1), http://protege.stanford.edu. Schneider, R., Marquardt, W. (2002), Information Technology Support in the Chemical Process Design Life Cycle, Chemical Engineering Science, 57, 1763. Zall, M. (2001), The Nascent Paperless Laboratory, Chemical Innovation, 31, 2. Zhao, C , Joglekar, G., Jain, A., Venkatasubramanian, V., Reklaitis, G. V. (2005), Pharmaceutical Informatics: A Novel Paradigm for Pharmaceutical Product Development and Manufacture, Proceedings of ESCAPE 15. Zhao, C , Jain, A., Joglekar, G., Hailemariam, L., Venkatasubramanian, V., Morris, K., Reklaitis, G. V. (2006), A Unified Approach for Knowledge Modeling in Pharmaceutical Product Development, Submitted to ESCAPE 16.

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Decentralized Supply Chain Dynamics and the Quantity Flexibility Contract Vijayanand Subramanian'', Joseph F. Pekny^ and Gintaras V. Reklaitis^ '^Oracle Corporation, 25 First Street, Third Floor, Cambridge, MA 02141, USA ^Purdue University, Forney Hall of Chemical Engineering, 480 Stadium Mall Drive, West Lafayette, Indiana 47907, USA

Abstract In a decentralized supply chain, i.e., one with multiple loci of control, the dependencies between the entities are formalized through contracts. The typical goal is to choose contract forms and values that are beneficial to both the contracting entities. Representation of supply chains at a level sufficient to capture the essential system interactions requires a high-level architecture that combines simulation (SIM) and optimization (OPT) techniques. In this framework, the simulation captures the systemwide uncertainties while the optimization model captures the combinatorial nature of manufacturing decisions. In this paper, we extend the application of the SIM-OPT framework to a two-level decentralized retailer-manufacturer supply chain described in Subramanian V. et al (2005). The application involves the quantity flexibility (QF) contract and its role in mitigating the bullwhip effect, an amplification of demand variability that can arise under certain conditions. In the earlier work, a case study with independent and identically distributed (i.i.d) demand was used to quantify the impact of a series of contract settings on these two entities. Preliminary investigation into the effect of the manufacturing function on the contract was also reported. In the present paper, the contract is explored under the more general case of non-stationary demand. The following features are examined: (i) taking advantage of non-stationary demand, the impact of the contract parameters on the manufacturer are verified (ii) the interaction between manufacturer's process investment decisions and the contract is generalized (iii) further studies into the effect of the production function on the contract are summarized. Through this paper, we continue to demonstrate the use of the SIMOPT fi-amework in studying decentralized supply chain dynamics, and, in particular, the utility of detailed manufacturing models. 1. Existing Literature and Motivation for SIM-OPT An extensive literature review of supply chain contracts in decentralized systems is available in Subramanian V. et al (2005) and thus will not be reiterated here. The emerging work in management science literature has theoretical appeal, but does not use the detailed production models necessary to capture the complexities of practical applications. Detailed models are necessary to rigorously test the contract for feasibility, investigate the contract from the supplier's perspective, and generate insights that were not previously possible. Embedding these combinatorial optimization models within a simulation environment allows for treatment of uncertainties with arbitrary distributions

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and also permits an unrestrictive representation of the application. It is for these reasons that this work relies on simulation in consort with optimization. 2. Problem Definition Consider a retailer-manufacturer system in which the retailer faces demand uncertainty. The retailer's problem is to choose inventory levels to meet a desired service level. The manufacturer's problem is to fulfil the retailer's orders at minimal cost. Under certain conditions, the manufacturer's demand variability can be much larger than the retailer's, and this has been termed the "buUwhip" effect (Lee et al. 1997a,b). In our application, we consider the role played by demand forecasting in causing the bullwhip effect wherein the retailer's continual updating of the forecasts and consequently the ordering policy, results in increased variability of his orders. This amplification of demand variability can significantly impact the manufacturer leading to increased inventory, production and out-sourcing costs. Curtailing this variability is the motivation for the QF contract. A finite horizon QF contract can be described briefly as follows. At period i, let aij be the fraction by which retailer may adjust period j ' s order commitment upward (or actual order if i=j, j>=i) and let aij be the fraction by which retailer may adjust period j ' s commitment downward (or actual order if i = j , j>=i). If Qt,t+i is the order estimate for period (t+i) at period t (i>=0, i=0 corresponds to actual order) then the following inequalities bound future order estimates (Eq. 1) and actual orders (Eq. 2) as a function of order estimates placed in the past: (1- at,t+i)* Qt-i,t+i (1- a,r

<= Qt,t+i <= (1+ ^t,t+i)* Qt-i,tH

Qt-i,t <= Qt,t <= 0+ ^r.t^* Qt-ij

i=l...N-t; N: total periods (1) (2)

Usually, QF contracts have the property that the flexibility offered for a future period depends only on the number of periods into the future. In other words, ati,ti+i = ^x2,\2+i and ati,ti+i=at2,t2+i3. Case Study In Subramanian V. et al (2005) a detailed description of the retailer-manufacturer case study is provided. In this paper, the focus is on the extensions. Previously, consumer demand was assumed to be i.i.d across time. While this simplification facilitates the interpretation of the results, it is not realistic. Furthermore, the forecasting technique (simple averaging) used in that work leads to accurate forecasts over the long run, which consequently diminished the bullwhip effect. In this paper, consumer demand is modeled as non-stationary. As a result, the forecasts will (at the very least) lag the true demand, and the forecasting error and bullwhip effect, in all likelihood, can be expected to fluctuate but not diminish over time. 3.1. Retailer The retailer supplies nine stock keeping units (SKUs) to consumers. It is assumed that the time varying demands are normally distributed. The retailer's problem is to choose periodic order quantities in order to meet a target average service level across the horizon (service level for a period is defined as fraction of demand fulfilled for that

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period). In the base case for contract analysis, the retailer needs to forecast the demand and the manufacturer experiences the bullwhip effect due to this demand forecasting. At the beginning of each period, the retailer uses single exponential smoothing to forecast the demand. 3.2. Manufacturer The manufacturer uses a 3-stage process to convert raw materials into two bases (in 2 towers), the two bases into three variants in a post-dosing mixer and three variants into nine SKUs on 3 packing lines. The manufacturer responds to the retailer's orders by periodically solving for the production plan and implementing the production schedule. The planning and scheduling mathematical programming formulations used for this purpose can be found in Subramanian V. (2004). The manufacturer's total costs are the sum of production, inventory and outsourcing costs. In the base case when the retailer is operating at the desired service levels, the retailer's inventory costs and manufacturer's total costs are noted. When the order bands are imposed as per the QF contract, the retailer continues to place current orders and provide order estimates within the restrictions of the contract. Under these constraints, the retailer re-optimizes the safety stock level for each SKU in order to reach the desired base case target service level. When the retailer re-attains the target service level, the change in costs for the retailer and manufacturer are noted. Naturally, the contract is feasible if there is a net reduction in system costs and the resultant savings can be appropriately divided. At the very least, the manufacturer can defray any potential increase in retailer's costs and leave him no worse off than in the base case. In such a scenario, the entire net savings is pocketed by the manufacturer. 4. Sample Results This section provides a sampling of results to illustrate the key extensions in this paper. Complete details are available in Subramanian V. (2004). The first subsection will explore a series of contract settings. The interaction between process investment decisions and the contract savings for a manufacturer will be generalized in the second section while the impact of the production function on the contract will constitute the third section of results. The following cost structure is assumed. The retailer's holding cost is $l/unit/SKU. The manufacturer's inventory cost is also $l/unit/SKU, his variable production cost is $10/unit/SKU, and the penalty or outsourcing cost is $50/ unit/SKU. The horizon of interest is 20 periods (or weeks) while the order and production planning horizons are assumed to be 6 periods each. For a given vector of safety stock levels such as the base case, 1000 simulations were performed, which consumed about 11 CPU hours on an Intel Pentium IIKD 864 MHz PC with 512 MB RAM. The simulations were repeated to obtain statistically significant values for the estimated service levels. For the base case, the average service level for SKU# SI was ~ 0.9708. The standard error in estimation was 0.0006 and a t-test placed the service level between 0.9696 and 0.9720 with 95% confidence. For a specified vector of safety stock levels and for a given contract setting, the same number of simulations is repeated with the identical random number seed. Recall that under the contract case, the retailer reoptimizes the safety stock levels and is left no worse off than the base case with respect to target service levels. To illustrate the bullwhip effect, a simulation run was made with

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system parameters identical to the base case but assuming the retailer knew the demand distribution. Figure 1 plots the true demand variance, the retailer order variance when the retailer knows the demand distribution, and the retailer order variance for the base case when the retailer has to forecast. One can observe that when the retailer knows the demand distribution, his order variance closely mirrors the true demand variance but in the forecasting scenario (base case), there is an amplification of order variability. In addition, one can also observe that this order variance amplification or bullwhip effect fluctuates but does not diminish across time (as was observed with i.i.d demand). ' Demand Variance • Order Variance (known demand) ' Order Variance (base case - bullwhip)

•c

Figure 1: Amplification of Order Variance: The Bullwhip Effect

4.1. Quantification of the Contract and Impact of Contract Parameters For an order horizon of 6 periods, and assuming equal upward and downward order revision flexibility, the QF contract is completely specified by 5 parameters: (ao, ai, a2, a3, a4). ao restricts the flexibility for the current order, ai restricts the revision of the order estimate for the next period, a2 restricts the revision of the order estimate two periods from now and so on. The first set of results analyzes six contract settings labeled R, A', B',C',D',E': (0.2,0.2,0.2,0.2,0.2), (oo,0.2,0.2,0.2,0.2), (0.2,oo,0.2,0.2,0.2), (0.2,0.2,00,0.2,0.2), (0.2,0.2,0.2,00,0.2) and (0.2,0.2,0.2,0.2,oo). Here, oo refers to zero restrictions on flexibility. For these six cases, the change in costs relative to the base case for the retailer, manufacturer, and the total system are shown in Figure 2. As can be seen, all six contract cases are infeasible as they increase the costs for the whole system relative to the base case. In Subramanian V. et al (2005), many instances of feasible, i.e., beneficial contract settings are reported, reinforcing the point that a rigorous quantification using detailed models is necessary as certain settings can prove to be highly unfavorable. In moving from R to each of these cases. A' through E', the manufacturer's costs increases. This is naturally due to the increased flexibility in the QF parameters and the resulting increase in the retailer's order variance. An analysis of

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retailer's order variance by period shows that it increases consistently on moving from E' to A', which suggests that as far as impact on the manufacturing costs is concerned, the contract parameters can be ranked in the following order of importance: ao> ai> a2> a3> a4. This was also observed in Subramanian V. et al (2005) but in that i.i.d study, the order variance continuously decreased with time, and the true impact of the contract parameters on the order variance was confounded. S Manufacturer 1 Retailer

100000 -

^ System 80000 S

60000 -

o o

40000-

•~

20000 -

1

0-

o

1 1i

liJ

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A

B

C

D

E

-20000 ^

Impact of Contract Parameters: Six Cases ~ >

-40000 -

Figure 2: Quantifying the QF Contract: Six Cases (R, A', B', C, D', E') Figure 3: QF Contract and Process Investments 4.2. Interaction between QF Contract and Process Investments: Formalism In Subramanian V. (2005), the same contract setting is found to yield very different savings for different manufacturing processes. Thus the contract setting can affect a manufacturer's decision to invest and make a change to the production process (say, reduce changeovers). Here, we formalize this interaction using figure 3. Let p and q refer to two production processes. Let M, M' refer to the manufacturer's total base and contract case costs. The contract case costs includes the payout to the retailer to leave him no worse off. Let Cpq capture all increases in cost in moving from process p to q (such as investments) apart from the change in the total manufacturing costs and Lpq denote all reductions in cost (such as reduction in labor costs on reducing changeovers) apart from the change in the manufacturing costs. If the manufacturer moves from p to q and signs the contract, his total costs at q's contract case can be calculated as: M'q + Cpq - Lpq Thus, (M'q + Cpq - Lpq) < M'p should hold for an improving move from process p to q. To yield a non-pathological scenario we need (Lqp - Cqp) <= (Cpq - Lpq), or else, the manufacturer can shuttle from p to q and q to p and improve indefinitely. 4.3. Effect of Production Parameters on QF Contract As part of the manufacturing process, sequence-dependent changeovers are prescribed on the post-dosing mixer (stage 2) and the packing lines (stage 3). In this case study, increasing the changeover matrix on stage 2 of the production process led to increasing

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contract savings. While a strong correlation was observed with i.i.d demand too, the relationship was inverted suggesting that the impact of the production function is case parameter specific. Apart from changeovers, other production parameters such as equipment downtimes and production rates across the 3 stages of the process were also varied in this case study. The results reinforced the conclusions that: (i) production function has a big impact on contract savings (ii) relationships between certain production parameters and the contract exist. As an illustrative example, Table 1 shows the trends observed on varying downtimes on a tower and two packing lines, PL_2 and PL_3. Table 1: Effect of Varying Downtimes Change in Contract Savings Equipment Name

Increasing Downtime

Decreasing Downtime

Tower

Decreases (-61%)

Increases (-49%)

PL_2

Increases (-38%)

Decreases (-23%)

PL_3

Increases (-40%)

Decreases (-38%)

The impact of the outsourcing/production cost ratio was also analyzed and a linear increasing relationship between the ratio and contract savings emerged. This leads to the counter-intuitive scenario wherein the least cost competitive manufacturer, i.e., one with the highest operating costs, extracts the maximal savings and could potentially woo the retailer. The general implications of relationships between production parameters and contract savings are discussed further in Subramanian V. et al (2005). 5. Conclusions In this paper, we extended our previous work applying the SIM-OPT framework to a decentralized two-level retailer-manufacturer supply chain. The following findings were made (i) using non-stationary demand, we verified the impact of the contract parameters on the manufacturer (ii) the interaction between manufacturer's process investment decisions and the contract was generalized (iii) further studies into the effect of the production function on the contract were very briefly summarized. The interested reader can refer Subramanian V. (2004) for detailed analyses, especially on the impact of the production parameters. References Subramanian, V., Pekny, J.F and G.V.Reklaitis (2005) A computational framework for decentralized supply chain analysis : The quantity flexibility contract, Under review (CIPAC report, School of Chemical Engineering, Purdue University, available on request) Subramanian, V. (2004) PhD Thesis, School of Chemical Engineering, Purdue University. Lee, H.L., Padmanabhan, V. and S. Whang (1997a) The buUwhip effect in supply chains, Sloan Management Review, 38, 93-102. Lee, H.L., Padmanabhan, V. and S. Whang (1997b) Information distortion in a supply chain: The bullwhip effect. Management Science, 43, 546-558.

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Planning and scheduling of multipurpose continuous plants C. Schwindf", S. Herrmann^, a n d N . Trautmann^ ^Institute of Management and Economics, Clausthal University of Technology, Julius-Albert-Str. 2, 38678 Clausthal-Zellerfeld, Germany ^Departement fiir Betriebswirtschaftslehre, University of Bern, Engehaldenstr. 4, 3012 Bern, Switzerland

A b s t r a c t . In this paper we consider the short-term production planning problem of multipurpose continuous plants. This problem quite naturally decomposes into a planning problem of optimizing the operating conditions and processing times of the continuous tasks and a scheduling problem, which consists in allocating processing units, input materials, and storage space over time to the resulting operations. The planning problem can be formulated as a continuous nonlinear programming problem of moderate size. Due to constraints on material availability and storage capacity for intermediate products, classical schedule-generation schemes cannot be applied to the scheduling problem. That is why we use a new two-phase approach dealing with the two types of constraints separately. 1. I N T R O D U C T I O N A multipurpose continuous production plant consists of several processing units and storage facilities for intermediate products, which are linked by flexible hoses or pipelines. Final products are produced through a sequence of continuous tasks being executed on processing units. A processing unit must be cleaned between the executions of different tasks, the cleaning times generally being sequence-dependent. Each task may be executed on alternative processing units, and for each task the processing time, the production rate, as well as the proportions of the input and output products may be chosen within prescribed intervals. Whereas certain intermediate products can be stocked in dedicated storage facilities of finite capacity, others are chemically instable and must be consumed instantly. The execution of a task on a processing unit during a specified processing time and with specified production rate and input and output proportions is referred to as an operation. Given a set of primary requirements for the final products of some family, the short-term production planning problem consists in first, generating an appropriate set of operations (planning problem) and second, scheduling the operations on the processing units of the plant (scheduling problem). Since the plant can only be reconfigured for processing the next product family when all operations have been completed, we assume that the objective is to minimize the makespan needed for producing the given primary requirements.

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A significant body of chemical engineering research has been focused on the shortterm planning of batch plants. Much less work has been reported on continuous process scheduHng even though continuous processes constitute an important component of process industries. Classical approaches to continuous process scheduhng assumed the demand for final products to be constant over time (see e.g., the model proposed by Sahinides and Grossmann in [1]). More recently, different types of MINLP and MILP continuous-time formulations have been developed for the general short-term production planning of continuous plants with demands at discrete points in time (such models have for example been devised by lerapetritou and Floudas [2], Mendez and Cerda [3], and Mockus and Reklaitis [4]). A relaxation-based branch-and-bound algorithm for solving the scheduling problem with given set of operations has been proposed by Neumann et al. [5]. Constraint propagation techniques for the scheduling of continuous material flows can be found in Sourd and Rogerie [6]. In contrast to the monolithic MINLP and MILP models, we follow a hierarchical approach, with the planning problem at the top level and the scheduling problem at the base level. This heuristic decomposition of the problem allows us to cope with instances of practical size within a short amount of computing time. 2. T H E P L A N N I N G P R O B L E M In this section we consider the planning problem in more detail. Let T be the set of all tasks r and V be the set of all (raw, intermediate, and final) products TT under consideration. For each task r G T we have to determine the processing time p^, the production rate 7^-, and the input and output proportions ar-K of all products IT ^ V consumed or produced, respectively, during the execution of task r , where we estabhsh the convention that a^Tj: < 0 for input products TT. By V~ and V^ we denote the sets of all input or output products of task r . Symmetrically, T^ := {r e T \ IT e V~} and TJ' := {T e T \ TT e V:^} are the sets of all tasks consuming or producing product TT. Moreover, let V^ be the set of all perishable intermediate products, let p^, be the given primary requirement plus an unavoidable residual stock minus the initial stock for product TT, and let a^ be the capacity of the storage facility for TT, where a^^ = 0 ii TT G V^. For simplicity, we assume that all alternative processing units on which a given task r e T can be executed are identical. As a consequence, the assignment of units to tasks may be deferred to the scheduling phase, and the planning problem (PP) can be formulated as follows: Min. s.t.

MTTT

Ir

—

^TTV

_

{T&T,

^TTT

TT €

V- U V+)

(1)

^P^
(reT)

(2)

^^^ <%

{reT)

(3)

= 1

(reT)

(4)


(TT

e V)

(TT

e pp,

(TT

€ VP)

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E

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rreVr ^TTX^TPT

^TTTTT

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=E

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(6) (7)

Planning and Scheduling of Multipurpose Continuous Plants

2161

The goal of the planning phase consists in defining the operating conditions and processing times in such a way that the total workload to be processed is minimized. Constraints (l)-(3) define the feasible domains of input and output proportions, processing times, and production rates. The mass balance constraint (4) says that for each task the amount of products consumed coincides with the amount of products produced. Constraint (5) ensures that the final inventories of the products after the execution of all tasks are sufficiently large to meet the primary requirements and that the residual stocks after having satisfied all demands do not exceed the storage capacities. Constraint (6) guarantees that each perishable product n G V^ produced by some task r can be simultaneously consumed at the same rate by any consuming task r ' G 7^~. Constraint (7) requires that the total production time of a perishable product TT equals the total consumption time, which together with (6) implies that the amount of TT produced coincides with the amount consumed. 3. T H E S C H E D U L I N G P R O B L E M As a result of the planning phase we have obtained a set O of operations i with durations Pi^ which must be scheduled on the processing units subject to material-availability and storage-capacity constraints. The scheduling problem can be modeled as a resourceconstrained scheduling problem with renewable resources k € 7^^, continuous cumulative resources / G 7^^, and sequence-dependent changeover times between operations. By Si we denote the start time of operation i, and S = {Si)i^o is the production schedule sought. We group identical processing units to a renewable resource k G 71^ whose capacity Rk is equal to the number of units in the group. If operation i is executed on a unit of resource /c, the resource requirement is r^^ = 1, otherwise we have Vik = 0. Between consecutive operations z, j G O that are carried out on the same unit of resource k a sequence-dependent changeover time arises for cleaning the unit. By rk{S) we denote the minimum number of units of resource k needed to implement schedule S with respect to all assignments of operations to units. Number rk{S) can be computed efficiently using network flow algorithms (see e.g., Schwindt [7], Sect. 5.2). Further renewable resources may be introduced in a similar way to model the limited availability of secondary resources like manpower or further equipment. In addition, we associate a continuous cumulative resource / G TV with each product TT G P , :S/ •= 0 and Ri :— a^^ being the minimum and maximum inventory levels of /. The (total) requirement of operation i belonging to task r for resource / stocking product TT equals rn — a^^'^rVi- Let Xi{S,t) be the portion of operation i E O that has been processed by time t given schedule S. The inventory of continuous cumulative resource / at time t is r/(S', t) — ^^^Q riiXi{S^ t). The scheduling problem (SP) now reads as follows: Minimize (SP) <^

subject to

mdiXi^o{Si-\-Pi) rk{S) < Rk

{k e 7^^)

Ri < n ( 5 , t)
(/ G n^,

Si>0

{ieO)

(8) t > 0)

(9) (10)

The objective function corresponds to the production makespan. Constraint (8) guarantees that there exists an assignment of the operations to the resource units such that

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for each renewable resource k the number of units used jointly does not exceed the resource capacity at any point in time. The inequalities in constraint (9) correspond to the material-availability and storage-capacity constraints, and (10) are the nonnegativity conditions for start times Si. For solving scheduling problem (SP) we use a randomized multi-start version of a twophase priority-rule based heuristic. A preliminary version of this method for the case of batch production can be found in Schwindt and Trautmann [8]. At first, we perform a preprocessing s t e p where we apply different constraint propagation techniques providing temporal constraints of type Sj — Si> 6ij between the start times of certain operations i, j G O. Those constraint propagation techniques have been adapted from methods proposed by Schwindt [7], Sect. 1.3, and Laborie [9] for discrete cumulative resources. It is customary to represent the temporal constraints as an operation-on-node network A^, which may contain cycles of nonpositive length. In the first p h a s e of the priority-rule based method, we relax the storage-capacity constraints. Using a serial schedule-generation scheme, in each iteration we schedule the start of an eligible operation j at the earliest point in time at which the renewable resources used and all input products are available during the entire execution time of j . An operation j is eligible if two conditions are met. First, all predecessors i of j with respect to a specified precedence order -< must have been scheduled. A classical precedence order is the distance order where i ^ j if the temporal constraints imply Si < Sj but not Sj < Si (see Neumann et al. [10], Sect. 1.4). It proves to be expedient to modify the distance order such that all operations from a strong component in N are scheduled one after another. The second condition says that the remaining storage capacities after the completion of all operations i scheduled in previous iterations must suffice to stock the output products of operation j . After the termination of the first phase we have obtained a schedule S satisfying the temporal, the renewable-resource, and the material-availability constraints. We then introduce, for each intermediate product TT, new temporal constraints of type Sj — Si > Sij which establish a FIFO p e g g i n g of operations producing with operations consuming product TT. The pegging ensures that no shortage of product TT can occur during the execution of any production schedule S satisfying the new temporal constraints. The set of temporal constraints needed can be generated as follows. For the cumulative resource / G IZ^ belonging to product TT, we construct two collections of time intervals / + := [t^_i, ^J[ and / - :=z [t~_i,t~[ {/uL = l , . . . , z / ) respectively referring to operations i G O^ :== {z G O | ^i/ > 0} producing and operations i G O f :— {i e O \ ru < 0} consuming product TT. Let A{S,t) := {i e O \ Si < t < Si -\-pi} denote the set of all operations which according to schedule S are in progress at time t. I^ and / ~ are chosen to be the largest intervals for which (i), IQ =tQ = 0, (ii), t'^
Planning and Scheduling of Multipurpose Continuous Plants

t'l^ -t~ - {Si - Sj),

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otherwise

for each pair {i,j) of operations i G A{S^t^-i) fl O^"^ and j G ^(5,^^-1) fl 0^~. It can be shown that the schedule S generated in the first phase is consistent with all new temporal constraints, i.e., Sj — Si > Sij for all above pairs {i,j). In the second phase, we re-perform the scheduhng of the operations, starting each operation j at the earliest point in time where the temporal constraints from preprocessing and pegging are satisfied and where during the execution of j the renewable resources and sufficient storage capacity for all output products are available. Since the pegging between consuming and producing operations guarantees the material availability at any point in time, the resulting production schedule S is feasible. During the first or the second phase, it may sometimes happen that due to temporal constraints between operations i already scheduled and the operation j selected, the latest time-feasible start time of j is strictly smaller than the earliest resource-feasible start time of j . Then no feasible start time can be found for operation jf, and the current partial schedule cannot be extended to a feasible schedule. Since already the problem of finding a feasible schedule is NP-hard, this kind of deadlock cannot be avoided in a scheduleconstruction procedure. To resolve the deadlock, we perform the following unscheduling step. We determine the set U of all scheduled operations i that must be delayed when increasing the latest start time of j , we increase the earliest start times of operations i GU beyond times Si, and we restart the scheduhng procedure with the modified earliest start times. 4. NUMERICAL RESULTS We have validated the hierarchical approach using an industrial case study of a fast moving consumer goods manufacturing plant with 22 tasks. This case study has been introduced by Schilling and Pantelides [11] and later on taken up by lerapetritou and Floudas [2], Zhang and Sargent [12], and Mendez and Cerda [3]. In the latter paper, it is supposed that any task has to be executed exactly once, which matches our problem setting. Having not yet included multipurpose storages into our model, we have considered Case I, where unlimited intermediate storage capacity is assumed. Mendez and Cerda have computed a schedule satisfying the given primary requirements for the 15 final products and maximizing the sales that can be produced within a planning horizon of 120 hours. Starting from Case I and this schedule, we have generated two sets of test instances for the minimum-makespan problem. Test set A contains instances referring to the original primary requirements. For the instances of test set B we have replaced the original primary requirements by the larger production amounts belonging to the maximum sales. The two test sets contain 10 instances z/ = 1 , . . . , 10 each, which have been obtained by including u copies of each task and multiplying the primary requirements by factor z/. The planning problems have been solved with NLP solver CONOPT 3 under GAMS 22.0. The priority-rule based method has been implemented in C-f-h with random as priority rule. The number of passes has been chosen to be equal to 50 z/. The experiments have been performed on an AMD Sempron personal computer with 1,8 GHz clock pulse

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and 884 MB RAM. Table 1 shows the best makespan Cmax found and the computation time tcpu in seconds for the 10 instances of test set A and B, respectively. For the first instance of test set B we have found a schedule with the optimal makespan of 120 hours. The results for i/ = 2 , . . . , 10 show that the method scales quite well. The computation times are rather modest. The largest instances including 220 tasks, for which 500 passes of the priority-rule based method are performed, are solved within less than 10 seconds. Table 1 Results for test sets A and B V

cpu

^cpu

1

2

3

4

5

6

7

8

9

10

71.29 143.57 213.86 284.16 354.43 425.25 499.00 569.29 642.58 711.86 0.12 0.17 0.52 1.35 3.10 0.30 0.86 2.06 4.35 6.15 120.00 244.01 367.04 489.80 614.18 713.90 857.27 978.71 1079.29 1223.73 0.22 0.27 4.52 6.24 0.63 0.97 1.48 2.19 3.20 0.40

REFERENCES [I] N. V. Sahinides and I. E. Grossmann, 1991, MINLP model for cyclic multiproduct scheduHng on continuous parallel lines. Computers and Chemical Engineering 15, 85-103. [2] M. G. lerapetritou and C. A. Floudas, 1998, Effective continuous-time formulation for shortterm scheduling: 2. Continuous and semicontinuous processes, Industrial Engineering Chemistry Research 37, 4360-4374. [3] C. A. Mendez and J. Cerda, 2002, An efficient MILP continuous-time formulation for shortterm scheduling of multiproduct continuous facilities. Computers and Chemical Engineering 26, 687-695. [4] L. Mockus and G. V. Reklaitis, 1999, Continuous time representation approach to batch and continuous process scheduling: 1. MINLP formulation. Computers and Chemical Engineering 26, 687-695. [5] K. Neumann, C. Schwindt, and N. Trautmann, 2005, ScheduHng of continuous and discontinuous material flows with intermediate storage restrictions. European Journal of Operational Research 165, 495-509. [6] F. Sourd and J. Rogerie, 2002, Continuous fiUing and emptying of storage systems in constraint-based scheduling, in: Proceedings of the Eighth International Workshop on Project Management and Scheduling, Valencia, pp. 342-345. [7] C. Schwindt, 2005, Resource Allocation in Project Management, Springer, Berfin. [8] C. Schwindt and N. Trautmann, 2004, A priority-rule based method for batch production scheduling in the process industries, in: D. Ahr, R. Fahrion, M. Oswald, G. Reinelt, (eds.) Operations Research Proceedings 2003, Springer, Berlin, pp. 111-118. [9] P. Laborie, 2003, Algorithms for propagating resource constraints in AI planning and scheduling: Existing approaches and new results. Artificial Intelligence 143, 151-188. [10] K. Neumann, C. Schwindt, and J. Zimmermann, 2003, Project Scheduling with Time Windows and Scarce Resources, Springer, Berlin. [II] G. SchilHng, C. C. Pantehdes, 1996, A hybrid branch-and-bound algorithm for continuous time process scheduling formulations, in: AIChE Annual Meeting, paper no. 171d. [12] X. Zhang, R. W. H. Sargent, 1998, The optimal operation of mixed production facilities: Extensions and improvements. Computers and Chemical Engineering 22, 1287-1297.

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Priority-rule based scheduling of chemical batch processes N. Trautmann^ and C. Schwindt^ ^Department of Business Administration, University of Bern, Engehaldenstr. 4, 3012 Bern, Switzerland ^Institute of Management and Economics, Clausthal University of Technology, Julius-Albert-Str. 2, 38678 Clausthal-Zellerfeld, Germany Abstract. We present a priority-rule based method for batch production scheduling in the process industries. The problem consists in scheduling a given set of operations on a multipurpose batch plant such that the makespan is minimized. Each operation consumes a given amount of input materials at its start and yields a given amount of output materials at its completion. An initial, a minimum, and a maximum stock level are specified for each material. Some materials are perishable and thus cannot be stored. The operations may be executed on alternative processing units. Processing units require cleaning between certain operations and before any idle time. Due to the constraints on material availability and storage capacity, classical schedule-generation schemes cannot be applied to this problem. That is why we propose a new two-phase approach dealing with the two types of constraints separately. 1. INTRODUCTION In batch production mode, the total requirements of intermediate and final products are partitioned into batches. To produce a batch, at first the inputs are loaded into a processing unit. Then a transformation process is performed, and finally the batch is unloaded from the processing unit. We consider multi-purpose processing units, which can operate different processes. The duration of a process depends on the processing unit used. The minimum and maximum filling levels of a processing unit give rise to lower and upper bounds on the batch size. Between consecutive executions of different processes on the same processing unit, a cleaning may be necessary. Moreover, a processing unit needs to be cleaned before any idle time, in order to avoid ongoing reactions of residuals. In general, storage facilities of limited capacity are available for stocking raw materials, intermediates, and final products. Some products are perishable and must be consumed immediately after production. Material flows can be hnear, divergent, convergent, and cyclic. The short-term planning problem considered in this paper consists in computing a feasible schedule with minimum makespan for given primary requirements. Various solution methods for the short-term planning of batch production are known from literature. Most of them follow a monolithic approach, which tries to solve the problem as a whole starting from a mixed-integer hnear programming formulation of the

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problem. In those models, the period length is either fixed (time-indexed formulations, cf. e.g., Kondili et al. [9]) or variable (continuous-time formulations, see e.g., Castro et al. [3] or lerapetritou and Floudas [6]). A disadvantage of these monolithic approaches is that the CPU time requirements for solving real-world problems tend to be prohibitively high (cf. Maravelias and Grossmann [10]). To overcome this difficulty, heuristics have been developed which reduce the number of variables (cf. e.g., Blomer and Giinther [1]). A promising alternative approach is based on the decomposition of the planning problem into interdependent sub-problems, as it has, for example, been proposed by Brucker and Hurink [2], Maravehas and Grossmann [10], and Neumann et al. [12]. Trautmann [15] decomposes the short-term production scheduling problem into a batching and a batch (production) scheduling problem. A solution to the hatching problem provides the numbers and sizes of batches for all processes and an assignment of the batches to the processing units subject to inventory balance and storage capacity constraints. The processing of a batch is called an operation. The hatch scheduling prohlem is concerned with the scheduling of all operations on the processing units subject to material-availability and storage-capacity constraints. Neumann et al. [12] develop a truncated branch-and-bound method for solving the batch scheduling problem. In this paper, we present a priority-rule based method for batch scheduhng. In Sections 2 and 3 we partly follow the presentation of Schwindt and Trautmann [14]. 2. P R O B L E M S T A T E M E N T Suppose that a solution to the batching problem provides n operations 2 = 1 , . . . , n to be scheduled. For notational convenience we introduce two fictitious operations 0 and n + 1 representing the production start and the production end, respectively. F := { 1 , . . . , n } is the set of all real operations, and V := F u { 0 , n + 1} is the set of all operations. Let Si > 0 he the start time sought of operation i. Then Sn-\-i coincides with the production makespan, and vector S = {Si)i^v with 5o = 0 is called a schedule. Each processing unit can be viewed as a unit-capacity renewable resource with changeovers (cf. Neumann et al. [13], Sect. 2.14). Let IZ^ be the set of all renewable resources and ki G IZ^ be the resource processing operation i. By Pi and Q we denote the processing and cleaning times of operation i, where we suppose that pi = Ci = 0 ior i = 0, n -\-1. The need for cleaning a processing unit generally depends on the operations sequence on this unit. By P^ QV xV we denote the set of operation pairs {i,j) for which passing from i to j always requires a cleaning of processing unit k. Recall that a cleaning of k after operation i also becomes necessary in case of an idle time, independently of the subsequent operation j . Given a schedule 5 , let 0{S) designate the set of all pairs (z, j ) with i y^ j , Sj > Si, and ki = kj. Schedule S is called process-feasible if Sj >Si+p^ Sj = Si+Pi

+ Q, if (2, j ) e 0{S) n Pk or Sj >Si+Pi + Ci, if (i, j ) G 0 ( 5 ) \ P^. )

(fc e no)

(1)

The s t o r a g e facilities can be modeled using so-called cumulative resources (cf. Neumann and Schwindt [11]). For each nonperishable product, there is one cumulative resource keeping its inventory. Let IV be the set of all cumulative resources. For each / G TV, a minimum inventory Ri and a maximum inventory Ri are given. Each operation i EV has a demand ru for resource I e TV. If r^/ > 0, the inventory of / is replenished by ru units at the completion time Si-\- Pi of i. If ru < 0, the inventory is depleted by

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2167

—Vik units at the start time Si of i Demand ru coincides with the amount of product I consumed or produced by operation i, which depends on the batch sizes and input or output proportions fixed during the batching phase, r^i represents the initial stock level of resource /. Let V^^ := {i ^ V \ ru > ^} and 1^~ := {i ^ V \ ru < 0} he the sets of operations replenishing and depleting, respectively, the inventory of / G IV. Schedule S is said to be storage-feasible if at any point in time all inventories are within the prescribed bounds, i.e.,

Ri<

Yl

^ii+ J2

i£V^^ :Si+Pi
^^^ ^ ^^

(^ ^ ^^' ^ ^ ^)

ieV-:Si
We refer to the left-hand inequalities as the material-availability constraints^ the righthand inequalities are called the storage-capacity constraints. The case of dedicated storages, where each storage facility is assigned to one product /, is modeled by choosing Ri to be the safety stock of product / and Ri to be the capacity of the corresponding storage facility. If several products share some common storage facility, we again assign one cumulative resource / to each of those products, where as before ^ is equal to the safety stock. The maximum inventories are set to Ri = oo. To account for the hmited storage space we introduce an extra cumulative resource /' with Rif = —oo maintaining the total inventory of all products stocked in the shared storage. Accordingly, the maximum inventory Ri^ is equal to the capacity of the storage facility. Usually, t e m p o r a l constraints of the type Sj > Si + Sij for (i, j ) G E with E CVxV have to be taken into account as well. 6ij is a minimum time lag between the start of operations i and j . If Sij < 0, then —Sij can be interpreted as a maximum time lag between the start of operations j and i. In case of Sij = Pi, the corresponding temporal constraint is referred to as a precedence constraint. For each operation i GV we set Soi := 0 and Si(^n-\-i)'-=Pi -\- Ci. For each perishable product, batching provides an assignment of producing operations i to consuming operations j . Defining the minimum and maximum time lags 5ij = —Sji = Pi between the start times of i and j ensures that the amount of the perishable product produced by i is consumed by j without any delay. Based on time lags Sij for (z, j ) G E we can compute distances dij between any two operations i, j G V. Distances dij coincide with the minimum time lags between operations i and j that are implied by the prescribed time lags (see e.g., Neumann et al. [13], Sect. 1.3). A schedule S satisfying Sj > Si-\- Sij for all (z, j ) ^ E \s called time-feasible. A schedule which is time-, process-, and storage-feasible is called feasible. The batch scheduling problem consists in finding a feasible schedule S with minimum makespan Sn+l-

3. S O L U T I O N M E T H O D Since more than forty years, priority-rule based methods are widely used for solving shop floor and project scheduling problems arising in manufacturing and service industries (see the early papers by Giffler and Thompson [4] and Kelley [8] or the review by Haupt [5]). Those classical methods, however, cannot deal with material-availability and storage-capacity constraints (cf. Trautmann [15]). The basic idea of our new priority-rule based solution method is as follows. The algorithm consists of two phases. During the first phase, we relax the storage-capacity constraints. Using a serial schedule-generation scheme, the operations are iteratively scheduled on the processing units in such a way

(2)

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that the inventory does not fall below the safety stock at any point in time. Deadlocks are avoided by means of a specific unscheduling technique. Based on the resulting schedule, precedence constraints between replenishing and depleting operations are introduced according to the FIFO rule. Those precedence constraints ensure that the materialavailability constraints are always satisfied. In the second phase, which again applies the serial schedule-generation scheme, the operations are scheduled subject to the storagecapacity and the precedence constraints introduced. In the remainder of this section we explain the schedule-generation scheme of the first phase in more detail. We then briefly sketch the modifications needed for using the scheme in the second phase. Let Pred{j) be the set of predecessors of node j with respect to the strict order {(i, j ) G VxV \ dij > 0 and dji < 0}. i E Pred{j) means that operation i must be started no later than operation j but that conversely, operation j may be started after operation i. Moreover, denote the completed set of operations i already scheduled by C and let S^ := {Si)iec be the partial schedule constructed. We say that an operation j ^ C is eligible for being scheduled if (i) all of its predecessors have been scheduled, i.e., Pred{j) C C and (ii), the inventory levels in all cumulative resources do not fall below the safety stocks after the completion of all operations from set C U { j } , i.e., n ( 5 ^ , oo) + Vji > Ri for all I e1Z^. The procedure is now as follows. At first, we initialize the earliest and latest start times ESi and LSi for all i E V. In each iteration of the schedule-generation scheme we then determine the set E of eligible operations j , select one operation j * G £ according to a priority rule, determine the earliest feasible start time t* > ESj* for operation j * , schedule j * at time t*, and update the earliest and latest start timep of the operations i not yet scheduled. Starting with partial schedule S^ where C = {0} and 5o = 0 we perform those steps until all operations have been scheduled, i.e., until C = V. Sometimes it may happen that due to prescribed time lags between scheduled operations i E C and the operation j * selected, the latest start time LSj* of j * is strictly smaller than time t*. Then no feasible start time can be found for operation j * , and S^ cannot be extended to a feasible schedule. In this case, we perform the following unscheduling step. At first, we determine the set ^ = {z G C | LS* = Si — dj*i} of all operations i that must be delayed in order to increase the latest start time of j * . Then, we increase the earliest start times of operations i from set U to time Si-\-t* — LSj*, update the distances dij accordingly, and restart the scheduling procedure. After having obtained a time- and process-feasible schedule satisfying the materialavailability constraints, we link producing and consuming operations following the FIFO rule (First In First Out). This means that for each I e 1Z^ we iterate the replenishing operations i G Vi^ in the order of nondecreasing completion times Si + Pi and allot the corresponding ru units to depleting operations j G Vf in the order of nondecreasing start times Sj > Si-\-Pi. For each pair (i, j ) G VJ^ xVf for which j consumes units produced by i, we introduce a precedence constraint between i and j by setting Sij := m.ax{Sij^Pi). Subsequently, we update the distances dij and proceed with the second phase of our procedure. When during the second phase we deal with storage-capacity instead of materialavailability constraints, we define the eligible set to be f := { j G ^ \ C | Pred{j) C C, ri{S^, oo) -h rji < Ri for a l l / G Tl^}. In the definition oi £, we use the predecessor sets Pred{j) from the first phase. This allows us to schedule depleting operations before the replenishing operations allotted to them have been added to the partial schedule. The peg-

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ging between producing and consuming operations guarantees that the resulting schedule is feasible. 4. EXPERIMENTAL PERFORMANCE ANALYSIS We have evaluated the performance of our approach based on two test sets. The instances of the first set have been obtained by varying the primary requirements of an example presented in Maravelias and Grossmann [10] (set MG). The second set has been generated by Blomer and Giinther [1] and refers to an industrial case study introduced by Westenberger and Kallrath (set WK, see Kallrath [7]). For each instance we have computed an optimal solution to the batching problem. This solution defines the batch scheduling problem that is (approximately) solved by the priority-rule based method. We have implemented a randomized multi-start version of our method in C under Microsoft Visual C++ 6.0. The tests have been performed on a Pentium III personal computer with 800 MHz clock pulse, where we have prescribed a CPU time limit of 60 seconds for each instance. The results obtained are compiled in Table 1, where n again Table 1 Computational results for the MG and WK test sets Instance MG-01 MG-02 MG-03 MG-04 MG-05 MG-06 MG-07 MG-08 MG-09 MG-10 MG-11 MG-12 MG-13 MG-14 MG-15 MG-16 MG-17 MG-18 MG-19 MG-20 MG-21 MG-22 MG-23 MG-24 MG-25 MG-26 MG-27 MG-28

Primary requirem. (3, 3, 7, 7) (3, 7, 7, 3) (7, 7, 3, 3) (3, 7, 3, 7) (7, 3, 7, 3) (7, 3, 3, 7) (5, 5, 5, 5) (6, 6, 14, 14) (6, 14, 14, 6) (14, 14, 6, 6) (6, 14, 6, 14) (14, 6, 14, 6) (14, 6, 6, 14) (10, 10, 10, 10) (9, 9, 21, 21) (9, 21, 21, 9) (21, 21, 9, 9) (9, 21, 9, 21) (21, 9, 21, 9) (21, 9, 9, 21) (15, 15, 15, 15) (12, 12, 28, 28) (12, 28, 28, 12) (28, 28, 12, 12) (12, 28, 12, 28) (28, 12, 28, 12) (28, 12, 12, 28) (20, 20, 20, 20)

n 32 27 26 33 25 27 26 50 44 49 47 47 46 46 61 63 64 59 61 51 61 81 87 88 76 84 72 69

Sn+l

20 18 16 20 18 17 16 29 28 28 28 25 25 27 40 37 40 34 34 29 34 52 52 CXD OO

43 38 40

Instance WK-01 WK-02 WK-03 WK-04 WK-05 WK-06 WK-07 WK-08 WK-09 WK-10 WK-11 WK-12 WK-13 WK-14 WK-15 WK-16 WK-17 WK-18 WK-19 WK-20 WK-21 WK-22

qTGH

36 42 42 48 44 48 52 48 54 60 68 60 64 66 148 124 112 124 208 184 184 214

n 24 30 27 30 27 33 33 30 41 36 52 43 49 51 82 71 77 75 100 87 94 98

Sn+l

36 44 45 41 42 49 43 44 49 50 65 52 56 61 97 78 90 88 136 105 124 105

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stands for the number of operations to be scheduled and Sn+i is the makespan of the best schedule found. Instance MG-07 corresponds to the original example, for which Maravelias and Grossmann [10] have obtained an optimal solution with the same makespan within a comparable amount of computing time. For two of the instances (MG-24 and MG-25), no feasible schedule could be generated within the time limit. Note, however, that already the feasibility variant of our problem is NP-hard. For the instances of the WK set, we have compared our results to the makespans S^^^ yielded by the time-grid heuristic of Blomer and Giinther [1]. The latter algorithm is based on a monolithic discrete-time formulation of the problem. The results for the large-scale instances of the WK set indicate that the decomposition method is able to find significantly better solutions within a very short CPU time. REFERENCES [I] F. Blomer, H.-O. Giinther, 1998, Scheduling of a multi-product batch process in the chemical industry, Computers in Industry 36, 245-259. [2] P. Brucker, J. Hurink, 2000, Solving a chemical batch scheduling problem by local search. Annals of Operations Research 96, 17-38. [3] P. Castro, A. P. Barbosa-Povoa, H. Matos, 2001, An improved RTN continuous-time formulation for the short-term scheduling of multipurpose batch plants. Industrial & Engineering Chemistry Research 40, 2059-206. [4] B. Giffler, G. L. Thompson, 1960, Algorithms for solving production scheduling problems. Operations Research 8, 487-503. [5] R. Haupt, 1989, A survey of priority rule-based scheduling, OR Spektrum 11, 3-16. [6] M. G. lerapetritou, C. A. Floudas, 1998, Effective continuous-time formulation for shortterm scheduling. 1. Multipurpose batch processes. Industrial & Engineering Chemical Research 37, 4341-4359. [7] J. Kallrath, 2002, Planning and scheduling in the process industry, OR Spectrum 24, 219250. [8] J. E. Kelley, 1963, The critical-path method: Resource planning and scheduling, in: J. F. Muth, G. L. Thompson (eds.) Industrial Scheduling, Prentice Hall, Englewood Cliffs, pp. 347-365. [9] E. Kondili, C. C. Pantelides, R. W. H. Sargent, 1993, A general algorithm for short-term scheduling of batch operations: 1. MILP Formulation, Computers and Chemical Engineering 17, 211-227. [10] C. T. Maravelias, I. E. Grossmann, 2004, A hybrid MILP/CP decomposition approach for the continuous time scheduling of multipurpose batch plants. Computers and Chemical Engineering 28, 1921-1950. [II] K. Neumann, C. Schwindt, 2002, Project scheduhng with inventory constraints, Mathematical Methods of Operations Research 56, 513-533. [12] K. Neumann, C. Schwindt, N. Trautmann, 2002, Advanced production scheduling for batch plants in process industries, OR Spectrum 24, 251-279. [13] K. Neumann, C. Schwindt, J. Zimmermann, 2003, Project Scheduling with Time Windows and Scarce Resources, 2nd ed.. Springer, Berlin. [14] C. Schwindt, N. Trautmann, 2004, A priority-rule based method for batch production scheduling in the process industries, in: D. Ahr, R. Fahrion, M. Oswald, G. Reinelt (eds.) Operations Research Proceedings 2003, Springer, Berlin, pp. 111-118. [15] N. Trautmann, 2005, Operative Planung der Chargenproduktion, Deutscher UniversitatsVerlag, Wiesbaden.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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A rigorous approach to coordinate production and transport scheduling in a multi-site system Carlos Mendez, Anna Bonfill, Antonio Espuna, Luis Puigjaner* Chemical Engineering Department-CEPIMA, Universitat PolitLcnica de Catalunya ETSEIB, Av.Diagonal 647, E-08028, Barcelona, Spain. E-mail: [email protected] Abstract The efficient coordination of production and distribution systems has received increasing attention as companies move towards higher collaborative and competitive environments. This work focuses on the operational level of Supply Chain Management (SCM) to efficiently coordinate short-term production and transport scheduling. Both systems are modeled as detailed scheduling problems using a novel MILP formulation based on a continuous-time representation. A hybrid MILP-heuristic approach is also presented to reduce the inherent combinatorial complexity of the problem. Keywords: Supply Chain Management, short-term scheduling, transport activities. 1. Introduction Production and transport issues constitute a central activity to be considered within any multi-site system. From the operational perspective, both problems have generally been treated separately and independent from any SC environment (Chandra and Fisher, 1994; Ertogral et al., 1998). On the one hand, numerous methodologies have been proposed in the literature to address the short-term production scheduling problem in the chemical industry. Extensive reviews can be found in Shah (1998) and Reklaitis (2000), among many others. On the other hand, the transport scheduling problem, usually referred to as pickup and delivery problem, has been extensively analyzed in the area of Operations Research. Numerous exact and heuristic algorithms have been proposed, focusing mainly on the individual and geographical aspects to reduce the delivery cost (Dondo et al., 2003). A review of heuristic solution techniques for vehicle routing and traveling salesman problems was presented by Marinakis and Migdalas (2002). The decoupled production and distribution processes rely on finished goods inventory to buffer both operations. However, inventory costs and the trend to operate in a just-intime manner are putting pressure on firms to reduce inventories in their distribution chain. Besides this, complex temporal and capacity interdependencies arising between production processes need to be also considered when dealing with SCM in the chemical process industry. The efficient coordination of production and distribution systems remains as an open and challenging area for research. Only few contributions have been reported so far in this direction, and most of them focus mainly on the integration at the strategic and tactical levels. This work focuses on the operational level of SCM to coordinate the short-term production and transport scheduling. The problem has been modelled using a new MILP formulation based on a continuous-time representation. Different heuristic rules are presented which can be easily embedded in the mathematical framework, thus exploiting some problem characteristics to significantly reduce the search space and, consequently, the computational effort.

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2. Problem statement The integration of production and transport scheduling is addressed from the perspective of a production plant of a multi-site system that owns a fleet of vehicles for its logistic needs. Particularly, the scenario considered concerns a multipurpose batch plant, which produces a number of products over time, and maintains an inventory of finished goods that have to be distributed to a number of delivery centers. Information related to the configuration of the plant, the production recipes, and the set of orders to be produced is specified for production scheduling. For transport scheduling, data is required concerning the set of interconnected locations with the distances matrix, the fleet of available vehicles, and the set of transport orders to be fulflUed at specifled due dates. It is worth mentioning that due dates are defined for deliveries at distribution centers and not for batch production as it is usually considered in the short-term scheduling of batch plants. It is assumed that each route consists of a sequence of visits to delivery centers, and must start and finish at the plant. The transport time between centers involves the travel time, a discharge time depending on the amount of material delivered, and a stop time. The integrated problem consists of identifying detailed production (batches to be produced, assignment of batches to processing units, sequencing and timing) and transport schedules (loads, assignment of vehicles to orders, routing and timing), so as to optimize some established criterion, while managing the material fiows and temporal interdependencies between production and delivery centers. Different criteria, ranging from time considerations to economical measures can be easily considered in the model. In this work, measures of flow-time, number of routes, earliness and tardiness have been aggregated into a single cost function. Production cost could also be incorporated. 3. The integrated production-transport sclieduling formulation The following notations are used in the proposed MILP-based formulation: Production sets: P (products); B (batches); t/(processing units); 7 (production tasks). Transport sets: C (delivery centers and production plant); O (orders); R (routes); V (vehicles). Production parameters: /^^ (initial stock of product/?); j^/^,^ (processing time of task t of product/?); ^^^ (batch size of product/?). Transport parameters: ddo (due date of order o); slo (slack time of order o)\ So (size of order o); ^^^ (fixed stop time in center c); ^v (capacity of vehicle v); ^/?v (average speed of vehicle v); Jc,c'(minimum distance between centers c and c'); ut (unloading coefficient); W/(weight coefficient). Production variables: TEbp^ti^riA time of task t of batch b of product/?); Lt^p,b,u (binary variable denoting that task t of batch h of product/? is performed in unit w); Wp^b,p',b' (binary variable denoting that batch b of product/? is manufactured before (after) batch A'of product/?'). Integration variables: ISKpo (initial stock of product /? allocated to order o); Ap^^^o (amount of product/? from batch b allocated to order o); Z^,^,^ (binary variable denoting that batch b of product/? is partially or totally allocated to order 6). Transport variables: ^^(earliness of order o)\ To (tardiness of order o); TSc^r.c (arrival time of route r of vehicle v at center c); r^v,r (departure time of route r of vehicle v); Tf^r (end time of route r of vehicle v); Xo,^' (binary variable denoting that order o is delivered before (after) order o'); Yo,v,r (binary variable denoting that order o is loaded in route r of vehicle v); i/v,r(0-l continuous variable denoting that route r of vehicle v is utilized).

A Rigorous Approach to Coordinate Production and Transport Scheduling

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On the basis of this notation, the detailed production and transport scheduling comprises the following constraints: Production scheduling constraints: General allocation, sequencing and timing constraints were taken from Mendez et al. (2003). However, other existing formulations can be used for production scheduling. Integration constraints: They link the initial stock and the batch production to the orders to be satisfied, since a single order commonly involves multiple batches. • Assign the initial stock of product p to the required orders

•T^isKp^,

ISr

yp

(1)

• Split batch b of product p between required orders bSp>J^Ap^,^,

yp,b

(2)

o

• Satisfy the order size using initial stock and batch production ISKp^, + Ya^p,,,o = So

^P^oe Op

(3)

b

• Determine if batch b of product p is partially or totally allocated to order o So^p,b,o ^ ^p,b,o

^P^b,oeOp

(4)

• Determine the earliest start time of each route based on batch assignment Tsy^r ^ TEbp^^j^^^i,,^^^ - M[2 - Zp^i,^^ - 7^^^,.)

V/?,Z?,o e O^,v,r

(5)

Transport sclieduling constraints: • Every order o must be assigned to a route r of a vehicle v

V

r

• Determine if route r of vehicle v is utilized 7,^,,, < / / , , ,

Vav,r

(7)

• Maximum vehicle capacity q, > Y,^J„^v,r

Vv,r

(8)

O

• Sequencing of consecutive routes for the same vehicle Tf,,r^Ts^^,^,

Vv,r

(9)

• Minimum visiting time for center c, from plant to the first visited center TSc,,,>Ts,, '

+-I!^^-M[\-Y,,,) sp^

y^oe

0^,v,r

• Sequencing and timing of visits to distribution centers

(10)

2174 TSc,

C. Mendez et al >TSc,^

+st,+s^ut +

^^-M[\-X^^^)-M[2-Y,^,^^-Y^,^,^^^ SPv

(11) \fc,c\oe

0^,o'e

0^.,v,r:o
yc,c',oe

0^.,o'e 0^:,v,r:o
•sPv

•

(12)

Overall routing time, from the last visited center to plant

Tf^ >TSc^

+st,+s,ut

+ ^^^i^-Mi^-Y^^^)

yc,oeO,,v,r

(13)

SPv

• Order tardiness T,>TSc,^,^,-dd,-M[\-Y„^,^)

\/c,oeO„v,t

(14)

\lc,oeO,,v,r

(15)

• Order earliness E,>dd,-TSc,^,^,-M{\-Y,^,^^)

Objective function: Minimize weighted total cost Min

Y,^\^v,r

+ ^ 2 ^fv,r " ^'^v^r ) + 2 ^ 3 ^ o + ^A^o

(16)

4. Embedding heuristic rules in the MILP formulation The proposed MILP formulation provides a general representation of the problem. However, the model size may quickly exceed the current capabilities of the MILP codes. In order to make the method more attractive for real-world applications, heuristic rules that take advantage of particular problem features should be embedded in the model in order to reduce the search space without compromising the solution quality. Some heuristic rules are introduced below: EDD-PR: The allocation of batches to orders (Z^^,o) should follow a first-produced first-delivered rule. Since the actual delivery sequence is unknown beforehand, the arrangement of orders according to the earliest due date or the minimum slack time (MSLT-PR) criterion is assumed. EDD-TR: Orders loaded in the same vehicle are delivered {Xo,o) following the earliest due date or the minimum slack time (MSLT-TR) criterion. CLT: Delivery centers are grouped into several clusters depending on the distances between them. Orders loaded in a route of a vehicle must belong to the same cluster {Yo,v,r)' Clustering of centers aims mainly at reducing transport cost. Based on problem characteristics, these heuristics predetermine the value of some discrete decisions and incorporate additional constraints that prune the feasible region. Their effectiveness will usually depend on each problem instance.

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5. Case study and computational results The proposed MILP approach is appHed to a SC comprising a single batch production facility manufacturing two different products that have to be distributed among eight retail outlets geographically spread around 300 km from the production site (Figure 1). The production recipe for both products involves three production stages and eight operations (see Figure 2). Two vehicles with an average speed of 50 km/h and load capacities of 500 and 700 units are available for logistic needs. Weight values of 50, 100, 5 and 20 are considered in the objective function for the flow-time, number of routes, earliness and tardiness, respectively. The first row of Table 1 reports the computational results corresponding to the full MILP. As can be observed, the solution generated is characterized by a high objective value and large relative gap even after Ih. CPU time. This poor performance arises mainly because of the large size of the model. Other rows of Table 1 show the results when the MILP is combined with different heuristic rules. It is worth noting here that the hybrid approach, which generates a sub-problem of the original full MILP, was able to find much better solutions in most of the cases with modest computational effort. This situation clearly reveals that the appropriate incorporation of process knowledge in the MILP not only improves the computational performance but also the solution quality. Table 2 reports the detailed transport schedule for the best solution, which was found in only 26.56 seconds. Figures 3 and 4 depict the detailed transport routing and the corresponding coordinated schedule, respectively. Andorra

PerDigti^h Procedure 1

Zaragoza

^ ^ ^

Vic

(Girona

Tank

load Procedure 2

hold ....,

Procedure 3

'••i^-(^i^0^ |;;,chaf^'"

Tarragona

heat

Reactor l / Reactor 2

discharge

*•

Products A/B

dean

Figure 1. Delivery center locations

Figure 2. Production process scheme.

Table 1. Computational results for pure MILP and hybrid MILP-heuristic approaches MILP model

O.F. (Earliness; Tardiness; Flow-time; Routes)

Const.; binary vars.; continuous vars.

CPU time*

Rel. gap

MILP (no rules)

8474.9 (16.14; 191.51; 79.28; 6)

3476; 537; 310

3600

0.64

EDD-PR

5129.9 (39.03; 1.24; 86.2; 6)

3140; 537; 262

3600

0.17

EDD-TR

No integer solution (CPU limit)

3476; 508; 339

3600

-

EDD-PR-TR

4931.8 (91.12; 0.21; 79.44; 6)

3140; 508; 291

3600

0.07

EDD-PR-TR & CLT

5157.7 (87.71; 3.66; 80.94; 6)

2936; 507; 280

3600

0.13

MSLT-PR

4938.6 (24.93; 0.0; 84.28; 6)

3105; 537; 257

3600

0.12

MSLT-TR

9317.1 (102.4; 204.96; 82.12; 6)

3476; 506; 341

3600

0.76

MSLT-PR-TR

4776.2 (47.24; 0.0; 78.8; 6)

3105; 506;288

203.2

0.0

MSLT-PR-TR & CLT

4776.2 (47.24; 0.0; 78.8; 6)

2901; 490; 292

26.56

0.0

* Seconds on Athlon 3000 with CPLEX 7.0 in GAMS 20.5 (maximum CPU time: 1 h.)

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Table 2. Detailed transport scheduling for the hybrid approach (MSLT-PR-TR & CLT) Route

Vehicle

Start

End

Order size

Site

Arrival

Departure

E

Rl

VI

57.9

63.9

300

GIR

60.0

60.8

0.0

0.0

R2

V2

60.6

71.2

200

LLE

64.2

64.9

2.8

0.0

400

TAR

67.0

68.2

0.0

0.0

400

VIC

103.9

104.8

4.1

0.0

300

PER

108.0

109.3

0.0

0.0

700

VAL

139.0

140.1

0.0

0.0

250

ZAR

146.7

148.4

40.34

0.0

R3

V2

R4

VI

102.5

132.0

114.1

155.6

R5

V2

132.0

148.7

700

VAL

139.0

140.7

0.0

0.0

R6

VI

183.0

193.3

350

AND

187.0

188.4

0.0

0.0

,

Figure 3. Transport routing.

.

.

.

.

.

.

.

.

time(d)

Figure 4. Production-transport coordinated schedule.

6. Conclusions A novel MILP-based approach has been presented for the efficient coordination of the production and transport scheduling. The integration exploits the flexibility of the plant leading to a better management of the inventory profiles and material flows between sites. Since even a simple SC configuration leads to a large-scale optimization problem, the MILP formulation was combined with various heuristic rules that allow reducing the search space and generating good integrated solutions with modest computational effort.

Acknowledgements Financial support received from the European Community (projects MRTN-CT-2004-512233; RFC-CR-04006; INCO-CT-2005-013359) from the Spanish Ministerio de Educacion y Ciencia, from the UPC, and the Generalitat de Catalunya (project 10353) is frilly appreciated. References Chandra, P., Fisher, M. L. (1994). European Journal of Operational Research. 72, 503 - 517. Dondo, R., Mendez, C. A., Cerda, J. (2003). Latin American Applied Research. 33, 129 - 134. Ertogral, K., Wu, S. D., Burke, L. I. (1998). Tech. Rep. #98T-010, Department of Industrial & Mfg. Systems Engineering, Lehigh University. Marinakis, Y., Migdalas, A. (2002). Combinatorial and Global Optimization, 1st Edition.World Scientific Pubhshing Company, ISBN: 9810248024. Mendez, C.A., Cerda. J. (2003). Optimization and Engineering, 4,1 - 22. Reklaitis, G. V. (2000). Latin American Applied Research. 30, 285 - 293. Shah, N. (1998). Foundations of Computer-Aided Process Operations. AIChE Symposium Series, 75 - 90, ISBN: 0-8169-0776-5.

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Multi-criteria Evaluation for the Chemical Industrial Parks Jia Xiaoping^, TianZhu Zhang, Lei Shi

Department of Environmental Science & Engineering, Tsinghua university, beijing 100084, China Abstract Providing useful indicators of sustainable production will play an important part in successfully implementing sustainable production strategies for chemical industrial parks. The aim of this paper is to develop a framework for the evaluation of sustainable production at the system level. It provides multi-criteria hierarchical structure. Multicriteria evaluation through combining qualitative and quantitative analysis is performed. It is expected that the framework will helpful for decision-making of chemical industrial parks. Keywords Chemical Industrial Park, Sustainable Production, Analytic Hierarchy Process, Multi-Criteria Decision-Making 1. Introduction Sustainability issues are becoming more important challenges for the chemical process industries nowadays. Chemical industrial park, a practical form of industrial symbiosis, emerged to achieve the goal of resource sharing among the participating firms leading to the maximization of environmental, economic, and social benefits. To successfully implement sustainable strategies for chemical industrial parks, the evaluation of sustainable production will play an important role. The quantitative and qualitative criteria of sustainability are classified and described as economic, environmental, ecological, thermodynamic, and socio-political factors in the literature. But a generic framework has yet not to be established. In the field of sustainable development, sustainable production "involves the creation of goods and services using processes and systems that are non-polluting; conserving of energy and natural resources; economically efficient; safe and healthful for workers, communities, and consumers; and socially and creatively rewarding for all working ^ Corresponding author. Tel:86-10-62796956, Fax: 86-10-62796955, E-mail: [email protected]

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people"[l]. According this definition, this paper will propose a framework and a procedure of multi-criteria evaluation, which is presented as a tool for decision-making, to support design, construction, operation and management of chemical industrial parks 2. Framework 2.1. Reviews of indicators A number of frameworks or approaches for assessing the environmental and economic performance of chemical processes and products have been developed in recent years [2]. The frameworks and tools include the following: life cycle assessment, materials flow accounting, energy analysis, environmental input-output analysis, exergy analysis, emergy analysis, life cycle costing, total cost accounting, and cost-benefit analysis [27]. Beyond individual process boundaries to a broader systems level (park level), the participating firms link through physical exchange of materials, energy, water, and byproducts. Many sustainability indicators have been proposed from different perspectives in the literature. Eissen et al. showed that new ideas on chemical industry with regard to global sustainability are required. And they indicated that assessment of chemical industry by the integration of economic, social and ecological dimensions is needed [8]. Hu proposed a series of evaluation indicators for assessing eco-industrial systems, such as eco-productivity index, exergy depletion index, environment index, and couple degree [9]. Lou presented extended emergy analysis to describe ecological sustainability of industrial systems [10]. Quantitative evaluation for an eco-industrial park remains an ongoing issue. This paper will presents a comprehensive framework considering resource, environmental, economic, and social aspects in subsequent section. 2.2. Design of the framework Table 1 shows the 3-level (overall goal~criteria~sub-criteria) framework. On the first level is the overall goal (integrated sustainability indictor). On the middle level are four criteria, including resources, economic, environmental and social sustainability performance. Each criterion contains a group of sub-criteria on the third level. Resource sustainability relates to energy, water, and raw materials input, mass of recycled materials, energy and water, and mass of by-products exchanged. Economic sub-criteria include investment, net profit, net present value, operation and maintenance cost, and environmental cost. Environmental sub-criteria relates to potential environmental impacts associated with releases (including global warming potential, ozone depletion potential, eutrophication potential, photochemical oxidation potential, acidification potential, human toxicity potential), mass of air and water pollutant released, and mass of solid waste disposed. Social sub-criteria concerns social impacts, social acceptance, and health and safety risk. 3. Procedure of multi-criteria evaluation In this context, the framework is comprised of not only economic but also resource, environmental and social aspects to enable a state analysis as shown in table 1.

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Obviously, decision-makers need simultaneously consider quantitative cost and noncost related as well as qualitative criteria. The criteria normally are incommensurable. For such cases, methods for multi-criteria decision-making (MCDM) are applied. Table 1. Framework of sustainability criteria Overall goal

Criteria

Raw materials input (CI =0.070) Energy input (C2=0.348) Resource sustainability (Sl=0.25)

Water input (C3=0.027) By-products exchanged (C4=0.348) Land use (C5=0.207)

Economic Sustainability (S2=0.25)

Sustainability (G=l)

Environmental sustainability (S3=0.25)

Social Sustainability (S4=0.25)

notes

Sub-criteria

Investment (C6=0.109) Cost (C7=0.236)

Net profit (C8=0.549) Net present value (C9=0.106) Global Warming Potential (CI 0=0.15) Photochemical oxidation potential (CI 1=0.05) Acidification potential (CI 2=0.15) Eutrophication potential (C13=0.15) Human toxicity potential (C14=0.15) Mass of air pollutant released (CI 5=0.15) Mass of water pollutant released (CI 6=0.15) Mass of solid waste disposed (CI 7=0.05) Social impacts (CI8=0.06) Social acceptance (CI9=0.151) Health risk (C20=0.596) Safety risk (C21=0.190)

Including recycled energy Including recycled water

Sustainable Process Index [11] Including operation, maintenance, environmental cost, etc

Ref. [12] Ref. [12] Ref. [12] Ref. [12] Ref [12] Ref [12]

Ref.m Refm Refm

LMU

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According to the framework, analytic hierarchy process (AHP) is presented for solving such MCDM problem. The AHP allows decision-makers break down a complex problem into a hierarchical structure, and it also allows them combine qualitative and quantitative aspects to fit realistic decision process. The detailed procedure for conducting AHP may refer to ref. [13, 14]. The following part provides a procedure for multi-criteria evaluation for chemical industrial park: Step 1: to determine appropriate system boundary and scope. The system under discussion contains multiple levels from process systems engineering viewpoint, including equipment unit, process, plant, firm, and park level. Step 2: to conduct inventory analysis. Input-output material and energy accounting are performed. Data of social aspects are collected from empirical studies and questionnaires. According to the criteria as shown in table 1, data are classified, characterized and quantified. Step 3: to model four-level hierarchy structure. On the first level is the overall goal. On the second level are the four criteria that contribute to the goal. On the third level are the twenty-one sub-criteria, and on the fourth level are the three alternatives that are to be evaluated in terms of the sub-criteria on the third level. Step 4: to assign relative important based on the preference of decision makers and experts. Pairwise comparison matrix is constructed according to the relative importance of each criterion (sub-criteria) based on Saaty's linguistic measures of importance. At this step, numerical comparison scales are assigned to each pair of criteria and sub-criteria. Step 5: to calculate the total priorities of alternatives. Priorities are calculated based on the judgment matrix. The consistency measure is performed. The different alternatives are ranked according to their priorities. Step 6: to perform scenario and sensitivity analysis to reach a final decision. Preferences of decision makers are modified, scenario and sensitivity analysis are performed according to variations in the weights of the criteria (sub-criteria). 4. Case study To illustrate the procedure discussed above, Ningbo Chemical Industrial Park of Zhejiang province is presented as a simple case study. There are three alternatives of infrastructure system for the expanded planning: Alternative 1 (Al) — natural gas based heat supply system and 25% water reuse in the park; Alternative 2 (A2) — natural gas based heat supply system; Alternative 3 (A3) — coal based heat supply system. Data of input-output accounting of the park are based on the report for environmental impacts of regional development for Ningbo chemical industrial park [15]. The questionnaires are performed to survey the public opinions.

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The matrices of pairwise comparisons are based on (1) operation data of the park; and (2) the knowledge of process engineers, environmental scientists and the public. In table 1, the values alone with criteria or sub-criteria are the local priorities of the factors on the basis of the principal eigenvector of the comparison matrices. Table 2 gives the local priorities of each sub-criterion. On the last line in table 2 are total priorities of the alternatives. To compare the alternatives pairwise with respect to how much better one is than the other based on the sub-criteria on the third level, there are twenty-one 3X3 matrices of judgments since there are twenty-one sub-criteria on level 3, and three alternatives to be pairwise compared for each criteria. All the judgments of relative importance are acceptable according to the consistency of judgments. All the pairwise matrices and procedure of calculation are omitted. Table 2. Priorities based on the twenty-one pairwise comparison matrices priorities ^ ^ ^ - ^ alternatives Al A2 A3 suE^^cnTi Raw materials input Energy input Water input By-products exchanged Land use Investment Cost Net profit Net present value Global Warming Potential

0.532 0.60 0.549 0.333 0.754 0.156 0.122 0.540 0.651 0.200

0.321 0.20 0.297 0.333 0.181 0.185 0.648 0.297 0.223 0.600

0.147 0.20 0.163 0.333 0.065 0.659 0.230 0.163 0.127 0.200

Photochemical oxidation potential Acidification potential Eutrophication potential Human toxicity potential Mass of air pollutant released Mass of water pollutant released Mass of solid waste disposed Social impacts Social acceptance Health risk Safety risk Priorities of the alternatives (ranking)

0.333 0.754 0.333 0.200 0.429 0.627 0.258 0.627 0.600 0.200 0.072 0.4065

0.333 0.181 0.333 0.600 0.429 0.280 0.105 0.280 0.200 0.600 0.650 0.3837

0.333 0.065 0.333 0.200 0.142 0.083 0.637 0.083 0.200 0.200 0.278 0.2141

From Table 2, we can see that Al is the preferred alternative for the sake of sound contribution to four criteria. This choice consists with the decision-maker's preference.

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5. Conclusions A greater concern of the concept of chemical industrial parks engendered by industrial ecology will lead to reach sustainability. It also increase the variation associated with multi-criteria evaluation. The comprehensive framework for indicators is proposed in this work. The detailed procedure of multi-criteria evaluation is presented to measure the sustainability. We have illustrated the use of AHP method that can be use as an aid to evaluate the sustainability for the alternatives of chemical industrial park. The results suggest that the process is easily implementable. There are some future work including further refinement of the data inventory and the linkage of AHP to expert systems in order to facilitate more robust evaluation. Acknowledgements The authors gratefiilly acknowledge the research funding of Natural Science Foundation of China (No. 40501027). 6. References [I] [2] [3] [4] [5] [6] [7] [8] [9] [10] [II] [12] [13] [14] [15]

Lowell Center for Sustainable Production. Sustainable Production: A Working Definition. Informal Meeting of the Committee Members, 1998. Wrisberg, N., Udo de Haes, H. A., et al. Analytical tools for environmental design and management in a systems perspective, Kluwer Academic Publishers, New York, 2002. Cano-Ruiz, JA., GT. McRae, Annual review energy environment, 23(1998) 499. Wang, Y., Xiao Feng, Comp. Chem. Eng., 24(2000) 1243. Hau Jorge L., Bakshi BR, Ecological Modelling, 178 (l-2)(2004) 215 Clark, J. and D. Macquarrie (eds.). Handbook of Green Chemistry and Technology, Blackwell Science, Oxford, 2002. Khan F.I., R. Sadiq, Journal of Cleaner Production, 12(2004),59. Marco Eissen, J. Metzger, Angew. Chem. Int. Ed. 41(2002), 414. Hu SY, Evaluation eco-industrial systems, in proceedings r world congress on recovery, recycling and re-integration (CD-ROM), Beijing, China, 2005. Lou helen, M. A. Kulkami, et al. Clean Technologies and Environmental Policy, 6(2004)156. Krotscheck C , M. Narodoslawsky, Ecological Engineering, 6(1996) 241. Young D.M, Cabezas H., Comp. Chem. Eng.,23(1999)1477. Saaty, T. L.. The analytic hierarchy process. McGraw-Hill, New York, 1980. Jia Xiaoping, Han FY, et al, Comp. Chem. Eng.,29(2004)243 Environmental science research & design institute of zhejiang province, technique report, 2004 (in Chinese).

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Scheduling with high accuracy at low maintenance costs: an approach using discrete event Simulation Martin Jung, Carsten Vogt BASF AG, D-67056 Ludwigshafen, Germany

1. Abstract Making schedules for batch controlled plants in the chemical process industry is a topic often discussed. A wide range of scheduling tools have a separated model and a mathematical optimizer. In this paper we discuss a solution where a event discrete simulation tool is connected to the DCS. With all necessary model information stored in the DCS the scheduling tools is plant independent, where a optimizer (a combination of heuristics and genetic algorithms) generates a proposal for the final schedule.

2. Introduction The proper scheduling of resources such as production equipment, storage capacity or raw materials become increasingly essential to maintain competitiveness of a company in the global economy. Key issues are fast and flexible response to dynamic market requirements as well as cost effective planning in order to keep IT costs low. This paper presents a new concept of detailed scheduling systems to achieve simultaneously four goals of detailed scheduling systems: • • • •

Low implementation costs Low maintenance costs High accuracy Automated optimal initial proposal of schedules

3. Current approaches to scheduling in the chemical industry Planning and scheduling with regard to problems of the chemical industry has been discussed in the past (Kallrath (2002), Sand and Engell (2004)). However these authors focused on improvements of the mathematical treatment of scheduling problems which occur in the chemical industry. However, the practical implementation of scheduling systems in particular with respect to the reduction of the high investment costs that are typically associated with those systems, have not been sufficiently considered yet. Especially in case of batch production processes individual modeling of each production plant is required for a sufficient planning quality and is one of the reasons for high costs. Additionally batch processes typically undergo a continuous improvement. As a consequence the process duration for a batch can shorten considerably, simply because the crew is gaining experience concerning the optimal operation of a process. According to our experience this process improvements can be very frequent and solely improve the batch or recipe time only to

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a small amount. Nevertheless those process improvements will add up to a considerable shortening of batch time (cf. fig. 1)

Advantage by process optimization i ' i ^ | [Ord«:.U . I

Yearl

Fig 1. Improvements of the process leads to shorter cycle times of batches This continuous diverging of reality and scheduling model will either result in increasing inaccuracy of the schedule or in a continuous maintenance effort and resulting high maintenance cost. To overcome these main disadvantages of currently available scheduling systems, a new concept for a scheduling architecture was developed and an operating solution was established for an example process.

4. The example process The examples process is a batch production plant, which can be characterized by the quantities given in table 1. Additional constraints are changeover costs between different products and the customer demand, which is taken from the product storages. With these constraints the focus of the plant was: • Maximization of delivery availability • Minimization of changeover costs • Maximization of production • Minimization of inventory costs Position Products Facilities relevant for process time Routings (ordered set of facilities) per product lots to be scheduled for each scheduling run storage restriction (only final products) Common (and limited) resources Table 1 - Description of the example process

Amount 100 280 3-9 300 2-10 lots

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5. A new concept for scheduling The above described scheduling problem arose a few years ago when none of the commercially available solutions could meet all required issues. Therefore it was decided to develop a new system that integrates as many standard software as possible. The main issue was, that the DCS (Distributed control system) is recipe-controlled, which means all production processes use recipes describing exactly what to do. In other words the DCS does not only open valves controls temperature but also run a complete batch which is defined in a recipe. Figure 2 shows a small example-recipe. Each square represents an operation, i.e. the smallest recognized step in this recipe. For example "start stirring" or "start transfer fi-om 1'^ stage to T"^ stage". An operation using the same unit is represented by a square of same color. The process starts at the top (operation 1) and ends at the bottom (beyond 7 and 8). Each connection is stored in a relation table. That means 1 is connected to 2 and 3. Where 3 is connected to 4, 5 and 6. The sequence 3-6 is an alternative branch and processed only if a special condition is ftilfilled. This recipe structure enables even modeling of complex production processes. In our example process a typical recipe has more than 100 operations and 20 to 30 units are involved. This recipe structure follows the Namur-convention and therefore is represented in many DCS. In the new approach, discussed in this article, the scheduling problem is solved by using the information, stored in the DCS for running the plant.

Panrilel activities

/Utamativ

Figure 2: Example of a DCS recipe

Operation Start End 1 2 1 3 2 7 3 4 3 5 3 6 4 8 5 8 6 8

Unit Start 1 1 1 2 2 2 3 1 3

End 1 2 1 3 1 3 2 2 2

Table 2: Basic operation relation-table

The recipe structure is accessible by an interface, namely a database. Within this database the - so called - basic recipes and also the feedback of running control-recipes is stored. For each operation the actual start- and endtime is in these interface tables.

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Since all information are provided by the DCS, it is possible to use this information for a simulation-engine that is connected to a GUI (Graphical User Interface) and to an ERP-System (e.g. SAP) and which can interpret the recipes. At the end of the simulation-run the results are written back to the DCS (that means the production plan). Optimization

Simulation

Event-Disorete Simulation Model GUI-Data-TableB

GUI(iC3antt)

GUI-Interface-Tables

GUI-Inter&ce-Service

Middleware-Tables

Middleware

PM-Tables

Prodnchan-MGnflger

^^H^ DCS

SAP

Data-Transfer

ItHuit:!^ Oracle-Database

Figure 3: IT-Structure of the Scheduling System

Figure 3 shows the result. The DCS is connected to a middleware which is responsible for the transfer of basic recipes, control recipes, tanks laves and so on in the surrounding systems. In opposite direction, the DCS accepts control recipes where units, materials and starting sequence are defined. The middleware interacts with the SAP and receives new customer demands. On the other hand it informs which products are finished, which amounts of raw-materials were used and so on. The core is the simulation which is capable to read the basic recipes, the control recipes (i.e. the processed production orders), tank and inventory levels and so on. Hence the simulation get all necessary information from the middleware-interface (a pass-through from the DCS) no (or in fact very little) further information is required for making the simulation run. With the iGantt the reported orders from SAP are visible in a order container where the orders can be either placed manually by drag-and-drop on the production lines or optimized by starting the optimizer, so that all required production orders are moved automatically to the adequate production line. 5,1. Advantages of the system One of the main advantages is the reduction of installation costs, because the system is based on either commercial available (i. e. the DCS, SAP, iGantt) or in-house solutions that are product and plant independent (the simulation + optimization). Only the middleware has to be fit to the specific data-transfer behavior. All components have (more or less) standardized interfaces. That means it is possible to put the simulation and the iGantt to another plant, plug it there to the database and let the interface tables to be filled correctly. The simulation read all recipes, processed orders, customer demands

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and so on. After simulation of all planned orders the result is written back to the interface. This reduces installation costs of fiirther scheduling-systems dramatically. Furthermore maintenance costs are reduced. When a new product shall be produced on the plant, a new recipe has to be programmed in SAP and DCS only. If the production capacity is increased by installation of ftirther units beside ensuring accessibility for the DCS-recipes no further changes are required. And even the in the introduction part mentioned required update of the cycle-times occurs automatically. Because all processed production orders are transferred to the interface. The simulation can look up all the old orders, learn all durations and select reasonable durations for each operation. And that means, the "older" the system is, the better it is. It can "learn" from the past.

6. Simulation and Optimization The simulation is based on an event discrete simulator (eM-Plant(D). The Ganttcomponent gets all its information from interface-tables. The content is transferred into datatables where it is possible to hold several "experiments". You can compare several schedules and release one to the interface-tables where it is transferred to the middleware and to the DCS. The database keeps a history of released schedules. The simulator is started by the iGantt and starts to check for new recipes, loads the current plant situation (e. g. the running production orders, inventory and storage tank levels) and can determine when all running orders will be finished. The optimizer itself is a combination of heuristic and genetic algorithms.

7. Supporting the scheduling by optimization Operating the described simulation based scheduling (manual scheduling) is convenient with respect to manual scheduling or manual changes to a given schedule. Additionally the scheduling process was enhanced with optimization algorithms. For the given example process the goals for the support by optimization algorithms were twofold: 1. To speed up the scheduling process by a good first guess which then might by modified later. Due to a scheduling process, which was carried out daily and therefore quite time consuming, this was seen to be beneficial. 2. To generate better solutions than manual planning, with regard to change-over costs, inventory cost, capacity and delivery capability. The optimization was realized by coupling a genetic algorithm to the simulation core. The output of the genetic algorithm determines the production sequence which then was fed into the simulation to determine start und end times of each lot. The quite tight storage situation of the example process led to constrains, resulting in infeasible results to a very high number: If several lots are scheduled early although customer demand arrives later, the storage volume is not capable of taking up enough product. The simulation in this case produces feasible results by delaying the remaining schedule. In addition the consideration of every single recipe step of the production recipe led to quite lengthy

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calculation times, which could not fulfill the requirements for the daily scheduling (<20 minutes time for generating a scheduling scenario, up- and download of data from a 2 GB scheduling database, calculations and visualization). Therefore the genetic algorithm was tailored with scheduling heuristics in order to speed up the scheduling process. An example for such a scheduling heuristic is the preference of campaign production or the preference of already delayed customer demand. Although not being able to find the global optimum anymore (due to the limited degrees of freedom), this tailored genetic algorithm provided a substantial betterment of the objective function within a short response time. References Kallrath, J. (2002) Planning and scheduling in the process industry. OR-Spectrum, 24, 219-250 Sand, G. and S. Engell. (2004) Modeling and solving real-time scheduling problems by stochastic integer programming. Comp. Chem. Eng., 28, 1087-1103.

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An integrated model for planning in global chemical supply chains A. Sundaramoorthy^, S. Xianming^, LA. Karimi^, R. Srinivasan^'^ ^Department of Chemical & Biomolecular Engineering, National University of Singapore, Singapore 117576. ^Process Science and Modeling, Institute of Chemical and Engineering Sciences, Singapore 627833. Abstract Multinational chemical companies are moving from a narrow, company-specific perspective to a broader and global supply chain perspective. Such a move is necessary in today's highly competitive environment so that any cost saving opportunity in the entire supply chain is utilized frilly. Supply chain efficiency and lean manufacturing are certainly important for the continued economic success of these companies. In this paper, we present a novel model for integrated production planning in global chemical supply chains. Our proposed multi-period linear programming model makes the sourcing, production, transfer, and distribution decisions in an integrated manner for globally distributed facilities of a multinational corporation to maximize its overall gross profit. Keywords: planning; chemical supply chains; supply chain management 1. Introduction Recent globalization has led to increased competition and opened new markets, forcing multi-national companies to develop new strategies to adapt to the ever-changing global business environment. Thus, integration (vertical/horizontal) and lean manufacturing are becoming the keys to success for these companies. Some of the major decisions that many pharmaceutical and specialty chemical companies face include make or buy, supply and distribution strategies, warehouse location, value-addition, logistics, transfer pricing, etc. both at the plant and enterprise levels. To improve customer service, these companies maintain significant inventories at plant sites and distribution centers. However, to compete successfrilly, they must operate their supply chain processes in a cost-efficient way, which requires more visibility and better co-ordination across different entities of the global supply chain. In a global enterprise, movement of materials between its entities located in different countries is quite common. Thus, the enterprise is exposed to several cross-border regulatory matters. Transfer pricing is one such important matter that draws a major attention from the enterprise-level management. It is obvious because a significant amount of savings can be tapped by setting these prices carefrxlly. However, there exists a conflict between the general goal of the enterprise in terms of overall profit and the individual entity's goal in terms of its own bottom line. Furthermore, the enterprise needs to be extra cautious while setting the transfer prices as local revenue authorities can penalize for underestimation / overestimation of the prices. However, there is a room for cost-savings that adds to the overall profit of the enterprise through intelligent and careful setting of transfer prices.

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Planning models for global supply chains must take into account these transfer prices (Gjerdrum et al., 2001) and other regulatory factors (Oh and Karimi, 2004). In the first two works, transfer pricing has been modeled as a variable with discrete choices of prices. However, the usual practice is that a single price is mutually agreed by both paying party and revenue agent. The above works have looked at the entities of the supply chain from a broader perspective. In this paper, we present a unified framework with a generalized recipe representation to model disparate entities and their activities in a global supply chain in a seamless fashion with a granularity at the level of production lines. Then, we use this framework to develop a powerful multi-period Linear Programming (LP) model to address this important problem with special focus on pharmaceutical and specialty chemical industries. 2. Problem Description We consider a global pharmaceutical or specialty company that has several primary and secondary production facilities and distribution centers in geographically distributed locations. Some external suppliers provide the raw materials to the primary facilities, which produce active ingredients (AIs) and their intermediates. The secondary facilities in turn receive these AIs from the primary facilities and formulate them into products of different forms and strengths. These products then go to various distribution centers based on their demands. Here, the distribution centers impose the demands on the secondary facilities, which in turn demand AIs from the primary facilities. Since the production is driven by end demands, we view it as a "Pull Process". The primary production that involves several stages of operation could occur at different sites, while the secondary production is usually carried out in a single site. For seamlessly representing various processing steps, we model each facility (whether primary, secondary, or distribution center) as a set of production lines. For instance, each distribution center is viewed as one production line. We use sets Lp, Ls, and L^ respectively to denote all primary, secondary, and distribution lines irrespective of their geographic locations. Note that the distribution lines are only customers and do not appear in production planning. As we see later in the formulation section, production and inventory constraints are imposed only on the primary and secondary lines. We model fiill recipes (primary plus secondary production) of fmal products using recipe diagrams. A recipe diagram (RD) is simply a directed graph (Sundaramoorthy & Karimi, 2004) in which nodes represent the recipe tasks, arcs represent the various materials with unique properties, and arc directions represent the task precedence. Figure 1 shows the RD for the production in a pharmaceutical company with two primary facilities and one secondary facility. One primary facility has two lines (/i and /i), the other has three lines (/s, I4, and /s), and the secondary has one line (Is). Distribution lines /y and /g are not shown in the RD. In this example, Wi (same as w = 1) is the raw material, and W2, W3, W4, and nis are intermediates for the active ingredient me, which is then formulated into various products (my, wg, mg, Wio, and rriu) in l^. These products are finally transported to both /y and 4. Note that we use single line to represent various stages of secondary production that involves weighing & dispensing, granulation, packaging etc. However, this should not limit the application of our model as one can always use separate lines for different stages. Using single or multiple lines for secondary production depends on the level of information that the user is interested

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in. This example clearly shows that an integrated model is necessary to coordinate the production plans of both primary and secondary facilities so that the customer (distribution center) demands are met satisfactorily.

/=1 m=l

Lp- {h

hhUh)

Ls={k} m = \\

Figure 1. Production Recipe Diagram for an example Pharmaceutical Company. Each production line / comprises multiple stages of non-continuous equipment and can perform a set // of tasks in the recipe diagram using long, single-product campaigns. We use / for tasks and m for materials. Each task / consumes or produces some materials. Let Mi denote the set of materials {m e M^ that task i consumes or produces. Note that Mi includes all the different states of raw materials, intermediates, and final products associated with task z. For each task z, we write the mass balance as, y ^ CJ^^ (Material w) = 0

(1)

where, (Jmi is analogous to the stoichiometric coefficient of a species in a chemical reaction except that it is in kg/kg units instead of mol/mol. Thus, Gmi < 0, if task / consumes material m G Mi and cr^/ > 0, if it produces m e Mi. Furthermore, for each task i, we designate a primary material jUj, with respect to which we define the productivity of a line for task /. We assume that wastes generated in the primary production are readily disposed and the excipients required for the secondary production are readily available. Note that the waste materials and excipients are not shown in RD for the sake of simplicity. However, it is understood that Mi includes them as pseudo materials to account for the material balance of task /. Given the above information, our goal is to determine for how long and on which production lines each task must be performed and the inventory and supply details of each material. We assume that the transfer prices between the entities are fixed and known a priori. The objective is to maximize the overall profit of the company.

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3. Formulation The planning horizon Hconsists of TVrperiods {t= 1,2, ..., NT), where we define the interval between due dates [Z)D(,_i), DDi\ as period t. Our supply chain model features two classes of constraints: intra-site and inter-site. The former determines the production campaign lengths and related issues in the primary and secondary lines, while the latter determines the inventory, supply, and demand fulfillment in primary, secondary, and distribution lines. Unless otherwise stated, we write each constraint for all the valid values of its defining indexes. 3.1. Intra-site Constraints Let CLiit denote the campaign length of tasks and Hu denote the total available production time. Then, the sum of campaign lengths of all suitable tasks on line / must not exceed its available production time. Therefore,

Y,CL,,
,yieLp^Ls

(2)

We now define DQnt as the differential campaign length of task i. It is the length of campaign / with a production rate in between the lower and upper rates. Thus, DQ„<(Rll,-Rli,)CL„

yieI„leLp^Ls

(3)

Now, the actual production amount, Pmit, is obtained from the differential campaign length and the campaign length on its lower production rate as given below. (Rl;,CL,j,+DQ,j,) ,ymeMjJeLp^Ls

(4)

^M^i

3.2. Inter-site Constraints The material produced in a line may not feed the same line. The lines that consume this material can be at the same or different location. The consumption of the material varies greatly with the locations of lines. For example, if the consuming line is in the same location, then the material can be consumed as soon as it is produced. On the other hand, if it is far off, then only a fraction of the total production can be consumed in the same period. To model this situation, we introduce a parameter rrtrit to denote the amount within the same period. Let IS^nt denote the internal supply of intermediate or AI between the lines within period t. This is different from the scheduled supply SS^rih which is the supply of material at the end of period t. Hence, ^/-/^ is zero for finished products FP. Clearly, internal supply exists only for the intermediates and AIs. /M/ and OMi represent the set of materials that are consumed and produced in line / respectively. Therefore, IS„,r,^V.„;P.n

yme

IM,,me

OM„l ^l'

(5)

If/„;, denote the inventory of material m for line / at the end of period t, then

X V^meIM,',l'^l

(/5„«.,+SV,)

(6)

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Now, a line can consume materials within a period only if they are available in the inventory at the end of the previous period or if it receives material during the same period. Therefore, >

/5„„, ,\/meIM,

(7)

r^meOMi'J'itl

If the secondary lines cannot meet the demand of a material m in any period t, then the shortage is carried over the next period, i.e., ^./r ^ ^./(.-i) + ^ . . -

Z

^^mnt Vme M^Je L^J

(8)

l'3meOMi'J'itl

where, D^/^ and/^/^ are the demand and backlogs of materials Furthermore, supply from a current period can fulfill the demand from previous periods but the cumulative supply up to a period should not exceed the total demand until that period. Hence, Z

^^™n,-^Z^»./.' , V w e M „ / e L ^

l'^meOMi',l'^l,t'
(9)

t'
Let /^^^ denotes the safety stock for material. In order to maintain the safety stock target, we penalize the deviations of inventory below the safety stock target. We obtain these deviations by using, liu^llu-Imit

ymsM„lGLp^Ls

(10)

Unlike AIs or their intermediates, finished products can share the available storage capacity. Hence, we impose the following storage constraint for the finished products.

^

/„,
yieLs

(11)

m€M,r\FP

The bounds for the decision variables such as campaign length, inventory, supply and backlog are CL,, < H,„ DQ„ < (RJI,-RIJ,)H, , I„„
t'
3.3. Planning Objective Let pm,pcmh hcmih ^//'» ^ud amh dcnotc the purchase, production, holding, transportation, and backlog costs of materials and gmi denote the selling price or transfer price of materials. Let GP denote the overall gross profit of the company, then the objective of our planning model is: Maximize GP =

^ m,reLg,leLD,t

SmiSS^i'u -^iPm ^PCmi)Pmu +

(12)

m,l,t

oml m,l,t

m,l,r,t

m,l,t

mlt

m,l,t

This completes our formulation for the integrated planning in the global supply chains. It comprises objective function (12), constraints (2-11), and bounds.

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4. Example We consider the example discussed in section 2 (See Fig. 1). We used GAMS (Brooke et al., 1998)/CPLEX 9.0 on a Dell PWS650 workstation with Windows 2000 to solve the example. The planning horizon considered in this example is three months (90 days) that comprises three periods, with the time interval of one month (30 days or 720 hours) each. Thus, the due dates for the finished products are DDi =30 days, DD2 = 60 days, DD2, = 90days. However, AIs and their intermediates can be transported within each period from one production line 1 to another production line /'. Appropriate transfer prices are set for materials ms and me for transfer of materials within the company. The optimal solution for this problem was found in 0.035 s. The gross profit of the company is $2,465,720.54. Later, we evaluated our model using a large problem that involves 4 primary, 6 secondary sites, and 8 distribution centers with 60 materials, 56 production tasks, and 27 lines. We solved the planning problem with one year of horizon that consists of 12 periods, with time interval of one month. It took 1.265 s for this larger problem to find the optimal solution. The gross profit of the company is $45,668,326.58. Even though the second problem is much bigger (76,841 vs. 495 variables, 32,101 vs. 409 constraints, 152,596 vs. 1,346 nonzeroes) than the first problem, it required very less computational time (< 2 s). 5. Conclusion We developed an integrated planning model that considers all production facilities in a global chemical company. Our model addresses numerous innovative features. Our multi-period, LP model allows complex production recipes with multiple AIs and their intermediates, material movement among different production/supply/demand facilities and so on. Given the demands of products, revenues and transfer pricing details, our model is able to decide which primary or secondary facilities should produce which intermediates or products in what quantity in each period and how much material should be transferred between the subsidiaries. Also, the plant management can decide how much inventory should be maintained for each material handled in individual facilities within each time period. Hence, our model can serve as a decision support tool for the plant managers to make optimal production plans for a given planning horizon. References J. Gjerdrum, N. Shah and L.G. Papageorgiou, 2001, Transfer Prices for Multienterprise Supply Chain Optimization, Ind. & Eng. Chem. Res., 40, 1650. H.C. Oh and LA. Karimi, 2004, Regulatory Factors and Capacity-expansion Planning in Global Chemical Supply Chains, Ind. & Eng. Chem. Res., 43, 3364. A. Sundaramoorthy and LA. Karimi, 2004, Planning in Pharmaceutical Supply Chains with Outsourcing and New Product Introductions, Ind. Eng. Chem. Res., 43, 8293.

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Information Sharing in A Distributed Enterprise: Impact on Supply Chain Performance and Decision-making LB. Owusu^ and S. Hauan^ * ^Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA In this paper we discuss a spatial decomposition approach for enterprise modeling, highlighting the need for information exchange between distributed enterprise units to solve an enterprise optimization problem. This approach shows potential for significant solution speedup in enterprise optimization in particular as the problem size increases. We motivate our ideas using a multisite scheduling problem and present a distributed computing framework for enterprise modeling, simulation and optimization. 1. Introduction In recent decades, companies in the chemical process industry are competing within an increasingly global marketplace. Globalization implies making capital investments in production facilities where production requirements can be achieved at significant reductions in operating costs. However, the pertinent business decisions remain how best to allocate resources to achieve cost reduction. In enterprise-wide optimization, stakeholders are also interested in integrating information and technology in a consolidated framework to guide decision-making [3]. For current optimization methods and algorithms in Process Systems Engineering to be readily extended to solving large-scale enterprise-wide problems, they must address the computing resource constraints. We present a modeling approach based on a spatial enterprise decomposition which appears effective for addressing the computing resource constraints but with minor reductions in solution quality. In our approach an enterprise comprising several business units is modeled as a collection of interacting subsystems each consisting of an aggregation of specific units. 2. Enterprise Modeling The collection of business units - production and storage facilities, distribution centers - connected through material and information flow defines the enterprise. Mathematical programming constructs are used to formulate enterprise models that capture decision options, specify the flow of material between business units subject to production constraints and objectives. The solution obtained from solving this optimization problem indicates the optimal allocation of resources, and the assignment of tasks that best satisfies the stipulated enterprise objective. * corresponding author email : [email protected]

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2.1. Enterprise Optimization Enterprise modeling research has progressed along two broad approaches classified as planning and scheduling [1]. Planning models prescribe long-term enterprise decisions including demand distribution between production facilities over extended time horizons, ranging from months to a few years. Scheduling decisions are short term and involve generating optimal sequencing and assignment of production tasks to units to meet demand requirements. Current approaches for enterprise modehng and optimization use either using Object-oriented programming (OOP) or Agent-based techniques in a simulation framework [6], or Mathematical Programming (MP) [3] techniques. Simulation methods employ a decentralized problem solution approach well suited for modeling collections of interacting systems. Apphcation of these methods to enterprise modeling has focused more on enterprise dynamics and less on optimization. MP methods provide a formal approach for enterprise representation and model formulation. However, their application to large scale enterprise optimization is constrained by problem size. Recently, spatial and temporal Lagrangian decomposition has been applied in MP models to solve and obtain good solution estimates for large scale problems in reasonable computational time. In this paper we present an approach combining MP based model formulations with simulation based techniques to solve enterprise optimization problems. This approach is based on spatial decomposition and simulated using a distributed computing framework. 2.2. Modeling Approach The simulation framework presented in this paper uses a OOP methods to model enterprise entities. The OOP design is based on the representation of the enterprise as a network where nodes represent sites for manufacturing, storage and product distribution while edges show material and information flow between units [8]. The simulation steps involve iteratively optimizing each subsystem separately and assessing the aggregate solution quality. This is in contrast to solving a single large-scale optimization problem for the entire enterprise system over an extended time horizon. While our approach directly addresses the computational requirements for modehng and simulation, information sharing between spatially defined enterprise subsystems becomes critical. In this context, information refers to model parameters and intermediate solutions needed to solve each optimization sub-problem. The solution to monolithic model is used as the benchmark for performance comparison. By employing this modeling approach the following question arises: what is the impact of information exchange on the aggregate solution obtained in our distributed solution approach? How does the distributed solution compare to monolithic solution? We present a motivating example, a multi-site scheduling problem, to assess the impact of information sharing between business units in a distributed enterprise. As depicted in Figure 3, each enterprise subsystem has specific local objectives; optimization routines are embedded within each subsystem. However, decisions in each subsystem are based on local parameter values and impacted by improvements in the global objective by other subsystems. 2.3. Computing Framework The schematic of our distributed computing framework is shown in Figure 2. The distributed platform is a database-centered system designed as a Blackboard System [2]. Major components of the framework are (a) shared memory, the database, which stores

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data from enterprise simulations {h) knowledge sources, represented by computing nodes and their assigned enterprise computations, and a (c) control module which monitors distributed processes and controls access to information in the shared memory . 3. Case Study : A Multisite Scheduling Problem We motivate our approach for enterprise modeling using a case study of a multisite scheduling problem described in detail by Shah et al [5] with modified parameters. In this study we aim to (a) demonstrate the feasibility of a spatial decomposition approach for enterprise modeling, (6) compare solution quality of decomposition approach to monolithic approach, and (c) assess the effect of information exchange in an enterprise optimization problem. 3.1. Problem Statement In this case study, the enterprise is a simple two-echelon system consisting of geographically distributed sites each generating locally optimal production schedules to meet a single customer demand as depicted in Figure 4. Each plant can produce any combination of products; production cost per product differ between plants. The multisite scheduling problem is stated as follows: given fixed product demands over a specified horizon, find the optimal demand distribution between plants to satisfy demands on time and at minimal cost. The solution to this optimization problem specifies total production costs and total unsatisfied demand. Demand levels in the problem are chosen to be close to the total production capacity across all plants. We present results for a 4 site system with demands specified in Table 1 and highlight initial results on solution speedup in Figure 6. This problem can be solved in a single large MILP for over the entire horizon; this is the simultaneous solution approach. In terms of decision-making and information sharing, this approach is analogous to a single decision-maker having complete information about all plants and allocating production requirements between plants based on this complete knowledge. As such, the simultaneous solution generates the best possible production schedule and yields the minimal total production cost. In our proposed decomposition approach, a single MILP is formulated and solved independently for each plant. Since a global objectives exists, plants collaborate to satisfy this objective by sharing specific information about individual local solution. We solve and compare the solution obtained using our decomposition approach with the benchmark simultaneous solution. Information exchange strategies are presented in Section 3.4. 3.2. MILP Formulation The Scheduling model is based on the State-Task Network formulation using a discrete time representation [4]. Each plant has 6 tasks, i, 4 process units, j and 9 material states, 5, comprised of raw materials, intermediates and products. MILP model constraints include {1) Allocation Constraints: one task per period per time interval, t; no task can occur within a unit, j , until the current task is completed {2) Material balance: at any time interval, inventory level, Sst, for each state is given by the sum of current inventory and production and external receipts less amount consumed and delivered externally (3) Capacity constraints: amount of material processed in each unit cannot exceed avail-

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able unit capacity; total inventory cannot exceed storage limits {4) Demand Constraints: product inventory at the end of scheduling horizon, tp, should match demand. In the decomposition approach, the MILP cost objective is given for each plant, pi = pi', as ^obj

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Crm, Cdl, Cil are raw material, direct and indirect labor costs costs respectively. The last term is a penalty for unsatisfied demand with Wg penalty weight and Slpi'^g as production slacks for s. SR = {51,52,53} and Sp = {57,58,59} are the set of raw materials and products respectively. For the simultaneous approach, the cost objective is given as Z = J2pi ^obj Vp/, which is Eq. (1) added up over all plants. The results are presented in Table 1 for a 52-week scheduling horizon with production demands close to overall production capacity for units present in the system. 3.3. Implementation We use our distributed computing framework for solving the multisite scheduling problem using the spatial decomposition approach. Optimization problems for each plant are solved on different computing and results are stored in a shared database. The database facilitates information exchange between plants using an administrative framework module access control. 3.4. Information Sharing Schemes As explained in Section 3.1, without explicitly exchanging production information between plants, the decomposition approach deviates significantly from the simultaneous solution. In fact, it would require knowing the optimal allocation of production demands to each plant a priori, and solving each plant level MILP with that information. Our proposed solution is to iteratively solve each MILP over the entire planning horizon multiple times, refining production demands for each plant at each iteration based on global information about production slacks, Slg. Plant scheduling objectives, as given by Eq. (1), are modified to include an additive term accounting for weighted impact of global unsatisfied demand at the end of each iteration. For such large scale MILP problems, standard solution approaches include hierarchical decomposition leading to planning and scheduling subproblems, or Lagrangian relaxation and decomposition methods [3]. These methods are iterative and at each solution step, there is a single problem solver with complete information about each enterprise unit. In contrast, each plant in our distributed simulation only has access to its local parameters and variables. How and what information is shared between plants at each iteration step affects the result of the distributed optimization. The simulation algorithm presented also involves iterative computation and communication steps. During computation, each plant schedules production tasks to meet local product demands. In the communication step, each plant posts local inventory levels and unsatisfied demands to a common database accessible to other plants. We compare the solutions obtained using three information sharing strategies with the simultaneous solution

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approach. Complete iteration steps for each strategy are shown in Figure 5. Key differences between the three strategies are {A) (z) no information exchanged between plants {ii) demands are naively partitioned between plants, {B) (i) external module initially partitions demands equally between plants {%%) plants exchange information about unsatisfied demand, Slg, in subsequent iterations, and {C) {%) each plant provides estimates of desired production share based on local capacities {it) plants exchange unsatisfied demand, Slg^ information in subsequent iterations. 4. Results 8z Conclusions Table 1 shows the results for the multisite scheduling problem. As expected, decomposition without information sharing, ^4, gives results with significant cost deviations from the benchmark simultaneous solution, A^. Of the two strategies for information sharing, A, gives the best solution, less than 1% deviation in cost from A^ but with 5x more production slack. Most significantly, however, this solution is obtained faster than the simultaneous solution, an approximately 3x speedup. As shown in Figure 6, speedup benefits using distributed solution approach become more pronounced as the number of plants in the multisite problem increases. In contrast with the simultaneous approach, our solution approach shows no significant increases in solution time with problem size. In this paper we have demonstrated the feasibility of a distributed solution approach for enterprise optimization using spatial decomposition. We first presented a distributed computing framework for enterprise simulations. Using iterative information exchange schemes, we obtained good solutions for demand allocation in a multisite scheduling problem comparable to a benchmark simultaneous solution. Results indicate the potential for solving large problems in reasonable time using a distributed computing framework. 5. Acknowledgments This work was funded by the National Science Foundation under grant NSF/ITR CTS03121771. We acknowledge John D. Siirola for development work on RPI [7]. Table 1 Comparison between simultaneous and distributed methods Approach Inventory Schedule Sls/Ds X 100% Slack Penalty Cost Cost S7 Sg 59 A^ 1.1 2134 594075 0 0 596209 A 616477 0 26 0.3 94544 711021 B 591731 0 10 24 81090 672821 C 577610 598541 0 5.8 0 20931

Solution Time (mins) 36 6 15 13

REFERENCES 1. B.M. Beamon. Supply chain design and analysis: Models and methods. Int. J. of Prod. Econ., 55:281-294, 1998. 2. D.D Corkill. Blackboard systems. AI Expert, 6(9):40-47, 1991. 3. I.E Grossman. Enterprise-wide optimization: A new frontier in process systems engineering. AIChE J, 51:1846-1857, 2005. 4. E. Kondili, C.C. Pantelides, and R.W.H. Sargent. A general algorithm for short-term scheduling of batch operations I. MILP formulation. Comp.Chem.Engng., 17(2):211-227, 1993. 5. N. Shah, C.C. Pantelides, and R.W.H. Sargent. A general algorithm for short-term scheduling of batch operations - II. computational issues. Comp.Chem.Engng., 17(2):229-244, 1993.

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W . Shen. D i s t r i b u t e d m a n u f a c t u r i n g scheduling using intelligent agents. IEEE Intelligent Systems, 1 7 ( l ) : 8 8 - 9 4 , 2002. J . D . Siirola a n d S. H a u a n . R P I : A R e m o t e Process Interface library for d i s t r i b u t e d clusters. Comp.Che.Engng., 29(8):1815-1821, 2005. B . E . Y d s t i e a n d A. Alonso. Process Systems a n d P a s s i v i t y via t h e C l a u s i u s - P l a n k inequality. Systems and Control Letters, 30:253-264, 1997.

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Integration of supply chain management and logistics: development of an electronic data interchange for SAP servers Laureano Jimenez^ and Raiil Munoz^ ^Department of Chemical Engineering, University of Barcelona, Marti i Franques 1-10, 08028-Barcelona, Spain. E-mail: [email protected] ^ Qualitas Information Systems, Diagonal 605, 08028-Barcelona, Spain. E-mail: Raul. Munoz@dmr-consulting. com Abstract In the global chemical industry it is of crucial importance to keep an up-to-date knowledge map of customers and providers in order to minimize supply chain inefficiencies. ERP (Enterprise Resource Planning) involves coordinating and integrating applications both, within and among companies. In this way, there is a centralized database to manage all the corporate information (financial, production or inventory). This paper describes the architecture of the integrate application and the development of the EBI (Electronic Business Integrator) to connect the Elemica network and SAP (Systems, Applications and Products in Data Processing) servers, market leader in this area. EBI is an integration tool of business applications (Enterprise Application Integration, EAI) based in the ESB technology (Enterprise Service Bus), oriented to the intra and inter company business processes. Elemica is a global network of industrial sectors, including worldwide chemical and pharmaceutical companies, created to expand ERP, which key advantages are connectivity, neutrality and security. SAP is an ERP platform software widely use to have an integrated overview of the business and help to take strategic decisions. The system performance was tested with different scenarios, all including logistic processes: integration of filial companies, relationships of different companies operating ant the same site and integration of pharmaceutical companies. Keywords: supply chain management; enterprise resource planning, electronic data interchange; electronic business integrator. 1. Introduction Most companies had to face the integration of several applications developed at different moments in the departments of their companies. Sooner or later, companies had to achieve the amalgamation of those applications with a minimum impact on their performance. To overcome the problems associated and facilitate scalability, the socalled soft integration of applications is a common approach. In this way, the independence of the different applications is maximized thought the use of the integration technologies that had appear in the last decade. Those applications had received a lot of effort, as they cover the core aspect for B2B (Business to Business), B2C (Business to Consumer) or EAI (Enterprise Application Integration). ERP (Enterprise Resource Planning) is used to supervise the supply chain management, as it allows a centralized database for corporate data that can be managed concurrently

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(store, consult, analyze, update and check for inconsistencies). In this way, there is a unique application to manage information, logistics and finances with all customers, from suppliers to manufacturers, including product planning, purchasing, maintaining inventories, interacting with suppliers, providing customer service and tracking orders. ERP can also include additional application modules, as human resources features. In this way, it is clear that ERP involves coordinating and integrating both within and among companies. 1.1. Systems, Applications and Products in Data Processing (SAP) Virtually all worldwide chemical companies use an ERP system, being SAP (Systems, Applications and Products in Data Processing) the market leader (SAP, 2006). SAP, or any other ERP system, supports the core business processes of many industrial sectors. For the chemical industry, SAP includes the commercial processes (demand and production planning, sales order processing), the manufacturing processes (production and procurement), administration processes (revenue and cost controlling), dangerous goods management, and product safety. As SAP covers a wide range of industrial sectors, there is no generic industry solution, although some of the modules can be reused. The advantage of SAP is that they help to inform companies about their most important areas in a way that they can make more informed (and therefore, better), strategic decisions. 1.2. Elemica Elemica (Elemica, 2006) is a global network for buyers and sellers looking for an ecommerce solution to improve supply chain efficiencies. Elemica was developed to expand ERP in many industrial sectors, including the chemical, pharmaceutical and related suppliers and customers. Elemica focus on three key aspects that are a requisite for the users: connectivity (browser-based ERP), neutrality (founded by 22 leader worldwide industrial partners) and security (confidentiality, encryption data). Their leading position has driven to develop standards for the chemical industry to develop ERP connectivity. The network objective is that this combination of advantages will attract additional buyers and sellers, resulting in a broad collection of potential connections for new customers and suppliers. The aim of Elemica is to be used as a SPOC (Single Point of Contact) for all chemical sectors (petrochemicals, inorganic, intermediates...). 2. Process integration Figure 1 illustrates the differences between EAJ and B2B, depending if the applications to be connected/integrated are inside the organization (EAJ) or outside the corporation (B2B). Companies provide their applications following different formats (xml, EDI, ebXML, e t c . ) , but do not include the interface to link these applications, and thus the presence of a middleware is required. The level of transformation, routing and the applications of rules to format and move the information between applications has to follow certain standards that are highly dependent of the application (and thus, very time consuming and routinely work). 2.1. Electronic Data Interchange (EDI) The EDI (Electronic Data Interchange) is a procedure for companies and customers to exchange any type of documents and data (sales, finances, accounting, inventory...). EDI extracts data from the different applications used, manages this information (formatting and routing), and therefore if replaces paper order by automatically generated electronic messages between applications (Cash and Konsynski, 1985).

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Using EDI, the information interchanged is very widely, covering from sales (tracking back any purchase order), to financing and accountability (notifications of payments and rejected demands for payment) passing thought inventory control (product planning). The objectives of all EDI systems is to improve the efficiency of the company by minimizing some of the most common problems associated to the supply chain management (out of stock, low inventories, achieve a certain product rotation...). 2.2. Electronic Business Integrator (EBI) To amalgamate processes, within and among companies, the EBI (Electronic Business Integrator) performs the integration at a higher level. To achieve this, the system architecture requires a message broker that controls and administrates the information (Figure 2), several adapters to communicate between the applications using the standard protocols (ftp, share files, MQseries, smtp/pop, http...) and the format required by the external applications (xml, EDIfact, text, database...). The EBI message broker is able to integrate different middleware, databases and applications. The business decisions is ruled using a decision tree, developed by the EBI manager, that is activated by certain events, and then introduced in the expert system that structure the actions to be taken. In this way, the EBI can access on the one hand, the SAP, where the corporate database is installed, and, on the other, Elemica network. 1

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Figure 3. Architecture of the integrated appUcation that manages the purchase orders and deUvery of products. The cases developed in this paper were implemented using EBI, developed by EDICOM to integrate EBI and SAP. The overall structure of the application developed is shown in Figure 3. The system has two external modules: Elemica and SAP. The process, transparent to the user, is performed using EBI and the software tool developed. In this way, many different users can consult many companies connected to the Elemica network, in order to find the best alternative to purchase a certain product. 2.3. Tool requirements The architecture developed allows synchronous applications (/. e., a unique event starts the local and the remote application, thus shortening the timing for the commercial transactions). The capabilities of the tool developed have to cope with the following aspects: • Reliability: the communications among the applications had to follow standard protocols, with special emphasis to management the abnormal situations when data are lost. The option selected is the storage of all data in a persistent memory (in this case, if there is any problem in the transmission of the data, there is no need to generate the data again). All communications are based in secure web services (secure socket layers or similar). • Efficiency and scalability: the system has to be designed specifically to be able to manage a high amount of information concurrently. The work is distributed among different servers, to improve reliability, redundancy, safety and scalability. To achieve this objectives simultaneously, the connection are developed using SLA (Service Level Agreement). • Diagnostic of problems: all the steps of the process are registered and therefore, if any problem occurs, the audit procedure is simplified, as any problem can be tracked easily. • Management and monitoring: the centralized environment developed favors the training and the learning curve of the users. In addition, the maintenance of the system is easier. 3, Case studies The cases developed had some common aspects, all of the related with real-time information available (purchase orders, confirmations of orders, good outputs...). The cases were selected in order to cover a wide range of the casuistic that can be found in the industry and to show the capabilities of the tool developed. All the scenarios tested to verify the advantages of the adapter developed are related with logistic processes. The three scenarios selected are:

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• Relationship of different companies operating at the same site that share some resources (utiHties, commodities...) and had a number of material flows that had to be included in the invoices, delivery notes, orders, etc... • Integration of different filial companies: this case is similar to the previous, except that the transport between the sites had to be considered. • Integration of pharmaceutical companies, providers and customers. In this case, the number of products and customers is very high, while there are just a few providers for each raw material. 3.1. EBI application adapter Each component or application needs an adaptor to link its input and output with the BUS (a BUS is any subsystem that transfers power or data between different computer components). The adaptor, cdXltA EBI application adapter, allows: • Transform the information from the application data into the desired output format. • Introduce and retrieve data from the BUS thought the EBI message broker. The internal communications of the input adapter with the application is done following one of the standard protocols (ftp, shared files, MQseries, smtp/pop, http...), and according to the format required by the internal application (xml, EDIT ACT, txt, database...). The output of the adapter communicates with the EBI message broker using web services. The SAP adapter developed was built in Delphi and C++ (Delphi, 2006). The use of this adapter involves several customizations of the SAP system. It is necessary to use the ALE (Application Link Enable) layer in order to connect both systems, SAP and EBI through the adapter developed. The connection was established using RFC (Remote Function Control) technology. A RFC is a document describing the standards that allows any transfer of information in any type of computer network (/. e. tcp/ip). The RCF used is the API (Application Programming Interface) provided by SAP and described in RFC API (2006). 3.1.1. EBI message broker The most common ways to link EBI with any other application (Figure 4) are: • EBI provides two generic adaptors {EBI application adapter and EBI web adapter) to link any web application with the EBI message broker. • Developing adaptors to link the specific database or application. The adaptors can be reused, without any need to develop new code, as they hide the complexity of the task by transforming (and formatting) the information in a transparent way to the user.

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References Cash, J., Konsynski, B. 1985, Harvard Business Review, 63 (2), 134-142. Edicom web page, available at http://wv^rw.edicomgroup.com/en/index.htm (acccessed February 2006). Elemica web page, available at http://vv^v^rw.elemica.com/ (acccessed February 2006). Delphi web page, available at http://wsvw.borland.com/us/products/delphi/index.html (acccessed February 2006). SAP web page, available at http://www.sap.com/industries/chemicals (acccessed February 2006). Remote Function Control (RFC) Application Programming Interface (API) for Systems, Applications and Products in Data Processing (SAP) web page, available at http://help.sap.com/saphelp_46c/helpdata/en/22/04287a488911dl89490000e829fbbd/frameset .htm (acccessed February 2006).

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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A novel combined approach for supply chain modeling and analysis Fernando D. Mele, Carlos A.Mendez, Antonio Espuiia, Luis Puigjaner Chemical Engineering Department, ETSEIB, Universitat PolitLcnica de Catalunya, Av. Diagonal 647, E-08028, Barcelona, Spain Abstract The framework here presented consists in representing the supply chain (SC) network of a company by means of a multi agent-based simulation model, which mimics the system dynamics responding to uncertainties through local dispatching rules, invoking when necessary local optimization modules, and solving conflicts through message exchange. The simulator involves a number of independent and well-defined agents, each of them modeling an entity or node of the real-world SC. A central agent coordinates the global operation of the system. The hybrid approach proposed in this work offers the advantages of the multi-agent system to model the SC together with the optimization capabilities of local not so large mathematical programming models to solve in an efficient manner the decision problems that the central agent faces along one simulation run. The results so far obtained are very promising. Keywords: Supply Chain Management, software agents, mixed-integer linear programming. 1. Introduction Supply Chain Management (SCM) involves the decision-making related to material and information management through the entire Supply Chain (SC), from the initial suppliers to the final customers (Shapiro 2001). Many of present SCM approaches consider operations research optimization models such as mixed-integer linear programming (MILP), in which the modeling and optimization tasks are tightly connected. In addition to the incapability of these approaches to efficiently deal with SC dynamics and uncertainty, the quality of the representation is relatively low so as to avoid a high degree of complexity in the resulting model. Therefore, the simulation-based optimization approaches seem to be a good way to undertake these shortcomings. Software agent-based systems have resulted to be an effective tool for solving complex problems. Since they are built by combining autonomous cooperative computer programs or agents, they typically perform significantly better than these individual programs operating alone (Siirola et al., 2003). In particular, SC networks are truly complex systems whose good performance relies on the coordination of a number of entities. These entities are dynamic, geographically dispersed and disparate in the tools used for planning and managing their operation. Therefore, an approach that considers

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SCs by means of agents and discrete event-based simulation will be better suited to tackle SCM problems. The framework proposed is aimed to represent the SC network of a company by means of an agent-based simulation model, setting aside, for the moment, the global optimization issues. This model emulates the dynamics of individual components as well as the emergent interacting behavior. Since it mimics the natural structure and interacting mechanism of a SC, it has the additional advantage of being able to easily reconfigure when the chain structure changes. The simulator includes a central agent that coordinates the global operation of the system in such a way that its behavior approximates as much as possible the real behavior of a SC with centralized management and a high degree of information sharing. A problem ever arising when representing SC networks using software agents is the nature of the tools utilized to model the different decisions the central agent has to make. For example, if the network has to supply some material to a given customer, it will be desirable to apply some criteria in order to allocate this production to one of the available manufacturing plants. Usually, these decisions are made on the basis of if-then rules, which imitates the reasoning of human managers, or techniques belonging to the field of the so-called artificial intelligence (neural networks, fiizzy logic, etc.). In this work, it is proposed to use a hybrid system which offers the advantages of the modeling capabilities of the multi-agent system together with the optimization capabilities of the local not so large mathematical programming models to solve in an efficient and more rigorous manner the decision problems the central agent faces several times along one simulation run. The methodology used is presented in next section. Then, a case study illustrates the approach by presenting a comparison with other kinds of modeling such as if-then rules. Finally, the conclusions and fiiture work are drawn. 2. Methodology A software agent-based system has been developed so as to implement a SC model. This system not only allows for SC modeling, but also for analysis and optimization (Mele et al., 2005). The foundation of this simulator is a unified agent-based modeling of the entire SC network in an object-oriented fashion. This SC model provides an environment where all business processes can be emulated. Figure 1 illustrates the modeling of the SC based on this idea. Materials and information are modeled as objects. Each SC entity such as manufacturers and third party-logistics are modeled as agents. The internal departments of these entities are modeled as sub-agents. The material and information flows are emulated by the exchange of the objects between these agents. In the figure it can be seen how the real system is modeled using agents (one for each site plus a central coordination agent), being each of them modeled, in turn, by a set of sub-agents. The agents in the system are emulation agents if they represent external customers, external suppliers, or an entity belonging to the SC network such as manufacturing plants, storage facilities, retailers, etc. The agents representing external customers or suppliers emulate the behavior of real or hypothetical entities in order to consider them in the simulation runs. These agents have specific models mimicking this kind of entities. Otherwise, the other emulation agents are called site agents. This is very

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important from the implementation point of view as it is possible to define a generic agent or class for every SC site and then generate from this class some children classes, by inheritance, with the specific features for representing factories, distribution centers or whatsoever entity of interest. -1—3

pap

Distribution center

Warehouse 2 Suppliers

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"-tailer2

j

ll

Factory 2

Real Supply Chain

I

I

Agent-basec!,.simu1ator

J_ ""^53

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Figure 1. Agent modeling of the entities in a SC network. The central agent is a non-emulating agent that acts as a coordinator for the system and whose mission is related to communication amongst the emulation agents as well as the integration with the analysis and optimization tools. These modular tools are: a forecasting module that provides accurate values to simulate the customer demand; a negotiation module for signing long-lasting contracts with external suppliers and customers; three modules that provide the value of SC performance indicators (economic indicators such as profit and costs, financial indicators and environmental impact indicators); and an optimization module based on metaheuristics, which takes into consideration the values of the performance indicators in order to improve the whole SC operation. The objective of this approach is to show the flexibility of the agent-based simulator to include an heterogeneous set of tools. In this case, the central agent heuristic rules are substituted by a mathematical program that dynamically optimizes the assignment of the request for materials (RFM) to different candidate suppliers. Every time, during a simulation run, that the central agent receives an RFM from some entity of the chain, it solves the MILP-based model that can be briefly describes as follows. The objective function takes into account a penalty for tardiness, and transportation, shortage, production and inventory costs (Eq. 1).

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F.D.Meleetal

tdcT + ^trcJ, /

+^'Pp^p+J^i p

Pn,pSl^p+i^i,pkp

-SfJ

]

(1)

i

where the subscripts / and/? account for the alternative supplier entities and the different products, respectively, tdc, trCj, spp, prip and iriip stand for the tardiness, transportation, shortage, production and inventory unit cost, respectively. The main inputs of this model are the required amount of product/?, r^, and the due date d. The outputs are Yf, binary variables denoting that plant / suppHes one or more products, Wi, binary variables denoting that plant / will manufacture one or more products, r, the tardiness, Sfj^p, the amount of product/? directly supplied from inventory of plant /, S^i^p, the amount of product p to be manufactured in plant /, S^i^p, the amount of product p that cannot be met (shortage) because of cost or operational reasons, and Dj, the production dehvery time from plant /. The constraints for this model are related to product inventory (Eq. 2), Slp
to production (Eq. 4), 5,^^
(4)

to the amount of product supplied from plant / (Eq. 5), 5,(p+5,^
(5)

to the delivery time of plant / (Eq. 6), and finally, to the requirement tardiness (Eq. 7). D,>rt,W, +5]^f^mr,,^ +tt, -M{1-Y,) \/i (6) p

T>d-Di \fi (7) The remaining parameters are rti, the time at which plant i will be available for manufacturing, mvip, the manufacturing rate of product / in plant/?, ttt, the transportation time from plant /, and U^p, the inventory of product/? in plant /. After each simulation run, the outputs are used to calculate the total SC profit. This economic indicator will be used to compare the performance of the SC using different criteria to allocate the REM. 3. Case study 3.1. Scenario description The approach has been tested on a SC network producing two products A and B and involving six entities connected as shown in Figure 2. The material flow moves from the three supplier plants to the customers of the two retailer facilities and the ordering flows move in the opposite direction. When Plant 4 needs raw materials X or Y, an order is sent to the central agent who has to decide which of the plants will satisfy the request. Two different situations has been considered. In the first one, the decision is

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based on an heuristic rule: the order is placed on the supplier with the highest inventory level. Figure 3 shows the states and transitions diagram representing this rule in the simulator. In the second case, the decision lays on the MILP-based optimization model described in the previous section.

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Figure 2. Supply chain for the case study. Control B ,

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[!nvB1 ===iv82j{OF^2=0;z83=0, control/ orderB=0; control_c2=t; orderB^law2(IB,HB_DC1 ,HB_Ret1 .SProdlB); vectorB=ml .max(ml .horzcat(ln vB1 ,lnvB2 ,lnvB3)); V_

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Figure 3. Implementation of the heuristic rule. 3.2. Results The agent-based simulator with all necessary embedded rules has been implemented using two toolboxes of Matlah/^: Stateflow and Simulink. The optimization model is solved using GAMS-CPLEX. GAMS is interfaced with MATLAB using the library developed by Ferris (1998). Several simulations have been made in order to test the performance of the two cases exposed. Figures 4 shows the inventory profiles for the nodes of the SC obtained in both cases. On the whole, the inventories are lower for the second case, but the difference is not so evident in the suppliers (Plants 1 to 3) probably because in the first case the heuristics succeed in keeping a low level in the suppliers (the central agent takes materialfi"omthe supplier with the highest inventory level). If the total SC profit is compared, the operation of the chain with decisions based on optimization (250 M$) clearly outperforms the operation based on heuristics (232 M$).

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Plant 2, product X

120 r heuristic-based decision

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'

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4, Conclusions The main advantage of modeling heuristic rules is that they imitate the actual behavior of the decision-makers in a company, however, the possibility of introducing an internal optimization model helps to obtain some guidelines to improve the operation such as it is shown by the results obtained so far. This work shows how flexible is this agent-based approach for SCM. It allows not only for detailed modeling and simulation but improving the operation by means of integration with optimization tools both at a local level (e.g. scheduling, orders allocation, etc.) and at a global level (Mele et al. 2005). It might be considered as an alternative to the huge monolithic mathematical models for SCM. Acknowledgements Financial support received from the "Generalitat de Catalunya" (FI programs) is fully appreciated. Besides, financial support from GICASA-D (10353) and PRISM (MRTNCT-2004-512233) projects is gratefully acknowledged. References M. C. Ferris, 1998, Matlab and GAMS: Interfacing optimization and visualization software. Technical report, Computer Science Departement, University of Wisconsin, Madison, USA. F. D. Mele, A. Espuna, L. Puigjaner, 2005, Supply Chain Management through a combined simulation-optimization approach, European Symposium on Computer-Aided Process Engineering (ESCAPE-15), pp. 1405-1410. J. F. Shapiro, 2001, Modeling tha supply chain, Duxbury Press. J. D. Siirola, S. Hauan, A.W. Westerberg, 2003, Towards agent-based proces system engineering: proposed framework and aplication to non-convex optimization. Computers and Chemical Engineering, 27, pp. 1801 -181.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 PubUshed by Elsevier B.V.

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A Hybrid Approach Using CLP and MILP Applied to Tank Farm Operation Scheduling S.L. Stebel," F. Neves Jr.," L.V.R. Arruda" ""UTFPR, CPGEl Av. Sete de Setembro - 3165, Curitiba/PR - 80230-901, BRAZIL Tel: +55 41 3310-4701 Fax.: +55 41 3310-4683 E-mails: {stebel, neves, arruda}@cpgei. cefetpr.br Abstract This work develops an optimization model to aid the operational decision-making in a real world tank farm scheduling. The short term scheduling of tank farm is a hard task because the specialist has to take into account issues concerning plant topology, mass balances, transfer policies, resource constraints, demand pattern, and changeovers. So this operational decision-making is still based on experience with the aid of manual computations. The main goal of this work is to reduce the difference between a theoretical model and the practical needs. In order to reduce this difference the formulation addresses a new aspect related to the operator procedure. When the operator executes the programmed activities many tasks are delayed or advanced for personal convenience. This fact can cause bottlenecks in the system operation. In order to avoid them, some considerations about qualitative variables are inserted in the model. So that, the generated scheduling tends to be more practical to represent the qualitative variables by means a fuzzy system. Moreover the scheduling problem is modeled in a unified framework, which uses Constraint Logic Programming (CLP) and Mixed Integer Linear Programming (MILP). This approach had a computational time smaller than only the MILP model, and is able to define a good solution in few seconds. The proposed model can be used to test and correct new operational conditions and scenarios rather than to just determine the scheduling of regular activities. Keywords: Scheduling, Constraint Logic Programming (CLP), Mixed Integer Linear Programming (MILP), Tank Farm, Fuzzy Systems. 1. Introduction The short-term scheduling of activities in refineries has received a special attention in the last years from academic and industrial communities. The main reason for this is a constant increase in the oil processing. According to Magalhaes (2004) in a typical refinery several activities are performed by the scheduler: crude receipt; process units operations modes; inventory management; blending. This work focus on the inventory management activities in a tank farm, because when the refinery load increases the total tankage, in general, remains the same. Most of the existing literature in the refmery scheduling problems are based on mathematical programming (Shah, 1996; Pinto et al., 2000), more specifically MILP (mixed integer linear programming). It also has been used CLP (constraint logic programming) for generical scheduling problems. More recently have appeared the integration CLP-MILP as a promising alternative (Hooker, 2000; Jain and Grossmann, 2000). In this context, this work develops a hybrid approach based on CLP and MILP techniques, with aims to reduce the CPU time. Moreover, to minimize the difference between the theoretical model and the practical needs a fuzzy system is used to represent qualitative variables. Such variables are associated to the exchange cost parameters, which are included in the scheduling model objective

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function. For a given scheduling solution these variables represent how the user executes the set of activities. This cost is explained in section 3.1. 2. Problem Statement The short-term scheduling of activities in the refinery tank farm involves a set of decisions that must be taken by the scheduler. Figure 1 illustrates the main aspects of the tank farm structure. The tank farm area is composed by raw materials and intermediate/final product tankages. The tankage scheduling must take into account issues concerning physical restrictions, initial amount of product, operational constraints, flow rates, and demands. Considering these issues the scheduler must determine the scheduling involved in all operations. By the way, it has been based on personal experience, with the aid of manual calculations.

Local Market

Tenninal (Crude Receipt)

Refmerj;

Figure 1. Overview of Tank Farm Structure

3. Integrating CLP-MILP One common limitation reported of scheduling models based on mathematical programming, especially MILP with discrete time representation, is the explosion of CPU time. In order to reduce it, recently, the CLP-MILP integrated approach has been used in scheduling problems (Magatao, 2005). According to Focacci (2000) the CLPMILP integration can be made for algorithmic and engineering approaches. In this work it is used the second one. It is established a separation between the modeling and problem solving.

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An MILP model of this problem was proposed by Stebel (2003) and it is used as a basic model to the CLP-MILP hybrid model proposed in this paper. Some constraints are rewritten in CLP, others are maintained in MILP, and, finally, a constraint set links MILP and CLP variables. The OPL language is used to implement and solve the models (Hog, 2002a; Hog, 2002b). The most fundamental concept in OPL for scheduling applications is the activity. An activity can be thought of as an object containing three data items: a starting date, a duration, and an ending date (Hog, 2002a; Hog, 2002b). The activity created in the model is represented by operationrj,„, which has an starting date TSrj,„, and an ending date 77v,/,n (equations 1 and 2). TS^. ^ = operation^ ,, .start Vr G R, i e I, n G N TF ^ ^ = operation^ .^ .end

(1)

Vr G R, i G I, n G N

(2)

Constraints (3 and 4) are used in the hybrid model. These constraints specify that operation A precedes B. Furthermore, they avoid the overlap among parallel activities.

fe,

>rF,, J&(rF,,„. ^^)SC[TF,,„. ^O)J^F4^^^,,,„^„, =I)

(3)

V(rl,r2)6R,iGl,(nl,n2)€N (^^j ^2 / «i ni - V ^operation^^. ^^ precedes operation^^. ^, V(rl,r2)GR,iGl,(nl,n2)GN

(4)

Other important OPL resource, called generate, allows that the search tree be explored from a specific discrete variable (Hog, 2002a; Hog, 2002b). In general, this resource allows the solver to obtain solutions faster than when no command is used. In this paper, the binary variable RDrj^ which indicates whether tank r is sending to unit i in the slot«, is used to generate the search tree (expression 5). generate[RD^.^) VrG R,iG I,,nG N

(5)

3.1. Fuzzy Systems Qualitative variables are normally not considered in scheduling models, for instance, how the system operator normally executes the activities*. These aspects are a model refinement, and represent variable imprecision. In a practical standpoint the activities will be delayed or advanced resulting in dynamic bottlenecks in the system. To avoid them and reduce the difference between the theorical model and the practical needs a fuzzy system is proposed to model qualitative variables. Fuzzy set theory, introduced by Zadeh (1965), is a generalization of conventional set theory to represent vagueness or imprecision in everyday life in a strict mathematical framework . Fuzzy interpretations of data structures are very natural to formulate and solve various real-life problems, with a good intuitive appeal. In this paper, the implemented fuzzy system is composed by one output variable (exchange cost) and three input variables (difficulty, maneuvers, and period). Figure 2 illustrates the fuzzy system. Exchange cost: this variable represents the set of all operations during each hour of the day. Is composed by five pseudo trapezoidal functions are used to represent the

Set of tasks, for instance, open / close valves and, turn on / turn off pumps.

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attributes that this variable can be assumed: very low, low, medium, high, and very high (see figure 3).

Fuzzy Inference Engine

high

0.5

0.75

very j^jgj^

Exchange cost

0

period

medium

0.25

1

Figure 3. Exchange Cost

Figure 2. Fuzzy system

Difficulty: difficulty to open/close valves. Is composed by three piece-wise linear (triangular) functions with values between 0 and 1. These functions make the mapping of the values that each variable can assume with the attributes low, medium, high (see figure 4). Maneuvers: number of actuated valves to execute an operation. Is composed by three piece-wise linear (triangular) functions with values between 0 and 1. These functions make the mapping of the values that each variable can assume with the attributes few, medium, and several (see figure 5). Period: the 24 hour clock is divided in three different periods of labor (first- 23:00 to 7:00 hour; second 7:00 to 15:00 hour, and third 15:00 to 23:00 hour). There is a period of time where an overlap occurs between two periods. Is composed by seven piece-wise linear functions with values between 0 and 1. These functions make the mapping of the values that each variable can assume with the attributes endTl, T2, T3, TRl, TR2, TR3, and iniTl (see figure 6).

Figure 4. Difficulty

Figure 5. Maneuvers

Figure 6. Period

The base rule has sixty three rules. For instance, //"difficulty is medium and maneuver is several and period is TR2 then the exchange cost is very high and the implemented system uses a Mamdani inference engine to compute the final exchange cost (Stebel, 2003). The obtained values are represented by a parameter matrix that is inserted in the model objective function. In order to represents it in the CLP-MILP model some constraints are added to the original model.

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The objective function in shown by expression (6). It defines the operational cost minimization. Such cost is influenced by three factors: the pumping cost, the electrical cost, and the exchange cost. The first one represents the use of the resources: pumps, valves, and tanks. The second one is the time period that a product is pumped during on-peak hours. The third one illustrates the manner that operator executes the activities. min

(6)

reR fe(/aU/Z,) «eA' feHF

4. Results This section considers an instance of a real Liquefied Petroleum Gas (LPG) tank farm scenario. This area has five tanks that receive product from process units and external pipelines and send it to local and external markets. The considered scenario has several parameters that are used in the data preprocessing. The state of all tanks at the initial time is known (sending, receiving, or waiting). Table 1 shows a comparative* between MILP and CLP-MILP approaches. In the same way that Magatao (2005) the obtained CLP-MILP model was, in average, faster than the MILP model. Therefore, a rigorous formulation considering practical issues in the execution of the scheduling could be modeled without prohibitive CPU time. The modeling and optimization tool ILOG OPL Studio 3.6.1 is used to implement and solve the models on a Pentium 4, 2.4GHz, 1 Gbyte RAM. Table 1. Comparative Results between MILP and CLP-MILP approaches Number of variables Number of constraints Solution [$] CPU time [s] - (first/optimal) solutions

MILP 5,441 15,170 104 (520/10,582)

CLP-MILP 5,232 7,465 104 (12/852)

Figure 7 illustrates a Gantt chart where it is possible to verify the influence of electrical and exchange costs in the reported scheduling. The HPj and HP2 labels represent the cost variation (on-peak demand hours). Normally this value is five times greater than the normal cost. It is possible to notice that only the receiving of continuous process occurs in that period. The TRl, TR2, TR3 labels indicate the exchange period. In these periods the initial times of the activities should be avoided. It is possible to notice that only one operation occurs in that period TR2 (see the hatched circle in the figure 7). If the exchange cost was not considered the obtained solutions could have been unpractical. So that, the system operator tends to advance or delay tasks for personal convenience. As a potential consequence, bottlenecks are created and operational costs increase. For simplicity the solver settings in both approaches are maintained in the default option. For details see Hog (2002b).

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TRl

T R 2 ^ ^ i TR3

ii

TRl

-r-rt

i i'if H i ! ^ \ 1 io ,

46 I 1

Send to local market Receive from external pipelines

Send to external market

Figure 7. Gantt Chart 5. Conclusions This work addresses the scheduling problem of an oil refinery tank farm. The approach, which combines CLP and MILP in a unified framework showed better results than a previous MILP model. Heuristic information was used to guide the search process and it contributed to reduce the CPU time. For this reason other aspects were considered in the modeling to reduce the difference between a theorical model and the practical needs. Results show that the fuzzy system is effectively able to represent the problem uncertainty. The developed model can be used to test and correct new operational conditions and scenarios rather than to just determine the scheduling of regular activities.

References Hog, 2002a, Hog OPL Studio 3.6.1 - Language Manual, ILOG Corporation, France. Hog, 2002b, Hog OPL Studio 3.6.1 - Users Manual, ILOG Corporation, France. Focacci, F., 2000, Solving Combinatorial Optimization Problems in Constraint Programming, PhD thesis, Universita Degli Studi di Ferrara, Ferrara, Italia. Magatao, L., 2005, Mixed Integer Linear Programming and Constraint Logic Programming: Towards a Unified Modeling Framework, PhD thesis, CPGEI/CEFET-PR, Brazil. Magalhaes, M. V. O., 2004, Refinery Scheduling, PhD thesis. Imperial College London, UK. Jain, V. and I. E. Grosssmann, 2001, Algorithms for Hybrid MILP/CP Models for a Class of Optimization Problems, INFORMS Joumal on Computing, 13(4), 258-276. Hooker, J.N., 2000, Logic Based Methods for Optimization and Constraint Satisfaction, Wiley Interscience Series in Discrete Mathematics and Optimization, New York, USA. Pinto, J. M., Joly, M. and L. F. L. Moro, 2000, Planning and Scheduling Models for Refinery Operations, Computers & Chemical Engeneering, 24, 2259-2276. Shah, N., 1996, Mathematical Programming Techniques for Crude Oil Scheduling, Computers & Chemical Engeneering, 20, S1227-S1232. Stebel, S. L., 2003, Technical Report, CEFET-PR/CPGEI, Brazil, December 2003 (in Portuguese). Zadeh, L.A., 1965, Fuzzy Systems, Information and Control 8(3): 338-353.

Acknowledgements The authors acknowledge financial support from ANP and FINEP (PRH-ANP / MCT PRHIOCEFET-PR).

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PRoduct ONTOlogy. Defining product-related concepts for production planning activities Diego Gimenez^, Marcela Vegetti^'', Gabriela Henning^, Horacio Leone^'' ""INTEC (UNL-CONICET), Gilemes 3450, Santa Fe S3000GLN, Argentina ^INGAR (UTN-CONICET), Avellaneda 3657, Santa Fe S3002GJQ Argentina 'CIDISI (UTN), Lavaise 610, Santa Fe S3004EWB, Argentina Abstract Current Internet-based technologies enable the operation of Extended Supply Chains (ESCs) and introduce new requirements on enterprise systems. There is a real need to manage product-related information in such ESCs, where product models are the fundamental information source. This work describes an extension of the product information framework specified by PRONTO (PRoduct ONTOlogy), providing the foundations for a Distributed Product Data Management (DPDM) system supported by Semantic Web technology. The property and property value concepts were introduced in the ontology with the purpose of formalizing the data aggregation and disaggregation processes required by production planning activities. Keywords: product model, ontology, production planning activities 1. Introduction Manufacturing logistics (referring to all planning, coordination and support functions required to carry out manufacturing and associated logistic activities) demands accurate and reliable information of different granularity levels about products in order to be efficient. Traditionally, product information is spread among several intraorganizational systems, especially ERP, Product Data Management (PDM), and, more recently, Product Lifecycle Management (PLM) systems, with many possibilities of data replications, redundancies and inconsistencies. Moreover, these systems do not provide support for product data articulation among the different granularity levels, which are usually related to the temporal horizon of the associated decision problems. To overcome some of these difficulties, several contributions have introduced common product models to be shared by an organization. However, the centralized approach is usually not feasible in the ESCs context due to the lack of efficient support for interorganizational business integration. One major obstacle is the low degree of automation in the exchange and integration of product-related data among business partners, mainly due to the use of different representations by each one. In order to avoid this problem. Web-based PDM systems arose. Although these systems are technically and syntactically integrated, they do not allow a common understanding yet. Research activities in this area are oriented towards the use of ontologies as a foundation for the "Semantic Web". An ontology is an explicit and formal specification of a shared conceptualization and provides a conceptual framework for communicating in a given application domain (Gruber,1993). In consequence, ontologies for Product Data Models provide a framework for sharing a precise meaning of symbols exchanged during communication among the many stakeholders involved in the ESC and allow the definition of agile and flexible DPDM.

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Vegetti et al. (2005) have made a contribution in this area with the proposal of an ontology called PRoduct ONTOlogy (PRONTO). This contribution extends PRONTO with new concepts related to the specification of mechanisms for aggregating and disaggregating different kinds of product related data needed for ECS manufacturing logistics, as well as representing such data, along the product concept hierarchy. Section 2 justifies the need for the proposed extension. Section 3 introduces the product data model extension itself and illustrates it by means of a few examples. Finally, Section 4 presents the most important conclusions. 2. Need for an extended model Due to computational limitations and forecasting uncertainties, aggregate information (e.g. about product lines, product families, aggregate units of production) rather than detailed product information is used at the planning level. For instance, aggregate planning develops tactical plans for total sales, total production, targeted inventory, and targeted customer backlog for substitute or fictitious products representing the aggregate information of a set of similar items. In contrast, coarse information is disaggregated to feed data for solving material or distribution requirements planning problems (e.g. when aggregate plans should be converted into detailed master schedules). It is possible to propose at least two model hierarchies to manage the complexities of product information. One of them, referred as the Structural Hierarchy (SH) organizes the knowledge related with product structural information. The SH is a tool to manage the information associated to the multiple recipes and/or processes available to manufacture a given product or group of similar products. The material requirements planning (MRP) system is a classical example of an application that handles data along the SH. Within this hierarchy a typical information handled is the Bill Of Material (BOM) representation, which specifies the subordinate components (as well as their required quantities) that are physically needed to make each final product or assembly. The other hierarchy, referred as Abstraction Hierarchy (AH), organizes products according to different levels of specifications: product family, variant family and product. The AH hierarchy is oriented to manage the complexity originated by the huge number of products that are manufactured nowadays by current industrial facilities. The AH also employs knowledge structures and mechanisms for keeping consistency among the related product data at different abstraction levels. Many examples taken from the specialized literature reveal how "forward" and "backward" links (associated to aggregation and disaggregation tasks) along the AH should be employed to coordinate those planning functions that are executed at different time horizons. There are some contributions that propose knowledge representations and ontologies for product AHs. Nevertheless, they only support the handling of structural information along this hierarchy and do not provide a knowledge framework to manage other types of information along the AH, such as data associated to costs, demands, inventory, labour requirements, lead-times, logistic cube, etc. 3. Product Data Model Extension PRONTO (Vegetti et al., 2005) formalized a product knowledge representation for both hierarchies: SH and AH; nevertheless, it is mainly focused on product structural information. Specifically, the ontology suggests three abstraction levels for representing product-related concQ^is: product family, variant family dinA product (or physical item). They allow managing nowadays sudden increases of products (giving rise to multiple variants or alternatives) by considering the existence of similar products and adopting

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the generative BOM philosophy, where a specific BOM is derived from a common product structure. As it is shown in the UML class diagram of Fig. 1, each group of alike products receives the name of product family, which can include simple products (atomic raw materials, acquired components) or compound ones (non-atomic raw materials, manufactured assemblies, final products). Each family is associated to one or more structures, which represent simple products (Simple) or define different ways of either combining component parts and raw materials to make compound products {Composition) or decomposing non-atomic raw materials {Decomposition) by means of composition or decomposition relations {CRelation and DRelation) established with the ProductFamily class. In turn, a subset of members of a given family, having similar characteristics, is classified under the concept of variant family, which also can be simple {SVariantFamily) or compound {CVariantFamily). Each compound variant family specifies the variants considered in the product family structure from which it is derived {Inclusion), as well as the rules {Change) to adapt {Modify) such structure to the one that the members of the variant family particularly have. Finally, the actual products are referred by means of the product concept and represent the finest level of detail in the AH. The BOM of a given product is entirely defined by choosing {Selection) the specific variants to be combined in the structure of the variant family of which such product is a member. ProductFamily

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Figure 1. PRONTO's overall view In order to enlarge the semantics associated to the AH, the property concept has been incorporated to the product model allowing it to manage all kinds of information (structural and non-structural). Besides, it formalizes the data vertical integration along AH levels {product family, variant family dind product) during information aggregation and disaggregation processes, which is not supported by the current product models. In turn, the value (or values) that a property assumes for a certain product representation, at a given level of the AH, is specified by means of the property value concept. The ontology extension achieved by the incorporation of these new concepts is depicted in the class diagram of Fig. 2. The abstraction levels already defined in PRONTO are represented as a specialization of the ProductAbstraction class. Additionally, this class is linked to the Property one by means of the PropertyValue association class. The Property class includes the value type (i.e. string, symbol, boolean, numerical types) associated to the particular concept, as well as a detailed description of its meaning. Moreover, the property can be a quantitative or qualitative one, depending of its value type. In the Qualitative subclass, the allowed set of values should be specified, which will generally be of Symbol or String type. On the other hand, in the Quantitative subclass the set of available units of measure should be defined. Figure 3 shows a simple example of these two situations.

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^

Property K }

/T

ExteriorForm V a l u e T y p e = String ^description = represents the packaged product form fallowedValues = {box20x20, box20xl0,box30x20,..

Quantitative

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GrossWeight V a l u e T y p e = Real ^description = represents the packaged product weight ^nitsOflVIeasure = {Kg,Gr,Oz,Lb,...}

Figure 3. Instances of the Qualitative and Quantitative property subclasses The proposed model decouples the property concept from the respective values that a given property can assume in the various product abstractions at same or different levels of the AH. As mentioned before, the property value concept is represented by means of the PropertyValue class. This class includes the range of allowed values in the context of a given association, and the unit of measure whenever it is necessary to specify it. Regarding to the value itself, it can be a single one or a set of them. In such a case, the value attribute of PropertyValue will be a multivalued one. 3.1. Property Value Classification The proposed extension takes into account the fact that property values can be obtained from different sources and by resorting to different calculation or retrieval mechanisms. Also, it allows representing those property values that can exhibit a high frequency of change. In such cases, the information is always out of date and the stored value must be disregarded. Then, this conceptualization considers two specializations of the PropertyValue class, in agreement with the physical existence or not of the property value. Thus, the Enduring class represents the situation in which the value resides physically as an object and the Derived one models the circumstance in which the value is obtained at the moment it is required. Moreover, a specialization of each of the two previous categories is made according to the nature of the property valuation process. In the case of enduring values, the following subcategories were identified: a) Computed: the value is calculated by using one or more CalculationMethods (see Fig. 2). b) Restricted: the value that a given property assumes in a certain product abstraction is constrained by the value set that the same property takes in another product representation included at an upper abstraction level, which is related to the original one by means of a member association. In this case, a range of possible values will not be defined because it will be determined by the

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condition expressed above, c) Local: the valuation process does not use any value linked to the same or another property associated to another product abstraction level. Regarding the first category, when the value of a given property of an aggregate product (a product family or a variant family one) is obtained by means of an aggregation process, the calculation is based on the values that the same property presents in ProductAbstraction instances, defined at lower level and related to the original one by means of member associations (e.g., the value of the annual production of a specific product family may be obtained fi-om the aggregation of the corresponding variant families values, which in turn can be obtained by aggregating the annual production values of their associated product instances). In this case, the CalculationMethod will use an aggregation mechanism. In contrast, the information disaggregation process has the purpose of generating detailed data, related to the members of a given product abstraction of an upper level, from information contained in such aggregate product. For example, the demand forecasted for a specific variant family can be used to estimate the demand of a given member of this product abstraction by disaggregating the information on the basis of the product market share. In this case, a disaggregation method will be used. On the other hand, the Derived class is specialized into two subcategories: a) Granted: the value is taken from the value of the same property at an upper abstraction level. Within this context, it might be interpreted as if the value were inherited, b) Inferred: the valuation process is carried out via a calculation method. As for the Computed subcategory, an aggregation or disaggregation method can be employed. With the intention of exemplifying these new concepts. Fig. 4 includes some instances of the different property value categories related to a candy industry case-study.

r Average Values

i

« I 0 Infer re d » SFC_GW ^ a l u e =14.73 ^ a n g e = undefined ^>unitOfMeasure = Gr V

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« I 0 Inferred» SFFC_GW ^ a l u e =|?:82j ^ a n g e = undefined ^nitOfMeasure = Gr

« I O Local» SFC NW ^ a l u e = (4.00, 4.50) ^ a n g e = undefined ^>unitOfMeasure = Gr

« I 0 ProductFamily» SingleFlavorCandy member \ A « I O VariantFamily» SingleFruitFlavor Candy member\ A

—

res trictedby | A « I 0 Restricted» SFFC_NW ^ a l u e = 4.50 ^ a n g e = undefined ^nitOfMe asure = Gr

« I 0 Product» Straw berry Candy gra nted_from 1 A from T 1 inferred^ «IOGrantEd» « I O Computed» 1 « I O Quantitative» « I O Quantitative » 1 SC_NW SC_GW 1 GrossWeight 1 NetWeight ^ a l u e = 4.50 ^ a l u e = 4.80 ^ a n g e = undefined ^ a n g e = [4.75,4.85] computed_from ^HinitOfMeasure = Gr ^>unitOfMeasure = Gr

•

10:

instanceof

•

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A A

Data Vertical Integration

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Figure 4. Some instances of Property Value class In this example it is possible to see that the Abstraction Hierarchy is composed of SingleFlavorCandy, SingleFruitFlavorCandy and Strawberry Candy. For each level, the values of the GrossWeight and NetWeight properties are shown. For the first property, the values at the product family and variant family levels (SFCGW and SFFCGW instances) are inferred from the computed value of such property at the product level (SCGW instance). Besides, the value of the NetWeight property at the highest level (*S'FC_A/^^ instance) imposes a restriction over the values that this property could take at the variant family level (SFFC NW instanco). Finally, the property value at the lowest

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level (SCNW instance) is granted from the SFFCNW instance. Due to lack of space, the attributes (effectiveDate and obsoleteDate) that establish the validity period of the data defined in Enduring subclass, were omitted. This data vertical integration allows to automate the data aggregation and disaggregation processes required at the time of making strategic, tactical and operative decisions, related with the logistics activities involved in the ESCs. In turn, PRONTO provides a standard definition and a shared representation of the product data from which it is facilitated the semantic integration of these activities (related to Sourcing, Production and Delivering), which take place in various organizational areas (e.g. Engineering, Manufacturing, Marketing, Finance, Sales, Planning). Furthermore, by means of automatic update mechanisms (that can be executed in predetermined periods) the product data (associated to real and substitute products) is all time available and updated in the DPDM system. This task would become very complex without the explicit communication among the different abstraction levels. 4. Conclusions Product-related data constitutes the fundamental information source for many activities performed in industrial enterprises. The definition of ontologies for product models establishes a common formal vocabulary to be used for each stakeholder of the ESC which commit to a given product ontology, accepting the terminology that it prescribes. Particularly, PRONTO considers different abstraction levels in relation to the product concept: product family, variant family and product. These levels permit handling information with different aggregation degrees. The product ontology extension presented in this article formalizes both processes of information aggregation and disaggregation that occur during production planning activities. The representation of the calculation method for the Inferred and Computed properties allows to document and register the knowledge associated with such computations. The separation of the property concept {Property class) from its value (PropertyValue class) allows products defined at different abstraction levels to share the concept of a certain property and to assign it distinct values at each abstraction level. An extra advantage of such separation is that the definition of properties (their meanings and the value types that are permitted) can also be shared by the many participants in a supply chain. Thus, all the instances of the Property class could be seen as a repository of "attribute definitions" used by several organizations during integration processess. It must be remarked that product information is not static. ESCs are subject to continuous internal restructurings due to changes in the offer-demand relations or due to the inclusion of new actors (suppliers, manufacturers, logistic providers, customers). Thus, future work will focus on the development of contextual ontologies to make compatible the product representations introduced by these new actors with the current product model shared by the previous participants of an ESC. References Gruber, T., 1993. A Translation Approach to Portable Ontology Specifications. Knowledge Acquisition, vol. 5, 199-220. Vegetti, M., Henning, G., Leone, H., 2005. PRoduct ONTOlogy. An Ontology for Complex Product Modeling Domain. Proceedings ofENPROMER 2005. Rio de Janeiro, Brasil. Acknowledgements This work has been supported by CONICET, UNL, UTN and ANPCyT (PICT 12628).

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Recipe Informatics to Shorten the Lead Time from Product Development to Production in Batch Processes Tetsuo Fuchino,^* Teijii Kitajima,^ Yukiyasu Shimada,^ Kazuhiro Takeda/ Susumu Hashizume,^ Takashi Hamaguchi/ Rafael Batres,^ Akira Yamada,^ Kouji Kawano/ Yoshihiro Hashimoto ^ ^ Tokyo Institute of Technology, 2-12-1 Oookayama Meguro-ku, Tokyo 152-8552, Japan * Tokyo University of Agriculture and Technology, 2-24-16, Nakamachi, Koganei, Tokyo 184-8588, Japan "^ National Institute of Industrial Safety, 1-5-6, Umezono, Kiyose, Tokyo 204-0024, Japan ^Shizuoka University, 3-5-1, Johoku, Hamamatsu 432-8561, Japan ^Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan ^Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan ^Toyohashi University of Technology, Tempaku-cho,Toyohashi 441-8580, Japan ^ Mitubishi Chemical Co. Ltd., 191-1, Ooaza, Shiohama, Yokkaichi 510-0863, Japan ^ Mitsui Chemicals, Inc., 30, Asamuta-cho, Oomuta 836-8610, Japan Abstract The profitabihty of the batch processes to produce high value added product for a respective use depends greatly on the lead time from the product development to the commercial production. To shorten the lead time, integrated information support environment which enables to fully utilize the past experiences on commercialization of new products is indispensable. However, the commercialization form product development is carried out implicitly in practice. Therefore, it cannot clarify how experience of the past commercialization should be represented as information model to shorten the lead time for a new product. In this study, an engineering activity of "commercialization of a new product in batch process" is modeled explicitly by using IDEFO expression. The sub-activities, which constitute the commercialization, and information necessary for these sub-activities can be defined hierarchically, and the required information model to shorten the lead time is able to be specified. Keywords: IDEFO, activity model, recipe design, batch process, informatics. 1. Introduction Accompanying the diversification of the needs and shortening of the product life-cycle, the batch process has been coming to be oriented toward the production of the high value added product for the respective use. The order of such a high value added product depends on whether the product with required quality and quantity can be delivered within a certain period. Therefore, the profitability of the batch process depends on shortening of the lead time from the development to the commercial production. Author to whom correspondence should be addressed: [email protected]

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The quality requirement for the product is generally specified by a client. However, the requirement is a necessary condition to maintain quality of the client's product, and does not directly specify the engineering quality target for the batch production. For example, concerning to polymeric sheet for computer LCD display, a client (computer manufacturer) may specify clearness and/or coloring. On the contrary, the polymer manufacturer should transfer such ill-defined requirements into objective engineering specifications such as molecular weight distribution and/or polymerization distribution. To adjust the ill-defined requirements and the objective engineering specifications, it is necessary to repeat the cycle of a sample distribution and client evaluation. The lead time to commercialization is dependent on this number of cycles to be necessary. On the other hand, when a new product, which satisfies the client's requirement to some extent, is developed in the laboratory scale facility, then the developed laboratory recipe is informed to the recipe engineering group to design the master and/or control recipe for the commercial production plant, in which the above mentioned sample production and commercial production will be performed. The production design is usually carried out by using a commercial production plant to omit various tests in the bench scale plant with the aim of the smooth switchover from the production design to the commercial production. However, the recipe provided in the laboratory must be a laboratory facility dependent master recipe, and what the recipe engineering group is going to design is just the commercial plant depending master recipe. Moreover, although it is concentrating on making a predetermined product itself in product development, in addition to the quality requirements, safety and economical efficiency are taken into consideration in the production design. For this reason, in a product development stage and a commercialization stage, the raw materials and/or the solvents to be used may often differ from each other. Therefore, the recipe engineer should translate the master recipe of the laboratory scale facility into the master and/or control recipe for the commercial production plant through the trial and error test operations in the plant. The lead time to commercialization is depending on the number of such trial and error operations. It is clear that there exists difference in necessary information between the client and the production design and between the production design and product design. That is why the above mentioned trials and errors are necessary to complement the difference. If this informational complementation was well managed, then number of trial and error cycles could be reduced and the lead time for commercialization could be shortened. For this purpose, it is necessary to utilize fully the past experiences on commercialization of new products, and integrate information between the client and the production design and between production design and the product design. However, the production design or the recipe design is carried out implicitly, and it cannot clarify how experience of the past commercialization should be represented as information model to integrate the information. To defme the necessary information, the engineering activity to commercialize a new product should be clarified explicitly. In this study, engineering activities to commercialization of a new product in batch process from the development recipe is modeled into IDEFO activity model, and the necessary activities, information, tools and their relations are defined explicitly. In order to clarify the engineering viewpoint in the production design (recipe design), the activity model is based on ANSI/ISA-S88.01 (S88.01) [1]. To identify the mechanism to represent, capture, and retrieve the necessary information, the PIEBASE template [2] is applied.

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2. Scope and Approach for Activity Modeling From the viewpoint of the process lifecycle, product design and production design are carried out in the batch process industry, and are interactively related with product design and production design of client industry as shown in Fig. 1. Zhao et al. [3] are planning to provide the integrated environment through the lifecycle in the field of pharmaceutical development and manufacturing. The portion surrounded by the dotted line in Fig. 1 shows commercialization of a new product in the batch processes, where the activity is modeled into IDEFO (Integrated Definition Language) as same as the previous study [4]. It is obviously that the commercialization is interactive with product design and client's activity, and sub-activities and information should be classified explicitly to provide integrated environment. sample and product

•defined specification development recipe

equipment requirement facility information

x>mmercializati(^ development • - requirement

^ facility ^ information

Fig. 1 Scope of commercialization of a new product in batch processes To commercialize a new product, production (or recipe) and commercial facility should be designed (or provided). The former is fiinctional design and the latter is physical structure design, so that the engineering viewpoint is deferent. ANSI/ISA-S88.01[1] divided a recipe and equipment clearly, and relate them using equipment requirement information. In order to identify the necessary information for commercialization of a new product, it is valuable to define activities according to the engineering viewpoint. Therefore, the recipe design is considered on the basis of ANSI/ISA-S88.01[1] in this study. On the other hand, to design production based development recipe satisfying the ill-defined client's specifications, it is necessary to fully utilize the empirical information on the past commercialization. To manage the past empirical knowledge, the mechanism to represent, capture and retrieve empirical information [5] is necessary. Therefore, the activity to provide the empirical information should be defined independent from the other activities hierarchically, to clarify the necessary information. The PIEBASE (Process Industry Executive for achieving Business Advantage using Standard for data Exchange), which is an international group of process industry, made an IDEFO activity model (PIEBASE model) describing enterprise activities of process industry. Prior to making the model, PIEBSASE defined a template [2] as the norm of modeling, which categorizes activities into three types; "manage", "do" and "provide resources". In this study, PIEBASE model is based on for activity modeling. The activities to provide production facility and necessary empirical information are defined in "provide resources" activity class, and the actual engineering procedure is investigated. Then IDEFO activity model based on S88.01 can be made, and hierarchical structure of necessary information can be clarified.

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3. IDEFO Activity Model IDEFO is a method to describe business and/or engineering process, and the information to perform the activities is categorized into four types; 'Input', 'Control', 'Output' and 'Mechanism', which is called the ICOM for short. In the model, the rectangle represents activity, and the arrows describe the ICOM, where 'Input' is the information to be changed by the activity, 'Control' is the information to constraint the activity, 'Output' is results of the activity and 'Mechanism' is the resources for the activity. Each activity is expanded into sub-activities hierarchically, and the ICOM is also made in detail. In this study, activity to commercialize a new product in the batch process is to be analyzed, and the top activity as shown in Fig.2 is defined. The "Master Recipe" is design from "R&D Information" to satisfy the "Product Quality Requirement" of client. To adjust the ill-defined client's specification and the engineering quality target, "Product Sample" distribution and receiving "Sample Evaluation" are repeated. Patent Information

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Fig.2 Node A-0 Top activity for commercialize a new product in batch processes. The top activity of "Commercialize a New Product" is developed into five sub-activities, i.e. "Al: Manage Commercialization of a New Product", "A2: Design Master Recipe", "A3: Perform Trial Production", "A4: Evaluate Feasibility of a New Product" and "A5: Provide Resources for Commercialization" as shown in Fig. 3. The "Manage Commercialization of a New Product" receives "Product Quality Requirement", "R&D Information" and "Existing Facility Information", and it converts the ill-defined specification into the engineering quality target and the facility target. These targets are informed to the activity of "Design Master Recipe" as "Requirement for Master Recipe Design". The conversion is carried our by correlating the ill-defined specification and past actual results. The results are informed to the activity "Provide Resources for Commercialization", and representation and capturing of the empirical information are done. The necessary information for the correlation is retrieved by the "Request of Resources for Manage Commercialization", and is informed "Resources (Information) for Manage Commercialization" as "Mechanism". As well as the Al activity, the

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activities A2 to A4 utilize the past experienced results. These activities send "Request of Resources" and receive the necessary information as "Mechanism". Hecm Revism Beouest Sample Evaluation Commercialtzation Request C3 Code & Standard II ^ C2 Patent Informatton I product QualHy Target from Existing Facftty Product Quality Requiremetit ^' _ -. _ . 16

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Fig.4 Developed model form "A2: Design Master Recipe". The "A2: Design Master Recipe" is further developed into seven sub-activities as shown in Fig. 4. As well as the developed model from the top activity shown in Fig. 3, the activity "A21: Manage Master Recipe Design" converts the process overall quality target and facility target into the quality target of each process stage and equipment target. The results are informed to A22 to A26 sub-activities as "Control" information. The conversion in the activity of "A21: Manage Master Recipe Design" is also

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performed by correlating the process overall quality target (and the facility target) and the past actual results on master recipe design. The results of master recipe design are represented and captured in the activity "A27: Provide Information for Design Master Recipe". The necessary empirical information is retrieved on the "Request of Resources for Manage Master Recipe Design". The A22to A26 receive the requirements from A21, and also perform correlation with the past actual results on master recipe design from All activity. In this study, The activity of "A24: Design Formula" is two levels further developed into sub-activities. Although, those models and explanations are omitted for the restriction of the paper here, the consistent IDEFO activity model can be provided in the similar manner on the basis of PIEBASE template. 4. Conclusion To shorten the lead time from product development to production (commercialization) in batch processes, integrated information support environment which enables to frilly utilize the past experiences on commercialization of new products is indispensable. However, the commercialization form product development is not clarified explicitly, and it was difficult to specify the information model to support engineering. In this study, the engineering activity of "commercialization of a new product in batch process" has been modeled explicitly by using IDEFO expression. In making IDEFO activity model, the international standard ANSI/ISA-S88.01 [1] was based on for the recipe design, and the PIEBASE template [2] was applied. The obtained model implies the contents of ANSI/ISA-88.00.03-2003 "Batch Control Part3: General and Site Recipe Models and Representation" [6]. According to the PIEBASE template, the activity to represent and capture the past actual results was separated from the other activities to perform commercialization, the necessary empirical information could be defined hierarchically related to the engineering activity. Therefore, it became possible to define the information model by using the previous study [7]. Finalizing the IDEFO activity modeling and making information model are friture works. References [1] ANSI/ISA-S88.01-1995 Batch Control Part 1: Models and Terminology, ISA (1995). [2] http://www.posc.org/piebase/execsum.htm [3] C. Zhao, G. Joglekar, A. Jain, V. Venkatasubramanian and G.V. Reklaitis, Proceedings of Europian Symposium on Computer Aided Process Engineering-15, 1561 (2005). [4] T. Fuchino ,T. Wada, and M. Hirao, Lecture Notes in Comput. Sci., 3214, 418 (2004). [5] W. C. Regli, X. Hu, M. Atwood and W. Sun, Engng, with Comput. 16, 209 (2000). [6] ANSI/ISA-88.00.02 Batch Control Part 3: General and Site Recipe Modules and Representation, ISA(2003) [7] T. Fuchino, T. Takamura and R. Batres, Lecture Notes in Comput. Sci. 3681, 162 (2005).

Acknowledgements The authors gratefully acknowledge fiinding support from The Japan Society for Promotion of Science.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Efficient MILP-based solution strategies for largescale industrial batch scheduling problems Pedro Castro^, Carlos Mendez^, Ignacio Grossmann^ liro Harjunkoski"^, Marco Fahl^ "DMS/INETl 1649-038 Lisboa Portugal ^Chemical Engineering Department-CEPIMA, UPC, E-08028 Barcelona, Spain ^Dep. Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA ABB Corporate Research Center, Ladenburg, Germany Abstract This paper presents two alternative decomposition approaches for the efficient solution of multistage, multiproduct batch scheduling problems comprising hundreds of batch operations. Both approaches follow the principle of first obtaining a good schedule (constructive stage), by considering only a subset of the full set of orders at a time, and then improving it (improvement stage) by applying a rescheduling technique. The core of both approaches consists on the solution of mixed integer linear programming problems that, on each step, are variations of the scheduling model with global precedence sequencing variables of Harjunkoski & Grossmann (2002). The results for the solution of a 30-order problem show that the proposed decomposition methods are able to obtain solutions that are 35% better than those obtained by the solution of the full problem, on a fraction of the computational time. Keywords: Decomposition methods, short-term scheduling, rescheduling. 1. Introduction The increasingly large literature in the scheduling area highlights the successful application of different optimization approaches to an extensive variety of challenging problems (Mendez et al., 2005). This important achievement comes mainly from the remarkable advances in modeling techniques, algorithmic solutions and computational technologies that have been made in the last decade or so. However, there is still a significant gap between theory and practice. New academic developments are mostly tested on relatively small problems whereas real-world applications consist of hundreds of batches, dozens of pieces of equipments and long scheduling periods. In order to make the use of exact methods more attractive for real-world applications, increasing effort has been oriented towards the development of systematic techniques that allow maintaining the number of decisions at a reasonable level, even for large scale problems. Manageable model sizes may be obtained by applying heuristic model reduction methods, decomposition or aggregation techniques. Once an initial solution is generated with reasonable CPU time, gradual improvement through optimization-based techniques can be achieved with modest computational effort. Although these techniques can no longer guarantee optimality, this may not be so critical in practice due to the following: i) very short time is available to generate a solution; ii) optimality is easily lost due to the dynamic nature of industrial environments; iii) implementing the schedule as such is often limited by the real process; iv) only a subset of the actual scheduling goals are taken into account.

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Recent work by Castro & Grossmann (2005) compared the performance of five different approaches for the solution of multistage batch scheduling problems. The results showed that all, besides a uniform time grid continuous-time formulation, can be a valid option depending on the objective function. However, the study was performed for relatively small problems and sequence-dependent changeovers were not considered. Handling industrial-sized problems may quickly lead to an intractable model size. Also, the inclusion of sequence-dependent changeover times may substantially reduce the efficiency of most of the existing models. Following a preliminary evaluation of the most adequate formulation, this paper presents two alternative decomposition strategies that are able to efficiently deal with multistage, multiproduct scheduling problems involving large number of batches, many processing units and substantial cleaning requirements, typically found in the pharmaceutical and fine chemical industry. 2. Problem definition The industrial multiproduct plant under consideration has the following characteristics: • 17 machines and 6 stages, allocated in the following way: 2+3+3+3+3+3. • Unlimited intermediate storage between stages is assumed • Sequence dependent changeovers in most stages, all of them machine independent. Changeovers are usually of the same order of magnitude or even larger than the processing times. • Some orders may skip some processing stages. • Two sets of problem data are considered: i) a 30-order and ii) a 50-order problem. • The objective is the minimization of the makespan. 3. Selection of the most appropriate formulation Discrete-time formulations are not appropriate for this specific problem. On the one hand, the objective of makespan minimization can only be achieved through an indirect procedure that involves solving several problems (Maravelias & Grossmann, 2003). On the other hand, sequence dependent changeovers lead to a significant increase in the model size, both directly and indirectly. Directly, due to the increase in the number of constraints to account for the cleaning times. Indirectly, as a result of the larger number of time intervals and the consideration of a finer time grid for an exact representation of the problem data, while keeping the model sensitive to different changeovers. Continuous-time formulations based on a uniform time grid are generally inefficient for multistage problems without shared resources (Castro & Grossmann, 2005). Continuous-time formulations relying on multiple time grids have more flexibility and have proved to be more competitive for this type of problem. However, sequence dependent changeovers make them less efficient. Continuous-time formulations using explicit sequencing variables are more valid approaches since they hardly require any changes to handle sequence dependent changeovers. The model used in this paper is the one by Harjunkoski & Grossmann (2002), which relies on global precedence sequencing variables. The authors also presented a constraint programming (CP) model, which can be easily adapted to the case of sequence dependent changeovers. One just needs to assign to each unary resource (the machines), the corresponding changeover matrix (using ILOG's OPL Studio). Table 1 shows the computational statistics for the solution of the 30-order problem with the continuous-time formulation (MIL?) and the CP model. As can be seen, the size of the problem is already too large to be handled efficiently. While the MILP finds a fair solution in 1 h (maximum computational limit), it has many binary variables and

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features a very large integrality gap (the best possible solution at the time of interruption is still only 13.325, which translates into an absolute gap of 34.657 and a relative integrality gap above 72%). CP, one the other hand, was also tested but it was only able to generate a very poor solution at the beginning of the search (in the very first second of calculation). No fiarther improvements were observed in the remaining hour. Hence, the continuous-time MILP will be the only one considered next. Table 1. Computational statistics for the solution of the full 30-order problem Model

Bin. vars.

Cont. vars.

Cons.

RMIP

MIP

Best possible

CPUs

Nodes

MILP

2521

2708

10513

7.449

47.982

13.325

3600

62668

CP

-

1050

1300

-

79.989

-

3600

1432

4. Solution Strategy Since it is impractical to solve the full problem simultaneously, effective decomposition techniques need to be developed. Of a few alternatives tested, two have shown to produce very good results in a relatively small amount of computational time. Both approaches follow a two-step approach. The first, is a constructive step, where the goal is to obtain a good schedule with low computational effort. The second, is an improvement step, where the previous solution is gradually enhanced by applying a rescheduling technique with low computational effort. Bear in mind that the improvement step is common to both approaches. 4.1. Constructive approach 1 (API) The first approach follows a two step decomposition method consisting of a design model and a schedule refinement model. For the design model, we need to define a priori the number of orders to be considered (up to the full set of orders) and the number of orders to be selected, which will be typically around 5 (i.e. for 30 orders, 6 iterations will be required) to ensure a small integrality gap and reaching a good solution very fast in the refinement model. Following the solution from one iteration, the times at which the several machines end their processing can be determined and used to fix the machines release dates for the subsequent iteration. Furthermore, the last order to be processed can be identified and the information used to account for the correct changeover time. The design model is very similar to the original model of Harjunkoski & Grossmann (2002) but minor changes were required: i) Introduction of a new set of binary variables, to identify the selection of a particular order from the subset of orders considered in the design problem; ii) Introduction of a new set of constraints, to enforce an even distribution of the most difficult orders (these are high-processing-time orders that can only be assigned to a single machine on a given stage) through the several iterations instead of getting them all in the last iteration, thus avoiding a significant increase in the value of the objective function, the minimization of the makespan. While the design model generally provides a good solution, it is usually unable to prove optimality within the specified computational limit. If we deal only with the selected orders, a better solution can typically be found. The objective function will minimize the makespan plus a term that accounts for the completion times in all stages, since we need to link two consecutive iterations efficiently. By doing this, we ensure that the resulting schedule is tight for all machines and not just for the ones belonging to the bottleneck path(s) that define the makespan.

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4.2. Constructive approach 2 (AP2) The second approach also divides the full set of orders into smaller subsets. Here, the process of assigning orders to iterations does not require any special method, since there is much more freedom to link consecutive iterations. We can choose the set of order(s) to be considered in each iteration by following the lexicographic sequence. Besides the elimination of the design step, there are two important conceptual differences to API. i) Orders that have been scheduled in previous iterations (typically 1 or 2 per iteration) are no longer removed from further consideration. Instead, they will be considered over and over again, although with fewer degrees of freedom, until the last iteration; ii) Unit availability is no longer limited in time by using release dates. Previously assigned orders can be left- or right-shifted in time in order to insert the new order(s) to be scheduled. While sequencing and allocation decisions are made for the latter set of orders, for orders belonging to the former set, the binary assignment (they remain allocated to the same machine) and sequencing variables (they maintain their relative position in the sequence) are fixed but the continuous timing variables may change. This simple method can be illustrated by a simple example. Suppose that the previous iterations resulted in order sequence 4, 3, 2, at a given machine, and that a new order needs to be scheduled (order number 6). Fig. 1 shows that there are only 4 allowed sequences. Allowed sequences

Some forbidden sequences

1 6 1 4 1 3 1 21

I 6 I 4 I 2 I 3I

1 4 1 6 1 3 1 2 1

I 3 t 6 I 2 I 4 I

1 4 1 3 1 6 1 2 1

r ~ 3 I 4 I 6 I 2~1

1 4 1 3 1 2 1 6 1

I 2 I 3 I 4111]

Figure 1. Illustration of the scheduling step whenever a new order (6) is being considered

4.3. Improvement step The proposed improvement technique is based on the main ideas of the rescheduling model proposed by Roslof et al. (2001) and Mendez & Cerda (2003) and can be combined with either of the two previous alternative constructive methods. It involves several iterations, with each being identical to the last iteration of AP2, where all orders except a very small subset of them (e.g. a single order) have fixed assignments and fixed relative positions. The selection of orders among iterations can be made following some kind of sequence or be made randomly. Every time a better makespan is achieved, the allocation (equipment) and position (sequence) tables are updated. The stopping criteria for the rescheduling algorithm can be either a maximum predefined computational limit or a maximum number of iterations without improving the objective function value. 5. Computational results The computational studies were performed on a Pentium 4-2.8 GHz running GAMS/CPLEX 9.0. Both methods have several parameters that can affect the final result. In API, the number of orders to be considered (NPD) and selected (NPS) in the design problem are critical. For AP2, NPS defines the number of orders scheduled simultaneously. Common parameters include the maximum CPU-time for the scheduling (actually in API two values are specified, one for the design step and other

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for the schedule refinement step) and rescheduling phases and the method of order selection in the latter (direct or random sequence). Table 2 presents some of the most successful runs. The results show that both approaches lead to very good solutions when compared to those obtained using the full MILP (see Table 1). Usually, a better solution from the scheduling phase (6^^ column), leads to a better final solution after the rescheduling step (7* column for the direct sequence and 8*^ column for the random sequence). Also, for a given phase, the more the computational time the better the solution (compare rows 3 and 4, and 10 and 11). It was found that AP2 is more robust since the solution obtained is less dependent on the model parameters and is often better. However, the number of orders to be considered increases steadily with the number of iterations, contrary to API, so while the first iterations take little time, the last ones can have significantly large integrality gaps when the resource limit is reached, which may eventually compromise the quality of the final solutions. Despite this fact, AP2 performs better for the 50-order problem than for the 30-order problem. Table 2. Computational results Makespan (h)

CPU-limit (s)

CPUs

Res. Dir.

Res. Ran.

Sch.

Total

32.439

30,480

30.562

176

276

(20;10);(10)

33.592

31.811

32.316

161

266

5

(10;10);(10)

35.578

33.084

33.610

112

212

10

5

(10;10);(10)

34.229

32.192

32.898

110

217

AP2

-

1

(10);(10)

33.149

31.175

31.676

86.4

188

30

AP2

-

2

(20);(10)

32.523

31.007

31.678

175

275

30

AP2

-

3

(30);(10)

34.447

31.787

32.485

255

355

50

API

30

6

(25;15);(10)

56.082

51.997

54.269

322

422

50

API

30

5

(20;10);(10)

53.740

53.409

52.800

283

391

50

API

30

5

(10;10);(10)

56.228

53.684

55.131

194

295

50

API

30

3

(10;5);(10)

53.453

51.466

51.581

173

275

50

AP2

-

1

(10);(10)

52.911

51.275

50.721

240

342

50

AP2

-

2

(15);(10)

52.964

51.080

51.019

321

429

50

AP3

-

3

(20);(10)

55.705

52.960

52.668

306

407

Orders

Ap.

NPD

30

API

30

6~~

(25;15);(10)

30

API

30

5

30

API

30

30

API

30

NPS

Per phase

Sch.

The best solution to the 30-order problem, featuring a makespan of 30.480 h, is given in Fig. 2. Note that the completion times of the last-stage machines (M15-M17) are very similar, with Ml6 setting the makespan. It is hardly a trivial schedule due to the following reasons: i) the length of changeover periods and idle times exceed by far the lengths of the production times, in the last three stages of production; ii) the order sequence changes often from one stage to the other (e.g. order 17 is processed prior to 27 in stage 1, Ml, while the opposite is true in the last stage. Ml6); iii) despite the plant consisting of the same number of machines per stage after stage 1 (i.e. 3), it is unreasonable to consider independent parallel production lines since the subset of orders assigned to a machine in a particular stage does not remain constant throughout the

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subsequent stages (e.g. of the 8 orders assigned to M13 in stage 5, 1 (20) goes to M15, 2 (25 and 29) go to M16 and 5 go to M17); iv) even after a thorough examination of the schedule, there is not a clear limiting stage, which means that the bottleneck probably shifts repeatedly between different stages and even between machines of the same stage. M1 M2 M3 M4

9 116 115 I 28 I 3 19 121 118 I 29 11712 I 27 I 1 I 24 114

> 110 I 8 Is

22 23 11

4

6

m

m

m " H " • " , « ' » • " • « IE

25^15 i i S i i 3 1 4 1 2 . 1 . 2 a 7 1 2 j ^ B 14 I T T

M5 M6 M7 M8 M9 M10 M11

:^^H a i S I B S i i [ i i S i K D i S i l S » S H D l ^ B B a 3 B 3 ^ H D H

M12 M13 M14 M15 M16 M17

" m\^m ^'

1^3" iirjglfca

24 1

"smim^mMB ^ 25

30

Figure 2. Best solution found for the 30-order problem 6. Conclusions This paper has described two alternative decomposition approaches for the efficient and fast solution of large industrial scheduling problems. Both use the concept of decomposing the ftill problem into several subproblems, each featuring a subset of the orders. The main difference lies in linking the consecutive subproblems. While the first approach completely freezes the schedule of the pre-assigned orders and ensures feasibility for the remaining through machine release dates, the second approach allows for more flexibility by only fixing the assignments and relative positions of the previously scheduled orders. The second approach was found to be more robust and seems better suited for the solution of this specific type of problem. References Castro, P., Grossmann, I., 2005. New Continuous-Time MILP Model for the Short-Term Scheduling of Multistage Batch Plants. Ind. Eng. Chem. Res. In press. Harjunkoski, I., Grossmann, I., 2002. Comp. Chem. Eng., 26, 1533. Maravelias, C, Grossmann, I., 2003. Ind. Eng. Chem. Res., 42, 6252. Mendez, C, Cerda, J., 2003. Comp. Chem. Eng., 27, 1247. Mendez, C, Cerda, J., Grossmann, I., Harjunkoski, I., Fahl, M., 2005. State-of-the-art Review of Optimization Methods for Short-Term Scheduling of Batch Processes. Submitted to Comp. Chem. Eng. Roslof, J., Harjunkoski, I., Bjorkqvist, J., Karlsson, S., Westerlund, T., 2001. Comp. Chem. Eng., 25,821.

16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering W. Marquardt, C. Pantelides (Editors) © 2006 Published by Elsevier B.V.

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Innovation and Knowledge Management: using the combined approach TRIZ-CBR in Process System Engineering. Guillermo Cortes Robles, Stephane Negny, Jean Marc Le Lann LGC (PSI- Genie Industriel), UMR-CNRS 5503, INPT-ENSIACET, 118, Route de Narbonne, Toulouse 31077, France. Abstract In this article, a TRIZ based model is proposed to support the innovation and knowledge capitalization process. This model offers a knowledge base structure, which contains several heuristics to solve problems, synthesized from a large range of domains and industries and, also, the capacity to capture, store and make available the experiences produced while solving problems. Keywords: TRIZ, Innovation, Chemical processes, CBR. 1. Introduction As enterprises attempt to improve their capacity to innovate and consequently, their business performance, their attention focused more and more in some intangible assets: their knowledge resources. According to Smith [1], innovation's outcome depends solely on the creativity and knowledge of talented employees and the effectiveness of the methods and processes that support their work. The knowledge dimension has been covered and supported by a new discipline: Knowledge management (KM). KM encompasses several mechanisms to systematically managing the knowledge that evolves within the enterprise. With regard to creativity, the problem's complexity is continuously increasing and the time for solving it, decreasing. In spite of this increasingly complexity, problems still have been faced using traditional psychological based approaches like brainstorming, the trial-and-error search method, among others [2]. Thus, an approach able to generate ideas for systematically solving problems is needed. Recently, a new approach that conceives innovation as the result of systematic patterns in the evolution of systems has emerged in the industrial world: the TRIZ theory. TRIZ or Theory of Inventive Problem Solving creates an environment where individuals can systematically solve problems and improve decision-making. In this paper, both elements -creativity and knowledgeare integrated in a model that combines several TRIZ advantages and the process issue from the Case-Based Reasoning, for creating a support for the innovation and knowledge capitalization process. This combined approach is applied in the Process System Engineering field. 2. TRIZ, the Theory of Inventive Problem Solving This new vision about technological problems and scientific technical evolution has its foundations in the Soviet Union in 1946; when Genrich S. Altshuller, a young employee was working in the patent department of the Soviet navy. He was convinced that methods for systematically innovate were available. With this objective in mind.

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Altshuller began the search for those methods and in the process; He created the first innovation knowledge base. The particularity of this new approach for solving problems, settles on its origins. While developing his theory, Altshuller analyzed and synthesized knowledge from four sources: (1) an analysis of over 3 million worldwide patents; (2) the examination of available tools and methodologies for solving problems, with the aim to create an entirely new approach; (3) an inventor's creative mental patterns analysis, with the objective to extract the most creative solutions and strategies to solve problems and (4) an extensive scrutiny in scientific literature that revealed an enormous knowledge body that could be applied for solving problems [3]. The psychological strategies and technical knowledge, extracted from the history of technological and social evolution, were transformed to be reusable and then, embodied in TRIZ. As result, TRIZ theory encompasses a set of fundamental concepts, a collection of tools and heuristics to solve complex problems and several laws or trends of evolution for technical systems. Among main TRIZ concepts are: • • •

The evolution of all technical systems is governed by objective laws. The concept of inventive problem and contradiction like an effective way to solve problems. This also means that any problem could be modeled as contradiction. The innovative process can be systematically structured [4].

Consequently, TRIZ has the capacity to considerably restraint the search space for innovative solutions and to guide thinking towards solutions or strategies, that have demonstrated its efficiency in the past in a similar problem and, in this process, to produce an environment where generate a potential solution is almost systematic [5]. In this paper, a TRIZ tool named Contradiction Matrix has a central role. This tool and its intrinsic concepts are described in the next section. 2.1. The Contradiction Matrix While exploring the patents database, Altshuller found a common denominator between several patents, in different technological disciplines: the fundamental problem that characterizes these inventions was the same, and was solved in the same way. He also found that a limited number of parameters - 39 Generic Parameters- and solving principles -40 Inventive Principles- could be used to characterize any problem. Consequently, Altshuller shows that knowledge from patens databases, could be extracted, transformed and arranged in such a way, that its reutilization was accessible by any person in any domain. This reflection guided the creation of several TRIZ tools and concepts, between those, the contradiction concept and the Contradiction Matrix. A contradiction occurs, when any tentative for improvement in a system parameter, has an undesirable degradation in a second one also useful; so an inventive problem is one that contains at least one contradiction and an inventive solution, is that which overpass totally or partially a contradiction. Those concepts determine one of TRIZ milestones: problems are solved without compromise or tradeoff [2]. The 39 Generic Parameters make possible to model any problem as a contradiction and the 40 Inventive Principles, permit to restrain the solution space for effectively direct the creative effort to solve problems. Those elements are organized in a 39*39 matrix, named Contradiction Matrix (Table 1).

* An extensive description is available at the TRIZ Joumal. www.triz-joumal.com

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^ ^ ^ ^ ^ ^ Degraded 2 ...39 1... A ^ ^ ^ ^ ^ Weight of moving object Weight of stationary object Productivity To improve ^^^^^_ 1. Weight of moving object 35, 3, 24, 37 39. Productivity

28,27

35,26

Table 1: A Contradiction Matrix fragment To use this matrix, firs the parameter "A" - in Unes-that has to be improved is identified and then, the parameter "B" -in columns- which is deteriorated. The intersection between line and column isolate a cell, that encloses the inventive principle or principles that have been successfully applied to resolve this particular conflict in analogous problems. Those principles are represented with a number and hierarchically organized in every cell. Example: This example concerns the performance of a low pressure chemical vapor deposition reactor (LPCVD) with a vertical configuration (figure 1-A). While analyzing its performance a contradiction was identified: to improve the quality of the silicium film in the wafer, the gap between wafers must be large, in consequence, the quantity of wafers inside the reactor is reduced affecting productivity. The problem was stated as "To increase the productivity in the reactor without radically modify its shape". Using the Contradiction Matrix and Productivity as feature to improve and Shape as degraded parameter, is possible to identify four inventive principles to solve this contradiction in a hierarchical order: 14 (spheroidality), 10 (Prior action), 34 (Rejecting and regeneration parts) and 40 (Composite materials). Principle 14 says: replace linear parts or flat surfaces with curved ones and cubical shapes with spherical shapes, replace linear motion with a rotating motion; utilize a centrifiigal force. The interpretation of this principle reveals that the useful working area should be conceived as a spherical one. This concept is showed in figure 1-B [6]. The obtained reactor has a 90 wafers capacity while the typical one has a 25 wafers capacity; consequently, the productivity is radically improved.

B Thermocouples Wafers External fiimace Gas injectors Pedestal

• -"^-X^ontrol ^/Thermocouples!

Quartz tube

Quartz tube Internal fUmace Wafers

O

Gas injectors

External furnace Nacelle

Figure 1: The proposed solution TRIZ practitioners have proved that applying common solutions for the resolution of contradictions, identified as effective when applied to parallel problems in the world patent base, radically improves the design of systems and products [4]. This fact implies that problems sharing the same contradiction are similar in nature and for this reason; one problem's solution could be exported to other problems containing the same contradiction. According to Teminko, 95% of inventive problems in any domain have already been addressed and solved in some other field [4].

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Nevertheless, even if TRIZ has in its structure the knowledge extracted from several technical domains and scientific disciplines, it does not have any component exclusively conceived to capture and reutilize the knowledge deployed or created while solving problems. For dealing with this lack, a synergy with another approach is desirable, the most indicate to accomplish this objective is the Case-Based Reasoning (CBR). 3. The Case-Based reasoning (CBR) In the CBR process, problems are solved by reusing earlier experiences [7]. In this process, new problems are compared with cases or specific problems encountered in the past, to determine if one of the earlier experiences can provide a solution. If a similar case or set of cases exists, their solutions must be evaluated and adapted to find a satisfactory one. This approach has proved its utility to support design activities, equipment selection and also knowledge management activities among others [8], [9], [10]. The CBR as methodology for problem solving encompasses four essential activities: retrieve, reuse, revise and retain [7]. In this process (figure 2), the problem solving process starts with an input problem description or new confi*onted case [10]. This description is used to -Retrieve- a problem or set of previous solved problems (cases), stored and indexed in the memory. Then if one or various stored cases match with the initial problem, the most similar case is selected to -Reuse- its solution. Subsequently, the derived solution must be -Revised-, tested and repaired if necessary in order to obtain a satisfactory result. Finally the new experiences which comprise failure or success, strategies to repair and implement the final solutions, among others particular features, are -Retained- for further utilization and the previous cases memory is updated.

Previous Cases General Knowledge Figure 2: The CBR cycle

Tested/Repaired Case Retain Learned Case

J

One of main disadvantage in a CBR system is intimately relied on its memory. In other words, a CBR system dealing with a problem that has never been faced up in the past will not be capable to offer an efficiency initial solution. To downgrade this inconvenient, a tool capable to define and propose some search directions for any kind of problems is advisable, in this case, the Contradiction Matrix. The synergy between those TRIZ tools and the CBR process is showed in next section. 4. Creating a synergy between TRIZ and CBR The complementary characteristics between TRIZ and CBR allow the creation of a synergy. In this process, the initial problem is described and modeled as a contradiction. Then, this contradiction and some other elements derived fi'om the problem description are used to retrieve a similar case in the memory. This search could offer or not a similar case. Consequently, at this state of the search, two sub-processes could have place:

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•

A similar case is retrieved. So, its associated solution is evaluated to decide if such initial solution will be reuse. • No similar cases are stored in the memory. Thus, the system will propose at least 1 inventive principle (and no more than 6 between the 40 that exists), that has been successfully used in the past, to solve this specific contradiction in some other domains. Afterward, the inventive principles which are in reality some standard solutions or strategies to solve problems, must be interpreted to propose a potential solution. Subsequently, both sub-processes converge and the proposed solution is then verified and repaired if necessary in order to obtain a satisfactory result. Finally the new experiences which comprise failure or success, strategies to repair and implement the final solutions, among others particular features, are retained for being reusable in the future and the case memory is updated. Abstract Domain Target Case and Historical Cases (Analogous Problem)

,:/.^ Associated analogous solution

......... w

••••'•.'^•••••Reiise:

Formulation

••.•••••..^:

Creative Effort

jRIZ

Problem description. Identified contradiction

Revise

Industrial Reality

Adapted Solution Confirmed solution

Figure 3: The synergy TRIZ-CBR [11] Example: to illustrate the use of our tool, a very mere chemical engineering example is presented. The purpose of this example is to demonstrate the interest and principle of operation of our tool. Consequently, the Simulated Moving Bed process (SMB) and its evolution are treated nevertheless this tool can be applied in the same way in an industrial case. The SMB is a chromatographic technique to continuously separate multi components mixture. The starting point is the True Moving Bed (TMB) (figure 4A). The TMB process has to be improved because of its main drawback: circulation of a solid phase. As explained earlier, this drawback is expressed in term of a contradiction: line 33 and column 19 of the matrix. The crossing cell does not give some previous similar case in the memory. Thus, a creative solution has to be formulated with help of the principle in the crossing cell: (1) Segmentation, (13) Inversion, (24) Intermediary. The first principle specifies that the object or process can be fragmented into independent zone. On of the sub-principle of principle 13 is "Make movable parts fixed and fixed parts movable". Having in mind that the circulation of the solid must be reduced, it can be fixed. Consequently if the solid becomes static, we have to perform the inlets and outlets ("fixed parts movable") in a rotating way. Combination of both principles 1 and 13 gives the solution (SMB) (figure 4B). In its evolution, the SMB process has to be improved because it is only limited to one function: separation. Here again, a contradiction is expressed. But now, the crossing cell gives us a previous case with its associated solution in the memory: make an object perform multiple functions like in reactive distillation. The solution is adapted to our problem to give the Simulated Moving Bed Reactor.

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Elueni

Eluent

[Liquid Extract Solid

^^ Extract

B Raffinate

Feed III Raffinate

Figure 4: the Simulated Moving Bed Reactor 5. conclusions The presented model offers a way to transfer the solution from an identified analogous problem to a new target problem, reducing effort and time in solving problems, because this approach combines the TRJZ ability to propose creative solving strategies applicable across-domains, and a framework that closely relates knowledge and action, besides one of the ways to drive the iimovation process, consist in reusing knowledge that has been acquired. Another important product of this model is learning, which is in fact an inherent to a CBR system, because a CBR system store in a memory passed experiences for later use and for that reason, an excellent way to share knowledge. This model has been implemented in a computational system which is actually in test at the Industrial Systems Engineering research group from the Laboratory of Chemical Engineering (LGC-PSI).

References [1] Smith H., 2005. The imiovator is a Problem Solver. Computer Science Corporation, June 2005 CSC World, (18-21). [2] Altshuller G., 1999. The Innovation Algorithm. TRIZ, systematic Innovation and Technical Creativity, Technical Innovation Center, First edition. [3] Cavallucci D., 1999. TRIZ: I'approche Altshullerienne de la creativite. Techniques de ITngenieur, A 5 211. [4] Teminko J, Zusman A, Zotlin B., 1998. Systematic Innovation: An Introduction to TRIZ. St. Lucie Press. [5] Hippie, J., 2005. Solve Problems Inventively. American Institute of Chemical Engineers in CEP Magazine April 2005 Vol. 101, No. 4, p 44-50 [6] Vergnes H. 1996. Etudes experimentales et modelisation du reacteur annulaire et de son modele reduit. Ph.D. thesis at the I.N.P. Toulouse. [7] Aamodt A and Plaza E., 1994. Case-Based Reasoning: Foundational Issues, Methodological Variations, and System Approaches. AI Communications. lOS Press, Vol. 7: 1, pp. 39-59. [8] Braunschweig B. and Surma J., 1996. Case-Base Retrieval in Process Engineering: Supporting Design by Reusing Flowsheets. Engineering Applications of Artificial Intelligence, Volume 9, Issue 4, August 1996, Pages 385-391. [9] Watson I., 2001. Knowledge Management and Case-Based Reasoning: A Perfect Match?. Proceedings of the Fourteenth International Florida Artificial Intelligence Research Society Conference, May 21-23, 2001, pp. 118-122 AAAI Press. [10] Avramenko Y., Nystrom L. and BCraslawski A., 2004. Selection of internals for reactive distillation column—case-based reasoning approach. Computers & Chemical Engineering, Volume 28, Issues 1-2, 15 January 2004, Pages 37-44. [11] Cortes Robles G., Negny S. and Le Lann J., 2004. Knowledge Management and TRIZ: A Model for Knowledge Capitalization and Innovation". World Conference: TRJZ Future 2004, Florence, It. ETRIA (The European TRIZ Association).

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Decision-making tool for scheduling of batch processes: the dynamic hybrid simulation kernel Nelly Olivier, Raphaele Thery, Gilles Hetreux, Jean-Marc Le Lann Laboratoire de Genie Chimique, 117, route de Narbonne, 31077 Toulouse, France

Abstract PrODHyS is a dynamic hybrid simulation environment, which offers extensible and reusable object oriented components dedicated to the modeling of processes. The purpose of this communication is to illustrate the potentialities of PrODHyS through the use of hybrid simulation for the scheduling of semi-continuous processes. Keywords: Object oriented software components, modeling and dynamic simulation of hybrid systems, scheduling of semi-continuous process.

1. Introduction Since several years, batch or semi-continuous process systems are the prevalent mode of production for low volume of high added value products. In this framework, scheduling is an important function for governing these plants. Such processes are composed of interconnected and shared resources, in which a continuous treatment is carried out. For this reason, they are generally considered as hybrid systems where discrete aspects mix with continuous ones. Otherwise, the recipe is more often described with state events (temperature or composition threshold, etc.) than with fixed processing times. As a consequence, the simulation of unit operations and physico-chemical evolution of products often requires the implementation of phenomenological models. On the other hand, scheduling of processes leads generally to NP-complete problems requiring simplifying assumptions to be solved with classical optimization methods. In some cases, these simplifying assumptions don't allow to exploit the entire flexibility of the process. In this framework, combining mathematical programming and rigorous simulation to solve scheduling problems would enable to take advantage of both approaches. In our approach, the scheduling module is coupled with a dynamic hybrid simulation module in order to design a computer aided decision-making system. This communication focuses on the environment PrODHyS, which supports the simulation aspects of this tool. Particularly, its potentialities are underlined by the modeling and the simulation of a complete semi-continuous process.

2. The environment PrODHyS Developed in our laboratory since several years, PrODHyS is an environment dedicated to the dynamic simulation of industrial processes. Based on object concepts, it offers extensible and reusable software components allowing a rigorous and systematic modeling of processes. The primal contribution of these works lay in the determination and the design of the foundation buildings classes. The last important evolution of PrODHyS is the integration of a hybrid dynamic simulation kernel (Hetreux and al., 2003, Ferret and al, 2004, Hetreux and al, 2004). This hybrid feature is managed with the Object Differential Petri Nets (ODPN) formalism, which combines a set of differential and algebraic equations (DAE) systems with high level Petri nets (defining

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the legal sequences of commutation between states) in the same structure. This environment offers an equation-oriented dynamic hybrid simulator, which can detect state and time events. PrODHyS is built as a library of classes (design with UML, coding in C++) designed to be derived through object mechanisms (polymorphism, composition, inheritance, genericity). Currently, it is made up of more than one thousand classes distributed into seven packages and two independent functional layers (simulation/modeling). 3. Process modeling with PrODHyS The simulation of a discontinuous process requires distinct modeling of the operative part {iht process) and the command part (the supervisor). Concerning the operative part, the specification of any device of PrODHyS is defined according to two axes: a topological axis and a phenomenological axis. 3.1. Topological axis The topological axis (see figure 1) defines the structure of the process: physical connections (material, energy, information) between the different parts of the process and hierarchical decomposition of the devices.

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Figure 1: Topological description of processes in PrODHyS

3.2. Phenomenological axis : the ODPN formalism The phenomenological axis rests on a mathematical model based upon mass and energy balances and thermodynamic and physicochemical laws. Figure 2 represents an operative sequence which permits the feed of a tank until a given volume is reached. The behavior of each equipment is represented by an ODPN. QQQ Q QQCHMUUMBU QHflWmjm

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The marking of the signal place of an active entity induces the evolution of its Petri net. This Petri net can itself induce the evolution of active or passive entities in cascade through the graph composed of the connection of the different material and energy ports. Thanks to the ODPN formalism, the size and the structure of the resulting DAE systems change all along the simulation, according to the actual state of the process. A more detailed description of the ODPN formalism can be found in (Hetreux and al., 2003). This mechanism is used to dissociate the model of material from the model of devices which contains the material. Object tokens are reusable and reduce the complexity of the devices Petri nets. Moreover, according to the ISA/SP88 norm (www.isa.org), the notion of macro-place has been added in the ODPN formalism. It consists in replacing a sequence of places/transitions relating to an operation or biphase by a single macro-place ( figure 5).

4. Simulation of a semi-continuous process The considered process (figure 3) is inspired by a process described in (Joglekar et al, 1985). It is composed of batch operations (reaction, transfer, startup of the distillation column) and continuous operations (distillation).

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Figure 3. Considered flowsheet 4.1. Description of the flowsheet This unit aims at manufacturing a product P from two reactants Rl and R2, respectively stored in the tanks ST1 and ST2. The reaction (R) is an endothermic equilibrium reaction (R : Rl + R2 ^-> P) with a rate law given by the equation (1). This operation is carried out in the reactors BR1 and/or BR2. As shown on figure 4, the reaction rate increases with the temperature. To optimize the reaction rate without vaporizing the reaction mixture, a preheating to 383 K is required (temperature kept constant during the reaction). Furthermore, the equilibrium limits the molar fraction of product P to 0.83. As an infinite reaction time would be necessary to reach this composition, the reaction is stopped when the molar fraction of product P is egal to 0.8. All or a part of the reaction mixture is transferred in the storage tanks ST5 or ST6 according to their availability. To satisfy the purity specification of the product P (98% molar), a

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separation step is required. It consists RT p RT •Xp (1) 'R =^r« '•^Rl • \ 2 - ^.1'^ in a distillation operation carried out in the column COL. The purified product P is recovered at the top of — - / the column while a mixture of / residual Rl and R2 are obtained at the |.. bottom. This unit operation works .^-^. ^ - ^---^ firstly in a discontinuous mode for the startup of the column and the ,^^ manufacturing of the first batches .-^ (Startup place) and in a continuous -.—.—-—__^--mode (FiniteRefluxJ for the or manufacturing of the following Figure 4 : Reaction rate as a function of T batches. Figure 5 represents the Petri net associated to the macro-place of the distillation operation and its coupling with the Petri net of the distillation column (which controls itself the Petri nets of its constitutive devices). The reboiler REB can only be filled with the content of the tank ST5. A detailed description of the startup of the distillation column is given in (Perret et al., 2003). When all the plates of the column are filled, the mixture stored in the tank ST5 is continuously transferred on the feed plate of the column with a constant flow rate (via the pump P4). This step enables the filling of the reboiler (place Filling Reboiler). Then an infinite reflux distillation step is performed during one hour (place Infinite Reflux) and finally, the stream divider VPA1 is opened to perform the finite reflux distillation step (r=1.2). The product P is recovered in the tank ST4 and sent to a packaging unit where it is stored in 10 Hters bottles. The residual Rl and R2 are recycled in the tank ST3. '-[HO This mixture, whose composition is close to the stoechiometry, is reused as raw materials to feed the reactor BR2. Figure 5. Macro-place "distillation" 0,9

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4.2. Process modeling The models implemented in this study take into account the total and partial material balance, the energy balance, liquid/vapor equilibrium, reaction kinetics, as well as the hydraulic phenomena. Indeed, except for the pipes working with a pump, the transfers between tanks are carried out by gravity. This induces that the outflows of the tanks are a function of the hydraulic pressure (and thus of the level of liquid). The calculation of the time required for this operation is not simple, particularly when feeds and drainings are authorized simultaneously. This problem of "operation time depending on state" is also found with operations such as chemical reaction or separation because their processing times generally depend on the initial physicochemical state of material

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(temperature, concentration, etc.). The benefit of state event simulation precisely rests on its ability to consider events which occurrence date is not known a priori. 4.3. Simulation parameters We assume that the geometrical characteristics of devices are fixed. The objective is to satisfy the delivery of various quantities of product P according to a calendar imposed by the conditioning operation. However, several parameters or policies must be estabhshed before the running of the simulation. Figure 6 gives a short part of the simulation results.

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Figure 6. Evolution of the composition in BR1

4.3.1. Transport policies and batch size The possibihty of accumulation of products in a tank (such as ST4, ST5 and ST6) offers a flexibility in the manufacturing control by uncoupling the draining and filling phases of two successive operations. A tank also enables to modify the volume of a batch between two operations, with a continuous range of volume value. However, the capacities of these tanks are limited and the mix of different products is obviously impossible. As the volumes to be manufactured are generally higher than the capacity of the reactors, the size and the transport policy of batches within the unit are significant characteristics which must be clearly defined. 4.3.2. Transition between production modes The presence of tank represents also the most common configuration for the transition between two zones of process having a different operating mode: continuous for one and discontinuous (or batch) for the other. When material circulates from the discontinuous zone towards the continuous one (such as tank ST5 toward the column COL), the activity of the continuous zone is guaranteed by a sufficient quantity of material in the tank. If the racking flow of the continuous zone causes a permanent decrease of the holdup in the tank, which is not compensated by the arrival of batches in the discontinuous zone, the occurrence of a rupture of production is unavoidable on a sufficiently long time horizon. The management of such situations is particularly delicate especially when the startup of a continuous installation can take time. On the opposite, if the continuous racking is too weak, an overflow of the tank capacity is possible and the activity of the discontinuous workshop has to be braked. In this context, the determination and the respect of the starting dates in the discontinuous zone become a dominating constraint. For a reverse direction (i.e. continuous towards discontinuous), the management of the tank obeys the same principles for different causes. The rupture of production of the continuous process happens when the capacity of the tank is reached (such as between column COL and tanks ST3 and ST4). Conversely, the absence of a sufficient quantity of material in the tank can differ the starting of a batch operation and decreases the productivity of the discontinuous chain (situations to be managed between tank ST4 and conditioner COND).

5. Proposed approach for scheduling of semi-continuous processes The satisfaction of all the constraints raised previously with only the assistance of simulation appears difficult. To solve this problem, the coupling with methods of operations research seems to be judicious. However, many methods suggested in the

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iterature apply under the assumption of constant and known processing times. This constitutes a severe restriction in front of the sensitivity of some unit operations to the adjustments of the operating conditions. The distillation column is an example where the operational duration is related to several factors: the quality of the initial load, the heating policy of the reboiler, the reflux ratio, the racking flows, the thermal losses, etc. Ignoring these parameters in the representation of the system induces an approximation of the processing times (to which a safety margin is often added). As a consequence, finding an "optimal" solution can appear useless. Indeed, what can we think about the optimality of a solution established on an over-estimated model of the system? Our goal is rather to find an approached solution in term of optimality but estabUshed on the basis of a representation closer to reality. The methodology suggested sets up a strategy on two levels in which a scheduling module is coupled with our dynamic hybrid simulator (figure 6). At the beginning, the scheduling module assumes over-estimated durations and establishes plans of production with a stochastic procedure of optimization for example (Azzaro-Pantel et al., 1998). The sequence of task on the resources, the assignment of the tasks and the volume of the batches are then transmitted to the simulator, which generates the corresponding recipe and simulates the dynamic operation of the process. The feasibility of the plan is established (or not) and the refined operational durations and indicators of performances (total time of execution, occupancy rate of the resources, etc) are transmitted to the scheduling module. This iterative procedure is carried out until the user considers the solution is satisfactory. Taking into account the work required to automate this procedure and to refine the link between scheduling and simulation modules, this system is still under development. 6. Conclusion The object oriented approach brings many advantages in terms of software quality (extensibility, reutilisability, flexibility), but especially in terms of modeling thanks to a hierarchical and modular description which is both abstract and close to reality. Based on these concepts, PrODHyS provides software components intended to model and simulate more specifically the industrial processes. The implementation of a formalism on high level of abstraction associated with powerful methods of numerical integration led to the construction of a robust hybrid dynamic simulator. In this communication, the potentialities of PrODHyS are illustrated through the modeling and the simulation of a complete semi-continuous process. The works in progress aim at integrating this simulation model within a decision-making tool dedicated to scheduling. References Azzaro-Pantel C, Bemal-Haro L., Baudet P., Domenech S., Pibouleau L., 1998. A two-stage methodology for short-term batch plant scheduling: discrete-event simulation and genetic algorithm, Computers and Chemical Engineering, Vol. 22, n° 10, ppl461-1481 Hetreux G., Perret J., LeLann J.M., 2003. Object hybrid formalism for modelling and simulation of chemical processes, ADHS'03, Saint-Malo, France Hetreux G., Perret J., LeLann J.M., 2004. Composant based approach for simulation of dynamic hybrid systems, Conference on Conceptual Modeling and Simulation (CMS '04), Genoa, Italy Joglekar G.S., Reklaitis G.V., 1985. A simulator for batch and semi-continuous processes, Computers and Chemical Engineering, Vol 8, No 6, pp 315-327 Perret J., Hetreux G., LeLann J.M., 2004. Integration of an object formalism within a hybrid dynamic simulation environment, Control Engineering Practice, Vol. 12/10, pp. 1211-1223 Perret J., Thery R., Hetreux G., Le Lann J.M., 2003. Object-oriented components for dynamic hybrid simulation of a reactive column process, ESCAPE 13

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Multiobjective Multiproduct Batch Plant Design Under Uncertainty A. Dietz/ A. Aguilar-Lasserre,*' C. Azzaro-Pantel,^ L. Pibouleau,^ S. Domenech^ ^Laboratoire de sciences du Genie Chimique - UPR CNRS 6811 LSGC - GROUPE ENSIC I rue Grandville - BP 451 - 54001 NANCY Cedex France ^Laboratoire de Genie Chimique- UMR 5503 CNRS/INP/UPS 5, Rue Paulin Talabot - BP1301 31106 TOULOUSE Cedex 1, France Abstract This paper addresses the problem of the optimal design of batch plants with imprecise demands and proposes an alternative treatment of the imprecision by using fuzzy concepts. For this purpose, we extended a multiobjective genetic algorithm developed in previous works, taking into account simultaneously maximization of the net present value (NPV) and two other performance criteria, i.e. the production delay/advance and a flexibility criterion. The former is computed by comparing the fuzzy computed production time to a given fuzzy production time horizon and the latter is based on the additional fuzzy demand that the plant is able to produce. The methodology provides a set of scenarios that are helpful to the decision's maker and constitutes a very promising framework for taken imprecision into account in new product development stage. Keywords: multiobjective optimization, genetic algorithm, fuzzy arithmetic 1. Introduction In recent years, there has been an increased interest in the design of batch processes due to the growth of specialty chemical, pharmaceutical, and related industries, because they are a preferred operating method for manufacturing small volumes of high-value products. The market demand for such products is usually changeable, and at the stage of conceptual design of a batch plant, it is almost impossible to obtain the precise information on the future product demand over the lifetime of the plant. However, decisions must be made on the plant capacity. This capacity should be able to balance the product demand satisfaction and extra plant capacity in order to reduce the loss on the excessive investment cost or that on market share due to the varying demands on products. The design of multiproduct batch plants has been an active area of research over the past decade (see (Shah, 1998) and (Pinto and Grossmann, 1998) for reviews). Most of the work has been yet limited to deterministic approaches, wherein the problem parameters are assumed to be known with certainty. However, in reality there can be uncertainty in a number of factors such as processing times, costs, demands, and not all the requirements placed by the technology of the process and the properties of the substances are defined. To cope with this, there has been increased interest in the development of different types of probabilistic models that explicitly take into account the various uncertainties (Sahinidis, 2003). For example, Wellons and Reklaitis proposed an MESfLP model for the design of batch plants under uncertainty with staged

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capacity expansions. Based on the structure of multiproduct batch plants, Straub and Grossmann (1992) developed an efficient procedure to evaluate the expected stochastic flexibility, embedded within an optimization framework for selecting the design (size and number of parallel equipment). Two-stage stochastic programming approaches have also been applied for design under uncertainty (lerapetritou and Pistikopolous (1996); Cao and Yuan (2002). It must be clearly said that the use of probabilistic models that describe the uncertain parameters in terms of probability distributions in an optimization framework is very greedy in computational time, either because of the large number of scenarios involved in the discrete representation of the uncertainty or the need to use complex integration techniques when uncertainty is modeled by continuous distributions. Besides, the use of probabilistic models is realistic only when a historic data set is available for uncertain parameters, which is rarely the case at the preliminary design stages in new product development. In this work, frizzy concepts and arithmetic constitute an alternative to describe the imprecise nature on product demands. Genetic algorithm optimization techniques were retained for both, MINLP and mutliobjective aspects of the optimization problem. For this purpose, we extended a multiobjective genetic algorithm, developed in previous works (Dietz, 2005a, b), taking into account simultaneously the maximization of the net present value (NPV) and two other performance criteria, i.e. the production delay/advance and a flexibility criterion. The paper is organized as follows. Section 2 is devoted to process description and problem formulation. Section 3 presents a brief overview of frizzy set theory involved in the frizzy framework within a multiobjective genetic algorithm. The presentation is then illustrated by some typical results. Finally, the conclusions on this work are drawn.

2. Process description and problem formulation The case study is a multiproduct batch plant for the production of proteins taken from the literature (Montagna et al., 2000). This example is used as a test bench since shortcut models describing the unit operations involved in the process are available. The batch plant involves eight stages for producing four recombinant proteins, on one hand two therapeutic proteins. Human insulin (I) and Vaccine for Hepatitis B (V) and, on the other hand, a food grade protein, Chymosin (C) and a detergent enzyme, cryophilic protease (P). In previous works (Dietz et al. 2005a, b), batch plant design was carried out minimizing the investment cost and the production system was represented using discrete event simulation techniques in order to take into account different production policies. Two strategies for campaign policies were tested, either monoproduct or multiproduct. In this work, only the monoproduct campaign policy was considered, so that the computation of cycle time can be easily implemented using the classical formulation proposed in (Montagna et al., 2000), involving size and time equations as well as constraints. A keypoint of the procedure is the computation of the so-called cycle time TLi for each product, which corresponds to the limiting time, i.e., the time between two consecutive batches of the product. The objective is to determine the number and size of parallel equipment units/storage as well as some key process variables in order to satisfy one or several criteria (see (Dietz et al. (2005a)) for a complete description of the problem). Although the minimizing investment (I) is most ofren considered in the dedicated literature, it is not the most adequate objective for the optimal design problem. In real

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applications, designers preferentially not only consider to maximize the net present value (NPV), but also to satisfy a due date. The corresponding mathematical expressions of the objective functions (considered as fuzzy with a ~-symbol) are proposed as follows: A£x (_NPV) = Max (fi)--

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Mi n (Advance / Delay)= Min(f2) =

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The penalization term is equal to an arbitrary value of 1/ co for an advance and co for a delay in order to penalize more delays than advances. A sensitivity analysis leads to adopt a value of 4 for co. Finally, an additional criterion was computed in case of an advance (respectively a delay), representing the additional production that the batch plant is able to produce. Without going further in the detailed presentation of the computation procedure, it can be simply said that a flexibility index (called criterion fs) is computed by dividing the potential capacity of the plant by its actual value. Max (flexibil^ index) = Max (£3) (4) 3. Overview of fuzzy multiobjective genetic algorithm approach 3.L Representation of fuzzy demands and time horizon due-date In this section, only the key concepts from the theory of fuzzy sets that will be used for batch plant design are presented (more detail can be found in (Kaufinann and Gupta, 1988). Different forms can be used for modeling the membership functions of fuzzy numbers. We have chosen to use normalized trapezoidal fuzzy numbers (TrFNs) for modeling product demand. Let us recall that the membership function values of a TrFN range from zero to one with the mode at one. The possibility distribution of TrFNs represented by a four-tuple [ai, a2, as, a4] with ai < a2 < as < a4 describes the more or less possible values for a demand. In other words, they can be interpreted as pessimistic or optimistic viewpoints of the designer. Figure 1 presents the typical values adopted in this work which correspond respectively to an imprecision of 10% with mode at one (respectively 15% with mode at zero). We also introduced in the model a fiizzy horizon time with a "rectangular" representation which may be viewed as latest and earliest dates to satisfy, with an imprecision of 10% (see Figure 2). I 1500 I 113501

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Figure 1 - Fuzzy representation of product demand (kg/year) 3.2. Fuzzy extension of a multiobjective genetic algorithm The multiobjective genetic algorithm presented elsewhere (Dietz et al., 2005b) was then extended to take into account the fuzzy nature of both demand and horizon time. Let us mention that the same encoding procedure was adopted since no fuzzy parameter is involved at that stage. The tunable parameters of the GA will also not be discussed here.

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Although the GA basic principles will not be recalled, it must be said that arithmetic operations on fuzzy numbers that will be used concern exclusively the objective functions and the constraints.

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Figure 2 - Fuzzy representation of horizon time (h) They involve addition, subtraction, taking the maximum of two fuzzy numbers (mainly at the selection stage and at the Pareto sort procedure), through the extension principle of (Zadeh, 1975). Although there exists a large body of literature that deals with the comparison of fuzzy numbers, the approach proposed by (Liou and Wang, 1992) was finally adopted here. Looking more closely at the selection stage, three cases were considered, as qualitatively shown in Figure 3, corresponding to either unfeasible solutions leading to unacceptable violations of a time horizon constraint (f3=0), or to acceptable solutions sharing a time domain with an horizon constraint (f3=l), or, finally, to solutions for which the computation of the additional demand that the batch plant is able to satisfy is interesting from a flexibility viewpoint (f3>l).In case C, the computed value of the total time necessary to manufacture all the products is shifted to the right so that the highest (respectively lowest) value of the four-tuple of the TrFN corresponds to that of the due date for time horizon. Fuzzy AG T Evaiuattort procediffe I

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Figure 3 - Fuzzy evaluation procedure in the GA 4. Typical results 4.1. Monocriterion case GA typical results obtained with NPV as the only criterion to consider are presented in Table 1. Ten runs were performed to guarantee the stochastic nature of the GA. Table 1 presents the mean value of the NPV as well as the right core and support deviation from the mean value. Symmetrical values are obtained since symmetrical values were considered for both product demand and horizon due-date. The order of magnitude of the results is of interest at the design preliminary stages.

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Run

AGOl AG02 AG03 AG04 AG05

Mean Right core Right Run Value deviation support [M€] dev. 4.64 4.65 4.59 4.61 4.29

12.5% 12.5% 12.6% 12.6% 13.5%

18.8% 18.7% 19.0% 18.9% 20.3%

AG06 AG07 AG08 AG09 AGIO

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Mean Right core Right Value deviation support [M€] dev. 18,9% 12.6% 4.60 19,4% 4.77 12.9% 18,9% 12.6% 4.61 18,8% 12.5% 4.65 18,7% 4.63 12.5%

Table 1 - Monocriterion (NPV) batch plant design 4.2. Tricriteria case The three criteria considered here are the NPV, and two batch plant flexibility criteria production Advance/Delay respect the due date and production flexibility criterion defined as the maximal production capacity respect the actual demand. Previous studies shov^ed their antagonist behaviour. An oversized batch plant gives more flexibility in terms of production but is penalising for the NPV criterion. A delay respect to the due data allov^s increasing the NPV criterion because the Investment can be reduced. Figure 4 displays the results when the three criteria are considered simultaneously after the final Pareto sort procedure over the solutions corresponding to each optimization run. Only, the average value of the involved criteria is reported here. Similar results can be obtained for the other couples of criteria. Although a thorough analysis was performed, only the guidelines that may be usefiil for the practitioner are given. For instance, this curve may be usefiil to detect unfeasible regions and to identify the promising regions from the viewpoints of NPV and flexibility index. In the illustrative example, we indicate some regions which may be interesting to explore since they involve high values for the net present value and exhibit a flexibility index greater than 1, corresponding to an acceptable advance in production (not reported here).

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Figure 4 - Tricriteria results: NPV-Flexibility results projection. 5. Conclusions In this paper, we have proposed a fuzzy approach to the treatment of imprecise demands in the batch design problem. Its benefits can be summarized as follows: - Fuzzy concepts allow us to model imprecision in cases where historical data are not readily available, i.e. for demand representation; - The models do not suffer from the combinatorial explosion of scenarios that discrete probabilistic uncertainty representation exhibit;

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- Another significant advantage is that heuristic search algorithms, namely genetic algorithms for combinatorial optimization can be easily extended to the fuzzy case; - Multiobjective concepts can also be taken into account. Finally, this framework provides an interesting decision-making approach to design multiproduct batch plants under conflicting goals. Nomenclature /= Investment cost (M€) fi= Working capital (M€) Vp= Revenue (M€/y) Dp= Operation cost (M€/y) Ap= Depreciation (M€/y); a= Tax rate

/= Discount rate; n= Number of periods P= Number of products to be produced M= Number of stages H= Due date (h) Hr Production time for product i (h)

Nj= Number of parallel units in stagey Vf= Required volume of a unit in stagey 0Cj= Cost coefficient for unit j Pj= Cost exponent for unity

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