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(1)
The multiobjective optimisation problem to be considered in the paper is as follows: min ||Tew||oo, subject to: ||Tuw||g < (3 and pGS(a, r, 9)
(2)
where e is the unit-feedback control error. This problem is ready to be solved by the offshelf software (Gahinet et al. 1995). The MATLAB function, hinfmix in the LMI toolbox, originally designed for mixed H2 f¥L problem has been slightly modified for the generalised-H2 /PLo problem in (2). Two multiobjective Pareto diagrams, the minimum ||Tuw||g against a-stability and the minimum ||Tuw||g against the minimum ||Tew||ooare to be produced by repeatedly calling the modified MATLAB function.
3. Case Study The approach for multiobjective controllability analysis is applied to a two-CSTR process. The process is schematically shown in Figure 2. A full description of the system and an eight-state model can be found in Cao and Biss (1996b). To focus on the control structure selection problem discussed here, constant volume assumption is applied to the process, which leads to a six-state model to be used in the paper. The control problem is to maintain both tank temperatures at desired values in the presence of cooling-water temperature fluctuations within ±10 [K], i.e. d=[Tcwi Tcw2]- Three possible control configurations to be considered are: SI: u = [Qn Q n], two feed flowrates and y = [Toi T02], two tank outlet temperatures. S2: u = [Qcwi Qcw2]» two cooling-water flowrates and y is the same as SI. S3: u is the same as S2, but y has two extra secondary measurements, cooling-water outlet temperatures, i.e. y = [Toi T02 T^woi TcwozlThe input constraints are 0.05 < QH-HQ 12 < 0.8 [mVs] and 0.05 < Qcwb Q cwz ^ 0.8 [mVs]. To cope with the constraints, Qn and Q 12 are converted to total flowrate, Q = Qii+Q 12 and flowrate ratio, R = Q12 /Q. The new constraints are: 0.05 < Q < 0.8 [mVs] andO
y = Cjx,
B2 = B3,
C, = C2 = the first two rows of C3
(3)
460 -17 9751 0 0207 0 0 0977 0 0
A =
[ B i B2
3]
0 0 0 0
-295 0 0 0 0 0
17.8996 -0.0131 0 17.8636 -0.0082 0 362.9950 0 0 0
8655 1889 3879 0617
-13 0 0 17 -0 0
0 0 -0 0294 0 0 0
7811 0101 8636 0082
0 0 327.5600 0
0 0 0 0
0 0 0 0 0 -0. 0235
0 362.9950 0 0
0 0 0 0 0.0589 -0.6220
0 0 0 -295.8655 0.0433 0.3787
0 0 0 -18 0088 0 0131 0
0 0. 0704 -0. 8000 0 0 0
0 0 0 0 0 0.0081
0 0 0.0137 0 0 0
0 0 0 335 4470
The effect of input constraints is normally assessed by the minimum singular value, which are 15.06 for SI and 5.13 for S2 at steady state, i.e. SI is better than S2 in terms of input constraints. S3 has two secondary measurements, thus this index cannot be directly applied. On other hand, SI is the only configuration, which has two unstable zeros (10.33 and 10.31). Physically, this is because the effect of feed flowrate on tank temperature has two opposite directions - positive via reaction and negative due to lower feed temperature. Therefore, S2 and S3 are better than SI in terms of unstable zeros. However, for overall performance, it is difficult to judge which configuration is the best only using these regular controllability measures. Therefore, the multiobjective controllability analysis approach described above is applied to this example. (b) Pareto Set: H,> ^ ^ o o
(a) Pareto Set: H^ v a-stability
P
S2 S3
0.8
^^
0.4
10
0.2 0
10
-0.5 a-Stability
S2 S3
0
~\
I0.6 ~
S1
S1
1
-1
V ^ 10
-2
10
0
Figure 3 Pareto Diagrams with Pole Region r=20, 6-77.6° and a=-0.5 in (b) The multiobjective problem is constructed by assigning z = [e u] with eGL2 the control error and UGLOO. The closed-loop poles region is defined as r=20 and 0=77.6° with a fixed to -0.5 or varying. To force zero error at steady-state, an integrator is inserted into each error channel and will be merged into the controller designed. Based on these conditions, the multiobjective Pareto diagrams are produced and shown in Figure 3. The
461 results in Figure 3 show that the achievable performance of SI (||Tew||=o) is Hmited by its unstable zeros. However, SI is still the best configuration when -a<0.6 and ||u||oo< 1. It is also shown that the nonsquare configuration, S3 does improve the controllability by introducing extra measurements into configuration S2. It can achieve almost the same performance as SI within the input constraints. If the input constraints were permitted to increase slightly, S3 would even be better than SI. This observation is verified by the simulation results (see Figures 4). The controllers used for simulation are designed to achieve |Tew||oo at the values corresponding to ||Tuw||g=l in Figure 3(b). The ||Tuw||g values predicted in Figure 3(b) match the maximum input deviations observed in the simulation (see Figure 5). (a) T^^ Response to T ^+10 [K]
A A
S2 S3
[\
( b ) T ^ Response to T^^+10[K]
n -
Tol(^)-Toi(0)
5 Time [s] (0) T^^ Response to T^^-10 [K]
fr \i
5 Time [s]
\^('y\^(^)
10 ( d ) T ^ Response to T^^-10[K]
-
^j^y^j^^ SI S2 S3
S2 S3 10
5 Time [s]
10
Figure 4 Output Response
4. Conclusion The proposed approach for multiobjective controllability analysis is able to identify performance limitation imposed by multi-factors, such as unstable zeros and input constraints. It is also suitable for more sophisticated configurations, such as nonsquare, cascade and two degrees-of-freedom control. The produced Pareto diagrams can be directly used for control design trade-off The generalised-H2 norm is better than Hoc norm to describe input with input constraints. The enforced closed-loop pole region makes the closed-loop time response more predictable.
462
References Astrom, KJ., 2000, European Journal on Control,Vol.6, pp. 2—20. Boyd, S. and Barratt, C , 1991, Linear Controller Design: Limits of Performance, Prentice-Hall, Englewood Cliffs. Boyd, S., Ghaoui, L.E., Feron, E. and Balakrishnan, V., 1994, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia. Cao, Y. and Biss, D., 1996a, Computers and Chem. Engng. Vol. 20, pp. 337—346. Cao, Y. and Biss, D., 1996b, Journal of Process Control, Vol.6, No.l, pp. 3 7 ^ 8 . Chen, J, 2000, IEEE Transactions on Automatic Control, Vol.45, pp. 1098—1115. Gahinet, P., Nemirovski, A., Laub, A. and Chilali, M., (1995), The LMI Control Toolbox, The Math Works, Inc. Chilali, M. and Gahinet, P., 1996, IEEE Transactions on Automatic Control, Vol. 41, pp. 358—367. Glover, K., 1986, Int. Journal of Control, Vol. 43, pp. 741—766. Havre, K. and Skogestad, S., 2001, Int. Journal of Control, Vol. 74, pp. 1131—1139 Rotea, M. A., 1993, Automatica, Vol. 29, pp. 373—385. Scherer, C , Gahinet, P. and Chilali, M., 1997, IEEE Transactions on Automatic Control, Vol.42, pp. 896—911. Skogestad, S. and Postlethwaite, I., 1996, Multivariable Feedback Control: Analysis and Design, John Wiley and Sons, Chichester. ( a ) S 1 : Q Response to T
( b ) S 1 : R Response to T
5 T i m e [s] (d)Q,
Response to T
-10 [k]
S2 S3
cwl
5 Tinne [s]
Figure 5 Input Response
10
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
^"^
Control of the Rotary Calciner for Soda Ash Production V.M. Cristea, P.S. Agachi " Babe§-Bolyai " University of Cluj-Napoca, Faculty of Chemistry and Chemical Engineering, 11 Arany Janos, 3400 Cluj-Napoca, Romania, e-mail: [email protected]
Abstract The paper presents the dynamic simulation results obtained by modelling and control of the rotary calciner for sodium carbonate production by the endothermic decomposition of sodium sesquicarbonate. The calciner model with distributed parameters consists of three zones, each described by sets of PDE's: the combustion zone, the reaction zone and the disengaging zone. Dynamic responses to typical disturbances are presented and compared with industrial and literature data. As main target variable, the sodium sesquicarbonate content control at the discharge end of the calciner is investigated using different control approaches. The classical PID control and the Nonlinear Model Predictive Control (NMPC) algorithms are compared in mixed feedforward and feedback schemes. Incentives are revealed for particular control structures using feedforward NMPC, becoming also a feasible approach for the control of similar processes taking place in rotary calciners.
1. Introduction Crude soda ash is one of the basic products in chemical industry that is used as raw material for a large variety of end products. Either obtained from natural trona ore or from decomposition of sodium bicarbonate obtained by carbonation of ammoniacal lye, the rotary calciner is one of the main processing units. The dynamic simulator of the rotary calciner proves to be a valuable tool for studying different construction design approaches, operating strategies and control system design configurations. The present work investigates different control approaches pointing out the benefits of particular control schemes able to be directly implemented on the industrial unit.
2. Research Results 2.1 Mathematical model description The rotary calciner consists of a cylindrical shell with its long axis having a small inclination from the horizontal direction. The trona ore is introduced at the highest end of the shell and moves towards the lower discharge end due to the rotation and the inclination of the cylinder. Inside the shell, in the feed end zone, the combustion of the natural gas flux is taking place, supplying the heat necessary for the endothermic reaction. Three zones can be distinguished along the shell in the direction of the cocurrent movement of the gases and solid material, figure 1. The first (bare) zone is the combustion zone, where the combustion flame is developed and raw material is heated up. The second (lifter) zone is the reaction zone, where solid
464
Figure 1. Schematic of the rotary calciner. material showers due to lifters that elevate the solid and spill it over the cross section, increasing the mass and energy transfer. The last (bare) zone is the disengaging zone where solid and gaseous products separate and are evacuated. The main assumptions for the mathematical model development are: all parameters in a radial cross section are constant, both gas and solids velocity is considered constant, heat transfer by conduction and radiation are negligible in the axial direction and coefficients of convection, emissivities, latent heat and heat of reaction are temperature independent. Elimination of the moisture entering the calciner with the trona ore is performed within the falling rate stage of the drying process, when the rate of drying is proportional to the free moisture content of the solids (moisture content less than the critical moisture content of 10%). The mass balance of the moisture flow is described by the equation (see Notation Section at the end of the paper for the list of variables):
^ > j = - | f e j -
(1)
QH
The mass balance equation for the total amount of sodium sesquicarbonate contained in the solid is: (2)
The mass balance equations for the solids and gaseous products are: 67 v^dt""'
dr^"'
100 v^" 226
'-l(Q.)-l(Q.H^^s.
_67j\ 226
hi0.1QJ
QH
" K(o.iQ^) ^^ a/^^^
(3)
(4)
where the last term in equation (4) is only present in the first zone of the calciner. The heat balance equations for the gas, solid, flame and calciner wall are presented in the following equations: Q,C, d ,^ , ^ „ d (5) y,
9'
(T,) = -Q,C^ -(T^ ) + Hf^+ //., -H^„ - W,„,
465
^—^^(T^m
(6)
+ ^ ..n + ^ f.. " ^ e v " ^ n
) = -QmCm J^i^J^^.n.
X^f^"^^ = -^^^4^^/^-^>-^^-^^^ ^ w ^ v v " ^ ^ ^ w >^ - ^gw
dt'
~ ^ K 7 n ~ ^wO
(7)
(8)
'
The heat fluxes considered in the above equations are: the heat flux for drying the solids Hev, the heat flux due to the reaction Hre, the heat flux from vapors to the gas H^g, the heat flux to the surroundings //wo» the heat flux from flame to solid material Hjm, the heat fux from flame to wall //yw and the heat flux from flame to calciner gas Hfg. The heat fluxes gas to solid material Hgm, gas to wall Hg^,. and wall to solid material H^^ are specified according to the different zones of the calciner. The mathematical model is based on the model described by Ciftci and Kim (1999) and presents a good fit with industrial data. The Finite Element Method was used for solving the set of partial differential equations, based on 150 finite elements (FEMLAB, 2000). 2.2 Simulation results First, the simulation reveals the steady state behaviour of some important process variables, for all of the three distinct zones of the calciner, as presented in figure 2. 1-400 1200 lOOO
"/ -/
,^
-/
X^
J \
\ T S
6 0 0
y
-j
"^,^
8 0 0
•^OO
S q
-\, \
Tm
i:
]
2 0 0 O
,_
lO
L
15 C a l c i n e r lengt*-» [ m ]
20
^
:
Figure 2. Steady state profiles of the sodium sesquicarbonate content Sq, solid T^n, gas Tg and wall T^ temperatures. The gas temperature Tg has a maximum placed in the first zone where the flame is developed and for which, a linear decay of the flowrate is considered. The wall temperature T^ also presents a maximum in this zone due to the substantial radiation flux of the flame. The solid temperature 7^ rises promptly at the beginning of the second zone where mass and energy transfer is intensified. Further, the solid temperature remains almost constant due to equilibrium between the heat transfer from the gas and the endothermic reaction sink. The sesquicarbonate content of the solid 5,^ decays only in the last two zones, reaching a value of about 7% at the discharge end, (further considered as the setpoint value for sesquicarbonate control).
466 Second, the dynamic behaviour of the calciner was also investigated in the presence of some typical disturbances. Some of the representative results are presented in figure 3 and figure 4. Figure 3 presents the response of the solid flowrate Q^ to a step upset of AQm=-^7000 kg/h at the feed end. Figure 4 shows the solid temperature T^ response to a step upset of AT^= + 10 ^K incoming also from the feed end. The presented results show the profiles of the process variables along the calciner length at six time steps (ti=0 h, t2=0.05 h, t3=0.10 h, t4=0.15 h, t5=0.23 h, t6=2 h,) until the steady state is restored.
10 15 20 Calciner length [m]
Fig. 3. Solid flowrate Qm dynamic response. Fig. 4. Solid temp. T^ dynamic response Inverse response was noticed for some of the process variables related (mainly) to the solid. The reason for this behaviour may be the fact that disturbance effects travel along the calciner with different velocity: a rapid one due to the high velocity of the gas flowrate and a sluggish one due to the solid inertia (mass or heat), (Cristea et al. 2001). The main simulated process variables are varying in the range of values reported by the industrial unit. The dynamic results confirm, by the time lags, the industrial process retention time exhibited by the solids in the different zones of the calciner. 2.3 Control of the decomposition process The main impediment in performing efficient control of the decomposition process is the lack of measured data obtained from different zones of the calciner, primarily needed for feedback control. This feature determines large time lags and dead time for variables measured at the discharge end, rising therefore an important challenge for the control system. As the concentration of the soda ash at the discharge end of the calciner or equivalent, the sodium sesquicarbonate content of the solid 5,^, is the target variable, the control of the latter was investigated. Both PID control and Nonlinear Model Predictive Control (NMPC) approaches have been implemented and tested in the presence of typical disturbances, such as: step increase of the solid inlet flowrate AQm=+500kg/h, step decrease of the solid inlet temperature AT^=-4 ^K and step increase of inlet sodium sesquicarbonate content of the solid ASg=-\-3%\ all disturbances applied at the time t=0.] h. Natural gas flowrate has been used as manipulated variable. First, direct control of the sesquicarbonate content S^ at the discharge end has been investigated. Three control approaches have been considered: feedback PID control, feedback NMPC control and combined feedback-feedforward NMPC control. The simulation results in case of 5,y control have presented incentives for the combined feedback-feedforward NMPC control. Although a sampling time of 0.05 h has been
467 considered in order to take into account for the delay of the required on line 5,y analyser, this control approach is not feasible both from practical and economical point of view. Second, different process variables have been inspected as possible candidates for indirect control of the sesquicarbonate content. Solid (at the discharge end) outlet, gases outlet and wall temperatures, as controlled (and measured) variables, have been considered and tested. The solid outlet temperature T^ control proved to be the most favourable control scheme, with direct influence on the target S^ variable. The solid temperature control in the presence of inlet (at the feed end) z\(2^ and AT^ disturbances, for both solid temperature controlled variable and sesquicarbonate target variable, are comparatively presented in figure 5 and figure 6, for all of the three investigated control approaches. 452.5 /I
452 451.5 S
451
A
A
- ' -
PID NLMPC NLMPC feedforward
PID NLMPC NLMPC feedforward
^- 450.5 450 449.5 449
h 2 3 Tlmefh]—
Figure 5. Solid temperature control and sesquicarbonate content dynamic response in the presence of the step decrease of the solid inlet temperature ATm=-4 ^K disturbance. 452 451 g450
—
E
PID [ NLMPC NLMPC feedforward [
—
PID NLMPC NLMPC feedforward
•" 449 44S 447
.
( 2 3 Time[h]
Figure 6. Solid temperature control and sesquicarbonate content dynamic response in the presence of the step increase of the solid inlet flow rate AQnt=+500kg/h disturbance. The control results show superior performance for the NMPC feedforward-feedback control structure compared to the other control approaches. This conclusion is attested by the shorter settling time and the reduced overshoot (low integral square error) of the sesquicarbonate content. Similar behaviour has been noticed for other disturbances. Although the gas temperature is affected by shorter response time, the control of this variable led to unacceptable offset in the sesquicarbonate control. The cascade control
468 of the solid outlet temperature, as primary controlled variable, associated with the gas outlet temperature, as secondary controlled variable, failed to accomplish improved control performance.
3. Conclusions The dynamic simulator with distributed parameters of the rotary calciner for soda ash production reflects the complex behaviour of the unit. Large time lags coupled with dead time and inverse response for some of the main process variables put severe restrictions on desirable control performance. Several process variables have been tested as possible controlled variables. The different control approaches investigated for the control of the sesquicarbonate content target variable, ranging from simple PID to NMPC control with mixed feedforward-feedback, proved the incentives for the solid temperature NMPC control in a combined feedforward-feedback control scheme. Further improvement of the control performance are expected to be obtained by the use of state observers based on spatial and temporal profiles of the main process variables.
Notation A =specific area, [mVm] Cc, Cg, Cfn, Cv =specific heat for combustion products, gas, solid and vapors, [kcal/(kg K)] ^fi eg> ^nu ^w =emissivities f^jhu Ffw =form factor for radiative heat transfer hf, hg, ho, hi, /zH;=heat-transfer coefficients, [kcal/(h m^ K)] Kr =frequency factor [h'^] L =length of the calciner, [m] L], L2, L3, L4, Lj, L3 = arc lengths for bare and lifter zones, [m] Lv =latent heat of water vaporization, [kcal/kg]
My^ =mass per unit length of the calciner wall, [kg/m] Qc, Qg, Qnv Qh QcH4 =flowrate of combustion products, calciner gas, solid, moisture, natural gas, [kg/h] S(j =sodium sesquicarbonate content of the solid, [%] 7}, Tg, T^ To, T^ =temperature of the flame, gas, solid, surroundings, calciner wall, [K] U = combined heat transfer coefficient [kcal/(h m^ K)] Vg, Vm =calciner gas and solid velocity, [m/h] AH =heat of reaction, [kcal/kg]
References Ciftci, S. and N.K. Kim, 1999, Control Schemes for an Industrial Rotary Calciner with a Heat Shield around the Combustion Zone, Ind. Eng. Chem. Res., 38, 1007. Cristea, V.M., S.P. Agachi and S. Zafm, 2001, Simulation af an Industrial Rotary Calciner for Soda Ash Production, IcheaP-5, Florence, 245. * * * FEMLAB, Users Guide, COMSOL AB, 2000.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
469
A simulation tool for the wood drying process Helge Didriksen and Jan Sandvig Nielsen dk-TEKNIK ENERGY&ENVIRONMENT, Gladsaxe M0llevej 15, 2860 S0borg, Denmark
Abstract A new software tool for the simulation of drying of wood in drying kilns can be used to support the kiln operators in running the drying process efficiently. The simulation tool predicts the moisture in the wood boards during the drying batch. Furthermore, internal tensions in the boards are predicted indicating the risk of board cracking, a severe quality reduction of the wood. The simulation tool can help optimise the drying process with regards to product quality, process capacity and energy costs.
1. Introduction The art of drying wood boards in a drying kiln is, in short, to dry the wood as quick as possible and as gentle as possible. A short drying time gives a high process capacity and low energy costs. A gentle drying gives a high product quality. The two overall objectives are, to some extent, contradictory. Checking of the wood, and other timber quality degradings, will occur if the wood is dried too quickly. On the other hand, an exaggerated caution will lead to a decrease in process capacity and an unnecessary highenergy consumption. The new simulation tool can simulate the drying batch and the tool can be used by operators and technical management to optimise the drying schemes with regards to product quality, process capacity and energy consumption.
2. Mathematical model The core of the simulation tool is a detailed dynamic model of the drying kiln and of the wood boards. Many researchers have been dealing with modelling of the wood drying process in order to explain and predict how the wood dries and also the appearance of quahty degradations such as cracking. Salin (1990), Stanish et. al. (1986) and Turner (1986) are some of the most important sources for the developing of the model in this work. The mathematical models consist of two distinct parts; a thermodynamical model and a tension/stress model. Detailed model equations of the model are described in Didriksen et. al. (2001) 2.1 Model structure The wood board is divided into a number of layers as shown in figure 1. There is massand energy exchange between the layers as shown in the figure. Deformations and tensions of each layer arises when the wood is dried down to a moisture content below the so called Fiber Saturation Point (FSP).
470 In each layer all process variables are assumed to be constant. In this way, the model is able to describe the moisture content profile and the tension profile of the board in the board depth direction. In other words, a one dimensional model structure is chosen. This is of course a simplification since the wood dries at a different speed at the corners and edges of the boards than in the middle, which also influences the tensions and deformations. However, the simplification is justified by the board's dimensions, they are long and flat, and the fact that the boards are laying closely side by side. A two dimensional of the wood board drying simulation is described e.g. in Turner (1986) and Perre and Turner (2001).
i i Svmmetrv' line
J Heat t
Thermal conductance
1
Evaporation
Transport of water - free water - tK>und water - w ater vapour
Figure 1. Model structure, wood board
2.2 Thermodynamic model The thermodynamical model is based upon the conservation of mass and energy and calculates how the water is ^'transported" through the boards and is evaporated at the board surface throughout the drying batch. The water evaporation and heat transfer from drying air to the boards depends upon the climate in the kiln and upon the drying air velocity. The calculations account for the drying properties of the type of wood in question. Wood is a complicated cellular porous material consisting of a number of tubular like cells as shown schematically in figure 2.
Figure 2. Wood structure, schematically
The water in the wood may exist in three different states; free water in the cell cavities, water vapour in the voids and bound water hygroscopically held in the cell walls. The transport mechanisms enabling moisture movement are capillarity liquid flow, r|f, water
471 vapour diffusion, r|vd, and bound liquid diffusion, riw, respectively. Heat is transferred between the layers by thermal conduction, QcondConservation of mass (water), m^, in layer no. i:
Conservation of energy, U, in layer no. i: u
}_dUj__( k .
+ ^bd, y.^C^-
/
V
[n^,^ )l,.apour. + Qcond^.X " Qcond,
^^^
2.3 Tension model Throughout the drying batch, the different layers of the board will have different moisture contents. When the moisture content reaches a certain value (the so-called Fiber Saturation Piont), the wood starts to shrink. Because the boards shrink uneven in the thickness direction, the board will be exposed to uneven shrinking and tensions in the board will arise. If the tensions exceed a certain limit, the board will check. Checking of the boards represent a serious degradation of the product quality. The tension model describes the deformations and induced stresses in the drying wood. The tension model is put "on top'' of the thermodynamical model. Four types of deformation are included in the tension model; shrinking z^, elastic strain Be, mechanosorptive creep e^s and viscoelastic creep Evs(3) When the values of the different deformations for each layer are calculated, an average deformation for the entire board, E , is calculated. n
7=1
Next, the tensions in the different layers of the boards, QJ, can be calculated
E is here the (temperature and moisture dependent) elasticity module and Ax is the thickness of each layer. The estimated tensions are indications of the risk of board cracking.
472
3. Simulation case The simulation tool is demonstrated by showing the drying of ash. Ash is a wood type, which is difficult to dry without quality degradations, and therefore, a very long drying time is required. The simulation case is based upon process data from a kiln at a Danish floor producer. The current dimensional data and wood species data is typed in the simulation tool. Furthermore, a typical drying scheme; consisting of dry and wet temperature of the drying gas, is typed in. The drying scheme is shown in figure 3. Notice the very sharp reduction in gas moisture at about t = 360 hours. When the operator measures moisture content below the FSP, he knows that the wood can stand a tougher drying because the wood is physically stronger at lower moisture contents. — Dry air temperature ("C) - - - - Wet air temperature ('C)
Drying sceeme
("1
80 75 70 65 60 55
1
1
50 45 40 35 30 25 20
0
20
40
60
80
100
120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420
^-Tet^K
€nn^i oq Mil*
1
]
Figure 3. Original drying scheme (dry and wet air temperatures).
40
60
80
100 120 140 160 180 200 220 MO 260 280 300 330 3«0 380 3
80
80
100 120 1
ie0
2a}22O24O2e02aD30033034O36O380400'2O
Figure 4. Moisture- and tension-profiles of the wood boards, original drying scheme.
473 The moisture contents and the tensions of the different layers of the boards are shown in figure 4. Internal tensions arise when the moisture content in the outer layers of the boards is below the FSP, in this case 28 % moisture. The outer layers will start shrinking at this moisture content. The other layers, however, have not started to shrink at this point. This is why a large positive tension appears in the outer layer and smaller negative tensions appear in the other layers. Later on, the moisture content in the second layer reaches the FSP, and a positive tension appears here also. This pattern repeats. When the inner layers starts shrinking, the outer layers have ''stiffened" in a stretched position. This can make the tension profile turn, resulting in positive tension values for the inner layers and negative tension values for the outer layers The values of the tension are indications of the risk of cracking of the boards. — Dry air temperature (*C) •-••Wetairtemperarjre (*C)i
20
40
60
80
180
3O0
200 220 240 260 Hours
320 340 360 390 400 420
Figure 5. Modified drying scheme (dry and wet air temperatures).
On the basis of the simulation results from the simulation tool, a modified drying scheme is proposed. The dry and wet temperature of the drying gas is shown in figure 5. The resulting moisture and tension profiles of the boards are shown in figure 6.
8O2O02a)2«26O2«O3O0J2O3«3
«
60
80
1M 1
3 360 380 400 <20
Figure 6. Moisture- and tension-profiles of the wood boards, modified drying scheme
474 The boards are dried to the same final moisture content value as with the original drying scheme (8.8 %) and within the same drying time (420 hours). The energy consumption (heat only) using the modified scheme is, however, reduced from approximately 35.1 MWh to 31.6 MWh. In other words, the modified drying scheme results in a heat energy reduction of about 10 %. The maximum predicted tension in the boards is lower in the modified scheme than in the original. This indicates that the risk of board cracking will be reduced using the modified scheme and the product quality of the boards will be, on the average, increased. Furthermore, the tensions in the boards are considerable lower at the end of the drying with the modified scheme. The final tension profile in the dried boards is a quality parameter. This is because cutting of boards with high internal tensions can result in problems like board bending. The tensions of the boards are reduced by a subsequent steaming. Nevertheless, a starting point with lower tensions before the steaming will be an advantage. The simulation program can also be used in a similar way to achieve a faster drying scheme without higher risks of cracking, resulting in increased production capacity of the drying process.
Acknowledgement The Danish Energy Agency and Junckers Industrier A/S have supported this work.
References Didriksen H., Nielsen J.S., Hansen M.W. (2001). ''Energibesparelser ved optimering af t0rreprocesser gennem anvendelse af modeller og effektiv regulering (report in danish)", dk-TEKNIK ENERGY&ENVIRONMENT Perre P. and. Turner W. (2000). An Efficient Two-Dimensional CV-FE Drying Model Developed for Heterogeneous and Anisotropic Materials. Proceedings of the 12* International Drying Symposium IDS2000 Salin J.G. (1990). Simulation of the timber drying process. Prediciton of moisture and quality changes. EKONO Oy, Helsinki Finland. (Doctor of Technology Thesis) Stanish M.A., Schajer G.S., Kayihan Ferhan, (1986). A Mathematical Model of Drying for Hygroscopic Porous Media. AIChE Journal Vol. 32, No. 8 Turner I.W. (1986). A two-dimensional orthotopic model for simulation of wood drying processes. App. Math. Modelling 1996, Vol. 20, January.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
475
Optimisation of a Methyl Acetate Production Process by Reactive Batch Distillation S. Elgue ^ L. Prat \ M. Cabassud \ J.M. Le Lann", J. Cezerac ^ " Laboratoire de Genie Chimique, UMR 5503, CNRS/INPT(ENSIACET)AJPS, 118 Route de Narbonne, 31077 Toulouse Cedex, France ^ Sanofi-Synthelabo, 45 Chemin de Meteline, B.P. 15, 04201 Sisteron Cedex, France
Abstract A general framework for the dynamic simulation and optimisation of global batch synthesis has been developed. In this paper, its application to the optimisation of a methyl acetate production process by reactive batch distillation is presented. Experiments performed on a batch pilot plant allow validating the dynamic model. Hence, optimal tuning of the operating parameters of the reactive batch distillation has been investigated by means of the dynamic optimisation procedure.
1. Introduction The synthesis of fme chemical or pharmaceuticals, widely carried out in batch processes, imply many successive reaction and separation steps. Synthesis optimisation is often restricted to the determination of the optimal operating conditions of each step separately. Therefore, such an approach does not definitely lead to the optimal conditions for global synthesis. For example, optimising the conversion of a reaction for which separation between the desired product and the by-products is more difficult than between the reactants, will involve an important operating cost, due to further difficulties in the separation scheme. Thus, necessity to simultaneously integrate all the process steps in a single optimisation approach clearly appears. For this purpose, recent issue in the dynamic simulation associated with efficient optimisation method may be exploited to accomplish this goal (Wajge and Reklaifis, 1999, Elgue et al., 2001). In reactive distillation process, the reaction and distillation steps simultaneously occur in the single unit. Unlike conventional configurations of reactors followed by separators, the reactants or products are continuously separated from the liquid reaction phase. This key feature allows an enhanced conversion in equilibrium limited reactions (Agreda et al., 1990). In this way, the application of our framework to a reactive batch distillation process appears obvious. The present paper is divided into two parts. The first part details the simulation model and presents the associated validation carried out through a process of methyl acetate production. The second part focus on the optimisation of this synthesis and the resultant benefits, according to the proposed methodology.
476
2. Process modelling A general modelling able to consider various configuration of reaction separation process has been developed (Elgue et al, 2001). In the purpose of the present study, the mathematical model results of the detailed and rigorous description of a batch reactor connected with a overhead distillation column. In this framework, a tray by tray approach has been adopted, the column condenser representing the first plate, the reactor the last one. Thus, the dynamic model is described by a set of differential and algebraic equations (DAE system) made around each plate. Ordinary differential equations (ODE) are due to energy balances, total and component mass balances. With regard to the reactor, it has to be noted that thermal modelling of the vessel and the jacket is particularly detailed. Algebraic equations (AE) consist of vapour liquid equilibrium relationships, summation equations, physical property estimations. In order to reduce the complexity of the model, the following typical assumptions have been made, on each tray: perfect mixing between vapour bubbles and liquid, equilibrium between liquid and vapour bubbles and introduction of the Murphree efficiency, negligible vapour holdup compared to the liquid holdup and constant volume of the liquid holdup. For models determination (enthalpy model, equilibrium constant, hydrodynamic relationship and bubble point temperature) mathematical model is connected to Prophy, a complete physical property estimation system with associated data bank. DISCo (Sargousse et al., 1999), a general solver of DAE systems based on the Gear method, obtains the solution of the mathematical model. Besides its accuracy and numerical robustness, DISCo offers substantial integration velocity thanks to the use of operator sparse and its automatic initialisation procedure.
3. Experimental Bonnaillie et al. (2001) considered a methyl acetate production process by reactive distillation. The batch pilot plant and the experimental results of their study have been exploited in the present work. The batch pilot plant involves a glass reactor with an overhead distillation column. The bottom plant is composed of a stirred jacketed glass reactor of 5 litres volume. The jacket is provided with a heat transfer fluid circulating in a boiler at around 160°C. The overhead distillation column consists of a multiknit packed column (50 cm in length and 10 cm in diameter) with a condenser and a distillate tank. The condenser involves a spiral coil heat exchanger provided with cooling water. An adjustable timer regulates periodic switching between distillate tank and reflux to the column, with a constant reflux policy. CH3-OH Methanol (65 °C)
+
^O CH3~C ^OH Acetic acid (118°C)
H2SO4
H20 Water (100°C)
+
CH3-C^ ^n—PR. Methyl acetate ('57°C)
(^)
Production of methyl acetate is ensured by the addition of acetic acid to methanol with sulfuric acid as homogeneous catalyst, as can bee shown on the reaction scheme (1).
477 Comparison of boiling point temperatures show the higher volatility of methyl acetate relative to the other components. Thus, methyl acetate removals, by distillation, enable the reaction to reach higher conversion than theoretical equilibrium value. Three experiments have been exploited: experiment with reaction only, experiment with distillation only (following a reaction phase), experiment of reactive distillation, i.e. coupling of reaction and distillation. The experimental conditions are listed in table 1. Table J: experimental conditions Experiment a b c
Reaction Distillation Coupling
Initial molar ratio 1.4 1.4 1.4
Reflux ratio
Catalyst amount
oo
2.5 oo,2.5
5 ml 5 ml 5 ml
4. Simulation validation In order to verify the mathematical model accuracy, the experiments carried out by Bonnaillie et al. (2001) have been simulated. The same thermodynamic (NRTL) and kinetic models have been used. It has to be noted that reaction rate model is a simple kinetic model (2), in agreement with data reported in literature (Smith, 1939).
r = K.er e x p ( ^ ) ( C ^ . . C , , , , - ^J12O£M^^
(2)
with : kester pre-expoueutial factor = 3300 l.mol'.mn ^mr'H2S04 EA activation energy = 41800 J.mol"^ Keq equilibrium constant = 5 For each experiment (reaction, distillation and coupling) the simulation are compared to samples withdrawn into reactor (fig. l.a to l.c). Figure l.b and l.c show, during coupling and distillation experiments, light differences between experimental and simulated compositions. This differences are explainable, on the one hand by the strong thermodynamic non-ideality of the mixture and on the other hand by the kinetic model simplicity. In fact complex kinetic models, taking into account the non-ideality of the mixture and so integrating thermodynamic models (Popken et al., 2000) would probably provide more accuracy. Nevertheless, the good agreement between experiments and their mathematical representations allows validating the simulation tool. Simulations also show, through coupling results, the advantages linked to reactive distillation and emphasize the necessity of an optimal operation policy.
5. Dynamic optimisation The purpose of the present study is to optimise the reactive distillation process of production of methyl acetate. Generally, operating time, reaction yield, energy consumption, safety and environmental constraints are the key points of reactive batch distillation processes. In the present process, the column is always operating with a total reflux policy during reaction phases and with a constant reflux policy during distillation
478
- Acetic Acid - Sim
Methanol - Sim
Acetic Acid - Exp
•
Methanol - Exp
Methyl Acetate - Sim •
Methyl Acetate - Exp
Water-Sim •
Water - Exp
§ 0.6
^
Case a
0.4
40
60 Time (mn)
i.u -
0.8 -
1 0.6
* ^-—'" o 0.4
-8"
Case b
1 —
0.2 ^ ^ ^ - ^ ^ *
"^
*
•"~---
•
^^"^^^^^iii;;;^
—x-—--:^.^
.~^-.__J_J]_^->", *-
on 40
60
80
100
Time (mn)
1.0
0.8
.2 0.6
Casec 0.2
- 1 - ^ ^ -
0.0 20
40
60
80
Time (mn)
Figure I: Variations of reactor composition with time for cases a,b and c
479 phases. Thus, the energy consumption (heat provided by the boiler, condenser cooling water) is only a function of the operating time. In this way, optimisation of the process only involves two criterions: operating time and conversion of reactants. Therefore, a function combination of operating time and conversion, with various weights, has been established in order to optimise the production process. Hence, a mono-objective optimisation method has been used, a successive quadratic programming method (SQP). All the different processes, reaction-distillation or coupling, have the same dynamic structure (fig. 2). In fact, according to the total reflux end time (tRoo), before or after the time of reaction equilibrium (teq), the process is respectively a coupling or a reactiondistillation process. Thus, the total reflux end time appears to be a main variable of the optimisation problems, as well as reflux ratio (R) and operating time(top). Distillation Reaction
E
v/////////////////////////m///////^^^^ 0
R:
R^"
^R<»
- • time R9t<
teq
uop to,
Figure 2: dynamic structure of batch reactive distillation process The total reflux end time is a very sensitive parameter of reactive batch distillation. In fact, if total reflux policy ends too early, conversion will be reduced owing to the distillation of reactants. On the contrary, a too long total reflux policy will lead to additional operating time and so waste of time. Experimentally, the appropriate total reflux time is very difficult to correctly estimate. Thus, its determination by means of optimisation represents a very challenging objective. Two main kind of optimisation problems have been studied (table 2). In the first one, the minimum operating time necessary to obtain the desired reactant conversion, is determined. In the second one, objective functions combination of operating time and conversion with various weighting are minimise. Table 2: Optimisation problems Kind Objective function 1. Operating time
2.
Operating time, acetic acid conversion
Variables Operating time Total reflux end time Reflux ratio Operating time Total reflux end time Reflux ratio
Constraints Model equations Acid Conversion Model equations
Several optimisation problems, of the two kinds previously presented, have been solved. The resultant solutions are detailed in table 3. The two first solutions present the optimal operating conditions for respectively 93 percent and 95 percent of acetic acid conversion (Conv). The three other solutions present the operating conditions leading to optimal criterions, criterions for which the conversion weight respectively decrease.
480 Table 3: results of the optimisation problems Objective Operating Constraint function time (mn) 80 top Conv > 93 % 118 top Conv > 95 % / 141 '-op'" \ ^ ^onv/ X 33 / 105 ^op' lA—i^onvj ^ 17 / 48 top"*" vt—^onvj X 1.7
Total reflux end (mn) 15 20 25 23 9
Reflux ratio 1.8 3.0 2.3 2.5 1.0
Acetic Acid conversion 93.0 % 95.0 % 95.9 % 94.5 % 85.8 %
Results of the first kind of optimisation problems show that a significant total reflux time (more than 15 mn) is required for high conversion of reactants. The second kind show that if conversion is privileged, total reflux time is almost invariable (around 23 mn) and only further operating time allows reaching higher conversion.
6. Conclusion The developed framework is found to be useful in the determination of optimal operating conditions for the methyl acetate production by reactive batch distillation. For different production conditions, this framework allows, in particular, determining the appropriate value of the total reflux period. For the continuation of the operation, a constant reflux policy has been adopted, due to the experimental device. Nevertheless, this program shows to be able to determine more complex optimal reflux policies (piecewise constant reflux profile, piecewise linear reflux profile, complex function of reflux profile) as part of other applications. In fact, this framework is not restricted to reactive batch distillation processes, but is also able to represent and optimise various processes integrating batch reactor and batch distillation column.
References Agreda V.H., L.R. Partin, W.H. Heise, 1990, High-purity methyl acetate via reactive distillation, Chem. Eng. Prog., Feb, pp. 40-46. Bonnaillie L., X.M. Meyer, A.M. Wilhelm, 2001, Teaching reactive distillation : experimental results and dynamic simulation of pedagogical batch pilot-plant, ISMR2, Nuremberg. Elgue S., M. Cabassud, L. Prat, J.M. Le Lann, G. Casamatta, J. Cezerac, 2001, Optimisation of global pharmaceutical syntheses integrating environmental aspects. Escape 11, Kolding, Denmark. Popken T., L. Gotze, J. Gmehling, 2000, Reaction kinetics and chemical equilibrium of homogeneously and heterogeneously catalyzed acetic acid esterification with methanol and methyl acetate hydrolysis, Ind. Eng. Chem. Res., 39, pp. 2601-2611 Sargousse A., J.M. Le Lann, X. Joulia, L. Jourda, 1999, DISCo: un nouvel environnement de simulation oriente objet, MOSIM'99, Annecy, France Smith H.A., 1939, Kinetics of the catalyzed esterification of normal aliphatic acids in methyl alcohol, J. Am. Chem. Soc, 61, pp. 254-260 Wajge R.M., G.V. Reklaitis, 1999, RBDOPT: a general purpose object-oriented module for distributed campaign optimization of reaction batch distillation, Chem. Eng. J., 75, pp. 57-68
buropean Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
481
Computer design of a new predictive adaptive controller coupling neural networks and kalman filter applied to siso and mimo control L. Ender^ R. Scheffer^ and R. Maciel Filho^ ^Chemical Engineering Department, Regional University of Blumenau, Rua Antonio da Veiga 140; CP 1507, Blumenau - SC, Brazil, CEP 89010-971 ^LOPCA/DPQ, Faculty of Chemical Engineering, State University of Campinas (UNICAMP), Cidade Universitaria Zeferino Vaz, CP 6066, Campinas - SP, Brazil, CEP 13081-970
Abstract
This work presents a predictive control algorithm based on constraint neural networks as internal non-linear model with a tuning algorithm based on the Kalman filter. The algorithm utilises a sequential quadratic programming algorithm to compute the next action of the manipulated process variables. The predictive control parameter, the suppression factor, is optimised on-line by a standard Kalman filter. The suppression factor is identified by a method based on the relative gain. The algorithm was tested on distinct chemical processes, a penicillin fermentation process (SISO) and a fixed bed catalytic reactor (MIMO). It shows that the suppression factor can be identified on-line, but a scaling factor has to be introduced because the process derivatives can become large. The proposed procedure still reduces the number of parameters to be adjusted in case of MIMO systems.
1. Introduction Model Predictive Control (MPC) concept has been widely accepted in industry applications and extensively studied by academia. The main reasons for such popularity of the predictive control strategies are the intuitiveness and the explicit contraint handling. Several versions of MPC techniques are Model Algorithmic Control (MAC), Dynamic Matrix Control (DMC) and Internal Model Control (IMC). Although the above techniques differ from each other is some details, they are fundamentally the same. All of them are based on linear process modelling (Zhan and Ishida, 1997). The most utilised model predictive control algorithm is the dynamic matrix control algorithm. Several modifications of the DMC algorithm were already proposed, such as identification by state space models and the use of non-linear MPC algorithms consisting of neural networks with a SQP optimisation algorithm (Economou et al. , 1986; Lee and Biegler, 1988; Piche et al. 2000; Ender and Maciel Filho, 2000). In this work the non-linear approach is used, as dealing with a variety of process conditions and disturbances demands a good process identification. All model predictive control algorithms minimise the following objective function,
J=mmarg Yj-.,J.ll,(yr.f.M-yjJ u
+Y"H^J
Sl!y(";.*
""M-/)"
^^^
where p is the number of controlled outputs, m the number of manipulated input variables, Np the prediction horizon, Nc the controller horizon and Xj the suppression factor of the corresponding manipulated input variable Uj.
482 The manipulated inputs and controlled outputs are subjected to the following constraints: }^_<5^(/:+ / ) < > ; _
i = lr".N^
(2)
u^^^
i = lr".N^
(3)
\u{k-^i--l)~-u{k-^i'-l}
/=:7,...,7V^
(4)
Adopting the receding horizon technique, only the first control action is implemented and all the calculations are repeated at each sampling time. The only difference in the different type of model predictive control algorithms is the way in which the output of the process is predicted. In this case a neural network is used and thus the output is predicted by yj,k =f{yk-n^k-nd,.j)+w,_j
(5)
To cope with process changes the neural network is trained on-line by a switching method as proposed in Ender and Maciel (2000). Due to the non-linear prediction model used, an sequential quadratic programming method has to be used. The suppression factor X assures that no exaggerated control action is calculated and influences the systems dynamics. A too small X results in large control actions, which can result in a instable response, while a too large X results in a sluggish response. Normally, the parameter is tuned manually until a desired process behaviour is obtained, which can be a very time-consuming procedure. Additionally, different values of X might be needed for other operating conditions. An automatic estimation procedure for X was developed in Ender et al. (2001), which still showed a small overshoot for some set-point changes. Therefore it was searched for another identification scheme of the suppression factor, which is presented in the next paragraph. The new identification schemes are compared to the identification scheme derived from the optimisation criterion as described in Ender et al. (2001) and are applied to two distinct chemical processes: the penicillin fed-batch process (SISO case) and a fixed bed catalytic reactor (MIMO cases). 1.1 Estimation schemes of the suppression factor The adjustment or tuning algorithm of the parameter X is based on the standard Kalman filter. To be able to adjust X a dynamical system has to be created which can observe the state of the parameter X as in:
/
X
(O)
where w^ and Vj, are random variables with a normal distribution of N(0,Q) and N(0,R) respectively. Zj, is the measurement related to the state X^,. Normally the noise of a parameter state is zero, but a small process noise results in a more stable filter.
483 The observation equation of the X was derived from the minimisation criterion J in equation 1 in Ender et al. (2001) by setting the derivative of J to the input change, Au, equal to zero. One of the shortcoming of this approach is that the adjustment of the suppression factor is based on the past and not on the present data. Thus the algorithm changes X in a feedback way. It would be desirable that the suppression factor is raised, when the process is changing rapidly. This assures that the process will not show oscillatory behaviour. It is used an intuitive approach, which is based on the definition of the relative gain.
abs\
^t
=
(7)
abs{u,-u,_,)X,{-^v,)
sample
The absolute value is taken as the suppression factor has a positive value. In case of the fixed bed reactor, the derivatives are very large and it was necessary to put a velocity factor, 0, which results in the following identification system:
(8)
abs\^ I absi
yjMJ
At yiMi
- yjM ^ sample
- y2.k
'abs(uj,-Uj,_j)
0
0
absiuj^.
i,k+i
-i^2.k-i)
'2,k+l
At sample
This approach leads to the introduction of a new parameter which still has to be tuned manually. But the number of parameters to be tuned will be reduced.
2. Results One of the case-studies considered is a large industrial fermentation process for the production of penicillin. The simulation is based on a model of Rodrigues (1999), which is validated with industrial data. The process is a fed-batch process and falls down in two parts, the growing phase and the production phase. In the growing phase a high sugar level is maintained, while in the production phase it has to be kept low as it inhibits the penicillin production. The emphasis is put on the production phase where the feeding strategy of the sugar substrate has to be chosen carefully to maximise the penicillin production. A single input, single output (SISO) system was set-up to verify the proposed process control scheme. The controlled variable was the dissolved oxygen concentration which was controlled by the rotation speed. Various constraints are applicable here as, maintaining the dissolved oxygen concentration above 30% and avoiding high rotation
484 speeds which destroy the fungi. A white noise was imposed on the dissolved oxygen concentration to simulate measurement noise. The other case-study is a fixed bed catalytic reactor for the production of acetaldehyde by the ethyl alcohol oxidation over a Fe-Mo catalyst. The effective control of those reactors is fundamental to obtain a safe operation, especially when high performance is desired. The control problem of such reactors is not an easy task since they are nonlinear, distributed and time-varying systems and show an inverse response due to differences in heat capacities from solids and fluid. The dynamic behaviour of the catalytic fixed bed reactor was modelled by the model proposed by Toledo (2000) and the control is done taking two temperatures at collocation points along the reactor tube. For both cases the servo problem as well as the regulator problem was undertaken. It is presented in figures 1 and 3 the servo and regulator control of the dissolved oxygen concentration. The evolution of the suppression factor is shown in figure 2 and 4. It can be seen that the dissolved oxygen concentration does not show any overshoot anymore in comparison with the identification method proposed in Ender et al. (2001). The suppression factor converges to about the same value when measure noise is present. Without noise the suppression factor does not converge to the same value in case of the servo problem. In case of the regulator problem the suppression factor converges to about the same value. In the servo case three perturbations are done, while in the regulator case 15 perturbations are done because of an optimal substrate feeding strategy applied (Rodrigues and Maciel Filho, 1999). This generates much more information content for the Kalman filter to adjust the suppression factor.
2
I O.7J c o.ioi ^ o
0.6^ J
•y
0.6(H
—
^
S
Q
•X^X with noise Aj,=2 with noise Aj,=0.04 with noise X = l without noise
\=\ Willi noise }^=2 with noise \=0.04 with noise A^=l witliout noise
0.8{H
J
1
0.5H
]
0.5(M O.45J
fWf^
0.4O
tiine (h)
Figure 1: Dissolved oxygen concentration of the penicillin reactor under servo control
Time (h)
Figure 2: Evolution of the suppression factor under servo control for the penicillin reactor
485 4CH
35-1 - ^ 1 . 0 witli noise - ^ 1 . 0 wiiliout noist
->,= 1.0 with noise • /,=1.0 witlK)ut noise
3M
1 5H lO-f
time (h)
Figure 4: Evolution of the suppression factor under regulator control for the penicillin reactor
Figure 3: Dissolved oxygen concentration of the penicillin reactor under regulator control
In figure 5 and 7 the servo and regulator temperature control is shown of the fixed bed reactor. If no velocity factor, ([), is used, then the suppression factor becomes very large. This makes the SQP algorithm unable to find a solution as the manipulated input variable is too much restricted. Still, the value of the velocity factor is not restricted to a very small range, which would make the tuning of this factor a difficult task again. It can be seen from figure 5 and 7 that control is not that much affected when the factor differs a factor of 10 as the suppression factors do not differ a factor of such value (figure 6 and 8). But it should be noted that in the servo control case the second suppression factor (not shown) differed a lot, which explains the difference in control observed in the servo control case. The suppression factor changes mainly when the process changes. From figures 6 and 8 it can be seen that large peeks occurs for the suppression factor when the set-point is changed. This is obvious from the applied identification scheme which is a function of the derivative of the process. In the penicillin process it was not needed a suppression factor as it has a slow dynamic behaviour. The fixed bed catalytic reactor has a very fast dynamics and thus it has high values of the derivative which result in a high suppression factor. Thus the velocity factor should be chosen in accordance with the gradient occurring under process changes. 453.51
24(H
'F^ 453.d
Q i
1 ^
-0=0.1 --0=0.01
20CH I80]
1 452.0^
1 451.0^ 450.5" 450 0-
c:
r
L
449.5]
•
>
—
-T
KI r>^ .
—0=0.1 —0=0.01
220]
/ Point 1 in reactor
1 —1
1
1
Reactor point 1
lOOj 80^ 6(H
Point 2 in reartnr
'
i4oJ
1
•
,
Time (h)
Figure 5: Temperatures at two points along the fixed bed reactor under servo control
40^
-H
0-]
.L_ —
, 1
—
1
i« —
T
Ir
••—
t
/
I ——
1 — ' — 1
Time (h)
Figure 6: evaluation of suppression factor of first manipulated variable under servo control of the fixed bed reactor
486 4036-
2824-
%
20-
= 0.1 = 0.05
r
/
16-
'1
4-j'
o4
time (h)
Figure 7: Temperature at second points along the fixed bed reactor under regulator control
,
,
.
,
.
1
.
1
.
1
tiine (h)
Figure 8: evaluation of suppression factor of second manipulated variable under regulator control of the fixed bed reactor
The introduction of the velocity factor means introducing a new parameter to be tuned, while it is wanted to eliminate the tuning of the suppression factor. While the number of parameters to be tuned for MIMO systems is reduced in this way, it is still not the ideal case where no parameter has to be tuned. Therefore it has to be continued the search for other ways to identify the suppression factors, which result in no introduction of another parameter.
3. Conclusions The proposed control scheme of a model predictive control algorithm with a constrained neural network internal model with auto-tuning capabilities results in a satisfactory control. The suppression factor is adjusted on-line by a standard Kalman filter through an identification method comparable to the relative gain. In rapid changing processes it was needed to introduce a new parameter called the velocity factor, otherwise the suppression can become too large prohibiting control of the process. It seems that this velocity factor is not restricted to a very narrow range and in case of MIMO processes the number of parameters to be tuned manually is lowered. Still, it has to be searched for better ways of identifying the suppression factor, which do not lead to introduction of new parameters.
References Economou, G.G., M. Morari, B.O.Palsson, (1986) Ind. Eng. Chem Process Des. Dev., 25,403-411 Ender, L. and R. Maciel Filho (2000), Comp. & Chem. Engineering, 24, 937-943 Ender, L., R. Scheffer and R. Maciel Filho (2001), Proceedings of ESCAPE 11, Denmark, 639-644 Li, W.C. and L.T. Biegler (1988), Ind. Eng. Chem. Res, 27, 1421-1433 Pinche, S., B.S. Rodsari, D.Johnson, M. Gerules (2000), IEEE control systems magazine, 53-61 Rodrigues, J.A.D and R. Maciel Filho (1999), Chemical Engineering Science, 54(1314), 2745-2751 Toledo, E. de V. (2000), PhD-thesis, State University of Campinas, School of Chemical Engineering, "Modelling and control of fixed bed catalytic reactors" Zhan, J. and M. Ishida, (1997) Computers and Chemical Engineering, 21, 2, 201-210
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Elsevier Science B.V. All rights reserved.
487
Comparison of Dynamic Approaches for Batch Monitoring N. M. Fletcher, A. J. Morris and E. B. Martin Centre for Process Analytics and Control Technology, University of Newcastle, Newcastle upon Tyne, NEl 7RU, UK
Abstract The first stage in the development of a performance monitoring tool is a representative model of the process. Traditionally batch processes are associated with non-linear, dynamic behaviour. One method of developing a non-linear model is through a local modelling approach. These local models can be combined to provide a global non-linear representation of the process. The paper develops and compares local models formed using linear Partial Least Squares (PLS), dynamic linear PLS and ARX structures with global models based on dynamic linear and non-linear PLS, as well as non-linear PLS using a simulation of a fed-batch fermentation process. Finally the use of a local linear dynamic PLS model for process performance monitoring representation was evaluated.
1. Introduction The use of multivariate statistical projection based techniques to model and monitor batch process have been recognised as one approach to enhancing process performance and contributing to an increased understanding of process behaviour. The key techniques include multi-way principal component analysis, multi-way partial least squares (Nomikos and MacGregor, 1994, 1995) and batch observation level analysis (Wold et al., 1998). However batch processes often exhibit non-linear, time variant behaviour with variables exhibiting serial and cross correlation. These characteristics challenge linear statistical multivariate batch monitoring techniques. To address these limitations, PLS has been extended to dynamic PLS to capture the process dynamics through the augmentation of the input data matrix with lagged values of the past input and output observations (e.g. Kaspar and Ray, 1993, Baffi et al. 2000). two possible approaches to removing the process non-linearities are through the removal of the mean trajectory or the application of non-linear PLS. Alternatively the batch process trajectories can be sub-divided into a number of distinct operating regions with a linear model being fitted to each operating region. These individual models can then be pieced together, thereby providing an overall non-linear global model (e.g. Foss et al, 1995). Such a local model based structure provides the potential for a novel approach to batch process performance monitoring. In the paper a number of techniques are used to develop the local model, including multi-way PLS, dynamic multi-way PLS and AutoRegressive with eXogenous (ARX) representations. The methodology is demonstrated by application to a simulation of a batch process and compared with the global modelling approaches of multi-way PLS,
488 ARX and non-linear PLS. Finally a process performance monitoring representation is developed from the local linear PLS models.
2. Local Models By partitioning the batch trajectories into a number of operating regions where linear approximations apply, a non-linear global model can be approximated through the piecing together of local models. The model developed for the local operating regime is valid for the process operating under specific conditions and gradually becoming invalid outside of that region. A validity function vector, p^, (0 < p < 1) defines the weight for a specific operating regime at each time point, 0, throughout the entire batch. In this initial study linear interpolation is used to provide a smooth transition between the operating regimes (Foss et al, 1995). The interpolation function is calculated as:
^;(*) = i r
^here^a;.(0)-l
V 0,
(1)
where N is the number of operating regions,;, within the batch process. A nominal PLS model is then built for each local region using cross-validation for latent variable selection. The loadings (pjk) for each latent variable, /:, and the weights (Wj), define the coefficients of each local model that are then stitched together using the interpolation function, (O^-, for the prediction of unseen batches: ^ r P^ = ^ p jj^ CO J
for /: = !,..., no. of latent variables
(2)
where j =1,..,N are the number of local models, P^t is the interpolated loading vector to be used in the prediction of the unseen batches for each latent variable and COj is the interpolation function for each operating regime calculated from Equation 1.
3. Simulation Example The simulation used to investigate this methodology was the fermentation of glucose to gluconic acid (Foss et al, 1995). One hundred batches were simulated for the nominal batch set and thirty for the unseen test batches. The inputs to the model were cell concentration (xO, glucose concentration fe), gluconolactone concentration (^3), dissolved oxygen concentration (X4) and oxygen uptake rate, OUR (xs) and the output was gluconic acid concentration (yi). The variables selected to define the different operating regimes were xi and Xs. Three operating regimes were defined. Operating regime 1 was identified to be where OUR and cell concentration increase. Regime 2 was defined as the levelling off of the OUR gradient and its subsequent decrease. The third regime is defined to start when cell
489 concentration begins to level off. Fig. 1 shows a plot of the five process variables. The operating range for each regime is also identified. From Fig. 1, the validity function p/ is calculated (Eq. 2). Fig. 2 shows the interpolation functions for the three operating regimes.
Fig. 1. Operating regimes.
Fig. 2. Interpolation functions for 3 local regimes
4. Results The 100 nominal batches were stacked to give a data matrix X with each variable being normalised to zero mean and unit variance. For dynamic PLS, the variable lags were determined by first fitting an ARX model to the data. From the weight of the parameters of the ARX model (Fig. 3), a lag for each variable was identified. In Fig. 3 it can clearly be seen that for variable one, a lag of four is more significant than previous lags, consequently 4 lags were included in the final model. The number of lags included for each variable differed between each of the global and local models. 4.1 Results for the Global Model. Cross-validation was applied to both the original data and the data set comprising the lagged variables. Four latent variables were selected for dynamic PLS (DPLS), two for PLS, three for non-linear PLS (NL-PLS) and two for PLS where the mean trajectory of each variable had been removed (PLSmean-traj)- This was confirmed by analysing the regression coefficients of the latent variables and their associated standard errors. Fig. 4 shows the regression coefficients ± the standard error for the global DPLS model. From Fig. 4 it can be concluded that from latent variables seven onwards are non-significant.
n . • 1
hi!
LkiiM
1
1234561234561234 5 61234 561234 5 6
Fig. 3. ARX parameters for the determination of variable delays (lags).
Fig. 4. Regression coefficients with standard error bars.
490 Table 1. Prediction of the output using global models ARX PLS r l^dmean traj
DPLS NLPLS NL-DPLS
RSS(x 10-^) 8.1 354.5 101.1 11.9 180.4 8.1
BIC(x 10-') 3.7 6.3 15 4.0 5.8 3.7
The Bayesian Information Criterion (BIC) and the residual sum of squares (RSS) were used to determine how well each modelling technique predicted the output in terms of model parsimony and the ability of the representation to model unseen data. Table 1 summarises the results of the different modelling approaches. As expected the prediction of the output variable improves significantly when the batch dynamics are included in the analysis. The models that exhibited the best prediction capabilities were the ARX and non-linear dynamic PLS (NL-DPLS) representations. The following plots show the prediction of the output for one batch using PLS and NL-DPLS, Fig. 5 and 6.
Fig. 5. Predicted and actual values for PLS model
Fig. 6. Predicted and actual values for NL-DPLS model
-t^
•'*!
Fig. 7. Residuals versus fitted values for PLS model
Fig. 8. Residuals versus fitted values for DPLS model.
An off-set can clearly be seen between the actual and predicted values when the model was based on linear PLS. In comparison, no off-set is seen for NL-DPLS. The residuals of the fitted model are plotted against the predicted values of the output to investigate whether structure remains in the model. The underlying structure of the data is not captured by PLS, Fig. 7. This is in contrast with the residuals from the DPLS model (Fig. 8). Some structure remains in the NLPLS model, whereas the NL-DPLS residuals are structureless (not shown). The residuals for the PLS model with the mean trajectory removed exhibit a high level of structure (not shown) similar to those shown in Fig. 7.
491 Table 2. Number of latent variables Model PLS DPLS
Local model 1 3 4
Local model 2 4 4
Local model 3 4 5
Table 3 Prediction of the output using local models Model ARX PLS DPLS
RSS (X 10') 8.1 351.0 9.9
BIC (X 10') 3.8 6.7 3.9
4.2 Results for Local Modelling PLS, DPLS and ARX models were selected for the local modelling stages. Table 2 summarises the number of latent variables selected for each local model using crossvalidation. Table 3 shows that there is a significant improvement in the prediction capabilities when a dynamic local model approach is used. No improvement is seen in the prediction of the output when using local PLS models compared to that achieved with the global PLS model.
Fig. 9. Residuals versus fitted values for local ARX models.
Fig. 10. Residuals versus fitted values for local DPLS models.
Fig. 9 shows the residual structure for the local model using an ARX model. It is evident that there is no structure remaining. The residuals of the local dynamic linear PLS model (Fig. 10) can be seen to exhibit less structure than those of the DPLS global model (Fig. 8). From the results it can be concluded that the use of local dynamic PLS models gave slighdy improved prediction capabilities over those of the global dynamic PLS model. Once again it can be seen that inclusion of the dynamics of the process into the analysis resulted in a reduction in the structure in the residuals. 4.3 Monitoring Charts The latent variable scores calculated from the nominal models were used to monitor the progression of a batch through time. The interpolation function, (O, was used to interpolate between the scores of each local model. The limits were calculated using ± 3 standard deviations of the average nominal score at each time point. Figures 11 to 14 show the monitoring charts for nominal batches for a DPLS global model and a DPLS local linear modelling approach. The scores can be seen to follow a similar trajectory
492 for latent variable 1 for both the DPLS global model and the DPLS local model (Fig 11 and 13). However, for latent variable 4, the DPLS local model approach shows a structured trajectory (Fig. 12), highlighting that information is retained in the lower order latent variables that is not detected in the global model (Fig. 14).
Fig. 11 Control chart for DPLS local model scores for latent variable 1.
Fig. 12. Control chart for DPLS local model scores for latent variable 4
Fig. 13 Control chart for DPLS global model scores for latent variable 4.
Fig. 14. Control chart for DPLS global model scores for latent variable 1.
5. Conclusions The use of a local modelling approach has been shown to result in a reduction in the structure of the residuals, leading to a more useful model compared to the overall global model when using dynamic PLS and non-linear DPLS approaches. Although the ARX model marginally exhibited the most accurate and parsimonious results, the PLS approaches have the potential to form the basis of a performance monitoring scheme using the latent variable scores and model residuals. An example is given of how the model can be used for through batch dynamic monitoring. This is an important area of research for the improved monitoring of dynamic, non-linear batch processes. Future research will address the modelling and dynamic performance monitoring of more complex non-linear processes, such as a large-scale industrial fermentation process.
Acknowledgements Miss Fletcher acknowledges the EPSRC and CPACT for financial support of her PhD.
References Baffi, G., E.B. Martin and A.J. Morris, 1999, Comput. Chem. Eng., 23, 395. Foss, B.A., T.A. Johansen and A.V. S0rensen, 1995, CEP, 3, 389. Kaspar, M.H. and W.H. Ray, 1993, Chem. Eng. Sci., 48(20), 3447. Nomikos, P. and J. F. MacGregor, 1994, AIChE Journal 40 1361. Nomikos, P. and J. F. MacGregor, 1995, Technometrics, 37, 41. Wold, S., N. Kettaneh, H. Friden and A. Holmberg,. 1988, Chemom. Intell. Lab. Syst., 44,331.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
493
Optimisation and Experimental Verification of Startup Policies for Distillation Columns Giinter Wozny and Pu Li Institute of Process and Plant Technology, Technical University Berlin KWT9, 10623 Berlin, Germany
Abstract Startup of distillation columns is one of the most difficult operations of chemical processes. Since it often lasts a period of time, leads to off-products and costs much energy, optimisation of startup operating policies for distillation columns is of great interest in process industry. In the last few years we have accomplished both theoretical studies and experimental verifications with the purpose of developing optimal startup policies for distillation columns. As a result, significant reduction of startup time period can be achieved by implementing the developed optimal policies. This paper summarises our recent results from these studies.
1. Introduction Due to its nature of phase transition, large time delay and strong interaction between variables, startup of distillation columns is one of the most difficult operations of chemical processes. Since the process is unproductive in the startup period, it is desired to shorten this period by optimising startup policies. The procedure of a startup from a cold, empty column to the required operating point consists of three phases (Ruiz et al., 1988): 1) heating the column by the rising vapour, 2) filling the trays by the reflux and 3) running the column to the defined steady state. The third phase requires the longest time and therefore has the potential to reduce the startup period by developing optimal policies. Despite its significance very few work has been done on optimisation of column startup policies. Total reflux (Ruiz et al., 1988) as well as zero reflux (Kruse et al., 1996) policies with a large reboiler duty have been proposed. The switching time from total or zero reflux to values at the steady state is determined by the criterion proposed by Yasuoka et al. (1987), i.e. at the time point when the difference between the temperature at the steady state on some trays and their measured value reaches the minimum. These are empirical approaches used for reducing the startup time of distillation columns. In the last few years we have carried out both theoretical and experimental studies with the purpose of developing optimal operating policies for distillation columns (Kruse et al., 1995; Flender et al., 1998; Lowe et al., 2000). Modelbased optimisation has been adopted in these studies. Three different models are proposed and their parameters validated on the real plants. Two optimisation approaches are used and modified for column startup optimisation. The developed policies are verified on different pilot columns. As a result, by implementing the optimal policies significant reduction of startup time can be achieved. This paper summarises our recent results from these studies.
494 Dy,
< A
v\
\L
FJ^
by direct setting by zero reflux switching point
B X,
_^
HU
Fig. 1: A two-stage model (left) and its reboiler composition profiles (right)
2. Modelling Column Startup Three different column models are formulated for column startup operations. A simple two-stage model is used to estimate the behaviours of startup and to gain a rough insight into the dynamics of distillation columns. As shown in Fig. 1 (left), the model consists of a total condenser and a rebolier. The component balance of the system is then
HU^=Fx,-Dy,-Bx, dt
f
^O
(1)
B
where F = D-\- B and y^ = Kxg. To study the trajectory of the reboiler composition x^ influenced by the reflux flow L, we assume HU,F,x^,V
are constant. Then the time
constant of the light component composition in the reboiler x^ will be
T
= •
HU F + iK- 1)(V - L)
(2)
It indicates that T will be reduced if the reflux is decreased. Fig. 1 (right) illustrates the reboiler composition profiles caused by the direct setting strategy (setting the steady state value of reflux during whole startup period) and by the zero reflux strategy. To achieve an optimal startup, a proper switching point t^ from zero reflux to direct setting is needed, such that the composition from JC^^ to xf along the arrow-pointed trajectory. The reboiler duty V has the same impact on x^ but in the opposite direction. In the same way, the influence of reflux and reboiler duty on the top composition y^^ can be analysed. As a result, the heuristic for startup is to run the column with a period of a maximal reboiler duty and zero reflux and then switch to their steady state value. The second model to describe startup behaviours is a detailed tray-by-tray model composed of dynamic component as well as energy balances, vapour-liquid equilibrium and tray hydraulics. Fig. 2 shows a general tray of this model, with the variables x,j,y^j,L.,Vj,Mj,Pj,Tj as component liquid as well as vapour composition, liquid as well as vapour flow, holdup, pressure and temperature on the tray. The whole model equations lead to a complicated large-scale DAE system.
495
Lj-i, Xiji
Top
V, y<j
Fj, Xfij
x.j,y.j,Li,Vj,Mj,Pj,T,
EM
Qj
Fig. 3: A general tray of the 2""^ model
y+1
...
y
EM
EM"4^1A:;
-^i^^
Fig. 4: State transition of trays during startup
With this equilibrium model, the startup behaviour is described from the first time point at which equilibrium is reached on all trays. Model parameters like tray efficiencies can be validated by comparison of simulated and experimental results. Based on this model, the third model proposed is a hybrid model that depicts column startup from a cold empty state (Wang et al., 2001). Each tray will be described from a non-equilibrium phase in which only mass and energy transfer are taking place to an equilibrium phase in which vapour-liquid equilibrium is reached. The switching point between these two phases is determined by the bubble-point temperature at the operating pressure. Fig. 4 illustrates the state transition of the trays: from the empty cold state (EM) —> liquid accumulation (LA) —> vapour-liquid equilibrium (VLE). Using this model the simulation of startup procedures becomes more reliable.
3. Optimisation Approaches In most cases, the aim of optimisation of distillation column startup is to minimise the startup period. It is a dynamic optimisation problem usually with reflux rate and reboiler duty as the decision variables. Our solution strategy to this problem is to use a modelbased optimisation. A general dynamic optimisation problem can be described as min
t.iartup
s. t.
g ( dxidt, x,u,t) = 0 h ( dx/dt, jc, w, r) > 0 X (0) = .^0
(3) (4) (5) (6)
where tstanup^ g and h are the objective function, model equations and process constraints, respectively, x and u are state and decision variables. Here g in (4) can be any of the above startup models and h in (5) the predefined steady state specifications. As noted in (6), an initial state XQ of the column should be given. We used two different approaches to solve such dynamic optimisation problems. The first one is a gradientbased sequential approach consisting two computation layers: an optimisation layer where the decision variables are solved by SQP and a simulation layer where the state variables and their sensitivities to the decision variables are computed by solving the model equations with the Newton method (Li et al., 1998). Collocation on finite elements is used for the discretisation of the dynamic system.
496
Time (min)
Fig. 5: Optimal policy (left) and composition profiles (right) for the packed column 1
I J
,
1
-o
0
0
O- •
•o
-
Bottom Purity -»0
K
Distillate Purity
•
0.2
0.4
0.6
Time[h]
0.2
0.4
0.6
0.8
Time[h]
Fig. 6: Optimal policy (left) and purity profiles (right) for the bubble tray column The approach is appHed to start up a pilot packed column for separating a mixture of two fatty alcohols. The aim of optimisation is to run the column to the top and bottom product purity specification in a minimum time period. The results of optimisation (Fig. 5 (left)) show that the reboiler duty should be at a maximum value and there should be no reflux for a period of time, and after that both decision variables should slowly approach their steady state value. Under this policy, both product compositions reach their desired value in around 22 min. Note that the top composition decreases in the first period and then returns to the specified value. The second approach we have used is simulated annealing (SA) which is a stochastic search method. The advantage of this method is that it does not require sensitivity information and thus can be connected directly to an available simulator (Hanke and Li, 2000; Li et al. 2000). Since commercial simulation software is widely used in industry, using SA is an easy way to conduct optimisation. The shortcoming of this method is its low computation efficiency, i.e. many runs of simulation are needed to reach the solution. SA is applied to the startup study of a pilot column with 20 bubble-cap trays for separating a methanol-water mixture. The equilibrium model is used in the problem formulation. The model was implemented in the software SPEEDUP as a simulator which is called by a file of SA. The problem definition is the minimisation of the time period from an initial state of the column to the desired steady state. Fig. 6 shows the computed optimal operation policy and the product purity profiles. Different from Fig. 5, both the reflux rate and reboiler duty shown in Fig. 6 (left) for this column should be high in the first period and decreased to the steady state value in the second period. The reason is that one is a packed column with very small holdup and the other is a tray-by-tray column.
497 ou — 78 76 -
0"*-: 3 70 -
b 68 i 66 64 82 -
^^ c r^__x^
~1 0' '
b
a
' 1 ' ' ' M' ' " r ' ' ' 1' • ' ' 1 50 100 150 200 25C Time (min)
Fig. 7: Temperature profiles of bottom (left) and top (right) of the tray column
4. Experimental Verification on Different Pilot Plants The first pilot column has a diameter of 100 mm and 20 bubble-cap trays for separating a water-methanol mixture under atmospheric pressure. The plant has an electrical reboiler and a total condenser. It is equipped with a control system and necessary measurements and electrical valves for flow control. Startup of the column to the steady state with a purity of 99.5mol% for methanol and water was studied. Fig. 7 shows the measured bottom and top temperature profiles by different startup policies: a - direct setting; b - zero reflux; c - optimal policy. All three experiments had the same feed condition (composition: 29 mol%, flow rate: 15 1/h, temperature: 60 °C). It can be seen that the time taken for reaching both bottom and top temperature at steady state was 220min, I70min and I20min by the three different policies, respectively. The second pilot plant is a heat-integrated column system (Fig. 8) consisting of a high pressure (HP) and a low pressure (LP) column, with 28 and 20 bubble-cap trays, respectively. The vapour from HP is introduced as the heating medium to the reboiler of LP. The plant is so constructed that it is possible to operate the process in downstream, upstream and parallel arrangements. Due to the heat-integration startup of the plant becomes complicated. A fully rigorous optimisation was not chosen, because modelling the build-up of the pressure of HP during startup is difficult and under investigation. Startup policies in parallel operation for this plant are developed from analysing the startup heuristic (Fig. 1) and the optimal results of the previous column (Fig. 6). According to Fig. 6, both the reflux flow and reboiler duty should be at a maximum value (about 1.5 times of the steady state value) and switch to their steady state value at a suitable time point. For the heat-integrated column system, HP should be started with the maximum reboiler duty and both HP and LP should have the maximum reflux. When the bottom temperature of LP reaches nearly its desired value, the pressure buildup of HP will be shortly finished. This means the reboiler duty of HP and reflux flow of HP and LP should be switched to their steady state value at this time point. This nearoptimal policy was applied to the plant. Fig. 9 shows the measured temperature profiles of the two heat-integrated columns in parallel operation. It took 3.2h for both columns to reach their desired steady state. The conventional direct setting strategy (both reboiler duty and reflux flow) was also tested for the same feed condition. The experimental result shows that the total startup time of the column system was 9h.
498
Fig. 8: The heat-integrated column system (left) and its flow-sheet (right) 160-1
J. i Botton 1
&i ll
'
Feed
1
; Top
— '" ^^ 0) J1 ra E «
1 IX)
i^l r
1 Bott(im
:::^ ^-^—^
J__Feed 1 Top
/I
time[h]
time [h]
Fig. 9: Temperature profiles of LP (left) and HP (right) by the near-optimal policy
5. Conclusions and Acknowledgements Model-based optimisation was used for developing time-optimal policies for distillation columns. The policies were verified on different pilot plants. Significant reduction of startup time was achieved by implementing the optimal policies in comparison to the conventional startup strategy. From these results some heuristic rules for column startup can be derived. These are now being further studied on startup of columns for reactive as well as three-phase distillation. We thank Deutsche Forschungsgemeinschaft (DFG) for the financial support in this work under the contract W0565/10-2-3.
References Flender, M., G. Fieg, and G. Wozny, Proceedings DYCOPS-5, Corfu, June 8-10, 1998. Hanke, M. and P. Li, 2000, Comp. Chem. Eng. 24, 1. Kruse, Ch., G. Fieg, and G. Wozny, 1996, J. Proc. Cont. 6, 187. Li, P., H. Arellano, G. Wozny and E. Renter, 1998, Ind. Eng. Chem. Res. 37, 1341. Li., P., K. Lowe, H. Arellano and G. Wozny, 2000, Chem. Eng. Proc. 39, 357. Lowe, K., P. Li and G. Wozny, 2000, Chem. Eng. Technol. 23, 841. Ruiz, A., I. Carmeron and R. Gani, 1988, Comp. Chem. Eng. 12, 1. Wang, L., P. Li, G. Wozny and S.Q. Wang, 2001 AIChE Annual Meeting, paper 85h. Yasuoka, H., E. Nakanisshi and E. Kunugita, 1987, Ind. Chem. Eng., 27, 466.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Published by Elsevier Science B.V.
499
Optimal Number of Stages in Distillation with respect to Controllability Marius S. Govatsmark, and Sigurd Skogestad^ Department of Chemical Engineering, NTNU N-7491 Trondheim, Norway
Abstract The central question to be examined in this paper is if it is optimal to have a large or small number of stages in a distillation column with respect to controllability when the objective is to have dual composition control. With multivariable controllers and without considering model uncertainty few stages shows somewhat better controllability than many stages because (i) the available manipulated variables have larger effect on the outputs (allowing larger changes in manipulated variables for few stages) and (ii) it is not necessary to reject the disturbances as fast as for many stages. However, in reality there will always be model uncertainty, and with uncertainty included the conclusion is reversed: it is better to have many stages. The reason is that with more stages the system is less interactive and thus less sensitive to uncertainty. Physically, with many stages a pinch zone develops around the feed stage, which tends to decouple the two column ends from each other.
1. Introduction We want to evaluate if it is optimal with a large or a small number of stages in a distillation column with respect to controllability. Economic objectives like design cost (connected to number of stages in the column and necessary dimensions for internal flows) and operation cost (energy cost connected to internal flows) are not considered here. The study is for dual composition where we have a given purity specification in the top and in the bottom of the column. The conventional LV-configuration is used for stabilizing the condenser and reboiler holdups. Skogestad (1997) claims that it is better to have many stages. He writes: How should the column be designed to make feedback control easier? In terms of composition control, the best is probably to add extra stages. This has rwo potential advantages: 1. It makes it possible to over-purify the products with only a minor penalty in terms of energy cost; recall the expression for V^in^l/(l-a)F which is independent of the purity. The control will then be less sensitive to disturbances. 2. If we do not over-purify the products, then with "too many" stages a pinch zone will develop around the feed stage. This pinch zone will effectively stop composition changes to spread between the top and bottom part of the column, and will therefore lead to a decoupling of the two column ends, which is good for control.
e-mail: [email protected]: phone: +47-7357-4154; fax: +47-7359-4080
500 However, this finding is disputed by Meeuse and Tousain (2001) who claim, based on optimal design of LQG controllers, that it is better to have few stages. The objective of this paper is to study this issue in more detail. Id e
Figure 1: D istillation column (One-feed two-product)
Figure 2. Block diagram for fu-analysis
2. Column data A distillation column separating two-component feed is studied, see figure 1. The feed is saturated liquid. There are two remaining degrees of freedom when we assume given column pressure and liquid holdup in the reboiler and condenser. The model details are described in Skogestad (1997). The controlled variables are composition of light component in top product (xj) and composition of light component in bottom product (Xb). With the indicated conventional control configuration for pressure and levels, the two remaining manipulated variables are reflux flow rate (L) and vapor boil-up flow rate (V). The disturbances are feed flow rate (F=l±0.2) and feed composition (ZF=0.5±0.1). The number of stages is varied between 25 and 51. The base case has 41 trays with the feed tray in the middle (tray 21). Column data is summarized in table 1.
Table 1: Column data Controlled variables (y)
X<1 Xd
Manipulated variables (u) Disturbances (d)
L
v F
ZF
Key hydraulic parameters
Tl Td Tb
iTi
Thermodynamic data Number of stages Feed stage number
A NT NF
0.99 0.01 21.5-2.2 22.0-2.7 1 0.1 0.02-0.2 0.02-0.2 0.02-0.2 0.5-10.2 1.5 25-51 13-26
kmol/kmol kmol/kmol kmol/min kmol/min kmol/min kmol/kmol min min min min
-
501
3. Analysis of the controllers Figure 2 shows the block diagram for the system where model uncertainty is included both as input and output uncertainty. This setup is based the setup previously used by Lundstrom and Skogestad (1995). G' is the plant model which consists of the disturbance gain Gd and the process gain G. G' has two outputs (Xd and Xb) and four inputs (L, V, F and Zp). The model is scaled with respect to acceptable control error (ymad= [0.01 0.01]^), allowed variation in manipulated variables (Umad = ^nominal) and expected disturbances (dmad= 0.2dnominai)- Maximum expected setpoint changes (rmad) are [0.01 0.01]'^. K is the controller. W^, Wd and W^ are weight matrices for setpoints r, disturbances d and measurement noise n. We and W^ are weights respectively on deviation from desired setpoints e and manipulated variables u. Model uncertainty is represented by WjAj which models input uncertainty, and AOWQ which models output uncertainty. Aj and Ao are any diagonal matrices with Hoo-norm less than one. The weighting matrices are diagonal with elements: Wr=rmad/ymad= 1 /(TrS+1 )= l / ( 3 0 s + 1 )
( 1)
Wd=l,w„=10-^w,=0.1
(2)
We=(Ti/ M , S + 1 ) / ( T I S + A ) = ( 0 . 5 S + 1 ) / ( S + 1 0 - ^ )
(3)
Ti (=lmin) is the closed-loop response time and Ms (=2) is the maximum allowed peak of the sensitivity function. In practice integral action is necessary when A is very small. We use A=10''^, except when analyzing the controllers with no integral action, for which we use A*=0.5. For the input uncertainty we use Wi=(TiS+Mi,o)/(Ti/Mi,,oS+l)=(s+0.2)/(0.5s+l)
(4)
Mi 0 (=0.2) is the relative gain uncertainty in the inputs with low frequencies, Mi,^, (=2) is the relative gain uncertainty in the inputs at high frequencies and the ij (=1.0 min) is the delay in inputs. For the output we use Wo=(ToS+Mo,o)/(ToMo.ooS+l)= S/(0.5S+1)
(5)
The relative uncertainty in the measurements are at low frequencies (Mo,o) assumed equal 0 and at high frequencies (Mi^o) assumed equal 2. TQ (= 1 min) corresponds to a delay up in each measurement. For the system in Figure 2, jiNp is the H^o-norm of the transfer function from the scaled inputs [r d n] to the scaled outputs [e^ u,,.], or equivalently tells us by which factor the performance weights must be reduced to have the scaled errors less than 1. [IRS tells by which factor the uncertainty (the A-blocks) must be reduced to guarantee stability. |iRP tells by which factor the uncertainty and performance weights must be reduced to give the worst-case scaled errors less than 1. In summary, |IRP, |IRS and |INP should be as small as possible, and preferably less than 1.
502 4. LQG-control We first follow Meeuse and Tousain (2001) and design a quadratic optimal controller (LQG) where we only consider disturbances and measurement noise, with no model uncertainty included. The design of the LQG-controUer is based on a scaled, linearized model of the plant: dx/dt=Ax+Bu+Wd=Ax+Bu+Bdd, y=Cx+n
(6)
The process noise (wj) and measurement noise (n) are assumed to be white noise with respectively covariances W and V. The LQG-problem is to find the optimal controller u(t) which minimizes T
J = E(\imjr f[x^(t)Qx(t) + u^(t)Ru(t)]dt)
(7)
0
where design parameters are Q=Q^ >0 and R = R S O . We design LQG-controllers for different number of trays. The inputs are weighted equal 0.1 (rii=0.01). Top composition and bottom composition are weighted equal l(qii=l). The remaining states have zero weights. The covariance matrix to the process noise (W) is selected as BdBd^. The measurement noise is assumed small and the covariance matrix for the measurement noise is selected as [0.0001 0.0001]. The results when using no integral action, are summarized in table 2.
Table 2. LQG- and jJL'Optimal controller analysis for different number of Stages NT/NF
25/13 27/14 29/15 31/16 33/17 35/18 37/19 39/20 41/21 43/22 45/23 47/24 49/25 51/26
V/F 22.0 9.69 6.63 5.25 4.48 3.98 3.64 3.39 3.21 3.06 2.95 2.86 2.79 2.73
LQG - No integral action lO^J* HNP ^RS ^RJ' 0.054 2.004 2.001 1.158 0.201 2.034 1.169 2.028 0.346 2.074 2.088 1.126 0.474 1.114 2.139 2.154 2.241 0.581 2.226 1.096 2.332 0.676 1.060 2.350 1.064 0.760 2.457 2.478 2.624 0.836 1.050 2.600 2.759 0.905 2.786 1.020 1.024 0.970 2.931 2.963 3.114 1.031 3.150 1.016 1.089 0.996 3.303 3.345 1.144 3.495 3.543 0.988 3.684 3.739 1.196 0.993
)i-optimal
LQG - Integral action tiNP
^Rs
MRP
^iNP
MRS
HRP
0.384 0.159 0.103 0.078 0.064 0.055 0.049 0.044 0.041 0.039 0.037 0.035 0.036 0.037
6.544 6.862 6.645 6.306 6.326 6.171 6.067 6.152 6.240 6.302 6.289 6.289 6.365 6.439
6.549 6.875 6.656 6.327 6.344 6.178 6.075 6.160 6.248 6.311 6.297 6.297 6.374 6.447
0.691 0.714 0.780 0.834 0.832 0.874 0.865 0.876 0.878 0.869 0.871 0.883 0.856 0.845
1.230 1.118 1.124 1.133 1.105 1.118 1.099 1.095 1.086 1.084 1.079 1.087 1.065 1.061
1.258 1.122 1.130 1.152 1.128 1.144 1.112 1.106 1.108 1.086 1.086 1.098 1.071 1.063
^NP*, ^RS* and |iRp* are computed with A* = 0.5, i.e. We*=(0.5s+l)/(s+0.5). We see that the value of the objective function J (and JIRP) is smallest for few number of stages, confirming the findings of Meeuse and Tousain (2001). Note that the internal vapor flow V in the column is larger with few stages. ^IRS shows that a large number of stages is best when we have model uncertainty. The uncertainties have small effect on the robust performance compared to other inputs (disturbances and reference tracking). If we increase the relative weights on the outputs (q=l,r=0.0001), the results change.
503 The results when using LQG-control with integral action included, are summarized in table 2. With increasing number of trays [i^p is decreased and the nominal performance is improved, JLIRP and JIRS are large and show that robust performance for the LQGcontroller with integral action is far from acceptable when we have model uncertainties, including delays. The LQG-controllers do not clearly indicate if it is optimal with few or many trays.
5. fi-optimal control The |i-optimal controller minimizes the structured singular value IIRP for the system. The ^-optimal controller is designed by DK-iteration (Doyle et.al.,1982). The results are summarized in table 2. With increasing number of trays JINP is decreased and the nominal performance is reduced. With increasing number of trays both |iRs and fiRp is decreased and both robust stability and robust performance are improved. This confirms the claims of Skogestad (1997).
6. Discussion In order to explain the contradictionary results, we will now look at the effect of the different number of stages with respect to some simple controllability measures: process gain (G), disturbance gain (Gj) and interactions (RGA). With few stages the manipulated inputs have larger effect on the outputs. One reason is that the internal flows, e.g. V, and thus the allowed changes in manipulated variables (=Unominai) are larger for few stages. From disturbance gains we see that many stages require somewhat faster control to reject the disturbances than few stages, though the column holdup is assumed larger for many than few stages. This may be explained by that with few stages the internal flows are larger. Figure 4 shows the 1,1-element in the relative gain array for 25, 31 and 41 stages. The RGA-values, and thus the two-way interactions, are much higher with few stages, especially at low frequencies.
3'
s ^
25
10^ 41
,^0°
10-^
.n-^ 10-^
01 f1/minl
10^
Figure 3: An for 25, 31 and 41 stages
Figure 4: Responses for ju-optimal (stable) and LQG-controller (unstable)
Interactions pose no problem when designing a multivariable controller with a perfect model, but may pose serious problems when uncertainty is included. A more detailed study is based on LQG-controller design. This study shows that nominally it is
504 preferable with few trays when we only consider disturbances and many trays when we only consider reference tracking. If we consider both disturbances and reference tracking, the controller tuning decides if it is optimal to have few or many trays. Nominally it is preferable with few trays when the weights on the outputs are relatively small (q=l, r=0.01), because the reference tracking is no significant problem (Wr=l). Nominally it is preferable with many trays when the weights on the outputs are relatively large (q=l,r=0.0001) or include integral action, because the reference tracking has significant effect. When the controller output weights are relatively large, the uncertainties have large effect on the performance. LQG-controllers give bad performance, but do not clearly indicate if it is optimal to have few or many trays. In the |Li-optimal controller design we consider uncertainty including delays, and the resulting controllers show better performance for many stages because of less interaction. Figure 4 shows simulations of step changes in the disturbances for 25 stages when using a LQG-controller with integral action included and when using a {^-optimal controller. We have taken into account some model input uncertainty including delay (e.g Gp=GGextraD where D=diag[0.8 1.2] and Gextra=diag[l/(0.02s+l)^]). The LQG-controller is unstable, which is expected since JIRP is much larger than one (see table 3).
7. Conclusion In conclusion, we find that a large number of trays gives somewhat better controllability than a small number of trays. The seemingly contradictory results of Meeuse and Tousain (2001) are correct, but only hold when having no model uncertainty including delay (and no reference tracking), which is of limited practical interest.
References Doyle, J.C., J.E. Wall and G. Stein , 1982, Performance and robustness analysis for structured uncertainty. IEEE Conf. on decision and control. Lundstrom, P. and S. Skogestad, 1995, Opportunities and difficulties with 5x5 distillation control. J. Proc. ConL 5(4), 249-261. Meeuse, F. M. and R. L. Tousain, 2001, Closed loop controllability analysis of process designs: Application to distillation column design. In: Escape-11. Skogestad, S., 1997, Dynamics and control of distillation columns - a tutorial introduction. Trans. IChemE 75(Part A), 539-562. Also Plenary lecture at Distillation and Absorption '97, Maastricht, September 9-10, 1997, C97-5, VI pp. 23-58.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
505
Dynamic Optimisation of Batch Distillation with a Middle Vessel using Neural Network Techniques M. A. Greaves\ I. M. Mujtaba'*, M. Barolo^ A. Trotta^ and M. A. Hussain^ ^Department of Chemical Engineering, University of Bradford, West Yorkshire BD7 IDP (U.K.) ^Department of ChemicalI Engineering, Engineering, University University of c Padova, via Marzolo 9, 35131 PadovaPD (Italy) ^Department of Chemical University of Malaya, hemical Engineering, Uni\ 50603 Kuala Lumpur (Malaysia)
Abstract A rigorous model (validated against experimental pilot plant data) of a Middle Vessel Batch Distillation Column (MVC) is used to generate a set of data, which is then used to develop a neural network (NN) based model of the MVC column. A very good match between the "plant" data and the data generated by the NN based model is eventually achieved. A dynamic optimisation problem incorporating the NN based model is then formulated to maximise the total amount of specified products while optimising the reflux and reboil ratios. The problem is solved using an efficient algorithm at the expense of few CPU seconds.
1. Introduction Batch distillation is frequently used in chemicals and pharmaceutical industries for the manufacture of fine and specialty chemicals where the product demand frequently changes. In regular batch distillation columns, the products and slops are withdrawn sequentially from the top of the column according to their relative volatilities and the heavier cut is recovered from the reboiler at the end of the batch (Mujtaba and Macchietto, 1996; Barolo ^ra/., 1998). Bortolini and Guarise (1970) and Hasebe et al. (1992) suggested a different column configuration that has a middle vessel attached to the feed tray, to which the feed is charged, with continuous recycling of liquid feed between the feed tray and the middle vessel. This column configuration is defined as a middle-vessel column (MVC), and over the last few years a lot of attention has been given to such a column (Skogestad et al, 1991 \ Cheong and Barton, 1999). One of the advantages of the MVC process (Figure 1) is that it can produce three products (on spec or off-spec or a combination of both) from the top, middle and bottom of the column simultaneously. Barolo et al. (1996) provided the first comprehensive experimental results on a pilot-plant MVC column. Batch distillation operation is a hard task to model because of the integrating behaviour of the system (Edgar, 1996). Optimising the performance of the column is even harder as the large system of differential and algebraic equations (DAEs) describing the
506 process makes the solution of the dynamic optimisation problem very expensive from a computational point of view, even with the computational power currently available (Mujtaba and Macchietto, 1996). Therefore, the aims of this study are to: (1) Model the process using neural network (NN) techniques rather than to base the model on first principles; (2) Evaluate the model against data generated by a rigorous MVC simulator (Barolo et ai, 1998); (3) Develop a dynamic optimisation framework using the NN based model to optimise operating parameters. This will avoid the need for time consuming experiments (or computation using rigorous optimisation techniques) in order to fmd the optimum operating conditions to satisfy a specified separation task (product amount and purity).
2. Pilot Plant and Rigorous Modelling Barolo et al (1998) developed a rigorous dynamic model for a pilot plant MVC column separating a binary ethanol/water mixture. The model was validated against a set of experimental data. The column consisted of 30 sieve trays (0.3 m diameter and 9.9 m total height), a vertical steam heated thermosiphon reboiler (capacity 90 L), and a horizontal water cooled shell and tube condenser. The reflux drum (capacity 40 L) was open to atmosphere. The middle vessel had a maximum capacity of 500 L.
3. Neural Network Based Modelling A feed forward network structure was selected as this structure is robust and performs well in approximating the complex behaviour of distillation processes (Greaves et a/., 2001). MD, XD VT
RD
MDE, XDE
MF, XF
Fn
MBE, XBE
Figure 1 - Schematic of Batch Distillation Column with Middle Vessel
* Author to whom all corresspondence should be addressed E-mail: [email protected]
507
Table I - Data Set 20 with parameters: RD = 0.855, RB = 0.727, Fu = 0.1789, DMOL = 0.00926 (kmol/min), BMOL = 0.03345 (kmol/min), RMOL = 0.0546 (kmol/min), VB = 0.089 (kmol/min) and Fu = 0.14 (kmol/min) Time (mins) 0 10 20
MDE (kmol) 0 0.093 0.185
0 0.865 0.862
MF (kmol) 4.921 4.532 4.143
0.322 0.331 0.343
120
1.111
0.856
0.251
0.672
XDE
XF
Since the purpose of this work is to propose a general framework for the optimization of batch columns, rather than to consider the solution of a particular case study, a simple separation task is considered; we are not claiming that an MVC is necessarily the best column configuration for this separation task. We consider an ethanol/water mixture and the objective is to obtain two products with different ethanol compositions, while minimizing the ethanol losses from the bottom of the column. The lighter cut is obtained via the distillate stream from the top of the column, while the heavier cut is segregated in the middle vessel and recovered at the end of the batch. In the MVC column, RD, RB, FLI, FL2 and VB (defined in Figure 1 and in the nomenclature section) are the governing parameters that will affect the separation tasks. For a given vapour load (VB) and liquid flow from the feed tray to the middle vessel (FLI) sets of data were generated using the rigorous model. A "grid" of reflux and reboil ratios was considered, and the operation was carried out, regardless of what the objective of the distillation was. Table 1 shows a typical set (set 20) of data generated. Four neural networks were used to predict the state variables MDE, XDE, Mp and Xp at any given batch time as a function of neural network input parameters as shown below: NN output (Pi) = f(RD, RB, DMOL, BMOL, ZO, t,)
where
PQ ={MDEQ,XDEO'
MFo,Xro},.o
andPf={M
(1) MF,XP|,.,^
are the time dependant state variables defining the separation achieved at time t. It is important to note that at any time t, MBE and XBE (Bottom Product amount and composition) can be calculated by overall mass balance, assuming negligible tray holdup. The multilayered feedforward network used is trained by a backpropagation method using a momentum term as well as an adaptive learning rate to speed up the rate of convergence (Greaves et al, 2001). Two data sets were used to train the neural networks and one for validation. Figures 2 and 3 show the results for data set 20. The figures compare the amount of product and composition for Overhead product tank (MDE, XDE) and the Middle Vessel (Mp, Xp) from the data set with those calculated by the NN based model. It can be seen that the results are an excellent match.
508
X X
Actual MDE Actual MF -NNel MF
=
Actual XDE
0.3
. NNel XDE
g.0.2
Actual XF - N N e t XF 60
80
80
too
120
140j
Time (mins)
e (mins)
Figure 2 - Overhead Product for Set 20
Figure 3 - Mid-Vessel Product for Set 20
4. Optimisation Formulation and solution The dynamic optimisation problem with an objective to maximise the amount of distillate and middle vessel products can be described as: the column configuration, the feed mixture, separation tasks optimal reflux ratio, reboil ratio and batch time the amount of distillate and middle vessel products equality and inequality constraints (e.g. bounds, etc.)
given: determine: so as to maximise: subject to:
Mathematically the problem can be written as: (PI)
Max
(MDE+MP)
RD(t), RB(t), t
subject to: Process Model (DAE/Hybrid/NN) XDE -£I ^ XDE ^ XDE +£I Xp -62 ^ Xp ^ Xp +£2 MF>0 MFOXPO-(MDEXDE+MPXP) > 0
^ PD ^ t^ < t < t^
PD
PD
»
PB
-
PB
-
PB
(Equality constraint) (Composition constraint) (Composition constraint) (Holdup constraint) (Component balance constraint) (Peflux and reboil ratio bounds) (Time bounds)
where XDE* and XF*are the specified purity of the overhead and middle vessel products; El and £2 are small but finite values (both set to 0.001); PD^ and PD^ are the lower and upper bounds of PD; PB^ and PB^ are the lower and upper bounds of PB; t^ and t^ are the lower and upper bounds of t. In the past, rigorous DAE model or hybrid (simple DAE coupled with NN) model based optimisation of regular batch distillation have been considered by Mujtaba and Macchietto (1996) and Mujtaba and Hussain (1998). However, these required the full integration of the DAE or Hybrid models several times during optimisation to evaluate objective function and gradients and were computationally intensive. In comparison, the proposed NN based model in this work is much faster and can evaluate the state
509 variables at any given time without requiring stepwise evaluation as in DAEs integration. The above optimisation problem was solved using an efficient Successive Quadratic Programming (SQP) based method (Chen, 1988). Forward Finite Difference method was used to calculate the gradients of the objective function and the constraints (required by the optimiser) with respect to the optimisation variables. The reason for choosing a fmite difference method as opposed to the analytical based method is purely due to difficulty in evaluating an analytical gradients for the NN based process model. Table 2 - Summary of Optimised and Simulated Results Simulated Run
r~
2 3 4 5 6 7 8 9 10 11 12 13
Data Set 6~~ 15 19 21 22 23 24 25 11 18 1 9 20
t ~~^D min 70 .639 .787 110 .849 150 .852 90 90 .884 130 .885 .885 160 .869 220 130 .661 200 .845 260 .880 100 .701 120 .855
RB
MDE+MF
kmol ^41 .726 .778 .642 .643 .727 .778 .844 .888 .857 .891 .735 .726
im
1.659 1.584 0.916 0.916 1.066 1.361 1.658 2.993 1.955 2.029 1.955 1.362
Optimised t RB ~ "~1R^ min .642 .639 67.18 .704 .802 72.90 139.74 .771 .843 .641 .847 87.88 .647 .865 91.26 .721 116.57 .876 .771 144.77 .872 .879 .821 176.93 90.94 .915 .758 .834 102.04 .881 .896 .883 246.68 .724 .729 84.45 .842 .715 98.62
MDE+MF
% Improvement
kmol 1.9476 2.2313 1.7030 1.2689 2.2326 1.6752 1.7245 1.9532 4.5509 3.5122 2.2911 2.2343 2.1108
in MDE+MF
7.84% 34.49% 7.51% 38.53% 143.73% 57.15% 26.71% 17.80% 52.05% 79.65% 12.92% 14.28% 54.98%
Table 2 shows the results obtained from the rigorous simulator and the neural network based optimised results in terms of batch time, reflux and reboil ratio, total distillate and middle vessel products. To get a rough idea of the effectiveness of the optimization routine, the %Improvement in the total amount of products obtained by optimisation over that obtained by the first-principle simulator are also reported in the table. In most cases significant reduction in batch time and increase in total products are observed. It was also verified that a good match exists between the results obtained by off-line optimisation and those obtained by implementing the optimal values of RD and RB into the rigorous simulator. These results are not included here for the sake of conciseness, and will be the subject of a subsequent report.
5. Conclusions A rigorous model of an MVC, validated against experimental data in a previous study, is used to generate a set of data, which is then used to develop a Neural Network based process model. An excellent fit between the first-principle model data and the data generated by NN was eventually achieved. A dynamic optimisation framework incorporating the NN based model has been developed to optimise operating parameters such as reflux and reboil ratios and batch time. It has been shown that the optimisation framework is able to improve on the values obtained through the simulator for Batch Time and Total amount of product collected. Performing experimental optimisation can
510 take many hours whereas computational optimisation through the proposed approach takes few cpu seconds, therefore allowing for more efficient design of experiments.
6. Nomenclature BMOL» DMOL
FLl FL2 LB MB, MD MBE, MDE Mp Np NT RB, RD RMOL
tf VB, VT XBE» XDE
Xp
= Bottoms and distillate molar rate (kmol/min) = Withdrawal rate from the column (kmol/min) = Feed rate to the column (kmol/min) = Bottoms residue molar rate (kmol/min) = Bottom and reflux drum holdup (kmol) = Amount of bottom and distillateproduct (kmol) = Holdup of the middle vessel (kmol) = Feed plate location = Total number of plates = Internal reboil and reflux ratio [0,1] = Reflux molar rate (kmol/min) = Final Batch Time (min) = Vapor boilup rate (bottom and top) (kmol/min) = Ethanol mole fraction of the bottom and distillate product = Ethanol mole fraction in the middle vessel
7. Acknowledgement The University of Bradford Studentship to M.A. Greaves and the UK Royal Society supports to M.A. Hussain and I.M. Mujtaba are gratefully acknowledged. M. Barolo gratefully acknowledges the financial support from the University of Padova (Progetto di Ateneo "VirSens"; # M1P012-1999).
8. References Barolo, M., Guarise, G.B., Rienzi, S.A., Trotta, A. and Macchietto, S. (1996), Ind. Eng. Chem. Res., 35, pp. 4612. Barolo, M., Guarise, G.B., Rienzi, S.A. and Trotta, A. (1998), Comp. Chem. Eng., 22, pp. S37. Bortolini, P. and Guarise, G.B. (1970). Quad. Ing. Chim. Ital., 6, pp. 1. Chen, C.L. (1988), PhD Thesis, Imperial College, London. Cheong, W. Barton, P.I (1999), Ind. Eng. Chem. Res., 38, pp. 1504. Edgar, T. F. (1996), / Proc. Contr., 6, pp. 99. Greaves, M.A., Mujtaba, I.M. and Hussain, M.A. (2001), In Application of Neural Network and Other Learning Technologies in Process Engineering (I.M. Mujtaba and M.A. Hussain, eds.), pp 149, Imperial College Press, London. Hasebe, S., Aziz B.B. Abdul, Hashimoto, I. and Watanabe, T. (1992), In IFAC Workshop on Interactions Between Process Design and Process Control (J.Perkins Ed), London, Sept 7-8, Pergamon Press. Mujtaba, I.M. and Macchietto, S.(1996), / Proc. Contr., 6, pp. 27. Mujtaba, I.M. and Hussain, M.A. (1998), Comput. Chem. Engng., 22, pp. S62L Skogestad, S., Wittgens, B., S0rensen, E. and Litto, R. (1997), AlChE / , 43, pp. 971.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
511
A Two-Level Strategy of Integrated Dynamic Optimization and Control of Industrial Processes - a Case Study* J. V. Kadam\ M. Schlegel', W. Marquardt\ R. L. Tousain^ D. H. van Hessem^, J. van den Berg^ and O. H. Bosgra^ ^Lehrstuhl fur Prozesstechnik, RWTH Aachen, Turmstr. 46, 52064 Aachen, Germany ^Systems and Control Group, Mech. Eng., TU Delft, Mekelweg 2, 2628 CD Delft, The Netherlands
Abstract This paper discusses a two-level strategy integrating dynamic trajectory optimization and control for the operation of chemical processes. The benefit of an online dynamic re-optimization of operational trajectories in case of disturbances is illustrated by a case study on a semi-batch reactive distillation process producing methyl acetate.
1. Introduction Increasing competition in the chemical industry requires a more agile plant operation in order to increase productivity under flexible operating conditions while decreasing the overall production cost (Backx et al., 2000). This demands economic optimization of the plant operation. However, existing techniques such as stationary real time optimization and linear model predictive control (MPC) generally use steady-state and/or linear representations of a plant model. They are limited with respect to the achievable flexibility and economic benefit, especially when considering intentionally dynamic processes such as continuous processes with grade transitions and batch processes. There is an evident need for model based process operation strategies which support the dynamic nonlinear behavior of production plants. More recent techniques such as dynamic trajectory optimization and nonlinear model predictive control (NMPC) are still subject to research, and often the size of the applicable process model is still a limiting factor. Moreover, the integration of model predictive control and dynamic optimization for an optimal plant operation is an open field of research, which is e.g. studied in the EU-funded project INCOOP*. Various strategies have been suggested to implement such an integration. In the so-called direct approach (Helbig et al., 2000) the two main tasks, economic trajectory optimization and control, are solved simultaneously repetitively on each sample time of the process. This corresponds to a single-level optimal
* This work has been funded by the European Commision under grant GlRD-CT-199900146 in the "INCOOP" project (www.lfpt.RWTH-Aachen.de/INCOOP). The authors gratefully acknowledge the fruitful discussions with all the INCOOP team members.
512 control strategy. However, for large-scale and highly nonlinear processes this approach is intractable e.g. due to computational limitations. In this paper, we employ a vertical decomposition approach: Here, the problem is decomposed into an upper level dynamic trajectory (re-)optimization, and a lower level (nonlinear) MPC which drives the process along the current optimal trajectory determined on the upper level. The interaction between the two levels is a key issue for the feasibility of such a decomposition. Data from the plant processed by a suitable estimation procedure enables nonlinear model-based feedback which can be utilized for several purposes: The dynamic optimization needs not to be performed each sample time but instead depending upon the nature of external disturbances. The feasibility of this approach is shown by means of a case study.
2. Problem definition The goal of an optimal process operation is to maximize profit. In the ideal case, a perfect model of the process exists, the initial state XQ at the beginning of the operation is known exactly and the process is not disturbed. Then the associated optimal trajectories for the operational degrees of freedom can be determined entirely off-line through the solution of an optimal control problem (PI): min<E>(jc,w,ro,^/^) s.t. 0 = f(x,x,u,dj), y=
(PI) x{tQ) = XQ
g{x,u,dj)
0>h(x,u,d) In this formulation, x denotes the system states with initial conditions XQ, U free operational variables and d given parameters. / contains the differential-algebraic process model, g maps the system state to the outputs y. Constraints such as path and endpoint constraints to be enforced are collected in h. O denotes an economic objective function to be minimized on the time horizon [to_ tf] of the process operation. In principle, problem (PI) can be solved by standard techniques for dynamic optimization (e.g. Betts, 2001) to determine an optimal u and optionally the fmal time of operation tf, e.g. in the case of batch operation or for minimizing transition time in continuous processes.
3. Decomposition of dynamic optimization and control The fact that the assumptions stated above are not fulfilled prevents an off-line solution of problem (PI) from being sufficient in any practical application. This is mainly due to model uncertainty and time-varying external disturbances d{t) and unknown initial conditions xo. To cope with this situation, a successive re-optimization of problem (PI) with updated models and initial conditions based on process measurements is required. However, the control relevant dynamics of typical processes will be too fast to enable realtime closed loop dynamic optimization. This is because current numerical techniques are not able to solve the problem (PI) for industrial-size applications involving large
513 models sufficiently fast on the sample frequency and given small prediction horizon. High sampling frequency generally demands shorter prediction horizons and this might cause feasibility problems as well. Alternatively, instead of solving (PI) directly, we consider a hierarchical decomposition. The two level strategy decomposes the problem into an upper level economic optimization problem (P2a) and a lower level control problem (P2a), as shown in Figure 1. (P2a) min ^(ly^^Jo^tf)
(P2b) mmVcv, -/'^Vay^ -/'^)Hu, -u'^fR{u, -u''^)
^
s.t. 0 = f{x,x,u'''^,dj),
'
3c(/o,) = -%
y''^ =g(xy'^Jj)
o>h{jy^ J) 4+1 = hi + Af, tj,^, = tj, + M
+(A'^-^^) P(X^-%)
s.t. 0 = f{x,x,u,dj), y=
J(%) = Jo/
g{x^ujj)
o>/7(j,w,J) 7^^^^ = 7oi + A F ,
F^^i = r^ + A?
The former is a dynamic real-time optimization (D-RTO) problem, which determines trajectories u^^^ ^y'^^^ for all relevant process variables such that an economical objective function O is minimized and constraints h are satisfied. Only economic objectives such as maximization of production or minimization of process operation time are considered in O . The problem is repetitively solved on the rest of the entire time horizon on a sample time A/ for an update of the previous reference trajectories. The sample time has to be sufficiently large to capture the slow process dynamics, yet small enough to make flexible economic optimization possible. The re-optimization may not be necessary at each optimization sample time, instead it can be done based on the disturbance dynamics. The process model / used for the optimization has to have sufficient prediction quality and should cover a wide range of process dynamics. On the lower level, an MPC problem (P2b) is solved in such a way that the process variables track the optimal reference trajectories in a strict operation envelope computed on the D-RTO level. The operation envelope, especially for controls w , is a small region around the reference trajectories; thus the MPC is referred to as delta mode MPC. The MPC sample time A? has to be significantly smaller than the D-RTO sample time At, since it has to handle the fast, control relevant process dynamics. One requirement for the process model / used on the MPC level, which might be different from the model / used on the D-RTO level, is that it has to be simple enough, such that problem (P2b) can be solved sufficiently fast. A good prediction quality of / is required for the shorter time horizon 1^0,,/^^J of (P2b). The initial conditions Xo,,Xo, and disturbances d,d for D-RTO and MPC are estimated from process measurements by a suitable estimation procedure such as an extended Kalman filter (EKF). A proper separation of disturbances on different time scales is crucial for the decomposition of control and optimization, since the actions on both levels are induced by some kind of disturbance. Besides physical disturbances acting on the process directly,
514 changing external conditions such as market and environmental conditions also can be viewed as disturbances, because they require an update of reference trajectories for the optimal process operation. For example, it is conceivable that prices or product specification requirements change during the process operation. The production should then be adapted to the new situation, which can be done by a re-optimization of the process operation. The estimator (cf. Figure 1) estimates disturbances for which disturbance models have been added to the process model. The time-scale separation decomposes slowly varying or persistent from stochastic disturbances with time constants smaller than the prediction horizon of the MFC. The decision for a possible re-optimization is based on a disturbance sensitivity analysis of the optimal reference trajectories. A reoptimization is started only if persistent disturbances have been detected and have high sensitivities. On the lower level, both types of disturbances are used in the nonlinear prediction of the process. Persistent disturbances are taken up as a bias on the corresponding variables in the MFC, whereas stochastic disturbances are accounted for via the disturbance models added to the process dynamics. In this fashion the MFC problem (F2b) can be solved to obtain the updated control moves (cf. Lee and Ricker, 1994). The structure in Figure 1 differs from the one suggested by Helbig et al. (2000): The sequence of estimation and time-scale separation is reversed. Both alternatives seem to have their merits and need to be investigated in the future. market and environment Ar
d,x,y,u Estimator
d.x y,w
Time Scale Separation
D-RTO
M
d ,x,y,u
L4U
MPC
Ar Process (model) (incl. base control)
j^f
u d(t)
Figure I. Vertical decomposition of dynamic optimization and model predictive control
4. Case study The concept introduced above has been implemented in prototype software tools and applied to an industrial-size test example. Details on the numerical algorithms used in the different modules are beyond the scope of this paper. The process studied is a semibatch reactive distillation process producing methyl acetate (MA) by esterification of acetic acid (AC) with methanol (MT) and byproduct water (W) (cf. continuous process described e.g. in Agreda et al., 1990). The process is started up with pure MT in the batch still and AC as a side feed stream. A gFROMS (gFROMS, 2001) model with 74 differential and 743 algebraic equations has been developed for this process. The dynamic optimization problems (F2a) have been solved using the optimizer ADOFT
515 (Schlegel et al., 2001). The objective is to maximize the amount of product (MA) for a fixed batch time of 4 hours (optimum, found by an off-line optimization with free final time) under strict enforcement of product purity of 0.95. The operational degrees of freedom are the reflux ratio and the reboiler vapor stream. Optimal profiles calculated by an off-line optimization run are shown in Figure 2 and 3 (solid lines) for the nominal case without disturbances. no disturbance reopt. after dist.
no disturbance reopt. after djst.
o.sl «06o S^'-^0.2
°0
1
2
time [h|
3
4
Figure 2. Control profiles for the nominal case and with re-optimization. The disturbance scenario considered in our study is a drop of 50% in the feed rate of the side stream, which occurs before 1.75 hours. This is a persistent disturbance which is direcdy measurable and effects the product significantly. The analysis of the sensitivity of the optimal solution to the disturbance has shown that the nominal optimal trajectories need not to be updated for disturbance values less than 25% and these are handled at the MPC level. This decision making strategy for considering re-optimization or MPC at a current sample time subject to disturbances is proven to be suitable for this case study. However, further research is needed in this area. The performance of the two level strategy is compared with using NMPC, delta mode MPC only and open loop operation. The product quality and the amount of product obtained using the above control strategies are depicted in Figure 3. If the original optimal trajectories would be followed (open-loop strategy) further, the disturbance prevents the required product quality of 0.95 to be met (* line in Figure 3) and leads to economic losses for this batch. A delta mode MPC that enforces a strict operation envelope around the reference trajectories and an NMPC without considering such an envelope are applied separately. The results depicted in Figure 3 (dash-dotted and dotted line resp.) show that these approaches are not economically viable (produces offspec and less amount of product) for the given disturbance scenario. The two level strategy of integrated dynamic optimization and control is then applied to the problem. A delta-mode MPC (constraints on control actions) is employed as a lower level MPC. The disturbance is recognized and a re-optimization of the trajectories is started (triggered by the sensitivity-based strategy). The new optimal operational trajectories are determined in order to meet the desired requirements. The re-optimization, which takes the changed state of the process due to the disturbance into account leads to changed optimal control profiles (Figure 2 -dashed lines). The profiles in Figure 3, left (dashed line) show that the product quality of 0.95 is met in the closed loop operation. Figure 3, right (dashed line) shows that more amount of on-spec product is produced. Thus the two level strategy guaranties an economical feasible operation
516 which is not guaranteed by the NMPC. Note that a rigorous nonlinear model is used at the MPC level which is the best option that can be considered. 0.4.^0.35-
no disturbance NMPC only ; reopt. and delta mode MPC
y
0.3^ |0.25Q.
o 0.2-
^0.921 o 0.91 }i Q g[^ —
•If
0.8!
no disturbance open loop delta mode MPC only NMPC only reopt. and delta mode MPC time [h]
i0.15-
(0
0.1-
yi^''
0.05-
°r^~
1
2 time [h]
3
Figure 3. Product quality and amount of product
5. Conclusion In this paper it has been explained that a more flexible plant operation to cope with changing market and operating conditions can be achieved by a systematic integration of dynamic optimization and model predictive control techniques. A vertical decomposition appears to be a viable strategy, which guaranties overall feasibility that might not be possible by an MPC only. With the help of a simulation study the benefit of the two level strategy, especially the dynamic re-optimization during the operation has been illustrated. Future research work in this area is required on a rigorous strategy for the separation of time scales, the relation of the process models used on the different levels, the choice of appropriate numerical algorithms on the different levels, etc.
References Agreda V. H., L. R. Partin and W. H. Heise (1990). High-Purity Methyl Acetate via Reactive Distillation, Chem. Eng. Prog., 86, 40-46. Backx, T., O. Bosgra and W. Marquardt (2000). Integration of Model Predictive Control and Optimization of Processes. In: IFAC Symposium on Advanced Control of Chemical Processes. Vol. 1., 249-260. Betts, J.T. (2001). Practical methods for optimal control using nonlinear programming. SIAM, Philadelphia. gPROMS User Guide Version 2.0 (2001). Process Systems Enterprise Ltd., London. Helbig, A., O. Abel and W. Marquardt (2000). Structural concepts for optimization based control of transient processes. In: AUgower, F. and A. Zheng (eds.). Progress in Systems and Control Theory, 295-311, Birkhauser, Basel. Lee, J. H. and N. L. Ricker (1994). Extended Kalman Filter Based Nonlinear Model Predictive Control. Ind. Eng. Chem. Res., 33, 1530-541. Schlegel, M., Th. Binder, A. Cruse, J. Oldenburg and W. Marquardt (2001). Dynamic Optimization Using a Wavelet Based Adaptive Control Vector Parameterization Strategy. In: Gani, R. and S. B. J0rgensen (eds.): European Symposium on Computer Aided Process Engineering - I J, 1071-1076, Elsevier.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
517
Bayesian Parameter Estimation in Batch Polymerisation Zhen Lu, Elaine Martin and Julian Morris Centre for Process Analytics and Control Technology University of Newcastle, Newcastle upon Tyne, NEl 7RU, UK
Abstract A Bayesian estimation framework is proposed for the tracking of time-varying parameters. The Bayesian approach is a statistical procedure that allows the systematic incorporation of prior knowledge about the model and model parameters, the appropriate weighting of experimental data, and the use of probabilistic models for the modelling of sources of experimental error. The interplay between these elements determines the best model parameter estimates. The proposed approach is evaluated by application to a dynamic simulation of a solution methyl methacrylate (MMA) batch polymerisation reactor. The Bayesian parameter adaptive filter is shown to be particularity successful in tracking time-varying model parameters.
1. Introduction The operating objectives in many batch polymerisation processes require the satisfaction of complex property requirements for the final polymer whilst simultaneously reducing production costs. Most mechanical and rheological properties of polymer products are directly, or indirectly, linked to the molecular structural properties of the polymer chains (e.g. molecular weight distribution (MWD), copolymer composition distribution (CCD), chain sequence length distribution (CSD), etc.), which are difficult (sometimes impossible) to measure on-line. Average polymer molecular weight properties (e.g. number and weight average molecular weights), that can be indirectly inferred from the on-line measurement of the solution viscosity or melt index of the polymer, are often selected as the major controlled variables that need to be maintained within well-determined limits so that desired product quality criteria can be satisfied. Control strategies require that pre-determined trajectories for key process variables (e.g. reactor temperature) are implemented during batch operation (e.g. Thomas and Kiparissides, 1984). However, the operation of the batch polymerisation reactor is influenced by both process disturbances and model parameter variations due to the inherent process-model mismatch or changing operating conditions. Unless the timevarying model parameter values and any subsequent optimal control trajectory is updated regularly during batch operation, the control strategy will fail to meet the product quality specifications and the operating requirements (e.g. Ruppen et al. 1997, Crowley and Choi, 1997, 1998).
518 Crowley and Choi (1997, 1998) provided a comprehensive study of the optimal control of the molecular weight distribution in a batch free radical polymerisation experimental reactor. They proposed a control scheme incorporating state estimation and optimisation that could follow the desired molecular weight distribution by manipulating the temperature profile in the reactor. An Extended Kalman Filter (EKF) was used to retrieve the entire state vector based on process measurements (e.g. polymerisation temperature, and conversion) and product quality variables (e.g. molecular weight distribution). A model predictive control algorithm was then implemented to track the sub-optimal temperature profiles. However, they did not consider model parameter uncertainty and process disturbances in calculating the 'optimal set point sequence'. Such a policy may fail to meet the product quality specifications under model parameter variations (e.g. termination rate constant due to gel-effect). In this study, an alternative to the EKF is proposed for the tracking of timevarying parameters through the introduction of a Bayesian estimation framework.
2. Bayesian Parameter Estimation Important information is lost when the model parameters are represented by a single point value, as in standard estimation and identification procedures, rather than by a full distribution as in Bayesian approaches. This is one of the main reasons for choosing a Bayesian approach to parameter estimation over conventional methods. It is assumed that the process has a state vector x(t) and is described by the continuous time process model:
x(/) = /(x(/),u(o,e,/)
(1)
where the measurement model takes the form: y{t,) = h{x(t,U,)
(2)
The problem is to estimate a vector of parameters, 9 , about which there may be some prior beliefs which can be expressed as a probability density function, p(9). This prior distribution may be arrived at either from using previously monitored data, or by subjective process engineering judgement where the density expresses the users knowledge before the experimental data was collected, which in most applications is somewhat subjective, y is a vector of model predictions, which depend on the parameters 9 . After obtaining the process observations, y , which have a probability distribution that is a function of the unknown parameters, the dependence of y on 9 can be expressed as the conditional probability density function p(y 19), where the experimental error is denoted as £(9) such that the following error model applies: £(9) = y - y
(3)
519 It is now assumed that the errors are modelled statistically such that there is a conditional probability density of £ for known 8 , p(e 10). Then if 9 is known, the conditional probability density p{y \ 9) can be represented as: p(y\^) = p(y-y\0)
= P(e\e)
(4)
To update the probability density of the unknowns 9 after new process data have been obtained, Bayes' theorem is used:
p(y) where
p(9|y)
is the Bayesian posterior distribution and contains all known
information about the unknown parameter(s) after the on-line measurement information has been incorporated into the prior information. The denominator, p(y) is the unconditional probability density of the observation data and from the law of total probability is given by:
p(y) = Jp(y|e)p(9M9
^^)
This value is a constant and acts as a normahsing constant. To construct the Bayesian parameter estimator, the posterior density is then expressed as: p ( e I y ) = —L_^(log P(y|e)-Mog /7(G)) ^ ^^-J
('^)
p(y) For a point estimate of the unknown parameter(s) 9 , 9 is used as the value that maximises the a-posterior density. That is, 9 maximises the probability that the estimate is correct. To maximise the posterior probability density p(9 | y ) , the function J can be minimised: y=-logp(y|9)-logp(9)
(8)
The above function sums new measured process data and prior information and gives the appropriate weighting according to the knowledge of the statistical error. In this initial study of this new approach to parameter estimation, it is assumed that the errors are zero mean Gaussian errors. The error probability density then becomes: p(e|9)=
—-cxp(~^e^a-'£) {2n\a\y
520 where a = E[£e^],
E[e] = 0 and a is the error covariance matrix. Thus, a working
expression for J is: (10)
1 1 ^ = - [ y - y ] or-^[y-y] + ~log(2;r|a|)-log/7(e) and the Bayesian parameter estimation problem is defined as: min
J
s.t.
BeQft
(11)
where QQ is a set determined by lower and upper bounds on the elements of 9 .
3. Results The process studied is the free radical polymerisation reactor of methyl-methacrylate (MMA) (Mourikas et ai 2001). A mathematical model describes the dynamic behaviour of an experimental pilot scale system (Figure 1). Heating and cooling of the reaction mixture is achieved by controlling the flows of a hot and a cold water stream, through the reactor jacket. The polymerisation temperature is controlled by a cascade control system consisting of a primary PID and two secondary PI controllers. The polymerisation is highly exothermic and exhibits a strong acceleration in polymerisation rate due to gel-effects. Batch duration is 120 minutes.
i-sp
T PID
PI Cold
—^-tx
Hot
Figure I. Plant Polymerisation Reactor In practice important kinetic parameters such as /c^, the propagation rate constant and kj , the initiator rate constant, cannot be determined accurately and may vary during the polymerisation. In this study, the propagation rate constant, k^, is represented by ^p = ^po8l,corr with the stochastic correction term
g'l,corr^ ^^^
^he initiator
decomposition rate constant k^ is represented by kj - ^jo
521 correction term g^^^,,. In the EKF, a random walk modle is assumed for the behaviour Sd^corr ^^^ of the stochastic state. In the process, the actual values of g'l^.^rr ^"^ assumed to vary linearly from 0.9 at the nominal point to 0.78 and 0.66 for the propagation and initiator decomposition rates respectively. 1 0.95
JOBS
'"^x
-^,
x
v^
O 08
^ra \
•
0.7
.
065
r
3.2
3
8
|28
|2.4
t
•*
6
"o S
^
/\ /
9
2 1.8
80
100
-N
/
•A
/
/
/
-r::?-
120
Figure 2. Performance of the adaptive EKF The most common case in industrial practice is where the available on-line measurements are taken to be monomer conversion from an on-line densitometer, the reactor and the jacket inlet and outlet temperatures. In this study these three measurements are assumed to be available. In the simulations that follows, Gaussian zero mean white noise was added to the measurements to simulate measurement noise. In all the plots the large-dashed line ( -) is the MMA process model, the dotted line represents the 'actual polymerisation process' ( ) and the solid line ( ) represents the estimated values. The EKF performance is shown in Figure 2 with plots of Number Average Molecular Weight, A//i, and Weight Average Molecular Weight, Mw. Clearly the EKF based parameter adaptation cannot follow the change in rate constants leading to severe model-plant mismatch and hence errors between the predicted and actual values of Mn and Mw. This relatively simple example reflects the EKF parameter adaptation concerns expressed by Scali, et al. (1997). Kozub and MacGregor (1992) indicated that the prediction of the molecular properties (number average molecular weight, weight average molecular weight and polydispersity) is vulnerable to model mismatch since these states are not observable. The main difficulty is the lack of any on-line measurements of the properties of a polymer molecular weight. The results shown above illustrate this problem. Instead of adapting the model and parameters off-line using the off-line measurements and analysis of the molecular properties from subsequent batch runs, Bayesian parameter estimation is used. The result is shown in Figure 3, demonstrates the significant
^22 improvement that can be achieved using a Bayesian approach that involves the estimation of a distribution based parameter value rather than a spot value. The updated model can be observed to track the real process tightly, resulting in good estimates of the polymer properties.
0.85
\
°- 0.8
1 i
^0.75
N
oc 0.7
% v
'N
32 3 = 2,8 2 2,6
-•v,.^
X\\\
5 2.4
<
/
i2,2 ^. 2
^-•-v
~^ ""^^
/
/
/\ / / / /
^ -'—-^^^^^.
1
..\
: V^
/
/
//
/ / 1
16
Figure 3. Performance of the Bayesian parameter estimator
4. Conclusions An alternative approach to EKF based parameter estimation using Bayesian methods is proposed. In this initial study, only the impact of parameter variation is considered. Although only two critical parameters were chosen, the potential of the approach is clearly demonstrated. Future research will address the combined on-line identification of important model parameters, model states and reactor initial conditions. This will provide a firm base on which to build robust on-line optimal reactor control policies.
Acknowledgements Mr Zhen Lu acknowledges CPACT and the University of Newcastle for the financial support of his PhD and Prof Kiparissides, CPERI, Thessaloniki, for the simulation.
References Thomas, I. M., and C. Kiparissides, 1984, Can. J. Chem. Eng., 62, 284. Ruppen, D., D. Bonvin., and D. W. T. Rippin, 1997, Comput. Chem. Eng., 22, 185-199. Crowley, T. J., and K. Y. Choi, 1997, Ind. Eng. Chem. Res., 36, 3676. Crowley, T. J., and K. Y. Choi, 1998, Chem. Eng. Sci., 53, 2769. Mourikas, G., P. Seferlis, J. Morris and C. Kiparissides, 2001, ESCAPE-11, Denmark. Scali,C., M. Morretta, and C. Semino, 1997, J. Proc. Cont., 7, 5, 357. De Valliere, P. and D. Bonvin, 1990, Comput. Chem. Eng., 14, 799.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
523
Super Model-Based Techniques for Batch Performance Monitoring Lindsay McPherson, Julian Morris, Elaine Martin Centre for Process Analytics and Control Technology University of Newcastle, Newcastle Upon Tyne, NEl 7RU
Abstract By combining mechanistic and empirical-based models, a process performance monitoring representation of a dynamic, non-linear process can be developed with the model-plant mismatch forming the basis of the monitoring scheme. In practice, the mechanistic model will not be perfect and therefore the residuals will contain structure. A modified model-based approach, Super Model-Based PCA (SMBPCA), is proposed which incorporates an additional residual modelling stage to remove structure from the residuals. The approach is evaluated on a simulation of a batch process using a number of residual modelling techniques including Partial Least Squares (PLS), dynamic PLS, ARX and dynamic Canonical Correlation Analysis (CCA). The out-of-control average run lengths for these techniques show that the SMBPCA approach gives improved process monitoring and fault detection compared to standard multivariate techniques.
1.Introduction Standard multivariate methods for the monitoring of batch processes do not take into account the non-linear dynamic behaviour associated with batch processes. One approach that has been proposed to overcome the limitations of traditional multivariate batch techniques is that of model-based process performance monitoring. By combining a mechanistic model and an empirical-based model, a process performance monitoring representation of a dynamic, non-linear process can be developed with the residuals calculated from the model-plant mismatch forming the basis of the monitoring scheme. Model-based Principal Component Analysis (MBPCA) has previously been applied to a simulation of an exothermic batch reactor and an ethylene compressor (Wachs and Lewin, 1998; Rotem, et al. 2000). The batch reactor was modelled using an exact first principles model of the process. However in an industrial environment, the assumption that a perfect first principles model can be built is generally not realisable. An evaluation of MBPCA was carried out by McPherson et al. (2001) on a simulation of an exothermic batch reactor (Wachs and Lewin, 1998) using a first principles model in which small parameter errors were introduced. The results showed that MBPCA did not perform significantly better than standard batch observation level PCA (Wold et al, 1998) with respect to fault detection. Returning to the original objective of removing the non-linear and dynamic components associated with process data, a reason for this can be established. Examination of the residuals, after the model-based technique had been applied, showed that serial correlation and non-normal behaviour was still present. A
524 modified model-based approach, termed Super Model-based PCA, is proposed whereby an additional residual modelling stage is incorporated to remove the remaining structure and to obtain a set of unstructured residuals prior to the application of batch observation level PCA (Wold et ai 1998).
2. Benchmark simulation The Super Model-based PCA approach was investigated on a simulation of an exothermic batch reactor (Wachs and Lewin, 1998). The two-stage simulation consists of five dynamic mass and energy balances representing three temperatures within the reactor and two reactant concentrations. Throughout the simulation four process variables were monitored, three reactor temperatures and the position of the coolant control valve. The reactant concentrations were not measured as it was assumed they would not be available on-line. Using Monte Carlo simulation, a nominal data set of 40 batches was generated. The first principles model of the process was built using the same five equations and, as discussed previously, parameter errors were introduced to emulate plant-model mismatch. Three different faults (Wachs and Lewin, 1998) were introduced, (i) a gradual decrease in the activation energy of 1% from time point 150 to time point 180, (ii) a gradual decrease in the heat transfer coefficients of 15% from time point 150 to 180 and, (iii) an increase in the initial concentration of component A by 5%.
3. Super model-based PCA In Model-Based PCA (MBPCA), unless a perfect mechanistic model of the process is available, serial correlation and non-normality will still be present in the residuals that are used as the basis of the monitoring representation. These attributes can mask the detection of abnormalities and deviations entering the process, thereby reducing the fault detection capability of MBPCA. Super Model-Based PCA (SMBPCA) includes an additional residual modelling stage prior to the application of PCA to remove the presence of serial correlation and non-normality (Figure 1). This results in a set of unstructured residuals to which linear statistical projection techniques can be applied. The SMBPCA algorithm is as follows: 1. Plant data is collected under normal operating conditions. 2. The model-predicted values are calculated and subtracted from plant-measured values, giving a set of structured residuals. 3. An error model is used to predict the values of the structured residuals from the original variables. 4. Predicted values of the structured values are subtracted from the original structured residuals, giving a set of unstructured residuals. 5. PCA or other monitoring methodologies are applied to the unstructured residuals and the confidence limits for the nominal representation are calculated. 6. The subsequent monitoring of the process is then based on the analysis performed on a nominal data set of unstructured residuals.
525
Plant
First Principles Model
Error Model
PCA
Fig. 1. Schematic of Super Model-based PCA Five different types of error model have been investigated to model the structured residuals, Partial Least Squares (PLS), Autoregressive with eXogeneous input (ARX), dynamic PLS, dynamic non-linear PLS and dynamic Canonical Correlation Analysis (CCA). 3.1 Super model-based PCA with Partial Least Squares Initially the error model was defined using a static PLS model, i.e. the residuals were modelled using the original variables. Comparing the normal probability plots of the original data and the residuals generated from the model-based approach (Figure 2) with the equivalent plot for the SMBPCA residuals that formed the basis of the PCA monitoring scheme (Figure 4a), it can be concluded that the SMBPCA approach reduces the non-linearity in the data. However, it is observed that the SMBPCA result is not significantly better than that obtained using the model-based PCA approach. To obtain an appreciation for the level of serial correlation in the data, plots of the partial autocorrelation function (PACF) were examined. Figure 4b shows the PACF plot for the residuals from the super model-based approach that can be compared with the equivalent plots for the original process data and the residuals generated from the model-based approach. Figure 3. For both MBPCA and SMBPCA with a PLS model, the level of serial correlation in the residuals is seen to approximately the same.
Fig. 2a. Normal probability plot (original data).
Fig. 2b. Normal probability plot of the residuals (MBPCA).
526
Fig. 3b. PACF plot of residuals (MBPCA)
Fig. 3a. PACFplot (original data).
!ioJQ8)Q6: 04-
JQ2;QO-
;-Q2'-04i .-06^
i-oe^-1.0^
Fig. 4a. Normal probability plot of the residuals (SMBPCA -f PLS).
1
1
1
1
2
-2
22
2
1 ^
1
1
2
Fig 4b. PACF plot (SMBPCA + PLS)
3.2 SMBPCA with Dynamic Error Models From the partial autocorrelation plots, it is evident that the process data contains serial correlation that can be modelled by an autoregressive model (Figure 3a). A number of different model structures were investigated to remove the serial correlation. Firstly the PLS residual model was replaced by an AutoRegressive with eXogeneous input (ARX) time series model. The ARX residual model was then replaced by a dynamic PLS representation (Qin, 1993). Dynamic PLS is a regression technique where PLS is applied to past values of the process variables and the residuals based on an ARX structure. By lagging the past values of the process variables and residuals, the impact of dynamic behaviour is addressed. Thirdly a dynamic non-linear PLS model (Baffi et al. 2000) was applied to the structured residuals. The non-linear PLS algorithm incorporates a polynomial (quadratic) function within the PLS algorithm through weight updating of the PLS inner and outer models, making it suitable for use in modelling non-linear systems. By integrating this non-linear PLS algorithm within an ARX framework, both the dynamics and non-linearity in the data are taken into account. Finally Canonical Correlation Analysis (CCA), a latent variable method that maximises the correlation structure between variable sets X and Y, was used to model the residuals. A set of orthogonal latent variables in X and Y that are most highly correlated are calculated. In this study, CCA was extended into an ARX-structured framework, dynamic CCA, so as to allow the serial correlation in the data to be taken into account.
527
4. Fault Detection Analysis The observation level scores for SMBPCA using a PLS residual model were used to develop a monitoring representation (Wold et al. 1998). The results are shown in Figure 5 for each fault type. Likewise the plots for SMBPCA with a dynamic nonlinear PLS error model are shown, Figure 6. Based on such plots, the out-of-control Average Run Length (ARL) was calculated from one hundred batches. The out-ofcontrol ARL is defined as the average number of samples between fault occurrence and fault detection. This measure was used to assess the fault detection ability of each technique. Figure 7 summarises the out-of-control ARL for each of the super modelbased techniques and the ARL for standard batch observation level PCA and MBPCA.
Figure 5: Observation level scores plots for faults 1,2 and 3 for SMBPCA + PLS. /"^ N
\
/ [ \
1
/
''^y
Figure 6: Observation level scores plots for faults 1,2 and 3 for SMBPCA + Dynamic Non-linear PLS. For each of the three fault types the ARL has been reduced through the application of the super model-based techniques compared to standard observation level PCA. SMBPCA with PLS is the least effective of the super model-based methods. Its inadequacy in dealing with structured residuals is because a dynamic, non-linear element is not included in the modelling. For the first two fault types, the decrease in activafion energy and the decrease in heat transfer coefficient, SMBPCA with dynamic non-linear PLS gives the best fault detection performance. Examination of the residuals (not shown) confirms that this is due to the removal of structure. For the third fault type, the increase in initial concentration of a reactant, SMBPCA -i- dynamic CCA gave the best performance, reducing the ARL by approximately 100 samples compared to standard PCA. Again this can be attributed to the removal of structure from the residuals. In general, all four super model-based techniques that involved dynamic components (ARX, dynamic PLS, dynamic non-linear PLS and dynamic CCA) gave approximately the same performance with respect to fault detection capability for faults
528 1 and 2. SMBPCA with Dynamic CCA and SMBPCA with ARX exhibited similar performance for the third fault type. Again the dynamic techniques showed significant improvement over the conventional methods.
standard
mbpca
sm+pis
sm+arx
sm+dypis
sm+dycca sm+nldypis
Figure 7: Average run length for faults 1 (\;^), fault 2 (M), fault 3 (U)-
5. Conclusions Super model-based PCA has been shown to be an effective tool for the monitoring of batch processes through the reduction in ARL compared to standard batch observation PCA. The limitations of standard observation level PCA and model-based PCA in dealing with the structure present in the residuals has been overcome by incorporating an additional residual modelling stage prior to the application of PCA. Of the different residuals modelling techniques tested, SMBPCA + dynamic non-linear PLS and SMBPCA + dynamic CCA exhibit the best fault detection capability. Examination of the residuals confirmed this result was due to the removal of structure and non-linearity from the data. Future work will address more complex processes and the need to differentiate between model-plant mismatch and process disturbances.
Acknowledgements LM acknowledges the EPSRC and the EU project PERFECT (Esprit Project 28870) for supporting her PhD studies.
References Baffi, G., E. B. Martin and A. J. Morris, 2000, Chemo. Intell. Lab. Syst., 52, 5. McPherson, L. A., Martin,E.B., Moms, A.J., 2001, IChemE, AFC 6, York, UK, 23. Qin, J, 1993, Proc. 32"^ CDC, Texas, USA, 2617. Rotem, Y., Wachs, A., Uwin, D.R. 2000, AIChE., 46(9), 1825. Wachs, A. and D. R. Lewin, 1998, Proc. Proc. IFAC DYCOPS-5, Corfu, Greece, 86. Wold, S., N. Kettaneh, H. Friden and A. Holmberg, 1998, Chem. Intell. Lab. Syst.44, 331.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
529
Closed loop indicators for controllability analysis F. Michiel Meeuse\ Yudi Samyudia^ and Johan Grievink^ ^Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft The Netherlands ^McMaster University, Department of Chemical Engineering, 1280 Main St West Hamilton, ON L8S 4L8
Abstract This paper presents an approach to analyse the controllability of process alternatives based on the optimal closed-loop performance. The approach presented by Tousain and Meeuse (2001) is extended to include the robustness properties in terms of two different uncertainty models: parametric uncertainty and co-prime factor uncertainty. The simulation results on a distillation column demonstrate that the optimal design parameters using the nominal closed-loop controllability will not alter after introducing the robustness criterion.
1
Introduction
It is nowadays generally accepted that the traditional sequential design of a chemical process and the control system can lead to plants that are difficult or even impossible to control, or to sub-optimal combinations of the process and the control system. Morari (1983) used the concept of perfect control to identify the aspects that inherently limit the achievable control performance of process systems: time delays, right half plane zeros, input saturation and plant/model mismatch. A large number of alternative controllability indices have been proposed, including the Relative Gain Array (RGA), Singular Value decomposition based indices and the Closed-Loop Disturbance Gain (CLDG). The basic idea of all these indices is to screen alternative designs on potential control problems afterwards. However the main limitation of these indices is that the relation with the closed-loop behaviour is often unclear. An alternative approach is the simultaneous design of the process and the control system by Mixed Integer Dynamic Optimisation (MIDO). In this approach first a superstructure is generated that contains a set of feasible process and controller options. Then optimisation techniques are employed in order to find the combination of process and controller that optimise the desired objective function under certain disturbance scenarios (Mohideen et al., 1996 and Bansal, 2000). The main advantage of these approaches is that in principle the optimum combination of the process and control system can be found. However detailed dynamic models are required and the solution is computational demanding. Moreover in all published cases the control structure was limited to multi-loop SISO PI controllers. In principle more general control algorithms could be considered, however this will increase the computational complexity even more. So we consider these methods as very useful in more detailed design phases, but too complicated for the earlier, screening phases. Tousain and Meeuse (2001) have presented an alternative approach based on optimal control: Closed Loop Controllability (CLC), which is a combination of the two approaches presented above. For a (linearized) model the optimal weighted closed loop variance is optimised by searching over the entire design space. For a linear model an
530 exact solution of the optimal controller can be attained, eliminating the need for dynamic optimisation. The main advantage compared with the more traditional controllability indices is that the screening can be compared based on the best achievable closed-loop performances. However the computational effort required is considerable less than in the simultaneous approach. There is no need for dynamic optimisation because the optimal controller is given by an exact solution Two limitations of this approach are that it relies on linear models and linear controllers and that no robustness properties can be guaranteed. Despite the fact that process systems are in general non-linear, a linear analysis often suffices e.g. by designing static nonlinear compensators (Morari, 1992). In this work we will extend this CLC approach to include robustness properties.
2
Basic modelling assumptions
We consider process systems that can be described by a set of Differential Algebraic Equations (DAE):
x{t) =
f{x{t),z{t),u{t),d{t),e{t),p),
0 = g{x{t),zit)Xt).d{t),e{t),p), y{t) =
(1)
h{x{t),z{t),0{t),n{t)),
where x(t) e R"^, z(t) e M"^, u{t) e M"" and
d(t) € R""' are respectively the state,
algebraic, input and disturbance variables, y{t) G W' are selected output variables, 6{t) e R"" are the uncertain parameters, n{t) G R"" is the measurement noise and p e P = P^x P^x...x P^ are the np design parameters, where Pare the parameter sets. These sets can contain real-valued and integer-valued numbers. Real-valued design parameters are for example dimensions, pressures and temperatures. Integer-valued design parameters are related to structural design decisions e.g. the number of trays in a column or the type of reactor. The behaviour of the plant in or close to its steady state operating point(s) can be described using linearised models. For all alternatives the non-linear model is linearised with respect to the states and input variables in it's operating point(s). We will use the standard state-space notation for these models: Ax = A{p)Ax -h B{p)Au -h d, A?/,, = C{p)Ax + n where Ax, Au and A?/„, are respectively the deviations of the states, the inputs and the measured outputs from their steady state values and n is the measurement noise. The system matrices [^,5,C] are functions of p only. The system can be augmented with disturbance models. For stochastic disturbances and measurement noise the augmented system becomes:
531
= y
+
.0 A,\
c o]
4
0
B 0
W+
w;, (3)
Wn 1
where [^^,5^ is the realisation filter for the disturbances, x^ and x,i are the states of the process and the filter, w^i and ^„ are Gaussian white noise stochastic variables with respectively covariance matrices R^i and R^^ and y,,^ are the measurements.
3
Closed loop optimal performance
Tousain and Meeuse (2001) have presented an approach for the CLC based on optimal performance. Their approach is based on the Linear Quadratic Gaussian (LQG) controller. The idea is to find the control u(t) that minimises: 1 '^
Yim—J{Ay'QAy
ch>ar
+
(4)
Au'RAu)dt
where J is the objective function, E denotes expectation, and Q and R are weighting matrices of appropriate dimensions. For linear systems an exact solution exists for this controller, given by combining the optimal state observer (Kalman filter) with the optimal state feedback controller for the deterministic Linear Quadratic Regulator problem (Kwakernaak and Sivan, 1972). Tousain and Meeuse (2001) define the closed loop controllability as the minimum of (4), optimised over all controllers: CLC = min controllers
1 '^
E\ Yim—J{Ay'QAy^Au'RAu)
dt
(5)
Meeuse and Tousain (2002) have presented two alternative ways in which both the closed-loop controllability and the process economics can be considered. In the multi objective optimisation formulation both the closed-loop controllability and the economics are considered as objectives. An alternative formulation, closed-loop economic optimisation, includes the closed loop variance in the constraints. In this paper we will focus on a revised formulation of the closed loop controllability, ignoring the economics.
4
Closed loop robustness
The controller designed in the previous section achieves optimal performance when the real plant equals the model. In general there will always be plant model mismatch. Take for instance only the fact the controller with optimal performance was based on a linearised model. Therefore any controller that will be implemented must also meet the design specifications despite the uncertainty in the system. Hence the controllers must be robust. However the LQG controller used in the CLC analysis presented by Tousain and Meeuse (2001) has no robustness properties. We will now show how this method can be extended with robustness properties. We will consider two types of uncertainties: parametric uncertainties and coprime factor uncertainties.
532 4.1 Parametric uncertainty Lets assume that the uncertain parameters in the system are indicated with 6. The parameter range is specified by:
e = [o\ej <e<e^].
(6)
The worst case, close loop minimum variance for a given design is then given by: 1 ^^
CLC^^ = max min E\ lim —J{Ay'QAy "^
eee
controller
+ Au" RAu)dt
(7)
where CLCRI is the closed loop controllability in case of parametric disturbances. Since the nominal values of 0 are included in 0 , the following property always holds: CLC^^^ > CLC 4.2 Coprime factor uncertainty An alternative uncertainty description, often used in the area of robust control is coprime factor uncertainty. Assume that the process transfer function, G(s), has a normalised left coprime factorisation (McFarlane and Glover, 1989): (8)
G{S)^M;\S)N,{S),
where both Mi(s) and N|(s) are stable. The perturbed plant can then be written as: G(5) = ( M , + A j - ( A r , + A j ,
(9)
as also show in Figure 1. This uncertainty model is very general. AM,
AN
N \*h4A M Figure 1. Coprime factor uncertainty. The robust stability margin is the maximum perturbation for which the closed-loop system is stable. McFarlane and Glover (1989) have shown that there exists an exact solution for the controller that maximises this stability margin. The stability margin, e, for a controller K is given by: (10)
{I~GK)-'MIn case of coprime factor uncertainty the CLC is defined as: 1 ^
limCLC,,.. = min E r^5c J" J St
6 > e
^
flAy'QAy-}-Au''RAu)dt '
533 Balas et al. (1998) state that values of e > 0.2 - 0.3 are generally satisfactory.
5
Example
Tousain and Meeuse (2001) have applied their CLC indicator on the distillation column system presented by Skogestad (1997). We will now include the two robustness measures, CLCRI and CLCR2 in their analysis.
System description The system under study is a binary distillation column. The feed with Xf = ^.5 should be separated in a top and bottom product with respectively x = 0.01 and x = 0.99. The vapour/liquid equilibrium is described with a constant relative volatility. A simultaneous disturbance in the feed flow rate and feed composition are considered. The design parameters are the number of trays (31 < NT < 51) and the feed tray locations {NF = centre ± 5). Parametric uncertainty The uncertain parameters considered are the feed rate and the feed composition: 0.9
Figure 2. CLC results. A maximised over 0, B nominal value of 6. Coprime factor uncertainty Figure 3 presents the stability margins for the various designs related to the coprime factor uncertainty. When this stability margin is added as a constraint to the optimisation problem, the feasible area is reduced. In line with Balas et al. (1998) we add the constraint that e > 0.2. The resulting feasible area is indicated in Figure 3. The values of CLCR2 in the feasible area are the same as the values of the CLC. Hence the
534 optimum design parameters in case of coprime factor uncertainty have not changed, compared with the nominal design. Number of trays 50r
ACi.-'-'
40r f---.,.
Feasible area
O
^
%
\\
>
' ^
<
.
•
'"~~^-^-^-•^
Feed tray 9ocation
^
Figure 3. Co prime factor uncertainty stability margin
6
Conclusions
We have shown how the closed loop controllability as introduced by Tousain and Meeuse (2001) can be extended to include robustness properties. Uncertainties can explicitly be taken into account. For parametric uncertainty this leads to a min-max-min formulation. Other types of uncertainty can be modelled as co-prime factor uncertainty where the stability margin can be added as robustness constraint.
Literature Balas, G.J., Doyle, J.C., Glover, K., Packard, A. and Smith, R. (1998) ^i-Analysis and Synthesis Toolbox, The Mathworks Inc. Bansal, V. (2000) Analysis, design and control optimization of process systems under uncertainty, PhD Thesis, University of London Kwakernaak, H. and Si van, R. (1972) Linear optimal control systems, Wiley, NewYork McFarlane, D.C. and Glover, K. (1989) Robust controller design using normalized coprime factor plant descriptions. Springer-Verlag, Berlin Meeuse, P.M. and Tousain, R.L. (2002) Closed loop controllability analysis of process designs: application to distillation column design, accepted for publication in Comp & Chem. Eng. Mohideen, M.J., Perkins, J.D., and Pistikopoulos, E.N. (1996) Optimal design of dynamic systems under uncertainty. AlChE Journal 42(8):2251-2272 Morari, M. (1983) Design of resilient processing plants III A general framework for the assessment of dynamic resilience. Chem. Eng. Science 38:1881-1891 Morari, M. (1992) Effect of design on the controllability of continuous plants, in: Perkins, J.D. (Ed) Interactions between process design and process control, IF AC Workshop, London Tousain, R.L. and Meeuse, P.M. (2001) Closed loop controllability analysis of process designs: Application to distillation column design, in Gani, R. and Jorgensen, S.B. (Eds) Computer Aided Chemical Engineering, 9, 799 - 804
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Published by Elsevier Science B.V.
535
Non linear dynamics of a network of reactors with periodical feed switching L. RUSSO\E. Mancusi^, P. L. Maffettone^ and S. Crescitelli^ ^Dipartimento di Ingegneria Chimica,Universita degli Studi di Napoli "Federico 11" Piazzale Tecchio 80, 80125 Napoli, Italia ^Dipartimento di Ingegneria, Universita del Sannio, Piazza Roma, 82100, Benevento, Italia ^Dipartimento di Scienza dei Materiali ed Ingegneria Chimica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italia
Abstract In this paper we first assess the symmetry properties of a periodically forced network of reactors. Then, by making use of symmetry properties, the dynamic behaviour of the system is characterised. Bifurcation diagrams are derived with a continuation technique based on a suitable map, and symmetric and non-symmetric regimes are detected and described Possible bifurcation scenarios and, in particular, symmetry-breaking bifurcations are discussed.
1. Introduction There is a growing interest in periodically forced reactors in the chemical engineering literature related to the possible improvement of heterogeneous catalysed processes. Dynamic regimes can be achieved in different ways. For example, one (or more) input variable is forced to vary continuously in time. Alternatively, Matros (1985) proposed several methods to sustain a dynamic behaviour by periodically changing of the feed position while keeping the feed composition and temperature constant. In this context, proposed implementations are: The reverse flow reactors (RFRs), where the flow direction is periodically inverted, and the network of reactors (NTW), where the forcing is obtained with a periodical permutation of the reactor order. Recently, for the treatment of lean waste gases, Brinkmann et al. (1999) have analysed a NTW of three catalytic burners operated with a valve system that allowed the cyclic permutation of their order. They have showed that the NTW is an efficient alternative to RFR to carry out autothermal VOC oxidation even at extremely low adiabatic temperature rise. From a practical point of view, the optimal regime of the above mentioned dynamically forced system is a periodic regime with the same period of the forcing. However, depending on the operating or design parameters, those reactor configurations can show a rich dynamical response as demonstrated by Reha^ek et al. (1992, 1998) and Khinast and Luss (2000) for RFRs. Indeed, it has been shown that multiperiodic, quasi-periodic or even chaotic regimes can be attained (Rehaeek et al.; 1992, 1998). RehaCek et al. (1998) have also investigated the symmetry properties of the regime solutions of RFRs. To our knowledge, a similar analysis is still missing for the case of an NTWs.
536 It is well known that a complete dynamical characterization of an industrial process is useful (and sometime required) to develop adequate control strategies and to safely operate and design the reactors. The discontinuous time forcing induces a spatio-temporal symmetry (Russo et al., 2002) into reactor model, and thus symmetric and asymmetric regime solutions are possible. The symmetry properties affect the bifurcation scenario: Some bifurcations cannot take place for the symmetry of the systems, independently of the intrinsic dynamics of the unforced system. It will be shown that solution symmetry cannot be lost through pitchforks as for RFRs, but through more complex phenomena such as resonance (frequency locking, e.g., Kuznetsov, 1998). We have exploited the symmetry properties of such systems to implement a numerical technique based on standard pseudo-arclength continuation methods (Faraoni et al., 2001). The numerical approach here used to perform the bifurcation analysis is based on the numerical construction of a suitable map that once iterated gives the Poincare map (Russo et al., 2002). To this end, public domain continuation software (AUT097, Doedel et al., 1997) was used. With this approach it is possible to carry out an accurate bifurcation analysis and to characterize symmetric and non-symmetric regimes.
2. Reactor systems In this work, we refer to simple reactor systems that are typically operated with a discontinuous forcing: A network of 3 identical CSTRs. In this case, every switch time the feed and the discharge positions are shifted according to a cyclic permutation. In each CSTR the irreversible reaction A-> B takes place following the kinetics: r=koC^exp
(1)
RT
The CSTRs are modelled with the two dimensionless ordinary differential equations, one for the mass balance and the other for the enthalpy balance, as follows: ^=-ai+g(t)a,,+Da(l-ai)'exp dt d9 ^=-ei+g(t)9,,+Da(l-a;)''exp dt
0
' yp9, ^
i+pe. J
+B(e„-e,)
(2)
(/0<-(mod3)
8(t)-
1
j/-(mod3)>l
In Eq. 2, the index i= 1,2,3 identifies tiie reactor, while g is a square wave function introduced to account for the discontinuous feed scheme. It is apparent that the vector field is changed discontinuously in time, and after a time 3T=T it recovers the initial configuration.
537 With some algebraic manipulation the system in Eq. (2) can be compactly represented as follows: du = F(u,i^,r) dt
(3)
where UGE"" (in our case u={ai, 9i...., Oj. , 6i ... (Xn, 6n}) is the state vector, -QeR^ (in our case i&={q, Da, B, y, (3, i, 6H}) is the parameter vector. Discontinuous forcing introduces symmetries in the dynamical system. In the present case, the symmetry is a spatio-temporal one due to the discontinuous vector field that obeys to the following invariance property (Russo et al., 2002; Lamb, 1995):
GF(u,7?,t) = F(Gu,i?,t-T)
with
fo 1 0^ G= 0 0 1 1 0 0
(4)
V
The matrix G is the generator of the symmetry group Z3, and thus this matrix has the properties that G^=I. The permutation is indeed: ^v.^ •^3
(5)
y"-^^
It is well known (Golubitsky et al., 1988) that symmetric systems can have symmetric solutions. Thus, periodic and symmetric solutions of system given by Eq. (3) are such that:
u{t) = Gu{t-T)
(6)
If a nonsymmetric solution exists, then the model symmetries imply the existence of other two nonsymmetric solutions. The property reported in Eq. 4 implies that if a nonsymmetric periodic solution exists, u(t), then the applications of G and G onto u(t) give the other two solutions. A peculiar property of systems with spatio-temporal symmetry is that the Poincare map is the iterate of another map (Lamb, 1995). In the case at hand, if P is the Poincare map, the following equation holds:
P=(G(pj'=tf
(7)
where (Px represents the evolution operator of system reported by Eq. (2) when it is not forced. Fixed points of P that are fixed points of H as well correspond to symmetric
538 periodic solutions. On the other hand, fixed points of P that are not fixed points of H represent nonsymmetric periodic solutions.
3. Results The bifurcation study was conducted by considering as bifurcation parameters the time switch (T). The symmetry property reported in Eq. (7) allows the continuation of symmetric solution branches by continuing fixed points of map H. This approach reduces by a factor 2/3 the computation time of solution diagrams with respect to continuation of fixed points of the P map. In fact, for the case of map H, the ODE set can be integrated for a time x whereas the map P not only requires an integration for a time T but also needs a permutations. The continuation has been implemented by using the software AUT097 (Doedel et al., 1997) as reported by Faraoni et al. (2001). In Fig. 1 the solution diagram for a network of 3 identical CSTRs is shown. The bifurcating parameter is the switch time i
a
0.9
i/
0.8
V
'•
\P
07 0.75
H
Figure 1 The solution diagrams report, as a representation of the regime solution, the dimensionless conversion (ai) of the first reactor vs. one third of the dimensionless period of the forcing (r). Solid lines represent stable T-periodic solutions; dashed lines unstable T-periodic solutions; dashed dot-dot lines stable 2T-periodic solutions; filled triangle Flip bifurcations; filled squares Neimark-Sacker bifurcations. Figure b shows details of the delimited rectangular region. The solution diagram reported in Fig. 1 does not contain pitchfork bifurcations that are the bifurcations through which RFR symmetric solutions (Zj symmetric) lose symmetry (Russo et. al, 2002). Indeed, that kind of bifurcations cannot be encountered in Z3 symmetric systems. Rather, loss of symmetry may take place through more complex phenomena as resonance (frequency locking). These phenomena can take place in the system under consideration since there are parameter regions in which it behaves as an oscillator when operated in unforced conditions. This means that the system is characterised by a natural frequency, which, in presence of an external periodic forcing, may couple with the external frequency giving rise to stable resonance regions (Arnold,
539 1983, Kevrekidis et al., 1986a, 1986b). It should be noted that the loss of symmetry implies that three different solutions are found, all sharing the same stability properties and bifurcations. By exploiting Eq. (4), once one solution is determined the others are easily calculated. In fact, they are calculated by applying G and G^ to the fixed point from which the continuation was started. For the sake of illustration, the symmetry properties of both the basic solution and that of the resonant isola are shown in Fig. 2.
(a)
(b)
Figure 2 Orbit projections in the phase space, (a) and (b) are obtained for the same value of the switch time (T=]) and starting from two different initial conditions, (a) is symmetric; (b) is nonsymmetric The solution reported in Fig. 2-(a) represents a symmetric solution (is a fixed point of map H). In fact, the projection is indifferent to any axis permutation. The resonant solutions relative to the frequency locking isola are nonsymmetric as it appears in Fig. 2-(b). In the case of periodic solution of the symmetric kind, the time series in the three reactors are identical but shifted in time of a time x as shown in Fig. 3-(a). In the case of nonsymmetric solutions the time series of the three reactors are completely different from each other as visible in Fig. 3-(b).
4. Conclusions This work has showed that reactor systems operated under periodic forcing posses symmetry properties effectively imposed by the forcing itself The presence of such symmetry can be exploited to ease the dynamical analysis since it determines possible bifurcation scenarios and suggests multiplicity when nonsymmetric solutions are found.
540 (b)
I
dimensionless time
S* 0-3
dimensionless time
Figure 3. Time series data for a symmetric solution (a) and a nonsymmetric solution (b) of period T.
5. References Arnold V.I., "Geometric methods in the theory of ordinary differential equations", Springer Verlag, (1983). Brinkmann, M., A. A. Barresi, M. Vanni and Baldi G., " Unsteady state treatment of a very lean waste gases in a network of catalytic burners," Catalysis Today, 47, 263 (1999). Doedel E. J., Champneys A. R., Fairgrieve T. F., Kuznetsov Y. A., Sanstede B., and Wang X., "AUT097: continuation and bifurcation software for ordinary differential equations", July (1997). Faraoni V., Mancusi E., Russo L. and Continillo G., " Bifurcation analysis of periodically forced systems via continuation of a discrete map", ESCAPE-11, 27-30 May, (2001), Kolding, Denmark Golubitsky, M., I. Stewart and D. G. Schaeffer, Singularities and groups in bifurcation theory. Vol II, Springer-Verlag, New York (1988). Kevrekidis I.G., R. Aris and L.D. Schmidt, 'The stirred tank forced", Chem. Eng. Sci., 41, 1549,(1986)a. Kevrekidis I.G., R. Aris and Schmidt L. D., "some common features of periodically forced reacting systems", Chem. Eng. Sci., 41, 1263, (1986)b. Khinast, J., and D. Luss, "Efficient bifurcation analysis of periodically-forced distributed parameters systems," Comp. Chem. Eng. , 24, 139 (2000). Kuznetsov, Y. A., Elements of applied bifurcation theory, 2nd ed., Springer Verlag, New York, (1998). Lamb J.S.W., "Resonant driving and k-symmetry". Physics Letters A, 199, 55, (1995). Matros Y.S, Unsteady Processes in Catalytic Reactor, Elsevier, Amsterdam, (1985). Rehacek J., KubiCek M. and Marek M., "Modelling of a tubular catalytic reactor with flow-reversal", Chem. Eng. Sci., 47, 2897 (1992). RehaCek, J., M. Kubi^ek and M. Marek, "Periodic, quasi-periodic and chaotic spatiotemporal patterns in a tubular catalytic reactor with periodic flow reversal," Comp. Chem. Eng., 22, 2^3 (199^). Russo L., Mancusi E., P.L. Maffettone and S. Crescitelli, "Symmetry properties and bifurcation analysis of a class of periodically forced reactors", submitted to Chem. Eng. Sci, (2002).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
541
Robust Model-based Controllers via Parametric Programming V. Sakizlis, N.M.P. Kakalis, V. Dua, J.D. Perkins and E.N. Pistikopoulos * Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.
Abstract In this paper a method is presented for deriving the explicit robust model-based optimal control law for constrained linear dynamic systems. The controller underlying structure is derived off-line via parametric programming before any actual process implementation takes place. The proposed control scheme guarantees feasibility under the presence of uncertainties by explicitly incorporating in the design stage a feasibility test and then by adapting on-line the origin according to the current disturbance realizations.
1. Introduction Contrary to conventional control design methods, model predictive control (MPC) is particularly effective for dealing with a broad class of complex multivariable constrained processes. MPC determines the optimal future control profile according to a prediction of the system behavior over a receding time horizon. The control actions are computed by solving repetitively an on-line optimal control problem over a receding horizon every time a state measurement or estimation becomes available. The capabilities of MPC are limited mainly by the rigorous on-line calculations that make it applicable mostly to slowly varying processes. This shortcoming is surpassed by employing a different type of model-based controllers the so-called parametric controllers (Pistikopoulos et al., 2000, Bemporad et aL, 2002). These controllers are based on recently proposed novel parametric programming algorithms, developed in our research group at Imperial College, and succeed in obtaining the explicit mapping of the optimal control actions in the space of the current states. Thus, a model-based feedback control law for the system is derived off-line, hence, avoiding the restrictive on-line computations. However, the inevitable presence of uncertainties, pertaining for instance to model inaccuracies, catalyst deactivation or heat-exchanger fouling, and the impact of persistent unmeasured disturbances, typically corresponding to slow variations in feed and utility conditions, have largely been ignored while designing the parametric controllers. Consequently, the performance of this novel control technique may lead to infeasibilities due to inaccurate forecasting of the process behaviour. These infeasibilities may result in situations such as off-spec production or hazardous plant * To whom correspondence should be addressed. Tel.: (44) (0) 20 7594 6620, Fax: (44) (0) 20 7594 6606, E-mail: [email protected]
542 operation. Hence, a modification of the explicit control law is necessary to ensure feasible and safe operation in the presence of disturbance and uncertainty. In this work a novel methodology is presented for designing robust model based parametric controllers for general dynamic systems. The optimal control policy is derived off-line as a function of the process states via our parametric programming based theory and techniques (Dua et ai, 2002; Sakizlis et al., 2001). The proposed control scheme manages to capture the effect of uncertainties and achieves satisfactory disturbance attenuation, thus avoiding the occurrence of infeasibilities. This is achieved (i) for the case of slowly varying disturbances, via suitably adjusting the state steady point based on disturbance estimates and implicitly adapting the control law to the current conditions; and (ii) for bounded uncertainties, by incorporating in the parametric controller design, an explicit feasibility constraint that ensures feasible operation for all possible uncertain parameter realizations.
2. Theoretical Developments 2.1 Parametric Controller For deriving the explicit model - based control law for a process system, the following receding horizon optimal control problem is formulated: ^fx,,o) = niin xj^^ Px^^^ + f^ [yl, Qy^^, + v,^, 7?v^„ J S.t. X^i^^i = A^X.^j^ ^ A2V,\^
(1)
yt\k =Bi^t\k'^B2^t\k ^^8(yk\t^^k\t^^k\i)
= Coyt\k +Q-^/t|r+C2^r|.t+Q
k = o...N
where XEXO ST are the states, ye ST, are the outputs and VE 9f^ are the controls; t is the time a measurement is available from the plant and k denotes the time element over the prediction horizon N. The outputs are the variables that we aim to control, i.e. to drive to their set-point, (temperatures, concentrations) whereas the states are the variables that fully characterize the current process conditions (enthalpies, specific volume), v^ denotes the sequence of the control vector over the receding horizon. The constraints g, which may pertain to product specifications or environmental and safety regulations, completely define the feasible operating region. By considering the current states jc* as parameters, problem (1) is recast as a multiparametric quadratic program (mp-QP). The solution of that problem (Dua et al., 2002; Bemporad et ai, 2002) consists of a set of affine control functions in terms of the states and a set of regions where these functions are valid. This mapping of the manipulating inputs in the state space constitutes a control law for the system. The mathematical form of the parametric controller is as follows: krio(^*) = ^c-^*+^r;
if CR^x* -hCR^
for c = l,...,N,
(2)
where A^^ is the number of regions in the state space, a^ CR\ and b^., CR c are constant vector and matrices and the index c designates that each region admits a different control law. Vector Vct\o is the first element of the control sequence, whereas similar expressions are derived for the rest of the control elements.
543 2.2 Disturbance compensation The model-based parametric controller described in the previous paragraph fails to address the impact oipersistent slowly varying unmeasured disturbances on the process performance. The incorporation of a disturbance regulator in a predictive controller is usually performed by estimating the disturbance values and updating the model accordingly to ensure nonzero output target tracking (Muske and Rawlings, 1993; Loeblein and Perkins, 1999). The equivalent concept is applied here, generating a mechanism for updating the model on-line and modifying accordingly the derived control law. A distinction is done between two systems (i) one being the real process plant and (ii) the other comprising a model that represents an estimate of the process behavior, as shown in Table 1: Table 1: Real vs. Prediction Model Real '^r|)t+l=A-'^r|it+M|)t+^% System 3^/|* = ^\^k + h^t\k + f'Qt\k 9:
Prediction model
real disturbance
^k^\
=A-^r|^+^2V^^*+^>^r
yt\k = ^l-^r|^ + ^2^t\k +
w:
t'^t
disturbance estimate
Vector w is modelled as a step disturbance. Once a disturbance enters into the system, at time k, there is an anticipated discrepancy between the measured output y and the predicted output y. Based on this discrepancy an estimate of the current disturbance is obtained via Kalman Filters or via a recursive or moving horizon least square estimator (Henson and Seborg, 1997). Then the disturbance estimate is employed for computing a new steady state point [x, v j . If the dimension of output variables y is equal to the dimension of control inputs v and no input and state constraints are violated in the new steady state, then the new steady state is obtained by solving the linear system: ( / - A i ) j c , - A 2 V , =W 'W\
B^x^ + BjV^ = -F'
w
(4)
Cases when ^=dim {v)> m=dim (y) or when there are active control/state constraints are treated in a similar fashion. Based on [x, vJ, the control law (2) is shifted as follows: kr|o(-^*) = ^c(-^*--^J + v , + ^ r ;
if CRlix* -x,) + CR; < 0 ; j
for c = l,...,N, (5)
This technique implies the construction of a state-of-the art adaptive parametric controller. This controller achieves output feedback by readjusting automatically its tunings to account for the presence of disturbances. 2.3 Robust Parametric Controller During the implementation of the parametric controller (5), constraint violations are likely to occur when the disturbance estimate is inaccurate or the process is subject to additional unknown uncertain variations. The traditional technique for designing a robust model based controller that avoids this shortcoming, relies on minimizing on-line the worst case cost, thus leading to a min-max optimal control problem (Campo and Morari, 1987). This can be (i) computationally prohibitive due to the complete exploration of the uncertainty extreme points and (ii) may also lead to a conservative controller since the worst-case cost is penalized in the objective. Recently, Bemporad et
544 ai, (2001) based on the work of Pistikopoulos et al., (2000) and Bemporad et ai, (2002) developed a technique for moving the computations off-line, but their approach merely minimizes the worst case oo- norm of the output/input deviations that may cause deadbeat control and may result in a conservative control policy. Here, feasibility analysis theory (Halemane and Grossmann, 1983; Bansal et al, 2000) is adopted for deriving the robust parametric controller (as in Kakalis et al, 2001). The system is considered to be perturbed by a vector of unknown input affme time-varying uncertainties co that are assumed to belong to a compact polyhedral set O) ei2 c ^T:
Note, that we distinguish between the input disturbance 0 and the uncertainty O) in the sense that the latter has a higher frequency of variation and it lacks any reliable estimate. The robust controller can be defined as the controller that provides a single control action that ensures constrains' satisfaction for all the possible uncertainty realizations over the complete horizon. This is mathematically translated to the following constraint: Vco,,^ G ^(Vje y[^/x,|^, v,w ,0).^) < 0]) <=> max min{w | u > gj^je (0
7} < 0
(7)
U
This flexibility test condition (7) is incorporated explicitly as a constraint in the optimization problem (1), aiming to derive a robust parametric controller. The solution of the inner infinite max-min problem is accomplished by recasting problem (7) subject to the dynamics (6) as a linear parametric optimization problem (mp-LP). This results a set of piecewise affine expressions (Bansal et al., 2000) for the feasibility function u in terms of the uncertain parameters, the controls and the current states: Ui=diO)t|k4-fiX*-i-piVt|k+hi, Vi=l,..I. Next, by exploiting the linearity of these functions and the convexity of the uncertainty set Q, it is proved that the critical uncertainties that bottleneck the plant operation lie on a finite subset cJ of the uncertainty space vertices. Thus, the feasibility constraint (7) is equivalently, replaced by a finite set of constraints corresponding to the critical uncertain combinations. This reformulation gives rise to the following multiperiod receding horizon optimal control problem:
^I=^i4+^2V.|, Xjp
_
-X
(8)
•
0>g(y^
i\k
,x' ,v^,J /:=0,..,A^; l = l,...,L; i\k
'
where !„ corresponds to the nominal scenario of the uncertainty w. The solution of (8) (i) ensures feasible operation without exploring the complete space of the uncertainty realization as in Campo and Morari, (1987) and (ii) corresponds to a less conservative control action since it minimizes the nominal and not the worst case quadratic cost. By treating the current states x* as parameters the optimal control problem (8) is recast as a
545 parametric quadratic problem (mp-QP) and its solution results in a robust parametric controller of the form of (2). This controller apart for accounting for the presence of uncertainties is also capable of rejecting slowly varying disturbances by performing origin rearrangement as described in section 2.2.
3. Illustrative Example The demonstrative example is concerned with deriving the explicit robust control law for an evaporator process studied in Kookos and Perkins, 2001. The controlled states are the pressure P2 and the concentration C2 of the product stream. Their nominal reference values are: P2=50.57KPa; C2=25%. The manipulating inputs are the pressure of the utility steam Pioo that heats up the feed and the cooling water flow F200 that condenses the by-products. The active output constraint when operating around the nominal point is: C2>25% whereas additional constraints are also imposed on the inputs.. The feed flowrate Fi is considered as a disturbance that can be estimated only within accuracy of ±0.25kg/s but varies within a wider range. For deriving the robust control law, first the feasibility problem (7), (6) is solved with a horizon N=2, considering Fi as a deterministic uncertainty co=Fi-Finom» where Finom= lOkg/sec is the nominal uncertainty value. The critical scenarios are: CL)I|O^' ' =0.25,0.25,0.25; (Otji^'^'^=0.25,-0.25,-0.25. A multiperiod MPC problem (8) based on those scenarios is formulated and solved as an mp-QP treating the current states as parameters. The solution features a robust parametric controller that is shown in Figure 1. The execution of the control law is shown in Figure 2. The system is initially perturbed and as it is driven back to the origin a sequence of step disturbances occur. The robust controller with origin readjustment generates a back-off from the nominal point to accommodate these variations, whereas the nominal controller although it does reject the disturbance at steady state exhibits severe constraint violations.
4 Conclusions In this paper a novel framework is presented for designing robust model-based parametric controllers of linear dynamic systems that are subject to input disturbances and uncertainties. The controller consists of piecewise affme expressions for the control variables in terms of the states. The implementation of the control action is achieved by simple linear function evaluations, thus avoiding any expensive on-line computations. The controller guarantees robustness by means of satisfaction of constraints and disturbance attenuation.
5 References Bansal V., J.D. Perkins and E.N. Pistikopoulos, 2000, AIChE, 46, 335. Bemporad A., F. Borrelli and M. Morari, 2001, Proc. Eur. Cont. Conf Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, 2002, Automatica, 38, 3. Dua v., N.A. Bozinis and E.N. Pistikopoulos, 2002, A multiparametric programming approach for mixed integer and quadratic process engineering problems. Accepted in Comp. Chem. Eng. (also appearing in ESCAPE-11 Proc. pp.979). Halemane K.P. and I.E. Grossmann, 1983, AIChE J., 29, 428.
546 Henson, M.A. and D.E. Seborg , 1997, Nonlinear process control, Prentice Hall. Kakalis N.M.P,, V. Dua, V. Sakizlis, J.D. Perkins and E.N. Pistikopoulos. 2002 A parametric optimization approach for robust MPC. Accepted in IFAC Cong. Aut. Cont. Kookos I.K. and J.D. Perkins, 2001, accepted in J. Proc. Contr. Loeblein C. A and J.D. Perkins, 1999, AIChE J., 45, 1018. Muske K.R. and J.B. Rawlings, 1993, AIChE J., 32, 262. Pistikopoulos, E.N., V. Dua, N.A. Bozinis, A. Bemporad and M. Morari (2000). Comput. Chem. Eng. 24, 183. Sakizlis V., J.D. Perkins and E.N. Pistikopoulos, 2001, Multiparametric Dynamic Optimization of Linear Quadratic Optimal Control Problems: Theory and Applications. Accepted in: Optimization and Control in Chemical Engineering. Editor: R. Luus. : e.g.:
8 Critical Region Fragments
80
Region 4
75
70
4.8<0.2C2 <6 -50.55<0.2C2-1.09P2 0.2C2-1.09P2<-49.54
65 QU ^
60
Control Law in Region 4
55
50
^^^^^^^^^attBOBM
45
Pioo= -I6.58C2+3.9P2+4I6.6
CR04
40 »
24J
25
25-5
26
26.5
27
^200=
-78.84C2+430P2-L95310^
Figure 1. Critical Regions of the state space Outputs 51
-
Nomiaal Parro Robust Paico
t (mlB)
Controls 230
400
220
^ tL"
350
200 190
tso 170
;•
300
[ I
250
i
200
J
n
ISO 20
40 t(nm)
Figure 2. Output and Control time profiles.
t (nii»)
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Published by Elsevier Science B.V.
547
Dynamic Trajectory Optimization Between Unstable Steady-States of a Class of CSTRs Andrea Silva B., Antonio Flores T*. and Juan Jose Arrieta C. Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880 Mexico D.F., 01210, Mexico.
Abstract In this work the computation of optimal dynamic transition trajectories between unstable steady-states was addressed. Using orthogonal collocation on finite elements, the optimal control problem was discretized and cast as a non-linear optimization program. Transition trajectories were computed for a CSTR operating around a nonlinear multiplicity region. With the proposed optimization formulation optimal transition trajectories involving unstable steady-states were successfully computed. 1. INTRODUCTION Design of transitions between steady-state operations is an important problem in the chemical process industry. For instance, in the polymerization industry transitions are carried out to switch from producing a certain polymer A to produce a different polymer B (A and B are the same polymer but with different characteristics such as molecular weight distribution). Another important problem for which transitions are sought occurs during plant start-up/shut-down. Since some of the economic attractive operating steady-states are open-loop unstable, it becomes important to be able to obtain optimal dynamic transition trajectories between such unstable operating points. When a mathematical model of the process is available it becomes feasible to compute optimal transition trajectories leading to an open-loop optimal control problem (OCP). Roughly speaking three forms of solving the OCP have been proposed [1]: (i) direct application of the Pontryagin's maximum principle giving rise to a two-point boundary value problem, (ii) parameterization of the control trajectory leading to a non-linear optimization program where both the objective function and the constraints are evaluated by integrating the model equations and (iii) full discretization of the OCP leading to a non-linear optimization program. In principle any of these methods can be used for computing optimal transition trajectories between open-loop stable steadystates (where stability is defined in the Hurwitz sense). However, when it comes to compute optimal transition trajectories between: (a) stable-unstable, (b) unstable-stable and (c) unstable-unstable steady-states the full discretization approach of the OCP looks as the methodology that should be used to cope with this sort of OCPs [2]. In fact approaches based on the control trajectory parameterization strategy could be used for state transitions between unstable steady-states by first achieving closed-loop
Author to whom correspondence should be addressed. E-mail: [email protected], phone/fax: +52 5 267 42 79, http://200.13.98.241/-~antonio
548 stabilization of the system and then computing the transition trajectory; such approach has been used in [3] to initially solve unstable transition problems. Direct application of the control trajectory parameterization approach is totally unfeasible because during model system numerical integration the states will not be attracted to an unstable operating point and will converge to a stable one [2]. As far as we know unstable dynamic transition problems have not been previously addressed in the open academic literature; hence we trust to make a contribution to the OCP area. For solving the openloop optimal transition trajectory problem between unstable steady-states, the full discretization approach was used in this work. To discretize the model the orthogonal collocation method on finite elements was used and the resulting OCP was solved as a non-linear optimization program. 2. PROBLEM DEFINITION In order to compute transition trajectories, the CSTR model as proposed by Hicks and Ray [4] was used. Because the original parameters set used by these authors did not lead to multiple steady-states, some of the values were modified in order to end-up with a multiplicity map. In dimensionless form the model is given by: dt
^ =^
+ ^.oexp[-%]y,-««G,-,J=/,
(2)
where yi stands for dimensionless concentration {c/Cf), y2 is the dimensionless temperature {T/Jcf), yc is the dimensionless coolant temperature {Tc/Jcf), yf is the dimensionless feed temperature (T/Jcf),3ind u is the cooling flowrate. Table 1 contains the numerical values of the parameters used in this work; this set of parameter values lead to operate around the multiplicity region shown in figure 1. Table 1. Parameter values.
e
J
9 a
20 100 7.6 1.95x10-^
Residence time (-AH)/(pCp) Feed Concentration Dimensionless heat transfer area
Tf few Tc N
300 300 290 5
Feed temperature PreexponenUal factor Coolant temperature EiARJcf)
3. PROBLEM FORMULATION 3.1 Orthogonal collocation on finite elements The dynamic mathematical model was fully discretized using orthogonal collocation on finite elements (OCFE). The approach consisted in forming the discretized version of the residual equation stated as:
r = r-f{y^u)
(3)
where r = [rj r2f, y'=[dyj/dt dy2/dtf, y=[yj y2f and f=\f]f2f- In the collocation method the residual is minimized forcing it to pass through zero only in a finite number of discrete points [5]:
549 jr5{x-Xi)dx
=0
(4)
where S(x-Xi) is the Dirac delta function. If the discrete points are chosen as the roots of an orthogonal polynomial then the discretization procedure is called the orthogonal collocation method. The way of discretizing the model consists in representing each unknown variable as a linear combination between a set of coefficients (}^)and a set of basis functions 0 / T J (commonly taken as Lagrange polynomials):
y{^,)=lyMri)
(5)
N+l '
(6)
k=OT k*j
where N is the number of internal collocation points and r, corresponds to the ithorthogonal collocation root. In terms of the discretized model the residual equation is rewritten as: /
X
^+1
• /
X
(7)
A^)-^yj
where h stands for the element length. The first derivative of the Lagrange polynomial is computed from:
0,(^,)=
n-
1
W+l
/V + 1
P=Oj
/=0
I
n(T,-T/)
(8)
For each kth-clcmtnt (except the last one) the discretized mathematical model is given by (notice that A^j = 0^(T-) ): •
Mass balance
yv+2
•
(9) Energy balance = 2^N + i
l^y2j~hj2{y>u)=oj •
(10)
Equations for solution continuity between elements (mass and energy) I
A'+2
1
N+2
I
N+2
1
A^+2
— X^N.2,,y2.,-7—EA,y2.,=0'^' = 2,/v + i • NE
(11) (12)
Total transition time (13)
k=l
where NE is the number of finite elements. For the last element only the mass and energy discretized equations are taken into account, but collocation is done including the end-point of the solution space. This means that for the last element i=2,N+2.
550
3.2 Dynamic optimization In terms of the discretized model the dynamic optimization problem can be formulated as: e
(14) ypyz
S.t.
Eqs. 9 - 1 3 are met yu ^ yi ^ >^u y2j
^ y 2 ^ >^2,.
w^ < u < w^ hi
1.4
o
-
y1 =0.1926, y2=0.6881, u=430 {u) y1 =0.2632, y2=0.6519, u=455 (u)
O.Qh
0,8 L-
^"""^^"•--^^
2
^^"^^-^ 3
0.7 ^
4
o.eh
50
100
150
200
250
300
350
400
450
Cooling fiowrate
Figure I. Multiplicity map (- stable solution, -- unstable solution).
500
551 Table 2. Definition of transition points. Transition A B
Initial state 2{s) 2{s)
^ ->
Final state \{s)
Transition C D
Initial state 3(w) 3(w)
->
Final state 2{s) 4(w)
In order to verify the correctness of our proposed optimization formulation a stablestable transition was computed (see figure 2) and compared to that obtained using commercial optimal control software [6]; both solutions were practically the same. Next, transitions starting/ending on an unstable steady-state were computed; they correspond to B and C transitions which actually are the same transition but in opposite direction. The optimal transition trajectories are shown in figures 3 and 4. Finally, a transition starting and ending on unstable steady-states was obtained (transition D). The optimal transition trajectory is presented in figure 5. It can be observed that for all transition cases, the concentration profile is always soft whereas the temperature and cooling flowrate trajectories present fast dynamics. Also, in all optimal trajectories the total transition time is significantly smaller than the residence time. Since transitions B, C and D involve unstable steady-states, it is unfeasible to test them on the open-loop system; they have to be tested under closed-loop control. It was observed that the proposed formulation was particularly sensitive to the finite element size; even though it was a decision variable it had to be strongly bounded to reduce the frequency of response discontinuities. Also it can be mentioned that in most cases the finite element length hit the upper bound, so at the end it was noticed that for the proposed formulation, the finite element size could have not been a decision variable. Nevertheless it was helpful in finding the correct finite element size. Due to the steady-states unstable nature strong numerical problems were expected when the OCP was transformed into an equivalent set of non-linear algebraic equations (i.e. singularity).However, for the transitions addressed in this work, such expected problems did not arise.
Figure 2. Transition A: stable-stable (2 -^1)
..1 KJ
/
Figure 3. Transition B: stable-unstable (2 -^3)
552 5. CONCLUSIONS The results of this work seem to indicate that the proposed optimization formulation is a feasible approach to compute dynamic transition trajectories that involve unstable steady-states. Contrary to what was expected, no significant numerical problems due to the steady-states unstable nature, emerged. We expect to apply a similar formulation to obtain optimal transition profiles in more complex systems (i.e. polymerization reactors).
Figure 4. Transition C: unstable-stable (3 -^l) -4 1
!
| 0 20(-
/
/
/
/
y Figure 5. Transition D: unstable-unstable (3 -^4)
6. BIBLIOGRAPHY [1] Sargent,R.W.H., 2001, Manuscript in preparation. [2] Biegler,L., 1984, Comp.Chem.Eng., 8,3/4,243-248. [3] Flores,A., J.Alvarez,E.Saldivar and G.Oaxaca, 2001, Dycops 6, Korea. [4] Hicks,G.A. and Ray,W.H., 1971, Can.J.Chem.Eng. 40-522-529. [5] Celia,M.A. and Gray W.G., 1992, Numerical Methods for Differential Equations, Prentice-Hall. [6] http://www.accescom.com/~adam/riots.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
553
MPC Control of a Predenitrification Plant Using Linear Subspace Models Oscar A. Z. Sotomayor^^\ Song W. Park^^^ and Claudio Garcia^^^ (1) LSCP-Department of Chemical Engineering (2) LAC-Department of Telecommunications and Control Engineering Polytechnic School of the University of Sao Paulo Av. Prof. Luciano Gualberto, trav.3, n.380, 05508-900 Sao Paulo-SP, Brazil Fax: +55 11 3813.2380, E-mail: [email protected]
Abstract In this paper, a model-based predictive control (MPC) technique is designed aiming to control the nitrogen (N)-removal from domestic sewage in an activated sludge wastewater treatment plant. The objective is to control the nitrate concentrations in both, anoxic and aerobic zones of the bioreactor and, therefore, to inferentially to control the effluent inorganic nitrogen concentration. The synthesis of the MPC controller is based on a linear subspace (state-space) model of the process, where an identification horizon is added to include a sequence of past inputs/outputs. This sequence can be used to estimate the model or the updated state of the process, thus eliminating the need for a state observer. Two different MPC control configurations are compared and the result shows the successful of the control application. The linear state-space model was obtained using subspace identification methods.
1. Introduction Activated sludge (AS) is the most widespread biological process for wastewater treatment. In this process, N-removal is performed in two stages: nitrification and denitrification, under aerobic and anoxic conditions, respectively. In comparison with conventional AS plants (chemical oxygen demand - COD compounds removal and, often partly, ammonium), N-removal AS plants are more complex. The co-existence of nitrification and denitrification processes is accompanied by new operational problems (Yuan, 1999). Increasing the efficiency of one process will always have negative impacts on the efficiency of the other one. Hence, automatic control of N-removal AS plants is necessary to achieve the adequate performance of the overall system. The control of N-removal AS plants is mainly focused on nitrate, ammonium and inorganic nitrogen (nitrate plus ammonium) removal. This paper is related to control of nitrate removal. In a pre-denitrification plant, the following approaches are possible: (1) control of nitrate concentration in the aerobic zone by manipulating the internal recycling flow rate, (2) control of nitrate concentration in the anoxic zone by manipulating the internal recycling flow rate, and (3) control of nitrate concentration in the anoxic zone by manipulating an external carbon source flow rate. Carlsson and Rehnstrom (2001) proposed two single-loop controllers to concurrently regulate both approach (1) and (3). Nevertheless, these approaches (1 to 3) are highly interrelated and
554 for the optimal control of the process they should be simultaneously performed in a multivariable control philosophy. Model-based predictive control (MPC) is currently the most widely implemented advanced process control technology for process plants (Qin and Badgwell, 1997), and they are commonly found in the medium level of a plant-wide control structure. The MPC formulation naturally handles multivariable interactions and constraints. On the other hand, an alternative to use polynomial models (e.g. ARMAX, CARIMA, Markov parameter models, etc), that can be quite cumbersome to obtain in the general multivariable case, are the state-space models (Viberg, 1995). In this case, a prediction model can be easily constructed making use of the extended observability matrix. In this paper, a MPC controller, based on a linear subspace (state-space) model of the process, is designed aiming to control a predenitrifying AS plant, used for COD and N removal from domestic sewage, as shown in fig. 1.
q„
Zone 1
Zone 2
Zone 3
__i
^
3^ Fig. 1. Layout of the AS plant The layout of the process is constituted by a bioreactor, composed of an anoxic zone (zone 1) and two aerobic zones (zones 2 and 3) coupled with a secondary settler. In the aerobic zones, the DO concentration is controlled in 2 mg O2/I by simple PI controllers. The objective here proposed is to control the nitrate concentrations in both, the anoxic zone and the last aerobic zone ( SJ^QX and 5'yvo,3» respectively), by manipulating the internal recycle flow rate (2int' ^"^ ^^e external carbon source flow rate, Q^^. Two control configurations are tested by simulation: one taking into account the influent flow rate Q.^ as a manipulated variable (in a 3x2 system) and another one considering it as being constant (in a 2x2 system). The state-space model is obtained by using subspace identification methods, with input/output data obtained from the ASWWTPUSP benchmark (Sotomayor et ai, 2(X)la).
2. A Prediction Model (PM) of the Process Consider that the process is suitably described by a discrete linear time-invariant (LTI) state-space model of the form: yk = Cx^
(1)
where x is the state vector, u is the input vector , y is the output vector, A is the state transition matrix, B is the input matrix and C is the output matrix, and the time index k denotes the sampling instant. A PM is defined to be of the form:
555
where L is the prediction horizon, yi^+m is a vector of future outputs, Aw^/^ is a vector of future input changes, and Pi(k) is a known vector that represents the information of the past, which is used to predict the future. This vector is a function of the number J (identification horizon) and the state-space model matrices. Fj^ is a constant lower triangular matrix, which is a function of the state-space model matrices. A simple algorithm to compute p^ (k) and F^ is given by (Di Ruscio and Foss, 1998):
PL(k)-O^A'0]y,_j,,,j+P,u,_j,,,j^, F,=[OLB
Ht]
(3) (4)
with the matrix P^ , which is related to past control inputs, defined as:
P,=0,ArU-0,A'0]Ht
(5)
where O^r^ is the extended observability matrix for the pair (A,C), with L block rows, Oy is the Moore-Penrose pseudo-inverse of the extended observability matrix Oy for the pair (A,C), with J block rows, ry_j is the reverse extended controllability matrix for the pair (A,B), with 7-1 block columns, and the H^ is the lower block triangular Toeplitz matrix for the triple (A,B,C), with L block rows and L-1 block columns.
3. Model Predictive Control As can be seen, this PM (2) is independent of the state vector. Hence, there is no need for a state observer. The MPC law is found by minimizing the following discrete time LQ objective: ^k =iyk^UL-h^ULYQ{yM/L-h^\fL)-^^4/Lf^^lhfL ^^hllP^klL (6) where ^^.+i/^ is a vector of future references and u^n^ is a vector of future inputs. (2, R and P are block diagonal weighting matrices. The problem can be formulated as:
min
Z^
(7)
subject to linear constraints on w^, Aw^. and y\ . The constraints can be written as an equivalent linear inequality of the form: «-AM,,,<)3,
(8)
It is convenient to find the relationship between Aw^./^ and w^./^ in order to formulate the constraints in terms of future deviation variables Aw^/^ by using: ^k/L =5-Aw^/^+c-w^_j (9) where 5 is a lower block triangular identity matrix and c is block rows identity matrix, of suitable sizes. The input constraints are of the form:
556 SAu,,,
A M , , , < A«™^
< u^l -cu,_,
F,Au,,, < y^l - p,(k)
-A«,,,<-A«™1'
-SAu,,,<-u^l+cu,_,'
-F,Au,,,<-yf;,+p,{k) (10)
The objective functional 3 , may be written in terms of A u , , , . yk+\/L ^^^ ^^ eliminated from 3 , by using the PM (2). M,/, can be eliminated from 3 , by using (9). Therefore, the LQ objective functional becomes: 3, =Aul,HAu,,,-^2f^Au,,, +3^
v^hcrc. H = R + F[QF,+S''
(11)
PS J, = FlQip,{k)-
r,,,,,)+ S"^ Pcu,_,.md
H is the Hessian matrix, a constant positive definite matrix. / ^ is a time-varying vector, independent of the unknown control deviation variable. 3 ^ is a known timevarying scalar, independent of the optimization problem. The problem can be solved by the following equivalent QP approach:
min
(AM[/^//AW,/^ + 2 / / A w , / J
(12)
subject to (8). When Aw^./^ is computed, the control signal to be applied to the process is w^ = w^_j -h Aw^ . (Note that only the first change in Aw^,/^ is used)
4. A State-Space Model of the Process The linear state-space model is obtained by using subspace identification methods, see Sotomayor et al. (2001b). This subspace model can be rewritten such that the measured disturbances d (Q.^, influent substrate concentration SSM and influent ammonium concentration Smjn) are considered as inputs and included in the input vector as:
-^it+i = ^ ^ + k ^ J -
0.9763 0.0199 03263 where, A = 0.0062 0.8818 0.0907 -0.0024 0.0072 0.9758 0.0368
B=h
(13)
, y^, = CXi,
C =
0.2259 -0.4026 -0.1810 , and 0.2664 0.2876 -0.4633
-0.0434 -0.1537 -0.0431 -0.0045 0.0234 0.0357 0.0283 -0.0044 -0.0100 -0.0091 -0.0003 0.0039
BA = -0.1505 0.0167
Aiming to incorporate feedforward/feedback action and integral error, these dynamics have be modeled and then be included in the state vector (13), as: ^k+l ^k^l ^k+\ ^K + \
A 0 -T.K^C 0
X"
B,
0
^a 0 0
0 0 \dk 0 + / T^K^ 0 0 \ 0 UK_
0
\^k
w
yk
[c 0 0 0
(14) ^k
557 where A^ and A^ are diagonal matrix such that J^^, = A^-^^ and
r^^i=A^r^,
respectively. K^ is a weighting matrix (diagonal positive definite), T^ is the sampling time and ^^ is the integral error vector of the form ^^^—^ = AT, (v^i ~''J • The augmented model (14) is used instead of (1) in the development of the MPC controller.
5. Simulation Results In order to test the performance of the MPC controller, two configurations are tested, both of them only for set-point changes (servo case). It is necessary to state that depending on the use or not of Q^^ as a manipulated variable, the dimensions of the matrices B^ and B^ change and, therefore, the objective function (12) and the constraints (10) are different for each control configuration. The tuning parameters of both configurations are not presented here. 5.1 3x2 system In this configuration, the influent flow is considered as a manipulated variable. In predenitrifying processes, the raw sewage is used as carbon source. Therefore, this configuraUon is applied to make good use of the influent COD concentration. In fig. 2 the responses of the process to set-point changes are shown. It can be observed that the system responds quite well. The effluent inorganic nitrogen concentration ('S'yvQ 3 + S^i^-x^) is maintained at low level. 30]
1^201 r. O
10'
0
^"^
£4 ,
10
-
^
20
X
20
30
A. ^
' 20
(a) Fig. 2. (a) controlled variables, (b) manipulated variables.
40
A
vV ^V
time(h)
5
W
30
(b)
5.2 2x2 system In this case, the influent flow is kept constant (i.e. previously controlled). Fig. 3 show the response of the process to set-point changes. The variables are well-controlled. Nevertheless, a higher control effort is required. Aiming to compare the performance of both configurations, here it is adopted the integrated squared error (ISE ), which is based on the response system, defined as:
558
Fig. 3. (a) controlled variables, (b) manipulated variables.
ISE = \{y,-rJdt
(15)
In table 1, it can be observed that the 2x2 system presents a better performance. Table 1. Numerical performance comparison for the two control configurations System 3x2 2x2
ISEi (SNO.I)
ISE3 (SNO.'*;
ISET = ISE, + ISE3
3.15 2.84
4.08 3.13
7.23 5.97
6. Conclusions In this paper, a MPC controller, based on a linear state-space model of the process, was implemented to improve N-removal capability of a predenitrication AS plant. The control was successful for set-point changes in both control configurations presented. In the 2x2 system, a higher control effort was required in the manipulated variables than in the 3x2 system. The 2x2 system presented a better performance in controlling the nitrate concentrations (see table 1), but the 3x2 system was better successful in controlling (inferentially) the effluent inorganic nitrogen concentration (see fig. 2). Acknowledgment: The authors gratefully thank the financial support from FAPESP under grant 98/12375-7. They are also grateful to Dr. David Di Ruscio from the Telemark Institute of Technology, Norway.
7. References Carlsson, B. and A. Renhstrom, 2001. Proc. ICA 2001, Sweden, 229-236. Di Ruscio, D. and B. Foss, 1998. Proc. DYCOPS 5, Greece, 304-309. Qin, J. and T. Badgwell, 1997, AlChE Symp. Series 93, 232-256. Sotomayor, O.A.Z., S.W. Park and C. Garcia, 2001a, Braz. J- Chem. Eng. 18, 81-101. Sotomayor, O.A.Z., S.W. Park and C. Garcia, 2001b, Contr. Eng. Prac, DYCOPS 6 special issue, (Submitted). Viberg,M., 1995, Automatica 31, 1835-1851. Yuan, Z., 1999, Proc. ECB 9, Belgium, 2217.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
5^9
Relational modeling of chemical processes for control logic verification Adam L. Turk^ Gary J. Powers^ Delft University of Technology The Netherlands Carnegie Mellon University United States of America
Abstract The area of formal methods offers improved fault detection for hybrid processes, such as chemical ones. Verification of a chemical process requires the construction of a finite state representation from the phenomena exhibited by the control system, physical process, operating procedures, and human behavior. The exhibited phenomena are modeled as a set of states, the transitions between these states, and their triggering events. In particular, the states and transitions are based upon landmarks and their relative position while the triggering events are based upon the actions in the system. This methodology capitalizes on the importance of relational properties instead of absolute ones in verifying a chemical process. The proposed methodology was applied to two industrial examples: a leak test procedure and a thermal oxidation process.
1. Introduction Detection of faults within control logic can lead to a more reliable process with respect to safety. Based upon current industrial practices, fault analysis of a modest sized process is difficult due to its large number of events or state space. Fortunately, improved fault detection for chemical processes may be found in the hybrid systems area where formal logic and mathematical methods are being developed for verifying sequential behavior of processes. In order to verify a chemical process using these techniques, a finite state representation of the control system, physical process, operating procedures, and human behavior must be built. This paper presents a method for synthesizing of a relational model that captures the relevant behavioral states that a process exhibits during operation.
2. Background Progress has been made in the area of verifying mechanical and chemical processes for safety. Several industrial processes, including an air traffic control system and an aircraft guidance system, have been verified using symbolic model checkers (Anderson et ai, 1996; Sreemani and Atlee, 1996). In these industrial examples, faults in the process were discovered and corrected, increasing reliability and safety.
560 Symbolic model checking has been used to verify the control logic of discrete control systems in chemical processes, including such complex systems such as a furnace standard and solids transport system (Moon, 1992; Probst, 1996). Formal methods, such as SMV, are used in the semiconductor and computer industries to uncover potential logic faults in microprocessors and software (Burch et aL, 1991; McMillan and Schv^albe, 1991; McMillan, 1993). One strength of SMV comes from its symbolic representation (binary decision diagrams) of the sequential and finite state space that describes the process. Another strength of this formal method is its efficient use of fixed point algorithms to search the state space for faults. These abilities allow SMV to completely verify large processes which can be typically on the order of 10^^ states or larger cite (Burch et ai, 1990; Clark et ai, 1994). With these strengths, SMV has shown that it may be a valuable tool for efficiently verifying the logic for any type of process. Verification with the symbolic model checker SMV requires the process to be represented as a finite state machine or non-timed discrete model. Chemical and mechanical processes are normally a combination of continuous and discrete systems. A key challenge to modeling these hybrid systems as a finite state machine is to represent the discrete domain with little modification while still capturing the dynamics and timedependent phenomena of the continuous domain.
3. Modeling In model checking, any process, such as a chemical one, must be represented as a finite state machine. A finite state machine is a set of states, triggering events, transitions, and initial conditions that characterize a process. The states within the representation describe the conditions of the process while the transitions identify the possible paths between these states. The events determine the time when transitions between states can occur. The key to an accurate and appropriate model is to capture the significant exhibited phenomena of the process with respect to the specifications as a set of states, transitions, and events. The state variables that describe the process, including the continuous ones, must be defined by a set or range of discrete values such as Boolean, integer, or enumerated type. The particular discrete values of the state variables are defined by landmarks in the physical chemical process, control system, operating procedures, and specifications. The landmarks from these different sources form the natural breaks or boundaries between displayed phenomena of the process. The values identified by these landmarks are relative to the state of the system and not an absolute numeric Figure I: Piping and valve diagram for value. The relative value of the variables a typical combustion system for the helps to reduce the size and complexity leak testing procedure of the state space while modeling the
561 appropriate phenomena of the physical process. The verification of a chemical process is concerned with the transition to critical or unsafe states rather than a change in the numeric value of a state variable. The landmarks determine the significant states that the discrete model needs to capture for verification. In addition to the states, the transitions between states and their triggers are also important for verification. These transitions and triggering events can be determined from observation, experimental data, theoretical mathematical equations, control logic, and operating procedures. These sources identify the possible sequence of states in a process and the actions that trigger the progression or movement along these paths.
4. Industrial Examples The synthesis procedure for a relational model was applied to two industrial examples: a leak test procedure and a thermal oxidation process. The modeling of these industrial examples demonstrates the procedure's ability to represent the physical system, control systems, operating procedures, operator behavior and their interactions. 4.1 Leak Test Procedure The leak test procedure is a structured approach for checking the seal across valves in a combustion system. In particular, the procedure tests for leaks across the valves located in a pipe network (fig. 1). The procedure pressurizes the pipe network by initiating the light up sequence of the combustion system which is followed by an emergency shutdown of the system. The pipe network is divided by the valves into eight pressure zones or pipe segments. The valves in each pipe section are tested by the presence of bubbling or no bubbling when a tap valve is opened. Once a leaking valve is detected, then the valve is repaired immediately. The procedure is begun anew after the repair of a leaking valve. The key operational issue of the procedure is the reliable testing and detection of a leaking valve. If a leaking valve is not properly diagnosed by the procedure then it can result in a fatal explosion. On the other hand, unnecessary replacement of a non-leaking valve wastes time and money. Both concerns are critical for the proper operation of the leak test procedure and form the basis of the specifications used to verify the procedure. The specifications also drive the construction of the finite state machine of the process. In particular, they highlight the need to include the steps of the procedure along with the operator's behavior in the finite state machine. The representation of the leak test procedure and the operator behavior as a set of states, transitions, and events was developed as follows. A state variable was created that moved though the steps in the procedure. At each given step, the necessary tasks were performed such as opening a tap valve to check bubbling. The bubbling at a given tap valve was linked to that pipe segment having pressure. The pressure in the pipe segments was defined as a Boolean state variable where zero described no pressure and one pressure. The pressure for a given segment was conditioned upon the opening of its associated valves and the pressure values beyond the opened valves. The pressure in a given pipe segment was also dependent on a separate Boolean variables that described the potential leaking of associated variables. This failure mode along with others were added to the finite state representation of the procedure and verified.
562 The leak test procedure was first verified by Probst with respect to the proper diagnosis of valves (Probst 1996). Building upon this work, we verified the procedure for the proper diagnosis of valves with faults generated by incorrect operator behavior and from the dynamics of a leaking valve. The computational results for the verification of several different leak testing models are presented in table 1 (Turk, 1999). The failure mode of the operator not testing or skipping a pipe section was added to the discrete model of the procedure. This failure mode shares characteristics with the dynamics of a slow leaking valve. Both faults lead to a positive result where one should not exist. Therefore, only one model is needed to see if either event will lead to a misdiagnosis. In addition to the representation with these faults, table 1 lists several other models that change the manner in which the position of the valves is determined. Originally, the position of the valves was determined by the step that the procedure was currently performing. This modeling assumption opened and closed valves even though the particular step associated with that action was not executed. The finite state machine was modified so that a particular step had to be performed in order for given valves to open and close. The values for the reachable states and transition relations in table 1 show the moderate complexity and small size of the discrete model, respectively. The result from the verification demonstrated that the leak test procedure would misdiagnose leaking valves if the operator skipped steps or the leak in a valve was too slow. Possible solutions for these failures might be a checklist for the operator and using pressure gauges instead of a bubble test. Table 1: Information on verified leak test procedure models by symbolic model checking (SMV) with dynamic variable ordering. Model Name
Boolean Variables
Base Case Segment Skip (1) Valves: Tap Valves: All Valves Macro-Pres Expand-Proc Segment Skip (2) 'Computations were performed on
Reachable States
24 5,944 24 14,611 30 14,611 38 14,611 32 14,611 32 14,552 32 5,153 15,984 32 a HP 812/70 workstation.
Transition Relations (OBDD Nodes) 11,495 14,205 28,102 95,100 54,064 37,632 44,688 49,601
CPU Time (sec)' 3 8 26 232 128 71 6 18
4.2 Thermal Oxidation Process A thermal oxidation process burns organic vapor in an effluent air stream from a metal casting pit before it is vented to the atmosphere (fig. 2). The casting pit is swept with air in order to remove any vaporized organic coolant before its concentration reaches the explosion limit and becomes a danger to operators. However, this air stream can not be released to the atmosphere due to environmental regulation. In order to vent the air stream to the atmosphere, the organic vapor is first burned off in a thermal oxidation unit. The specifications were derived from the operating procedures of the thermal oxidation unit. In particular, the specifications described the existence of a flame from the main burner and the stability of the temperature in the unit. The existence of a flame is
563 represented by a Boolean variable where the zero Exl^ust describes no flame present while the one indicates a flame. The transition Thermocoupl —o between these values is Qgaak dependent on the flow of fuel, flow of oxygen, and an Flame ignition source. The Detecto ^ temperature of the thermal Fan oxidation was divided into three discrete values based Natural m • ^ '^tj. _ Gas Make-Up upon the control landmarks Air Damper "^ "^ of high temperature at 1700° Combustion System F and low temperature at 1500° F. These two Figure 2: Diagram for the thermal oxidation setpoints from the control system create three discrete temperature region: below operating temperature, at operating temperature, and above operating temperature. In the finite state machine, these temperature regions are represented as an integer range from zero to two. The triggering events for the transition between these temperature states are the presence of the flame and the amount of energy generated by the flame. In turn, the energy generated is dependent on the flow rate of the effluent air, the flow rate of the natural gas, and the caloric value of the gas. The energy contribution from the organic vapor is assumed negligible. Several different models were built for the thermal oxidation unit. Each model represents the thermal oxidation process along with a failure mode that could potentially impact the stability of the system. The number of Boolean variables, reachable states, nodes in the transition relation, and computational time for these models are presented in table 2 (Turk, 1999). The value for reachable states indicates a simple system while the number of transition nodes shows its large model size with respect to the leak test procedure. Casting Pit Air Damper
d=>z
1 ^
Table 2: Information on verified thermal oxidation process models by symbolic model checking (SMV) with conjunctive partitioning Model Name
Boolean Variables
Reachable States
Base Case 66 23 Fail Pit Damper 66 57 Fail Gas Pres. (High) 66 59 Fail Over Temp. 66 69 Fail Under Temp. 66 58 Drift Temp. 66 34 Fail High Temp. 66 67 Fail Flame 66 81 Fail Flame Detect 66 82 ""Computations were performed on a Sun Ultra 5 workstation.
Transition Relations (OBDD Nodes) 865,280 1,342,126 1,376,861 1,558,914 1,869,591 1,477,692 1,838,378 1,682,027 1,499,612
CPU Time (sec)' 28 57 112 72 112 49 162 185 181
564 The verification results from the different failure modes showed that the thermal oxidation system would appropriately extinguish the flame when it could not recover from the instability. However, the thermal oxidation model could recover from several of the failure modes but only if the gain of the disturbance was small. The only fault that the system failed to correct was the drifting of the temperature given by the thermocouple. The drifting of the temperature would cause the system to act incorrectly. This fault can be fixed by adding a secondary thermocouple and associated logic against which the temperature of the thermal oxidation unit can be checked. This discrete model of the thermal oxidation unit proved the safeness of the thermal oxidation process with respect to its specifications.
5. Concluding Remarks The modeling methodology was used to represent two complex chemical processes. The significant exhibited phenomena of these systems with respect to the specifications were captured as a set of events, transition, and triggering events. This representation capitalized on the importance of relational properties instead of the absolute values in describing the chemical process. These discrete models were verified using SMV against a given set of specifications. The results from these industrial examples showed that each system had potential failure paths. The behavior of the operator in the leak test procedure could lead to misdiagnosed valves while the thermocouple in the second example would not function properly due to the drifting of the temperature value. Overall, the modeling methodology demonstrated its ability to create an accurate and appropriate representation that can be used to verify the safeness of a complex chemical process.
6. References Anderson, R.J., P. Beame, S. Bums, W. Chan, F. Modugno, D. Notkin, and J. Reese, 1996, Proc. of the Fourth ACM SIGSOFT Symp. on the Found, of Software Eng., Assoc, of Comp. Mach. Burch, J.R., E.M. Clarke, and K.L. McMillan, 1990, In the Proc. of the iT^ ACM/IEEE Design Automation Conf Burch, J.R., E.M. Clarke, and D.E. Long, 1991, In the Proc. of the 28'*' ACM/IEEE Design Automation Conf Clarke, E.M., O. Grumberg, and D.E. Long, 1994, ACM Trans, on Prog. Lang. 16(5). McMillan, K.L. and J. Schwalbe, 1991, In the Proceedings of the 1991 International Symp. on Shared Memory Multiprocessors. McMillan, K.L., 1993, Symbolic Model Checking, Kluwer. Moon, I., 1992, Autmatic Verification of Discrete Chemical Process Control System, Carnegie Mellon University. Probst, S.T., 1996, Chemical Process Safety and Operability Analysis Using Symbolic Model Checking, Carnegie Mellon University. Sreemani, T. and J. Atlee, 1996 Technical Report CS96-05, Department of Computer Science, University of Waterloo. Turk, A.L., 1999, Event Modeling and Verification of Chemical Processes Using Symbolic Model Checking, Carnegie Mellon University.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
565
Development of Dynamic Models for Fixed Bed Catalytic Reactors E. C. Vasco de Toledo, E. R. de Morais and R. Maciel Filho Faculty of Chemical Engineering, State University of Campinas (UNICAMP) CP 6066 - CEP 13081-970 - Campinas, SP - Brazil. email: [email protected] - FAX +55-1937883910
Abstract Heterogeneous and pseudo-homogeneous dynamic models, for fixed bed catalytic reactors are presented in this work. They allow to consider variations in the physical properties of the fluid and in the heat and mass transfer coefficients, as well as the heat exchange through the jacket of the reactor. The models permit a study on the dynamic behaviour of the system including the predictions of the inverse response phenomena. This work also allows to understand the model differences and their prediction capabilities.
1. Introduction The catalytic chemical reactor exhibit complex dynamic behaviour resulting from the non-linear distributed features which, among other things, give rise to inverse response resulting in catastrophic instabilities such as temperature runaway. The non-linearities are a consequence of heat generation by chemical reaction, and the inverse response arises from the presence of different heat capacities of the fluid and solid as well as the bulk flow of fluid causing interactions between heat and mass transfer phases. This causes differential rates of propagation of heat and mass transfer which influence the heat generation through reaction on the solid catalyst, (McGreavy and Maciel, 1989). Reliable models depend on the insight of how the dominant physic-chemical mechanisms and external factors which affect the overall performance. However, when on-line applications are required, simplified models have to be used which can keep the essential characteristics of the system. In this work, models for fixed bed catalytic reactors, based on heterogeneous and pseudo-homogeneous approach were developed. The proposed heterogeneous dynamic models for fixed bed catalytic reactors consist on mass and heat balance equations for the catalyst particles as well as for the gas phases, include the resistances to mass and heat transfer at the gas-solid interface and consider the resistances inside the catalyst particle. The heterogeneous dynamic models are used in applications where computational accuracy may be more emphasized than computational speed, for instance, reactor design, planning of startups, shutdowns and emergency procedure, (Martinez et al., 1985, Pellegrini et al., 1989). For real time implementation, as control and optimisation on-line, it is required to have the ability to overcome computational burden with a faster and easy numerical solution when compared to rigorous heterogeneous models. Bearing this in mind, reduced heterogeneous and pseudo-homogeneous models were developed. The reduced models
566 was done through mathematical order reduction, which eliminates the spatial coordinate of the catalyst particle and promote radially lumped-differential formulations. The pseudo-homogeneous model were developed based on the approach which incorporates the thermal capacity of the fluid and solid, (pCp)f and (pCp)s, respectively, (Vasco de Toledo, 1999).
2. Reduction Techniques The solution of diffusion/reaction multidimensional problems present difficulties associated with a large analytic involvement and also request considerable computational effort. Thus it is convenient to propose simpler formulations for the original system of partial differential equations, through the reduction of the number of its independent variables. Therefore, one or more independent variables can be integrated, leading to approximate formulations that retain detailed local information in the remaining variable as well as medium information in the directions eliminated by the integration. This information comes from the boundary conditions related to the eliminated directions. Two different reduction approaches, to know Hermite and Classic, generating differentiates lumped formulations were investigated. These techniques generate models that describe the axial profiles as a function of the time for the convenient explicit elimination of the dependence in the radial variable in the case of the fixed bed catalytic reactor. 2.1 Hermite Reduction Hermite developed a way to approach an integral based on the values of the integrating, y, and their derivatives, y, on the limits of the integration. The technique makes use of the approach Hii and simultaneously of the theorem of the mean value for spherical coordinates, leading the generation of the radial medium variables, (Correa and Cotta, 1996).
[]m = 3j;r^[]dr
^^^
where [ Jm defines a mean radial value of the amount inside of the left bracket. H,. = J V(x)dx = i-[y(0) + y(l)]-f-L[y (0) - yd)]
(2)
2.2 Classic Reduction The technique makes uses of the theorem of the mean value, given by equation (1) to generate the reduced models.
3. Dynamic Models The models of the reactor were generated under the following considerations: variation of physical properties, mass and heat transfer coefficients, along the reactor length; the plug flow profile of velocity; intraparticle gradient negligible (pseudo-homogeneous and heterogenou II models); mass transfer resistance between the gas and the catalyst
567 surface is neglected (pseudo-homogeneous model); axial dispersion was neglected. In this equations the following notations are used: Fluid, F; Solid, S; Mass Balance, MB; Energy Balance, EB; Coolant Fluid Equation, CFE; Continuity Equation, CE; Momentun Equation, ME. 3.1 Heterogeneous Model I - Rigorous Model
^ ax at
R;
r dv
G
ax
e
L
2 ax dr
D 1 a R ' Tp' dr
at
k a p„
eG C
pC 8 - = -^ r—^ ' " at R r dr dr
ax
ax
p^ L az
dr
'
'
ai ^ + h^a^P3(T;-T) dz
PMp^R^,(X^,T)
R ^ Tp' dr
-(pvj=0 ai, ^
at
u,
(FEB)
(SMB)
P.
p^(-AHJR,(X^,T) ^^ " at
(FMB)
(R + 1)
(SEB)
(CE) d\
2U
(4)
(5)
(6)
(7) (T(l,z,t)-TJ
L az • R. p, c^
|£ =- ^ l i ^ f az p D g
(3)
(CFE)
(8)
(ME)
(9)
with the following boundary conditions:
ax
ai
ar
ar
' = ^ " T ^ = T ^ = ^'
r =l
r =1
ax
ax
^p=^ ^
ax
=^
ar ar
ax
— ^ = 0,—^ = Bih(X,-X(l,z,t)) dr ar
3X.
_
k R , ax h, R 8 {X-X^]^ = !1I^{T-T) D ^ ^ ^^ar X ^'
=^
(symmetry)
for all z
'^
z = 0 X^ = X^ = 0,X = X„,X = X^,,X, = X,,, p = p,
(10)
(11)
forallz
(12)
for all r
(13)
Xhe Equations 7, 8, 9, 10, 11 and 13 are valid also for the remaining models. Xhe following notations is used, a^ is the heat transfer area for catalyst (mVkg catalyst); Bih, number of Biot; Cp, calorific capacity (kcal/kg.K); D, radial effective diffusivity (m/h); Dp, particle diameter (m); f, friction factor; G, mass flow velocity (kg/m .h); gc conversion factor; hf , heat transfer coefficient particle to fluid (kcal/m .h.K); hw,
568 convective heat transfer coefficient in the vicinity of the wall (kcal/m^.h.K); kg, mass transfer coefficient particle to fluid (m/s); L, length of the reactor (m); p, pressure of the reactor (atm); PM, the mean molecular weight (kg/kmol); r, dimensionless radial distance of the reactor; rp, dimensionless radial distance of the reactor; R, air/ethanol ratio; Rt, reactor radius (m); Rp, particle radius (m); Rw, rate of the oxidation (kmol of reactant mixture/h.kgcat.); T, reactor temperature (K); Tfo, feed temperature (K); Tgo, feed temperature (K); Tso, catalyst feed temperature (K); T(l,z,t); wall temperature of the reagent fluid (K); Tro, coolant feed temperature (K); t, time (h); u,velocity (m/h); U, global heat transfer coefficient (kcal/m^.h.K); V, velocity (m/h); X, conversion; z,dimensionless axial distance. A, conductivity (kcal/m.h.K); AHR, enthalpy of reaction molar (kcal/kmol); p, density (kg/m^); pe, catalyst density (kgcat/m^); ps, catalyst density (kgcat/m^); E, porosity; Subscripts: ef, effective; f, fluid; g, gas; i, interstitial; o, feed; R, refrigerant; s, solid; Superscripts: s, condition at external surface. 3.2 Heterogeneous Model II - Reduced Models
at
ii.
"rar
ax^
~ar
p C e—- = -^ ' •* at R;rar
G. ax^ P, L dz
+ Bihm-!^^-^^(X^-XJ
eG C a i L
dz
+ Bihth,a,„p„(T-T)
at
a-e)pS.f
(FMB)
(14)
(FEB)
(15)
(16)
p^
(SEB)
= -l.,.„^pJT-THBM^h^^
Classic Reduction HermiteReduction
Bihm = l 4 Bihm = 4 + KR /D t
P/
s
(17)
Biht = 1 4 Biht = 4 + h,R,A.
(18)
3.2 Pseudo-Homogeneous Model
^ = 2^11 at R; r ar ar
G. a x ^ a - e ) P M p . ^ ^ ^ ^ ^ ^ ^ ^ p, L az ep^
(8(pC ) + (l-e)(pC ) } ^ = A ^ i A
.9T
ar
(MB)
(19)
s G i C ^ a T ^ (l-£)p.(-AHJR,(X.T) L az (R+1) ^^^> (EB)
As a case study, the catalytic oxidation of tiie ethanol to acetaldehyde over Fe-Mo, Rw, catalyst was considered, Vasco de Toledo (1999). It is a strongly exothermic reaction, representative of a important class of industrial processes.
569
(21) Where P02, PET» PH2O. PAC are partial pressure of oxygen, ethanol, water and acetaldehyde respectively, and the Kj are the kinetic constants in the Arrhenius form. It is worthwhile mentioning that the rigorous heterogeneous model takes into account all the possibles resistances that may be important which are neglected in simpler mathematical models. Such models may be useful for different applications, depending upon the need of the prediction accuracy and required computer time.
4. Results and Discussions 446-j
St*p Pwturtoation of 5% In Tfo at Z s 0.13 m
445-1
— Hataroganaoua Modal I -- H«t«rog«n«ous Mo<M I - Het«rog«neou* Mo<M II • Clastic Raduction - Hataroganeoua Mo<M II • Harmlta Raduction - PaauckHHomoganaous Modal
— Hataroganaoua Modal II - Clasaic Raduction — Hataroganaoua Modal II • Harmlta Raduction — Paaudo-Homoganaoua Modal
Figure 1 - Reactor temperature response Figure 2 - Reactor temperature response for step perturbation in T^. for step perturbation in 7}^. The numeric solution of the models was obtained using the method of the lines in conjunction with the orthogonal collocation which showed to be an effective procedure for the space discretization (radial and axial directions) in conjunction with the LSODAR algorithm for the integration in time. For the numerical solution are used 5 collocation points for the radial direction (fluid and particle) and 7 points for the axial direction. Figures 1 and 2 show the dynamic behaviour of the heterogeneous and pseudo-homogeneous models. In these figures the inverse response phenomena is observed in the profile of the reactor temperature due to a step perturbation in the reactant feed temperature, 1^. This phenomenon is typical in fixed bed catalytic reactors. In Figure 3 is represented the temperature dynamic behaviour of the reactor due to a step perturbation in the mass flow velocity, G. It can be seen in these figures, that the dynamic behaviour of the heterogeneous and pseudo-homogeneous models is qualitatively similar, but quantitatively different. It is worthwhile mentioning that the reduced models require about half of the computer time to be solved when compared to the rigorous heterogeneous model. About the same relation is obtained when reduced models and pseudo-homogeneous model are compared, which indicates the potential of reduced models for real time implementations. These three figures show that the use of a certain model type (heterogeneous or pseudo homogeneous) to represent the reactor dynamic behaviour may lead to different reactor predictions. The temperature dynamic
570 behaviour, Figure 4, observed for a step perturbation in coolant feed temperature, TRO, allows a better understanding of the reactor behaviour along the reactor length, including the hot spot. The used model was the Heterogeneous Model I. Heterogeneous Model I Step Perturbation of 5% in Tro
- Heterogeneous Model I - Heterogeneous Model II - Classic Reduction - Heterogeneous Model II - Hermite Reduction - Pseudo-Homogeneous Model Step PerturtMitlon of - 70% in Q at Z - 0.13m
Figure 3 - Reactor temperature response Figure 4 - Reactor temperature response for step perturbation in G. for step perturbation in Tj^o.
5. Conclusions The proposed models, specifically the heterogeneous, have shown to be able to predict the main characteristics of the dynamic behaviour of fixed bed catalytic reactors, including the inverse response phenomena. This knowledge is essential to design and control of the reactor. However, the computational time demanded for the solution of the Heterogeneous Model I is high in comparison to the simplified models, making possible the application of this models for applications where computational time is not limited. However, when on-line applications are required simplified models have to be used. The models based on reduction techniques and pseudo-homogeneous approach overcome computational burden with a faster and easy numerical solution, as well as other difficulties found in rigorous heterogeneous models, especially related to the large number of parameters and sophisticated numerical procedures required to the solution.
6. References Correa, E. J. and Cotta, R. M., 1994, Improved Lumped-Differential Formulations of Transient Heat Conduction Problems, presented at III Congress of Mechanical Engineering on North-Northeast, PA, Brazil. Martinez, O. M., Pereira Duarte, S. J., and Lemcoff N., O., 1985, Modeling of Fixed Bed Catalytic Reactors, Comput. Chem. Eng., 9, 5, 535-545. McGreavy, C. and Maciel Filho, R., 1989, Dynamic Behaviour of Fixed Bed Catalytic Reactors, presented at IFAC Dynamics and Control of Chemical Reactors, Maastricht, The Netherlands. Pellegrini, L., Biardi, G. and Ranzi, E., 1989, Dynamic Model of Packed-Bed Tubular Reactors, Computers Chem. Eng., 13, 4/5, 511-518. Vasco de Toledo, E. C. V., 1999, Modelling, Simulation and Control of the Fixed Bed Catalytic Reactors, Ph.D. Thesis, University of Campinas, Sao Paulo, Brazil.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
571
Improving the Control of an Industrial Sugar Crystalliser: a Dynamic Optimisation Approach T. T. L. Vu and P. A. Schneider James Cook University, School of Engineering Townsville Queensland 4811 Australia
Abstract The controllability of an industrial sugar crystalliser is investigated. Using a simultaneous integration and optimisation approach, the optimal control problem is formulated in GAMS, based on a dynamic model of the vacuum pan validated against plant data. MINOS 2.50 solves this open-loop model, yielding: the minimum batch time, set-point trajectories and the optimal switching from high to low purity feed. These results are implemented within double-loop PI controllers. Alternative control variables are proposed, replacing traditional process outputs, which are not fundamentally process-relevant. Dynamic simulation results show that the proposed control schemes are satisfactory in the face of errors in growth rate correlations and variations in feed properties and initial batch conditions. Preliminary full-scale investigations have shown promise and will be extended into next year's sugar cane crushing season.
!• Introduction The number 6 fed-batch evaporative crystalliser, also known as a vacuum pan, located at CSR's Macknade Sugar Mill (North Queensland, Australia) operates through two distinct phases, composed of many sub-steps. The existing conductivity-based control scheme, developed several decades ago, dictates the feed policy to this unit operation. Increasing production capacity, while maintaining process flexibility and quality, is of paramount importance to CSR and so they made this unit available for testing purposes. In the first phase of the batch Vu and Schneider (2001) propose two control schemes, using state-based control variables to dictate the feed and steam rates. Our investigation into the second phase of the batch is presented in this paper. The second phase of the batch is composed of three sub-steps and is more complex than the first phase, since this phase includes a switch from one feed material to another. Previous authors (Frew, 1973 and Chew and Goh, 1989) concentrated on solving the optimal control problem, using an open-loop model. Their solutions could not handle uncertainties in initial conditions, variations in feed properties and plant-model mismatch. Their proposed optimal control solutions were never implemented. This paper first briefly describes the process at hand. A brief overview of the solution technique is then given. The keynote is the comparison between two different control schemes to determine the feed and evaporation rate policies of the fed-batch process. Finally closed-loop responses within vacuum pan dynamic simulations and preliminary implementation data from the factory are presented for discussion.
572
2. Process Description A schematic of a vacuum pan is presented below in Figure 1, showing the steam and various feed inlets. The starting and full levels of the vessel are indicated.
Figure 1: Schematic of a Vacuum Pan The second phase of the crystallisation process includes three main sub-steps. 1. Foundation: steam heating is resumed with approximately 50 tonnes of footing material, known as massecuite (M), which is composed of sugar crystals {x^) suspended in mother molasses {Mol). Molasses {Mol) is a solution comprising dissolved sucrose (JCJ), dissolved non-sucrose impurities (^2) and water (.^3). Liquor, a high purity feed, is simultaneously introduced to enrich the mother molasses, depleted due to sugar crystallisation. 2. Boilback: liquor feed is switched to A-molasses, a lower purity feed material, until the pan reaches its full volume. 3. Heavy-up: the pan is full. Feeding is stopped, but steam continues, leading to exhaustion of the mother molasses. After this step the massecuite is centrifuged, in order to recover the product sugar crystal. Rigorously determining the schedules for feed and steam rates, as well as the feed switching point, requires a more sophisticated solution method than is presently available to industrialists. The essential steps leading to batch vacuum pan optimisation are described next.
3. Methods The following sections present the problem formulation, including discussion on the constraints acting upon this system, followed by a brief description of the implementation.
573 3.1 Problem formulation The process dynamic model of the vacuum pan can be fully described by mass, energy and population balances. However, the dynamics of the crystal size distribution are simplified, while the energy balance is written in the simplest form possible. In other words the evaporation rate (E) is assumed to be proportional to the steam rate to the vessel. The mass balances of water, impurities and sucrose in the pan are straightforward. Readers should consult Vu and Schneider (2000, 2001) for more details. Three main constraints involve the crystal content (CQ, the solution oversaturation (OS) and the target purity of the mother liquor at the end of Heavy-up. The crystal content is defined as the mass fraction of crystal in the vessel, according to CC = xjM
<0.55
(1)
The solution sucrose oversaturation OS is the driving force for crystallisation. The higher the OS, the faster the crystal population will grow. However, the OS has an upper limit, termed the critical oversaturation or OScht- Beyond this point, nucleation of new unwanted crystals occurs, resulting in downstream processing inefficiencies. This critical level has been previously defined by (Broadfoot and Wright, 1972). At all times the fractional oversaturation, known as FOS, should be kept below unity. (2)
F0S{=0S/0S,,,)<1
The last constraint determines the switching point from fresh liquor to A-molasses feed in order to achieve the desired final molasses purity, or Pty {= x^ /xj + ^2), defined as mass ratio of sucrose to total dissolved solids in the molasses. Pty < 0.75
at Heavy - up
(3)
The upper limits of massecuite, crystal content and purity also serve as termination conditions for the second phase. In order words, when M reaches 100 tonnes and Pty drops to 0.75, feeding is stopped, but steam continues until the crystal content reaches 0.55. Challenges facing the batch pan problem include uncertainties in feed properties and initial conditions, such as purity, Brix{= jcj + X2 +X4/M), CC, footing seed diameter, etc. Due to upstream process fluctuations, the ranges of these variations can be large at the beginning of the batch. However, during the second phase, these variations become less important. After Boilback, the crystal content increases to a high level, reducing the natural circulation in the vessel and, consequently, the rate of heat transfer. This loss of circulation and the subsequent reduction in heat transfer are not presendy modelled, which must be taken into account for factory implementations. The optimal control problem formulated for the second phase is based on the nominal conditions. The objective function, OF, presented in (4) contains two different targets: minimum batch operating time and optimum trajectories of M, CC or FOS set points and, importandy, the switching point from the Foundation to Boilback sub-steps.
574 i=nfe
i=nfe
(4)
i=nfe
+
WfY[FOS{i)-FOS,Ji)]'
This type of dynamic optimisation problem can be solved using a simultaneous integration and optimisation technique, which approximates the state variables by interpolation polynomials (Vu and Schneider, 2000, 2001). These are differentiated and then back-substituted into the state equations, converting them to a set of algebraic equations. The scaling factors w^, w^ or Wf in the above equation force M, CC or FOS to follow the desired optimum trajectories of M,^, CCsp or FOS^p. The time variable in (4) is actually replaced by the summation of the element lengths. It is important to note that the number of finite elements (nfe) during Foundation, Boilback and Heavy-up are specified, but their lengths are decision variables in the optimisation problem. Thus the optimal switching points from one sub-step to another can be determined. 3.2 Implementation Instead of using feed and steam flow profiles, set-point trajectories will be implemented within two PI controllers. Two different control schemes are proposed. Both schemes select the mass-controller as a primary loop because mass can be directly and accurately measured and it is one of the ending conditions stated above. Another advantage is that the mass set-point trajectory will not be affected by the presence of errors in the growth rate expressions. This dominant loop controls the mass in the pan by adjusting the feed rate. First Scheme: M-F/CC-Steam - The second loop controls CC by adjusting steam. CC cannot direcdy be measured, but can be estimated using either a state estimation scheme or by combining other process outputs. Since CC is the second ending condition, this loop is simpler to be implemented. The greatest disadvantage of the first scheme is that the FOS remains uncontrolled. This might result in constraint (2) violation due to variations in feed properties and initial conditions. The gains and reset times of these two controllers therefore must be tuned against a worst-case scenario, requiring an iterative dual-level optimisation, as discussed in Vu and Schneider (2001). Second Scheme: M-F/FOS-Steam - To avoid these tuning problems, the second loop could control FOS by adjusting the steam flow rate. The gains should be set at the highest values, without leading to oscillations in the controlled or manipulated variables. Therefore they must be tuned against the fastest situation and tested with other cases. At high values of CC, FOS control is safer and more robust but this loop pairing also has some disadvantages. First, the secondary loop in this case must include a logical operation to stop the batch, using the ending condition of CC. Second, FOS is derived from many other correlations, which might bring in a large error in FOS determinations. Dynamic simulations obtained from two control schemes implementation results are compared in the following section.
and
preliminary
575
4. Results The scheme M-Feed/FO^-Steam was selected as the option to test under factory conditions. Figure 2 shows FOS, CC and M profiles in the pan. Solid curves representing factory implementation are plotted against broken lines for simulation. A significant deviation between these two curves especially at the end of the batch is due to heat transfer limitations on evaporation rate from the pan. It should be noted that the mass variable, M, was indirectly controlled, but it nonetheless follows the path prescribed by the optimisation results. This factory trial was necessarily run at a low FOS set point. Future experiments at the factory will determine how much higher this set point can be, before the system suffers. Dynamic simulation results using both control scheme options applied to the second phase of operation are shown in Figure 3: CC-Steam in the left and F(95-Steam on the right, showing the controlled (M, CC, FOS) and manipulated variables (Feed and Steam). In terms of batch time reductions, the FOS-Sieam loop should be used over that of the CC-Steam, since it maintains a higher level of sucrose oversaturation throughout the batch, and therefore sustains crystal growth at a higher rate.
5. Conclusions The controllability of an industrial sugar crystalliser has been investigated through the second phase of operation in a similar fashion used in the first phase investigation. New controlled variables: M, CC and FOS can replace the traditional output-based conductivity control. A double-loop control strategy is also proposed to replace the existing single loop controller. Based on the preliminary implementation, results from the factory indicate that the loop pairing of M-Feed/F05-Steam functions well in the second batch phase. Future work will evaluate alternative implementations of both proposed control schemes across both phases of this fed-batch crystallisation. Further more the pan model will be augmented with some forms of correlations for the overall heat transfer coefficient across the heating section.
6. References Broadfoot, R. and P.G. Wright, 1972, Nucleation Studies, Proceedings of The Australian Society of Sugar Cane Technologist 39* Conference, 353. Chew, E. P. and Goh, C. J., 1989, On Minimum Time Optimal Control of Batch Crystallization of Sugar, Chem. Eng. Comm. 80, 225. Frew, J. A., 1973, Optimal Control of Batch Sugar Crystallization, Ind. Eng. Chem. Process Des. Develop 12, 460. Vu, T.T.L. and P. A. Schneider, 2000, Operability Analysis of an Industrial Crystalliser, CHEMECA Proceedings 2000. Vu, T.T.L. and P.A. Schneider, 2001, Controllability Analysis of an Industrial Crystalliser, 6* World Congress Proceedings 2001.
7. Acknowledgements Both authors would like to acknowledge CSR Ltd for their strong and enthusiastic support of this project. Dr Vu would like to acknowledge the SRDC for funding under their CP2002 project program.
576
30
40 Time(min)
50
Figure 2: FOS, CC and M Profiles During Phase One of the Batch (M-F/FOS-Steam) (Solid line: implementation; broken line: simulation) M-F/FOS-Steam
M-F/CC-Steam
100
Feed
/ f
I
601 / 0 Iy
1 ^°
_
/
ol^
. Massecuite Foun^tion to Boilback
•
steam .*'''
20
^
i
0
20
40
0
20
40
Boilback to Heavy-up
60
^nd
80
100
120
60 80 Time(min)
100
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Figure 3: Dynamic Responses Using Different Control Schemes in Phase Two of the Batch.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
577
A Scheme for Whole Temperature Profile Control in Distributed Parameter Systems T Wahl, WE Jones and JA Wilson School of Chemical, Environmental & Mining Engineering University of Nottingham, UK
Abstract A control strategy is proposed for distributed process systems in which the whole measured temperature profile through the system is held as close as possible to a desired steady state profile defined externally via some global optimisation (not considered here). To do this the concept of a minimum error profile (MEP) is introduced which represents the closest feasible approach to the desired profile. The MEP then becomes the set-point or target for regulation. The implementation involves both state and parameter estimation, via extended Kalman filtering, and dynamic optimal control via LQR. Performance of the approach is illustrated with an example based upon a cascade of heated stirred tanks.
1. Introduction Distributed process plant items are often instrumented with an array of measurement sensors. For example an array of temperature probes is often installed to measure bed temperatures along a fixed bed catalytic reactor or tray temperatures along a distillation column. However, control in many such situations is based upon use of only one or two sensors selected from the complete set, the remaining sensors being used to provide monitoring information for the operator. Here we propose a strategy for control of the whole measured profile with the objective of holding it (i.e. the measured profile) as close as possible to a desired profile (defined externally to meet some global optimum). Temperature profile control is mentioned by Edgar et al. (2001) but no more detailed explanation or specific mathematical solution is proposed. In the present work the minimum error temperature profile (MEP) is introduced. This combines Edgar et al.'s (2001) objective with standard mathematical derivations as performed in Sage (1968). The MEP can be contained in a complete control scheme linking steady state optimization, system identification and dynamic optimal control. The supervisory-level global optimization of operating conditions is performed intermittently via general nonlinear programming. The results of this calculation are introduced via the MEP calculation which iterates much faster at the sub-level and provides a real-time shortcut approach to the optimized solution. Identification, e.g. via extended Kalman filter (EKE), is performed to obtain estimates of system state and parameter changes. The controller, e.g. a linear quadratic regulator (LQR), provides control action to be undertaken. Calculations performed on the sub-level are executed every sample instant and results from the global optimization are input when they are available.
578 Disturbance compensation is performed via changes in MEP where controller error is its difference from the actual profile.
2. Control Scheme An extended Kalman filter (EKF), driven with on-line measurements of the temperature profile, provides state and parameter estimates as input to the MEP control strategy. These estimates also feed the supervisory global steady state optimization, which delivers, on a slow intermittant timescale, the desired steady state temperature profile, Ax^^ expressed here as a vector of temperature perturbations around a fixed steady operating point. The MEP control objective is to hold the system as close as possible to the desired profile in a squared error sense by minimizing
e = (Ax-Ax,„^f(Ax-Ax,„^)
(})
where Ax is the corresponding perturbation in system steady state temperature profile. The dynamic behaviour of the temperature profile can be represented, as for example by Ackermann (1985), in discrete time as
Ax(k+1) = A^ Ax(k) + B^ Au^^^ (k) + B^ Au(k) + Ab(k) + Aw(k)
(2)
where Ax(k) is the actual current profile, A is the discrete state transition matrix, B^ is its associated driving matrix, AU is the control action component needed to achieve a desired steady state profile while ^u ^ (k) is the additional transient action contributed by the LQR to drive optimally to the steady state. Unknown process disturbances are represented by Aw(k), a set of white noise inputs, and Ab(k), a set of biases representing unknown deterministic inputs. Values for Ab(k) come as parameter estimates from the EKF. At any time instant k the steady state temperature profile predicted using this model will be
Ax = f,Au(k) + f,(k)
(3)
where f ^ = ( I - A J ~ ^ B ^ and f^(k) = (I - A^)"^ Ab(k). To minimize e the required control action Au is given as
Au = Au^,, = (f T f,) f,^ (f^ (k) - Ax,„^)
(4)
which will take the system to steady state at the minimum error temperature profile Ax^^p- Au^ ^00 is then the current, transient action superimposed by the LQR in order to drive the current error Ax(k)-Ax^^^ optimally towards zero.
579
3. A Case Study Example To demonstrate the performance and application of the approach we present a simple distributed process case study involving a single liquid stream flowing through a long cascade of mixed stages each with an unknown, independently varying heat F,To_ Tr ^2„ input, as shown in Figure 1. The result is formation of a temperature profile along Q2 Qi the cascade. Both the flow rate and Figure 1: A simple distributed system temperature of the feed stream can be with a temperature profile - a long manipulated to control the profile against cascade of heated vessels with unknown disturbances in the heat inputs. heat inputs. Normal operating conditions are given as: Inlet temperature To=30°C; Flow rate F=5000kg/h; Mass of liquid per vessel Q mi=3000kg; Number of simulated vessels n=7; Specific heat capacity c=4kJ/(kg K). The overall heat input is chosen to be lO^kJ/h, raising the temperature by 50°C. -| \ 1 1 1 r The heat is introduced as bell curve along 50 100 150 200 250 300 350 400 450 the vessels with its centre at the middle Time, min vessel. The resulting temperature profile (a) is sigmoidal. Two test disturbances are considered to demonstrate the system response. First, Q the temperature in the middle vessel is perturbed by 10°C, see the noise-free response in Figure 2(a). It can be seen that the effect of the temperature perturbation n 1 \ 1 r is flushed downstream through the system 100 150 200 250 300 350 400 450 until all temperatures return to their initial Time, min steady state. (b) Second, the heat input, AQ, is increased vessel e*** vessel in the middle vessel by 30% of its normal vessel T"* vessel value, see Figure 2(b). This leads to a steady state perturbation from the 4^ Figure 2: Open loop responses to vessel on. disturbances in middle vessel temperature (a) and heat input (b).
4. Estimator, MEP and Controller Tuning for Case Study Both process and measurement noise are included when considering closed loop control. Process noise. Aw, implemented as in Equation 2, and measurement noise, Av, have standard deviations of 0.5°C in temperature and 50kg/h in flowrate.
580 Heat inputs AQ are estimated as parameters in an augmented filter state (i.e. the bias term in Equation 2 is handled as Ab = C^ AQ)- An error covariance of (5xlO~'*Q)^ I is assigned. It should be noted that this covariance matrix can also be viewed as a tuning parameter. If the factor SxlO"* is increased then parameter estimates will converge faster. The drawback is that due to noise the estimates will fluctuate more. Under closed loop control this leads to stronger variations in in manipulated variables. Similar effects attach to assigning uncertainty to the initial guess of system state and paramteres ^x • However, these assignments have no great influence to the observers performance since representive presentation are taken when error covariances converged to steady values. The LQR's tuning parameters are chosen to be Q^ = io'I and R ^ = I , its objective function weighting matrices, Q^ for the state and R^ for the manipulated variables. When operating on its own the LQR's state is taken to be Ax^ = Ax . Here it is assumed to utilize the steady state gain which is to be re-calculated each sampling interval.
5. Steady State Profile Control To demonstrate the response of the MEP to an extreme disturbance, the peak heat input is shifted from the middle to the first vessel in the cascade. The overall heat input is kept constant so that the same system outlet Open LQR MEP + Initial temperature is reached, see the steady state Loop LQR 0 511 164 2230 ^MRP open loop response in Figure 3. Following 30 12 30 1 To F 5000 5000 4252 2360 this disturbance the LQR alone and, in a second simulation, the MEP plus LQR are o applied for system control. The smaller final steady state deviations from the 3 a u desired initial profile under MEP+LQR S control are apparent, with a squared error advantage of 68% over the LQR in the table. The MEP-based profile crosses the desired profile twice instead of once, as Vessel with the purely LQR-based strategy. Final Figure 3: Steady state profiles reached steady state values for manipulated under various control strategies after a variables are also given in the table in large disturbance in heat input profile. consistent units.
6. Dynamic Profile Control Temperature Disturbance: The system response to the 10°C disturbance, as in Section 3, is presented in Figure 4(a). A moving target temperature profile, Ax^^.^, can be observed. This is due to the delay and inaccuracy of parameter estimation. Due to noise and the disturbance injection the EKF produces varying heat input estimates. This leads to re-calculation of a new target temperature profile which the controller tries to follow. This is predominantly seen in vessels three and four where a shift in target comes back into it's initial range of noise induced changes. The influence of noise is even more evident in the manipulated variable movements in Figures 4(b) and 4(c). The target for
581 32.0
o 31.5
80757U65-
I 31.0
" " • —
/ '
\
Feed Temperature To MEP Target
g 30.5 a 30.0 •
V
I 29.5-
• ^
60-
T^^—r-
"" 29.0-
I
50
55^
100
200
150
50-
Time, min
45-
(b)
40oouu -
35in
1
50
1
100
1
150
1
200
Time, min
6000-
-
i 1
i i
i
5500-
'—1
5000l'^vessel 2'"^vessel S^'^vessel i^^vessel
vessel 6* vessel 7^^vessel MEP Target
40UU -|
Flow Rate F MEP Target
L . _ •:
i
rj
ZTJ !:^
L._j
1
1
50
100
1
1
200
150
Time, min
(a)
(c)
Figure 4: Closed loop response to temperature disturbance in middle vessel by lOK. Presented are vessel temperature and their MEP targets (a), inlet temperature (b) and feed flow rate (c). the flow rate returns much earlier to its initial range than the inlet temperature does. However, the deviation is also more distinct than for the feed temperature case. In both subfigures it can be seen that their actual values approach their targets until they would meet in no-noise steady state. Heat Input Disturbance: Response to disturbance in middle vessel heat input, as in Section 3, is presented in Figure 5(a). Due to the sustained change in heat input the target state does not go back to it's initial range. Again, most distinctive changes in target state can be observed in vessels three and four. Transient changes of state can be most clearly seen in vessels four and five. It takes the control scheme approximately 150min to bring the system to it's new target. The arithmetic averages of the MEP are given
by
avrg(Ax^^^)|'^'2^ =[0.48°C
-0.09°C
-1.18°C
1.22°C 0.17°C
-0.26°C
-0.34°cr
and avrg(Au^,,)|;^'^'2 =[0-68°C 510kg / hf • Observing changes of inlet temperature in Figure 5(b) shows a distinct shift towards the approximate target given by 30.68°C. A similar pattern occurs for the feed flow rate as presented in Figure 5(c). It increases towards 5570kg/h.
582 85 80 75 70 65 3
60
200
55 S
50 45 40 35 30 50
100
I
150
200
Time, min
.
6500
Ss
6000 H
B
5500 5000
•- I'^vessel - 2"'^we5»e/ •• 3^'^ves8el -•
6^^vessel
4500 200
l^^veasel
MEP Target
4*''t;e»se/
(a)
Time, min
(c)
Figure 5: Closed loop response to heat input disturbance in middle vessel by 30%. Presented are vessel temperature and their MEP targets (a), inlet temperature (b) and feed flow rate (c).
7. Conclusions The concept of the minimum error profile (MEP) is presented for control of whole temperature profile in distributed parameter systems faced with parameter drift. The approach minimizes in a least squares sense the error from a predefined desired profile and forms the target for application of the linear quadratic regulater (LQR). Application of the proposed contol scheme to a simple example, a cascade of heated stirred tanks, has demonstrated the feasibility and performance relative to a conventional LQR implementation. The approach has the potential for quite direct application to control of fixed-bed catalytic reactors and counter-current mass transfer processes.
8. References Ackermann, J. (1985), Sampled data control systems: analysis and synthesis, robust system design. Communication and Control Engineering Series, Springer Press Edgar, T.F., Himmelblau, D.M. and Lasdon, L.S. (2001), Optimization of chemical processes. Chemical Engineering Series, second edn, McGraw-Hill Sage, A.P. (1968), Optimum systems control, first edn, Prentice-Hall
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
583
Dynamic Plantwide Modelling, Flowsheet Simulation and Nonlinear Analysis of an Industrial Production Plant R. Waschler^*, A.Kienle\ A. Anoprienko^ T. Osipova^ Max-Planck-Institute of Dynamics of Complex Technical Systems Sandtorstr. 1, 39106 Magdeburg ^Donetsk State Technical University, Artemstr. 58, 340000 Donetsk, Ukraine ^AZOT, Sewerodonetsk, Ukraine
Abstract A detailed dynamic model of the Monsanto process for the production of acetic acid is presented. The nonlinear behaviour of the process is investigated and implications different basic control structures have on its dynamic behaviour are pointed out.
1. Introduction Plantwide control issues have recently been attracting increasing interest within the process control community, as can be seen by the publication of several review articles during the last couple of years (see, e.g., Skogestad and Larsson, 1998, and references therein). While promising great potential for rendering processes more economic, it is widely recognized that the control of highly integrated plants must be regarded as a most challenging task. It therefore seems appropriate first to gain insight into the behaviour of complex plants and also the associated control problems by considering suitable case studies. However, apart from the classic Tennessee Eastman benchmark problem (Downs and Vogel, 1993), and the case studies provided in Luyben etal. (1999), only few examples of industrial scale are found in the literature, a fact that is most likely to be attributed to the enormous efforts that come along with rigorously modelling an entire production plant. The purpose of this contribution is to present such an industrial example, the so-called Monsanto process for the production of acetic acid. It captures features that are believed to be typical of many chemical plants and therefore of common interest beyond this particular system. As our point of view is absolutely consistent with Luyben's, who stated that "the primary mathematical tool in the solution of the plantwide control problem is a rigorous, nonlinear model of the entire plant" (Luyben etal. 1999), we regard the model presented in this article as a sound basis for our forthcoming investigations on plantwide control. Emphasis in this contribution, however, is on the model itself, the numerical analysis of the nonlinear behaviour of the process and the impact basic control structures have on its dynamic performance.
* Corresponding author, phone: ++49-391-6110-375; e-mail: [email protected]
584
2. The Process Nowadays the most common and most economic way of producing acetic acid on an industrial scale is according to the Monsanto process, labeled by leading experts in the field as "one of the triumphs" of modern homogeneous catalysis (Maitlis etal 1996). Fig. 1 presents a simplified flowsheet that captures the main ingredients of the real plant. Peripheral equipment is left out for reasons of clarity. Inerts Makeup
:l
v.y
.-r'^
:-<.:
v^
\
/ MeOH CO
n
rl-_
L_ _ ^
Catalysi
V
~Tl^
Fig. I: Simplified flowsheet of the Monsanto process The liquid educt methanol and the gaseous educt carbon monooxide are fed to an adiabatic continuous reactor where the exothermic carbonylation reaction to acetic acid according to
CH.OH + CO -^ CH.COOH,
AH^= -138,6 kJ/mol
(1)
is carried out in homogeneous aqueous liquid phase at boiling point conditions of about 30 bar and 185 C. The reaction is homogeneously catalyzed by means of a rhodium (or iridium) carbonyl catalyst (denoted as Rh) which in turn has to be activated by a promotor methyl iodide (Mel). The heat of reaction is removed by evaporative cooling, with mainly the excess CO leaving the process. The liquid reaction product enters a flashdrum F operated at about 1.5 bar. This separator serves for the recovery of the extremely expensive catalyst, which - as essentially being nonvolatile - is recycled to the reactor along with a huge portion of the reactor effluent, while the catalyst-free vapour product is fed to a first distillation column CI. The main purpose of this column is the recovery of promotor Mel, accomplished by means of a liquid-liquid phase split in a decanter atop the column. While Mel and a water-rich stream are recycled to the reactor, the main product stream is withdrawn from an intermediate tray and fed to a second column C2 where the remaining inert component water is separated and recycled. Almost pure acetic acid is obtained as bottoms product of column C2.
3. The Model A detailed dynamic model of the entire plant has been implemented in the simulation environment DIVA (Mangold etal. 2000). Here, we restrict ourselves to the key characteristics of the system that both distinguish it from other case studies and which are the origin of its most intriguing traits.
585 3.1 Reactor At the heart of our overall model we have the reactor model which is of hybrid nature as it has to allow for switching between boiling and non-boiling conditions as consequence of a disturbance or during startup. Note that at standard operating conditions the temperature inside the reactor is the boiling point. For a detailed calculation of the vapour-liquid equilibrium we apply the UNIQUAC method to obtain activity coefficients of the liquid phase and the Redlich-Kwong correlation for the prediction of vapour phase fugacities. One special characteristic of our system originates from CO being supercritical at the given standard operating conditions. To approximately describe its solubility we follow Prausnitz etal. (1986) and apply the concept of ideal CO solubility in the reaction mixture. Dynamic component material balances and a liquid phase summation condition give the mole fractions of the five components we consider, i.e., methanol, CO, acetic acid, methyl iodide and water. The catalyst XRH is assumed to be a system parameter. We neglect side reactions and restrict ourselves to the brutto reaction (1). It has been intensely studied (e.g. Maitlis etal. 1996) and the underlying complex reaction mechanism giving rise to (1) seems fairly well understood. In particular, the reaction was distinguished as being of approximately first order in the catalyst and the promotor and zeroth order in the educts as long as the standing concentration of these reactants is sufficiently large. Yet, for small concentrations - and as there is almost 100% conversion of methanol this applies to the given process - the reaction order will shift to one in the limiting component, a fact we account for by introducing Michaelis-Mentenlike dependencies for the reactants in the rate law, yielding
r = 0.4986 • g-^^^°'^-^' • c^f • c'f, • ^^^"^ ' ""^^"^ Rh
Mel
\ I v
^co'^^co
1_i_Jt^
^2)
Exact values for the constants KMCOH and Kco are not known, however this is no problem as we could show that the sensitivity with respect to the Kj is small as long as they are given physically reasonable values. For our simulation studies we chose KMCOH = Kco =100. 3.2 Separation system and basic control structure The flashdrum is operated adiabatically, and for all units of the separation system we use detailed dynamic vapour-liquid equilibrium models including component material and energy balances. We assume that basic inventory control is guaranteed for all units and is incorporated in the models of all single units in terms of constant holdups and constant pressures. 3.3 Steady state simulation results Let us mention briefly that agreement between steady state simulation results of our entire model and actual measurements from a real plant - located in Sewerodonetsk, Ukraine, with a capacity of about 150000 t/a - is very satisfactory, given the fact that our model is purely predictive.
586
4. Nonlinear Behaviour We will now briefly investigate some nonlinear features of the process and try to emphasize the implications they have on possible control problems, always bearing in mind that our eventual goal is directed towards plant wide control. 4.1 Isolated Reactor As the key to understanding the overall plant dynamics is a thorough understanding of the reactor dynamics we will set out looking at the stand-alone reactor. Suppose all feeds and recycles to the reactor are fixed and we vary the amount of catalyst inside the reactor. Computationally, this can be done using the method for continuation of steady states in DIVA (Mangold etal. 2000). Fig. 2 illustrates that over a certain range of the parameter XRI, (abscissa) -j three steady states can occur. It can be shown I that this type of behaviour is a general feature ] of any two-phase reactor where one of the i reactants is by far more volatile than the other -j components (Waschler etal. 2001). In such "" cases the temperature will decrease with 1 increasing reactant concentration, leading to a • decrease of reaction rate. This can be I interpreted as a (nonisothermal) self-inhibition \ of the reaction. It is a well known fact that any J kind of self-inhibition is a potential source of o' multiplicity and instability. In the Monsanto process, CO (lower subplot) represents this by Fig. 2: Multiple steady states far most volatile component. Fig. 2 depicts the direct and inverse correlation between dissolved CO and reactor temperature (upper subplot). Note that these considerations are extremely important from an operating point of view as there is the problem of catalyst deactivation that could lead to a sudden extinction of the reaction once a lower threshold for catalyst concentration is crossed. Another typical example exhibiting this type of characteristics is the ethylene glycol reactive distillation system (Gehrke and Marquardt, 1997). 4.2 Reactor-separator system Let us now approach the realm of plantwide operation by closing the first recycle loop from the flashdrum back to the reactor. It has been widely recognized that control structure selection is the decisive task related to the plantwide control problem and this becomes evident already for the simple reactor-separator system (see also Pushpavanam and Kienle, 2001). Here, we are particularly interested to see how two different control structures cope with a disturbance as it may be encountered during the operation of a plant. Let the setup where reactor holdup, or liquid level, respectively, is controlled by manipulating the reactor effluent be denoted by control structure 1 (CSl, see Fig. 3). In
587 control structure 2 (CS2) the reactor effluent (CS1) itself is flow-controlled and one recycle flowrate from a downstream separation unit is manipulated to control the reactor level. Again assume all feeds and recycles to be fixed, except for the flash recycle, of course, and the manipulated recycle in CS2. Then assume that after one hour of standard operation there occurs a 20 K drop in methanol feed temperature TF,Me0H (subplot Upper left in Fig. 4) lasting for two hours, e.g. due to disturbances in the utility system. As can be seen, the effect of this disturbance on CSl (solid lines in Fig. 4) is drastic: the drop in energy supplied to the reactor decreases the reactor temperature TR (upper right), which in turn causes the reaction rate to decrease. Thus, less CO is consumed, resulting in an increase in Xco (lower right) and an additional Fig.3: Different control structures decrease in temperature because of the mechanism described above. Due to the material and energy integration with the flash, this effect is further amplified: the cold flash recycle increases as soon as the flash inlet temperature TR drops. This larger and colder recycle decreases the reactor temperature even further and simultaneously increases the reactor effluent LR (lower left), and is therefore identified as the origin of the accumulation of flowrates in the coupled system. In other words, we observe some sort of nonlinear snowball effect, similar to the one described by Luyben etal. (1999). To avoid the obvious shortcomings of CSl we follow Luyben's heuristic guidelines in terms of control structure selection and apply CS2 (dashed-dotted lines in Fig. 4), which turns out to be far superior in rejecting this disturbance. Fixing reactor effluent to some degree decouples the two subsystems. In particular, the flash will only be subject to a slight disturbance in composition and temperature, as opposed to the large flowrate disturbance encountered for CSl. As a consequence, the state of the flash will vary to a much smaller extent and the amplification of the self-inhibitory reactor 0.0095 I f characteristics is prevented. Thus, in CS2, the reactor can almost settle to a new steady state even within two Fig. 4: Disturbance in feed temperature hours after the disturbance.
588 4.3 Overall plant As can be expected simulation of the entire plant shows many of the features commonly attributed to highly integrated processes with several recycles. Most noticeably, we observe an increase in the system's time constants and inverse response behaviour to load disturbances. In contrast to other reported case studies, where it is crucial to purge out inerts and avoid accumulation of both inerts and reactants, it turns out that for the Monsanto process keeping track of the inert water and promotor methyl iodide, i.e., controlling their material balances using makeup streams, is a key requirement in order to run the complete process in a stable way.
5. Conclusions We have presented a dynamic nonlinear model of an industrial production plant and highlighted the nonlinear behaviour of the underlying process. Basic concepts for stable and robust operation at the nominal operating point were discussed. Future work will also focus on the design of supervisory control strategies for startup and the efficient performance of load changes, two problems representing big challenges in terms of plantwide control. Obviously, the efficiency of all new strategies will eventually have to be compared to well-established control techniques as, e.g., model predictive control. Acknowledgement: The authors gratefully acknowledge the financial support of the International Bureau of the German Ministry of Education and Research (BMBF) through grant UKR 00/004, supporting scientific technological cooperation. References Downs, J.J., and E.F. Vogel, 1993, A Plant-Wide Industrial Process Control Problem, Computers Chem. Engng. 17 (3), 245-255. Gehrke, V., and W. Marquardt, 1997, A singularity approach to the study of reactive distillation, Comp. Chem. Eng. (21, Suppl.), 1001-1006. Luyben, W.L., B.D. Tyreus and M.L. Luyben, 1999, Plantwide Process Control. McGraw-Hill, New York. Pushpavanam, S., and A. Kienle, 2001, Nonlinear behavior of an ideal reactor separator network with mass recycle, Chem. Eng. Science (56), 2836-2849. Maitlis, P.M., A. Haynes, G.J. Sunley, and M.J. Howard, 1996, Methanol carbonylation revisited: thirty years on, J. Chem. Soc, Dalton Trans. 2187-2196. Mangold, M., A. Kienle, and K.D. Mohl, 2000, Nonlinear computation using DIVA Methods and applications, Chem. Eng. Science (55), 441-454. Prausnitz, J.M., R.N. Lichtenthaler, and E.G. de Azevedo, 1986, Molecular thermodynamics of fluid-phase equilibria, Prentice-Hall, Englewood Cliffs, N.J. Skogestad, S., and T. Larsson, 1998, A review of plantwide control, Internal Report, Dept. of Chem. Eng., Norwegian Univ. of Science and Technology, Trondheim. Waschler, R., S. Pushpavanam, and A. Kienle, 2001, Multiplicity features of two-phase reactors, in preparation.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
589
Flatness-based optimization of batch processes M.E. van Wissen, S.Palanki* and J. Grievink, Delft University of Technology, Department of Chemical Technology, Julianalaan 136, 2628 BL Delft, The Netherlands. * Florida A&M University- Florida State University, Department of Chemical Engineering, Tallahassee,FL,32310-6046, USA.
Abstract A flatness optimization framework is proposed, to deal with optimization of batch processes under uncertainty. Via the concept of differential flatness we transform the problem, such that the terminal-cost optimization problem can be solved in a cascade optimization scheme. The results have been tested on an example of a batch bioreactor.
1. Introduction A wide variety of specialty chemicals are made in batch reactors. In batch process operations, the variables change considerably with time and thus there is no constant setpoint around which the process can be regulated. Because there is no steady state, the objective is to optimize some objective function, which expresses the system performance. The optimal operating policy for a given batch process is usually calculated under the assumption of a perfect model. However, realistic applications are subject to uncertainty in initial conditions, model mismatch, and process disturbances, all of which affect the optimal solution. This provides the economic drive for on-line calculation and implementation of the optimal operating policy. Normally, the optimization of batch processes leads to a piece wise, discontinuous solution. A cascade optimization scheme is proposed by Visser et al. (2000) to implement such an optimal trajectory despite disturbances and uncertainty. It is very often the case, that the optimal solution of such a problem lies on the input bounds, or state constraints, or a combination of both. In the proposed method, we make explicitly use of this information. Also uncertainty is treated in the framework of cascade optimization. The classical approach is via the Hamilton-Jacobi-Bellman (HJB) formulation. In batch reactor modeling, a lot of systems have been found, which have the so-called flatness property. For this class of systems, we propose a new theory which does not rely on the HJB-formulation. Via the concept of differential flatness, which is basically a method to dispose of the differential equations, we have developed an approach for which flat systems can be treated in the cascade optimization scheme. Moreover, the feedback controller, needed in the cascade optimization is now easy to obtain. The proposed scheme is illustrated on a simulation of an optimization problem in batch bioreactors.
590
2. Optimization Problem Problem formulation without uncertainty Batch optimization problems typically involve both dynamic and static constraints and fall under the class of dynamic optimization problems. It is true, that in most batch chemical processes, the inputs are flow-rates that enter the system equations in an affme manner (Srinivasan et al. (1997), Srinivasan et al. (2000), Visser et al. (2000)), but quite a few processes have been found for which this is not the case (Rouchon and Rudolph(2000)). The terminal-cost optimization of general dynamical systems can be stated as follows: P:
minJ(x(tf),u(tf)) u(t)
(la)
subject to x(t) = f(x(t),u(t)),tG[to,tf), 0 = x(to)-Xo.
(lb)
0 < c ( x ( t ) , u ( t ) ) , t G [ t o , t , ) , 0
(Ic)
where jc is the n-vector of state variables with known initial conditions XQ, U the mvector of control variables. We exploit the degree of freedom in u(t) to minimize an (economical) objective function P subject to path and endpoint constraints in x(t) and u(t). The right-hand side of the nonlinear dynamical system is defined as the n-vector/ The constraints to be enforced during the process and at the final time tf are given by c and Cp respectively. For a more detailed description of this problem, see Kumar and Daoutidis(1995). The classical solution of (1) can be found by solving the well-known Hamilton-JacobiBellman (HJB) formulation, which is easier to solve when the system is in controlaffine form, (Palanki et al. (1991), Visser et al. (2000)). Problem formulation with uncertainty For many realistic applications, we can assume that the model structure is known, but the model parameters are unknown or only known within bounds. For this case, the terminal-cost optimization of general dynamical systems is as follows: P:
minJ(x(t,,e),u(t,,e)) u(t)
(2a)
s.t. x(t) = f (x(t),u(t),e) + d(t), 0 = x(to) - X,
(2b)
0
(2c)
in which 6 is the /?-vector 0 of uncertain parameters and d is the function representing the unknown disturbances. Here, we choose to have probabilistic parametric uncertainty for 6 , in which the objective function P is the expected value of a random variable, i.e.
591 P: minE[J(x(tf,0),u(tf,0))].
(^^
u(t)
In realistic cases, one can think of the expected product quality or quantity to be maximized, or the expected loss to be minimized. The resulting optimization problem can be found in Srinivasan et al. (2000b), p. 14.
3. Differential flatness Differential flatness has been introduced by Fliess et al. (1995) in their studies of the feedback linearization problem in the context of differential algebra. A system is flat if we can find a set of outputs (equal in number to the number of inputs) such that all states and inputs can be determined from these outputs without integration. More to the point, if the system has states x (n-vector), and inputs u (m-vector), then the system is flat if we can find outputs y (m-vector) of the form y = y(x,u,u,...,u''*),
(4)
such that, x = x(y,y,...,y^''^),
(5a)
u = u(x,u,ti,...,u^''^).
(5b)
The outputs y are called/Zar (or linearizing) outputs. Differentially flat systems are useful in situations where explicit trajectory tracking generation is required. The reason for this, is that the behaviour of flat systems is determined by the flat outputs, and hence we can plan trajectories in output space, and then map these appropriate inputs. This property can be quite useful, when dealing with only a few flat outputs in comparison with the number of states and the number of inputs. For optimization problems the main advantage lies in the reduction of the computational effort, as there is no need to numerically solve the sensitivity differential equations in the nonlinear program (NLP). For the dynamic optimization problem P, we replace the equations (1) (or (2)) with the expressions containing all the flat outputs and higher derivatives. Another useful property of flat systems that we can use, called endogenous dynamic feedback (Fliess et al. (1995)) which is a dynamic feedback of the form (v is the mvector of new inputs) z = a(x,z,v),
(6a)
u = b(x,z,v),
(6b)
592 such that z satisfying equation (6a) can be expressed as a function of x and u and a finite number of their derivatives: z = a(x,u,...,u'),
(7a)
The endogenous dynamic feedback can be used in the cascade optimization scheme as proposed by Visser et al. (2000), which combines notions of feedback control with notions from the field of optimization under uncertainty, and hence can be seen as a practical implementation strategy for problems containing uncertainty, see Visser et al. (1997). For a detailed description of trajectory planning of differentially flat systems and applications, see Faiz et al. (2001) and Veeraklaew and Agrawal (2001).
4. Cascade optimization scheme In the cascade optimization framework proposed by Visser et al. (2000), the optimization problem is transformed into a tracking problem by use of so-called invariant signals obtained via the HJB-approach. In short, in the cascade optimization a 'low level' tracking controller is used, which guarantees that the system stays close to the optimal trajectory. After that, 'a high level' optimizer is used to guarantee optimality despite disturbances. We use the same framework, but instead of Visser et al. (2000), obtain the tracking controller for free, using the differential flatness approach.
5. Results Illustrating example The proposed methodology will be applied to a nonlinear model of fermentation of whey lactose to lactic acid by Lactobacillum bulgaricus in a batch bioreactor (Agrawal et al., (1989)). The mass balances are given by:
x, =^i(x)Xi - u x i / x ^
(8a)
X2 =u(Sf-X2)/x4-|a(x)Xj/Yx/s X3 =(a^i(x) + P)Xi -UX3/X4
(^^) (8c)
x,=u
(8d) K^^+x^+x^'/K. '
0
(8f)
^93 Table 1. Initial states and parameters Symbol State/parameters Biomass concentration Xl Substrate concentration X2 Product concentration X3 Volume X4 Substrate feed cone. Sf Substrate saturation const. Km Substrate inhibition const. Ki Product inhibition const. Cell mass yield Yx/s a Growth assoc. prod, yield Non-growth assoc.pr. yield P Max. spec, growth rate Final time
_tf
Value lg/1 Og/1 0.5 g/1 21 15 g/1 1.2 g/1 22 g/1 50 g/1 0.4 g/g 2.2 g/g 0.2 h-^ 0.48 h*^ 11.6h
The states in the model (8) are related by the following equation: (9) x,(x, + Y^,{x, -S,)) = C = x°,(xj + Y^3(x^ -S,)) In Mahadevan et al. (2001) the flat output for this problem has been derived, using (9). This has been used in the optimization problem of maximization the amount of product formed at the end of the batch, such that equations (8a)-(8e) are satisfied. In Figure 1 the trajectories and inputs of the optimal solution are plotted. Note that the input does not change too much.
Conclusions We proposed an optimization approach of general batch processes using the concept of differential flatness. Especially, in the cascade optimization scheme, the construction of the feedback controller is easy, once the flat outputs are obtained. In comparison with classical methods (e.g., the JB-formulation), the proposed method could also lead to reductions in computing time on-line, because of the transformation to a NLP. A disadvantage of the proposed method is that, beforehand, one does not know if a system is differentially flat. Moreover, even if one knows that a given system is flat, the flat ouputs are not always easy to obtain and might involve long algebraic manipulations. However, for a large group of reactor models for instance, it has been shown, see Rouchon and Rudolph (2000), that these are differentially flat and the flat outputs have been calculated. It is interesting to see if for all these models, the terminalcost optimization problem under parametric uncertainty can be dealt with using differential flatness. Another subject under investigation is the robustness of the method against parametric uncertainty.
594 state 5
^^^-^
4 3
x4
2
x3^^
X
^.,^-^'5(2
1 0
Xl
()
2
4
6 t Input
8
10
1
-
0.14 0.12
-
0.1 008
1
_j,^_
, _ _ _^
—
_
1
10 Optimal solution for batchbio :
12
objective: 11
Figure I. The states and inputs of the optimal solution of the batch bio reactor.
References Agrawal,P.,G.Koshy and M.Ramseier, 1989, An algorithm for operating a fed-batch fermenter at optimum specific growth rate, Biotechn. Bioengg., 33, 115. Faiz, N.,S.K. Agrawal and R.M. Murray, 2001, Trajectory planning of differentially flat systems with dynamics and inequalities, J. of Guid., Cont. Dyn., 24(2),219. Fliess, M., J. Levine, P. Martin and P. Rouchon, 1995, Flatness and defect of non-linear systems: introductory theory and examples. Int. J. Cont. 61(6), 1327. Kumar, A. and P. Daoutidis, 1995, Feedback control of nonlinear differential-algebraicequation systems, AIChE J. 41(3), 619. Mahadevan, R., S.K. Agrawal and F.J. Doyle (2001), Differential flatness based nonlinear predictive control of fed batch bioreactors, Cont. Engg. Pract 9, 889 Palanki, S., C.Kravaris and H.Y. Wang, 1991, Synthesis of state feedback laws for endpoint optimization in batch processes, Chem. Eng. Sc. 48(1), 135. Rouchon, P. and J. Rudolph, 2000, Reacteurs chimiques differentiellement plats: planification et suivi de trajectories, to be published. Srinivasan, B., E. Visser and D. Bonvin, 1997, Optimization-based control with imposed feedback structures, IF AC ADCHEM'97, 635. Srinivasan, B., S. Palanki and D. Bonvin, 2000a, A tutorial on the optimization of batch processes: I. Characterization of the optimal solution, to be published. Srinivasan, B., S. Palanki and D. Bonvin, 2000b, A tutorial on the optimization of batch processes: II. Handling uncertainty using measurements, to be published. Veeraklaew, T. and S.K. Agrawal, 2001, New computional framework for trajectory optimization of higher-order dynamic systems, J. of Guid., Cont. Dyn., 24(2),228. Visser, E., B. Srinivasan, S. Palanki and D. Bonvin, 2000, A feedback-based implementation scheme for batch process optimization, J. Proc. Cont. 10(5), 399.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
595
Modeling and optimization of a sugar plant using hybrid Batch Model Predictive Control Michiel E. van Wissen, Joost F.C. Smeets*, Ad MuUer** and Peter J.T. Verheijen Delft University of Technology, Department of Chemical Engineering Julianalaan 136, 2628 BL Delft, The Netherlands. *TNO-TPD, Department of Control Engineering PO Box 155, 2600 AD Delft, The Netherlands **COSUN Food Technology Centre Oostelijke Havendijk 15, 4704 RA Roosendaal, The Netherlands
Abstract This paper presents the modeling and optimization of a sugar plant regarding heat management. The optimization has been done using Batch Model Predictive Control, which takes into account the interactions between the batch and continuous units of the plant. On the basis of the results, the development of an advice system to assist operators in the plant has been started.
1. Introduction There are two reasons why model-based optimization is of high importance in the sugar industry. Using model-based optimization, one is able to predict a regular sugar quality and one can handle the large amount of energy involved. Uncertainty reduction and process restrictions motivate the choice of Model Predictive Control (MPC), a control technique that calculates a sequence of control signals in such a way that it minimizes a cost function over a prediction horizon. In this work, the interactions between both batch and continuous units play a role, hence an adapted version of MPC is proposed, the so-called hybrid Batch-MPC. This is to distinguish from Lee et al. (1999), who used the term Batch-MPC. An existing mass-balance model of a beet sugar factory has been extended with energy balances, recycles and variable cycle times for batch units. It is a hybrid model, consisting of batch and continuous vacuum pans and intermediate storage buffers. The product flows and volumes are calculated in the mass components: water, sugar, nonsugar and crystal. Relevant pan-program stages such as filling, seeding, boiling-up and emptying have been included. For the extended model, control variables and process restrictions have been identified. The model has also been validated with data from last year's campaign. The model was then extended with hybrid Batch-MPC in order to optimize the energy management system. The objective was to reduce the fluctuations in the energy system which consists of the evaporators and steamsupply for the vacuum pans. All of the preceding is going to be used in an operator-support system, which helps the operator in making decisions, i.e. the operator-support system is able to predict the
596 consequence of an action taken by an operator (the system can deal with a what-if scenario).
2. Modeling The sugar house and evaporation section of Suiker Unie in Groningen consists of the following units (see Figure 1). In the A, B and C-pans (also called the vacuum pans) the crystallization takes place, which are being fed by the S-1, S-A, S-B and S-C pans (also called the seed-station), in which the seed crystals used in the vacuum pans are grown. The purified juice stream passes through the evaporators (in which the solids content is typically increased from 15%-18% (thin juice) to 68%-74% (thick juice)), and subsequently through the vacuum pans. The thick juice is mixed with remelted B-and C-sugar to obtain so-called Standard Liquor which is processed in the A-pans (batch process) to end up with a solids content of typically 92-96 %, which after centrifugation leads to white sugar and A-syrup. The syrup is processed in the continuous B-pan to yield B-sugar and B-syrup which in turn is processed in the continuous C-pan and separated into C-sugar and molasses (C-syrup). Molasses, with a solids content of 80%85%, is used in cattle-feed and in the production of alcohol. The B-and C-sugar are remelted in thin juice and returned to the A-pans. Between the different pans, there are intermediate storage buffers, in order not to overflow a level in a certain pan. The B- and C-pans are mainly used to dispose of pollution in the juice stream. Most of the crystallization is done in the A-pans, and this happens in a batchwise manner, to have a better control of the crystal size and high supersaturations. The A- and S-pans are operated in a batchwise manner, which consists of the following steps: filling, concentrating, seeding, stabilising, boiling up, emptying steam cleaning and ready for the next strike. It should be added that the processing time of the A- and S-pans is variable, which is generally the case for batch operations (Nott and Lee, 1999). The function of the evaporation section is twofold: -It thickens the juice stream. -It is used as a heat-generator for the sugarhouse and other parts of the sugar factory. The process is modeled in Matlab/Simulink on basis of mass-balances, as in Bubnik et al. (1995). The product flows and volumes are calculated in mass components: water, sugar, non-sugar and crystal. This leads to a model with approximately 25-30 variables (e.g. levels, flows, program counters and dry substance content (brix)). Heat management In the original model, only mass-balances were taken into account. We introduced vapour streams between the different S-pans, A-pans and B- and C-pan, and with this we modeled the heat demand.
597
S-1 pan
S-C pan
S-B pan i
i
t S-buffef* Evaporators Thin juice
buffer
buffer 4^^
buffer
t
Mixed juice
_itt
V S-buffer
S-buffer
S-buffer
Bpan
buffer
Molasses Cpan
buffer
buffer
A-sugar
Thick juice Syrup/sugar
"^ Seed crystals
••> Heat demand
Figure 1. Units and streams of a sugarhouse and evaporation-section. Note that the heat demands of the B-, C- and S-pans are not indicated in the Figure (Bubnik et al. 1995)). Urbaniec (1989) and Chung (2000) present a detailed description of how to model heat streams in a sugar plant.
3. Validation For the model-validation we used three measurement-series (approximately with the length of one day, one week, and one month). With these data, we performed modelvalidation in two ways: 1. Model validation via mass balances. For the modeling we used the 'one-week' measurement-data. We used checks on mass balances to verify that the 'one-day' measurement data were correct within experimental error, but not so good, when we compared it to the 'one-month' measurement-data. The main reason for this were shutdowns during that period, and it is unsure if there were losses during these shut downs 2. Model validation via measurement data. For the three available measurements we compared the data with the model output. In order to this correctly one can compare the model results with the measurement data, just by plotting them in one plot. By doing so, one gets a fast overview if there are any discrepancies between the measurements and the model. First, we compared the datasets, and if necessary, we used a second order filter to obtain an 'average' value of a certain variable. After that, we used the obtained data set for the validation.
598 A wide array of validation methods is available in literature, and most of them depend on the characteristics of the model and its intended use (Rao et al., 1998). One of the measures, that establishes that the simulation response and the data response match with a certain ^desired' tolerance (Law and Kelton, 1991) is given by yn
(Yi-Zj)^ 2
(1)
in which y is the data response, z is the simulation response, n is the number of variables, ai,i=I...n, is the desired tolerance for the i-th variable. The value of (1) should be close to 1 if the desired tolerance is chosen correctly. For our data we used a in the range of 5% to 15% depending on the estimation of the actual error.
4. Model analysis After analysing the model, we came to the conclusion, that the optimization and control problem for the sugar house is not trivial because of the amount of interactions within the process: -The interaction between the heat demand of the sugarhouse and the evaporation section is reflected in the dry substance content (brix) of the thick juice. -Dependent on the shapes of the heat-demand-brix curves for the evaporation section and sugarhouse, the development of the thick-juice brix is intrinsically stable or unstable. An illustration of this is given in Figure 2. Assume we have a constant available heat in the sugarhouse and evaporation-section for varying values of the brix. Suppose at a given time in the process, there is a certain heat demand to the sugar and a certain brix, represented by point A. To this point corresponds a point B, which represents the amount of heat available to the evaporation section. Depending on the equilibrium curve of heat vs. brix in the evaporation-section, the brix of the thick juice will tend towards point C or point D. This tendency repeats until the amount of heat available to the evaporation section matches the amount needed to realise the actual thick juice brix (for example point E using the heat vs. brix-curve 1, leading to a stable situation). In practice, variations in brix can be controlled by adjusting the amount of of heat to the evaporation section, the brix of thin juice to the evaporation section, or the heat demand to the sugarhouse. Hence, one of the optimization variables should be the brix of the thin juice. - The behaviour of the process in the B- and C-pans is to a great extent dependent on the operational frequency of the A-and S-pans. - Brix and flow of the thin juice determine the operational frequency of the A-pans and S-pans. This is important for the optimization, because the brix and flow of the thin juice are a handle to schedule the A-pans and S-pans.
599 Available heat Heat-demand Equilibriumcurves
To evaporators C
B
D
H
To sugarhouse
Brix Figure 2. Brix vs. heat-demand - A change in operation-frequencies of the A-pans leads to variations in brix of the mixed juice through. This also needs to be taken into account when developing an optimization scheme. -The control of the flows to the B- and C-pans is not easy, because of the supply of these buffers to the S-B and S-C pans (see Figure 1). For an operator this is difficult to control manually, because every control-action taken might be good within 1-2 hours, but can lead to unforeseen actions later (say, after 8-10 hours). This is because of the long delay times in the buffers.
5. Proposed optimization using hybrid Batch MPC The widespread use of Model Predictive Control (MPC) in chemical industries is because it has several nice features (see Garcia et al. (1989), Morari and Lee (1999)). MPC can be used for handling multivariable and/or constrained problems and its concepts are easy to understand for operators with only a limited knowledge of control (Bordons and Camacho, 2000) .To our knowledge no MPC-application combined with a scheduling-algorithm applied to a mixed batch/continuous system is applied. The case of scheduling a problem in a sugar factory, used to illustrate a hybrid scheduling approach, has been done by Nott and Lee (1999), but no MPC was involved here. Applications of MPC to continuous problems in a sugar plant have been reported by several authors, e.g. Bordons and Camacho (2000), Prada et al. (2000), to name but a few. In our approach, we have reduced the complex problem to a simpler problem (with less A-pans and only the relevant inputs and outputs), with all the characteristics of the more complex problem, such as its hybrid character. For this problem, the proposed objective function (OBJ) to be maximised is OBJ= profit - batch costs - idle penalty - change flow penalty
(2)
in which 'profit' is the turnover for the production through the system, 'batch costs' are the costs associated with beginning a batch, 'idle penalty' is the penalty for scheduling
600 idle batch units and 'change flow penalty' is the penalty for changes in the continuous flowrate. The constraints under which the objective function OBJ (2) is maximised are 1. The model is limited to stable domains (for the thin juice brix, see Figure 2). 2. Physical constraints, such as bounds on the flows and levels in the vacuum-pans and seed-station. 3. Bounds on the (variable) cycle times of the A- and S-pans. 4. The start-frequency of the batch-section (A-, S-1-, S-A-, S-B- and S-C- pans) is chosen in such a way that the waiting time between pans is constrained. Using this objective function and constraints, the ratio between vapor input from the evaporators in the sugar mill and the steam demand of the sugarhouse is optimised. The results of applying hybrid Batch MPC are satisfactory for the simple problem. For the more complex model we expect a significant reduction in fluctuations and heat peaks in the heat supply to the sugarhouse. Hence, we expect an increased on-spec time with tighter specs. But, more importantly, a much more stable process-operation is achieved within a time-horizon of 6-8 hours.
6. Conclusions In the present paper we modeled the sugar house and evaporation-section of a sugar plant with the purpose of determing the heat streams in the process. The modeling has been done on basis of mass- and heat balances, which were checked on real plant data. After that a reduced model has been used for hybrid Batch MPC, which is a novel methodology combining scheduling and predictive control. The results are now going to be applied to the more complex model.
References Bordons, C. and E.F. Camacho, 2000, Applications of model predictive controls in a sugar factory, Proc. ADCHEM 2000, 329. Bubnik, Z. P. Kadlec, D. Urban and M. Bruhns, 1995, Sugar Technologists Manual. Chung, C.C. eds., 2000, Handbook of Sugar Refining, John Wiley, New York. Garcia, C.E., D.M. Prett and M. Morari, 1989, Model predictive control: theory and practice - a survey, Aut, 25(3), 335. Lee, K.S., I.-S. Chin and H.J. Lee, 1999, Model predictive control technique combined with iterative learning for batch processes, AIChE J., 45(10), 2175. Nott, P.H. and P.L.Lee, 1999, Sets formulation to schedule mixed batch/continuous process plants with variable cycle time, Comp.Chem.Eng. 23, 875. Prada, C , C. Alonso, F. Morilla and M. Bollain, 2000, Supervision and advanced control in a beet sugar factory, Proc. ADCHEM 2000, 341. Law, A.M., and W.D. Kelton, 1991, Simulation Modeling & Analysis, McGraw-Hill. Rao, L., L. Owen and D. Goldsman, 1998, Development and application of a validation framework for traffic simulation models, Proc. 1998 Winter Sim.Conf.1998, 1079. Urbaniec, K., 1989, Modern energy economy in beet sugar factories, Elsevier, Amsterdam.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
601
Development of an Internet-based Process Control System S. H. Yang*, X. Chen*, D. W. Edwards^ and J. L. Alty* *Department of Computer Science, Loughborough University, Loughborough, Leicestershire, LEI 1 3TU UK "iDepartment of Chemical Engineering, Loughborough University, Loughborough, Leicestershire, LEI 1 3TU UK
Abstract This contribution presents an approach for the design of Internet-based process control. Uniquely, it addresses multi-user collaboration and Internet transmission latency. Video feedback, text based chatting, and a whiteboard are embedded in the system and shared by multiple communicating users. Virtual supervision parameter control is implemented to overcome dynamic delays caused by the Internet traffic. The experimental results from the test bed show that the Internet-based control system can have a similar behaviour to the local control system if properly designed.
1. Introduction In the last decade, the most successful network developed has been the Internet and its users are worldwide. Internet technology offers unprecedented interconnection capability and ways of distributing collaborative work, and this has great potential for the high-level control of process plants. As a basis for the possible next generation of control systems, the concept of Internet-based process control has been introduced in recent years which would allow managers and/or operators remote access for monitoring and making adjustments to process plant operation. For example they could adapt to quick changes in the markets. It also allows collaboration between skilled plant managers situated in geographically diverse locations and the possibility of single group support for multiple installations. To date, most research work on Internet-based process control has resulted in smallscale demonstrations (Atherton, 1998; Cushing, 2000). Some researchers in this area, from higher educational institutions, focus on developing web-based virtual control laboratories for distance learning purposes (Aktan, et al., 1996; Ko, et al., 2001). They allow a remotely-located user to conduct experiments in their control engineering laboratory via the Internet. Unfortunately none of them discusses the limitations caused by Internet environment features such as Internet transmission latency and user isolation. However, Internet time delay and multiple-user collaboration are two essential issues which must be addressed in the design of Internet-based control system for industries and for web-based control experiments. This paper aims to develop an Internet-based process control system for a water tank in our control laboratory and use this system as a test bed for investigating the effect of
602 Internet time delays, concurrent user access and the nature of the communication between multiple users. The rest of this paper is organized as follows. In the next section the system architecture is described, including hardware structure and software structure. The Internet time delay and the virtual supervision parameter control approach (VSPC) are discussed in section 3. Section 4 provides a possible way of avoiding the conflict between multiple users. The system implementation and some experimental results are given in section 5. Section 6 provides the conclusions.
2. System Description 2.1 Hardware structure As shown in Figure 1, the whole system consists of five parts, which are a water tank, a data acquisition (DAQ) instrument, a web server, a web camera, and several web clients. The tank is filled by the inlet flow controlled by a hand valve and is emptied into a drainage tank through a connection pipe and a pump. The outlet flow is controlled by a local control system located at the server to maintain the liquid level of the tank at a desired value. The local control system of the tank is located in the server machine. The server machine and the DAQ instrument are connected and wired by RS232c serial cables. Through the serial cable, the real-time data is exchanged between the server machine and the instrument. A web camera connected to the server machine provides visual information to wHeOai the users through a video server. Because the web camera is independent from the DAQ, it can be considered as an extra sensor. The server provides the standard control functions as well as the Internet services, and acts as the video server. The Internet service is implemented mainly based on the LabView G-Server (NI, 2000). In addition to the standard Internet service, the server also needs to establish OGH the connections between the clients and the local controller. Using a web browser, Figure 1. Hardware structure of the several remotely-located users are Internet-based controls allowed to simultaneously monitor and control the tank.
2.2 Software structure The system software can be divided into two parts, the client side and the server side. Whilst the client side interacts with users, the server side is not only a web server, but also includes the control and data acquisition program to achieve the control task. From a functional perspective, there are two programs in the client side as shown in Figure 2, for controlling and monitoring functions respectively, which are the control panel and the monitor panel. The control panel responds to interactions from users. The
603 users can use it to issue commands and/or change the parameters of the controller. Through the TCP protocol, the control panel establishes the connection with the server. In addition to sending information to the server, it is also necessary to receive information from the server. If any client changes the parameters of the controller or issues a command, the server will broadcast the change to every registered user. The control panel deals with this information in order to synchronise the change and indicate the correct status of the controller. The monitor panel provides two functions, the dynamic image and the video & chatting system. The dynamic image consists of graphic information including the process flowchart and the dynamic trends of process variables, which provide the essential information of the current system status. The video & chatting system is designed to provide the visual information for monitoring the facility and the communication channels for multi-users. Multi-users can chat to each other by sending a text message and/or share a white board. On the server side, the service can be divided into two parts, the command service and the data service. The command service handles incoming requests, and interprets the received information to parameters and commands for the controller and the data service. It also broadcasts the incoming information to every registered client in order to synchronise the client information. In addition, it also handles multiple client conditions such as concurrent user access. The data service is designed mainly to generate an image according to the client requirement, and to send the image embedded in an html page to the clients. The mediator establishes a bridge between the controller and the instrument. The controller deals with standard automatic set-point and manual control. Microsoft NetMeeting is used to support T.120/H323 video standard facilitates for the video & chatting function.
Figure 2. Software Structure of the bite met-based control system
3. Virtual Supervision Parameter Control 3.1 Internet time delay One of the difficulties in Internet-based process control is the Internet transmission latency. Luo and Chen (2000) have repeatedly tested the transmitting efficiency of the
604 Internet by sending 64 bytes data every time from their Web server to different remote Web servers. The resulting statistics of the experiments show that the latency of the Internet contains the serious and uncertain time delays. A block diagram of the Internetbased control system is drawn in Figure 3. The total time of performing an operation (a control action) per cycle is t^ +^2+^3+^4 where the four types of time delay are: ti time delay in making control decision by a remote operator. t2 time delay in transmitting a control command from the remote operator to the local system (the web server). t3 execution time of the local system to perform the control action. t4 time delay in transmitting the information from the local system to the remote operator. If each of the four time delays is always a constant, the Internet-based control has a constant time delay. Unfortunately, as shown in Luo and Chen's experiments (2000) this is not the case. The Internet time delay, i.e. t2 and t4, increases with distance, but the delay depends also on the number of nodes traversed. Also the delay strongly depends on the Internet load. It is somewhat unreasonable to model the Internet time delay for accurate prediction at every instant. Therefore a control architecture, which is insensitive to the time delay, is needed for the Internet-based control system.
Delay t^
Internet Delay 1
Figure 3. The block diagram of the Internet-based control. 3.2 Virtual supervision parameter control The Virtual Supervision Parameter Control (VSPC) strategy is one practical approach for Internet-based control, which is insensitive to the time delay. As shown in Figure 4, the detailed control functions are implemented in the local control system. Internetbased control over VSPC is invoked only when the updated parameters like setpoints and Proportional-Integral-Derivative (PID) parameters are required to be sent to the local control system. The new set of VSPC parameters is used as input for the local control system until the next set of parameters is received. VSPC offers a high safety level because the local control system is working as its redundant system. Furthermore it is likely not to be greatly affected by the Internet time delay because the Internet time delays t2 and t4 are excluded from the close loop of the control system as shown in Figure 4.
605 VSPC
JDela^t^ Oe}a)[j2
Parameters
ill
Setpoir^^
Internet _Delayt^_^ variables
Figure 4. Virtual supervision parameter control.
4. Concurrent User Access Compared with the traditional DCS system, the special feature of the internet-based control system is multiple users. Although the DCS system allows several operators and/or engineers to operate the system at the same time, they normally sit in the same operation room. Therefore, coordination amongst them is not a real problem. In the Internet-based control system, the operators cannot see each other, or even meet previously. It is likely that multiple users may try to concurrently control a particular process variable. For example, in VSPC, if multi- users try to change the set-point simultaneously, the system may be unable to operate. The set-point of the controller fluctuates from point to point. The coordination among multiple users becomes very important. One of the promising ways is to assign users with different priorities. The user with a high priority can immediately overwrite the commands issued by users with lower priorities. After a new command is accepted, the system will be blocked for a certain period of time and refuse to accept any further command from users with equal or lower priorities.
5. System Implementation And Experimental Results The system is implemented using Java applets and LAB VIEW virtual instruments (VI). Figure 5 illustrates the remotely-located users interface. The left-hand side column is the control panel, and the right-hand side column is the monitoring panel. The control panel is a Java applet where Web users can issue the control command and/or change parameters of the controller. All the information in the control panel will be updated immediately once any other registered user has made any change on them in order to indicate the correct status of the controller. The monitoring panel is switched between the process flowchart, the process trends, and the video & chatting panel. The process flowchart indicates the current status of the process. The dynamic trends show the process responses under a control command. The experimental results show that by using VSPC, the Internet-based process control system can have a similar behaviour to the local control system even with some Internet traffic delay. Figure 5 illustrates that how the video provides the remote users with the visual information of the process. Text chatting and whiteboard pop windows are invoked by pressing a corresponding button below the video, which provide users with a communication channel for co-operation.
6. Conclusions Internet technologies can provide web clients a platform not only for remotely monitoring the behaviour of the process plants, but also for remotely controlling the
606 plants as well. In this paper an Internet-based control system for a water tank in our process control laboratory has been developed. The issues in the design of the Internetbased control system, concerned with the Internet time delay, multi-user cooperation, and concurrent user access have been addressed. The concept differs from other approaches in that it provides a way for communication and conflict resolution between multiple users, and the VSPC control strategy excludes the Internet time delay from the close loop of the control system and is likely not to be greatly affected by the Internet traffic. The experiment results show that the Internet-based control system may have a similar behaviour to the local control system under the VSPC scheme.
Figure 5. Video & chatting panel
Acknowledgement The contribution is part of the work of the EPSRC (Grant No. GR/R13371/01) funded project "design of Internet-based process control".
References Aktan, B., Bohus, C.A., Crowl, L.A., and Shor, M.H., 1996, Distance learning applied to control engineering laboratories. IEEE Trans, on Education, 39(3), pp. 320-326. Atherton, R, 1998, Java Object Technology can be Next Process Control Wave. Control Engineering, 45(13), pp. 81-85. Cashing, M., 2000, Process control across the Intemet, Chemical Engineering. May, pp.80-82. Ko, C.C, Chen, B.M., Chen, J., Zhuang, Y. and Tan, K.C., 2001, Development of a web-based laboratory for control experiments on a coupled tank apparatus. IEEE Trans, on education, 44(1), pp. 76-86. Luo, R. C. and Chen, T.M., 2000, Development of a multibehaviour-based mobile robot for remote supervisory control through the Intemet. IEEE Trans, on mechatronics, 5(4), 376-385.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
607
On-Line Optimal Control of Particle Size Distribution in Emulsion Polymerisation J. Zeaiter*, J.A Romagnoli, G.W. Barton and V.G. Gomes Laboratory for Process Systems Engineering Department of Chemical Engineering University of Sydney NSW 2006 Australia
Abstract This paper concerns itself with the question of the on-line optimal control of the particle size distribution (PSD) from an emulsion polymerisation reactor, using the monomer feed rate as a manipulated variable. A recently developed (and validated) reactor model was employed in the formulation of this control strategy. The model was implemented within the gPROMS software package and used as a "soft sensor" for the on-line estimation of the PSD within a model-based predictive control (MPC) formulation, so as to determine on-line the optimal trajectory for the monomer feed rate. For control purposes, the distributed nature of the PSD was represented by the leading moments, with the control objective being formulated as a function of both the breadth of the distribution and the average particle diameter. Off-line sample analysis, carried out using capillary hydrodynamic fractionation, was also incorporated into the control strategy (as irregular measurements), so as to update the on-line model predictions. Experimental studies on a laboratory scale styrene polymerisation reactor showed that such an approach was able to accurately predict the behaviour of the reactor, as well as improving its performance.
Introduction Control of the PSD in emulsion polymerisation by means of closed-loop strategies is a challenging problem (Dimitratos, 1994). Difficulties associated with the on-line measurement of the PSD, together with the complex mechanisms involved in emulsion polymerisation systems in general, limit operational options and make the control problem a formidable task. In such cases, conventional control strategies fail to ensure a consistent product quality, with the result that industrial practitioners have to rely on traditional "recipes" and experience. In recent times, however, advances in process understanding, mathematical modelling, soft-sensor technology and model-based control techniques have been such as to offer polymerisation reactor operators the chance of achieving major improvements in process operation and product quality. A viable approach here would seem to involve combining these powerful tools into an effective model-based control strategy - and this is presented in detail in this paper.
* Corresponding author. Tel.: +61-2-9351-4337; fax: +61-2-93512854 Email: [email protected]
608 A validated mathematical model (Zeaiter, 2001), based on the kinetic mechanisms of free-radical emulsion polymerisation, was used in this study. This model includes diffusion-controlled kinetics at high monomer conversion, and comprises a set of rigorously developed population balance equations. Using this model, the problem of achieving on-line optimal control of the PSD in the semi-batch emulsion polymerisation of styrene was investigated for the case where the final product has a predefined distribution. To achieve this goal, an input/output model predictive controller (MPC) was developed to calculate the optimal trajectory on-line, with the full dynamic model being run in parallel and used as a "soft sensor" to provide an on-line estimate of the PSD. The motive behind choosing MPC is the fact that it is the only strategy that can incorporate a number of performance criteria and is capable of utilising any available dynamic process model. A comprehensive review of MPC theory and applications is available in the literature (Garcia, 1989).
Simulation Model A detailed dynamic model for a perfectly mixed, semi-batch reactor has previously been developed (Zeaiter, 2001) for the styrene emulsion polymerisation system. This model is population balance based and solves a series of integro-partial differential equations coupled with a set of differential and algebraic equations. This equation set describes all known physical and chemical mechanisms that occur within both the particle phase and the aqueous continuous phase, including diffusion-controlled kinetics at high monomer conversion where the transition from the zero-one regime to the pseudo-bulk regime occurs. Within this model, both particle growth and nucleation are assumed to occur in discrete intervals of the particle size. Values for all physical constants and kinetic parameters were obtained from the open literature. The resulting set of equations describing the polymerisation of styrene (initiated by potassium persulfate and stabilised by sodium lauryl sulfate) are numerically solved using the commercial gPROMS package {Process Systems Enterprise Ltd). Model validation against experimental data has been carried out over a wide range of operating conditions (principally combinations of reaction temperature and monomer feedrate) with model predictions able to adequately describe the entire polymerisation process.
Control Strategy In this study, the objective of the control algorithm was to ensure the production of a polymer with a pre-specified PSD in the minimum reaction time. A particle size polydispersity index (PSPI) was chosen as the optimisation objective function, with particle concentration densities being set as constraints. The objective function to be maximised was, thus, defined as: Aiax[PSPI{r,tj.,^,)\
(1)
where tfi^ai is the total processing time. For operational reasons, the manipulated variable {ie the monomer feedrate) was specified with the following lower and upper bounds:
609
S X ~ OrnoleslsecI -~ F,,, 5Sx104 nioleslsec
(2)
The final PSD "shape" was included in this optimisation formulation in the form of endpoint inequality constraints, given in terms of the final molar concentration density of particles:
In this equation, the limits nminand ti, are specified so as to match the required PSD. Since we are dealing with a semi-batch (or fed-batch) process here, the maximisation of the PSPI is subject to additional constraints accounting for the total amount of monomer available in the "recipe", N,,,T, and the permissible total run time. These constraints are defined as follows:
N,,,T =1.6moles and t,,
5 t
(4)
In this problem, the population balance equations were discretised with respect to the particle radius, r. The required PSPI profile was calculated off-line through an interface to the gOPT dynamic optimisation code (also from Process Systems Enterprise Lfd).
On-Line Optimal Control In order to predict on-line the optimal trajectory for the monomer addition, a nonlinear model-based predictive controller (MPC) was used. The MPC algorithm uses an inputoutput model (Clarke, 1994; Camacho, 1999; Kanjilal, 1995) whose step response coefficients are determined as follow: Using the dynamic model, solve for the output using a constant monomer feedrate. The calculated output is designated by y"'. Introduce a consecutive set of step changes into the monomer feedrate, and solve the dynamic model 'into the future'. The resultant output is designated by ysrep.
Time
Figure I: Input-output model identification
610 The step response coefficients are then calculated as follow: /'''{k^i)-y''{k^i) Au(k-\-i)
a: =
(5)
where u represents the model input (ie the monomer feedrate in this case). This inputoutput model identification technique is illustrated in Figure 1. Model predictions can then be calculated at every sampling time from the calculated step-response coefficients using the following:
y(k + l) = ^aiAu(k
+1 - 0 + y''(k + l)-hd(k -f 1)
(6)
/=i
Modelling error and the impact of unmeasured disturbances are included in the last term on the right hand side of Equation 6. Once the process output has been measured at a given point in time, d(k) can be estimated by assuming all future d values are equal, as illustrated in Equation 7:
y''{k) = d{k + l) =
= d(k-^N)
(7)
In this formulation, the controller has to calculate the set of control moves (Au) into the future that allows the system to follow a pre-defined set-point trajectory. However, only the first control move is implemented on the process, with the entire optimisation being repeated at the next sampling time.
Experimental Validation A major problem in attempting to implement any form of advanced control strategy is the lack of appropriate on-line sensors for the measurement of the PSD. In practice, such measurements require sampling, dilution and off-line analysis and data processing, typically using capillary hydrodynamic fractionation (CHDF). 3iE-04
3.0E-04 2iE-04
^••>\ i U f f i T T M T ] g,f4«^n <'> o.
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1
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50
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100 110 120 130 140 150 160 170 ISO 190 200
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Figure 2: Control of the PSD via the PSPI. ^Set-pointy flKMeasurement
OOE.OO
0
10
20
30
40
SO
60
70
10
90
100 110 120 130 140 ISO 160 170 110 190 200
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Figure 3: Monomer feedrate calculated online by MFC.
611 Typical analysis time by CHDF is of the order 10-15 minutes, making this method impractical for on-line monitoring and control applications. To overcome this problem, a "soft sensor" approach was employed, with the full mechanistic model (Zeaiter, 2001) being used to provide on-line estimates of the polymer PSD. This approach requires the use of real-time gPROMS execution. All relevant operating conditions (such as monomer flowrate and reactor temperature) are taken from the reactor at discrete time intervals as on-line measurements, and used by the dynamic model to estimate the PSD. Note that a commercial SCADA system interfaces the polymerisation reactor to the computer running the MPC package, the latter being written in MS-Excel and Microsoft Visual Basic. PSD measurements are obtained from this model-based "soft sensor" every 110 seconds and fed to the MPC which calculates the appropriate control action (ie the monomer feedrate for the next time interval). Off-line samples are also taken from the process every 20 minutes, and the CHDF measurements compared to the (model) predicted results. Any modelling error was handled within the MPC algorithm as an unmeasured disturbance, as previously described. In this MPC formulation, the PSPI was calculated using the step response method, also described previously. 100
»fimtj-^ -
i
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\ \
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\^
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r1 \
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1 ,
Figure 4: Optimal particle size distribution of the final product. • Estimated (gPROMS model), MMeasured (CHDF) In this MPC application, the prediction horizon, the control moves, and the sampling time were set at 30, 5 and 470 seconds, respectively. The (relatively large) magnitude of the sampling time is related to the speed of the process dynamics. As the average volume growth rate for particles in emulsion polymerisation is very small (of the order 2x10'^^ L/s), then a relatively long sampling time is needed for discernible changes to be observed. Setpoint tracking of the PSPI by manipulating the monomer feedrate was investigated by first solving off-line the optimal control problem for the case where a specified (and quite broad) PSD was required for the final product. An optimal trajectory was thus generated for the monomer feedrate so as to give the required PSPI profile. This profile was then used on-line as the setpoint for the MPC. The ability of the MPC to hold the controlled output as close as possible to the setpoint is illustrated in Figure 2. Although the PSPI did increase throughout the run (and, consequendy, the PSD continually
612 broadened), an offset of variable magnitude was apparent at every sampling time. The reason for this behaviour lies in the tuning of the MPC, as the controller was set up to avoid large changes in the manipulated variable (so as to prevent the monomer flowrate hitting a bound). The weightings employed favoured this course of action at the expense of closely tracking the setpoint. As shown in Figure 3, the monomer feedrate profile was found to decrease with time after the batch pre-period (ie where no monomer was added) of 25 minutes, with no constraint violation observed for the manipulated variable during the entire run. The final shape of the PSD was in good agreement with the estimated "soft sensor" result (see Figure 4), with a broad distribution being obtained, as originally defined as the objecUve of the control strategy.
Conclusions This paper reports on the successful development and experimental implementation of an advanced control scheme for a styrene emulsion polymerisation reactor. This scheme was developed with a particular focus on achieving control over the full PSD of the polymer product (rather than the more simple option of controlling the mean particle size). Using a recently developed (and experimentally validated) detailed dynamic model of a semi-batch polymerisation reactor as an on-line "soft sensor" for the PSD, and as the means for calculating dynamic step response coefficients, it was possible to develop a model-based predictive control scheme whose objective was to provide a specified PSD for the final product. The success of the work reported indicates that this methodology could be readily extended to simultaneously provide fight control over a number of product attributes (eg in terms of both the particle and molecular weight distribufions).
References Camacho E.F. and Bordons C. (1999), Model Predicfive Control, Springer, NY. Garcia C. E., Prett D. M., and Morari M. (1989), Automafica, 'Model Predicfive Control: Theory and Pracfice-a Survey', 25, 335-348. Clarke D. (1994), Advances in Model-Based Predictive Control, Oxford ; New York : Oxford University Press. Dimitratos J., Elicabe G. and Georgakis C. (1994), AIChE Journal, 'Control of Emulsion Polymerizafion Reactors', 40, nl2, 1993-2021. Kanjilal P.P. (1995), Adapfive Predicfion and Predicfive Control, Stevenage, U.K. Process Systems Enterprise Ltd. (PSE), gPROMS Advanced User Guide, Ver.2.0, United Kingdom. Schork F.J., Deshpande P.B., Leffew K.W. and Nadkarni V.M. (1993), Control of Polymerization Reactors, M. Dekker , NY. Zeaiter J., Romagnoli J.A., Gomes V.G., Barton G.W. and Gilbert R.G., 'Operation of Semibatch Emulsion Polymerisation Reactors: Modeling, Validation and Effect of Operafing Condifions', submitted to Chemical Engineering Science Journal.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
613
Cyclic Production and Cleaning Scheduling of Multiproduct Continuous Plants Alessandro Alle\ Jose M. Pinto^* and Lazaros G. Papageorgiou^ (1) Department of Chemical Engineering, University of Sao Paulo, Av. Prof. Luciano Gualberto t. 3 n. 380, Sao Paulo, SP, 05508-900 Brazil. (2) Centre for Process Systems Engineering, Department of Chemical Engineering, UCL (University College London), Torrington Place, London WCIE 7JE, U.K.
Abstract The objective of this paper is to extend previous scheduling models (Pinto and Grossmann, 1994; Alle and Pinto, 2001) based on continuous time representation to include cleaning considerations. The proposed mixed integer non-linear programming (MINLP) model aims at simultaneously scheduling production and cleaning tasks of multiproduct multistage plants with performance decay. The resulting mathematical model has a linear objective function to be maximized over a convex solution space thus allowing global optimum to be obtained with an Outer Approximation algorithm. A case study demonstrates the applicability of the model and its potential benefits in comparison with a hierarchical approach.
1. Introduction This paper addresses the problem of cyclic scheduling of multiproduct multistage continuous plants (one unit per stage) with performance decay. Performance decay is a serious problem in many chemical processes and has several causes such as catalyst deactivation, fouling etc. As performance decreases, processing units require cleaning to restore productivity to initial state. The decision of stopping a unit for maintenance interferes with the production schedule. Therefore, it is important to consider both production and cleaning scheduling decisions simultaneously. Several analytical methods for determining the optimal cleaning schedules for equipment items have been proposed in the literature. The main drawback of these approaches is that they are restricted to a single equipment item. Only recently, Jain and Grossmann (1998) studied the scheduling of single stage with parallel units whose performance decreases with time and therefore shutdown for maintenance after regular intervals is required. Georgiadis et ai (2000) and Georgiadis and Papageorgiou (2000) have presented cleaning scheduling approaches applied to heat exchanger networks under fouling conditions based on discrete-time mathematical formulations. This work aims at extending previous scheduling models (Pinto and Grossmann, 1994; Alle and Pinto, 2001) for multiproduct, multistage continuous plants based on continuous time representation to include cleaning considerations. To whom correspondence should be addressed. j [email protected]
Fax: +55 11 3813 2380, E-mail:
614
2. Problem Description The plant is composed of M stages with one unit per stage, m, with no inventory limitations (see Fig. 1). The plant processes NP products in the same order in all stages (permutation flowshop plant). For every product, /, a fixed demand rate, dt, should be satisfied.
Intermediate Storage Tanks 2
Intennediate Storage Tanks M-1
Fig. 1. Multiproduct multistage cyclic continuous plant. Process yields at some stages may decrease with time. Therefore, units must shutdown for periodic cleaning. These cleaning tasks require both specific time duration and cost depending on the units and products processed. A typical production and cleaning schedule comprises product campaigns with subcycles, as shown in fig. 2. Cyzh time Tc A
A
A
A 1
s
-t
1 X
\ •
•
*
-
^
_
.
Hh-
*"^r
?
\
1 ""•
\ \
H h
A
--J
\
\ •-—.
"—r
\
A^m '
] 1
TP.
tune (days)
Subc3rcles
Fig 2. Cycle and subcycles in a unit with decay performance. The yield decay for product / at stage m is assumed to be an exponential function of processing time, r, as in Jain and Grossmann (1998). ^/.(0 = c,,+«,>,exp(-Z?,,;) (1) The overall objective of the planning model is to maximize profit over a given cycle time. The profit is given by the difference between revenues (sales of the final products) and costs (raw material, cleaning and transition) divided by cycle time duration. Transition times and costs are incurred when the line is changed from one product to
615 another. As transition times and costs are sequence dependent, there are trade-offs concerning the product sequence. A sequence that minimizes total transition costs may not be the one that maximizes the availability of the plant because transition times and costs are not necessarily proportional. There are two extremes in terms of cleaning policies. The units may be cleaned frequently and therefore units run at large yields. Consequendy, the productivity per unit of processing time is larger and raw material consumption is lower. On the other hand, the cleaning cost increases and the availability of the units decreases due to the time spent for cleaning. At the other extreme, when units are not cleaned frequently, the cleaning cost gets lower and the availability higher. In addition, the raw material cost increases while performance falls due to increased production losses. Thus there are significant trade-offs among cleaning and raw material costs as well as availability and yields. From the above discussion, it is evident that production and cleaning scheduling decisions should be considered in the same framework. Overall, the production and cleaning scheduling problem can be stated as follows: Given are cycle time {Tc), performance decay functions, final product prices (P/j, raw material (C/), cleaning {CUm) and transition costs {Ctr^j), cleaning (T,>„) and transition times {Tijr^, feeding rates (G,>„), maximum number of subcycles ( R"l^ ); determine product sequence (Z^), start times {TSim), processing times (TPim), number of subcycles (Rim), amounts of raw material (F,) consumed, final products (Wim) so as to maximize overall plant profit over the cycle time.
3. Mathematical Model The mathematical formulation is based on a previous continuous-time scheduling model for continuous muldproduct plants (Alle and Pinto, 2001) that is extended here to capture cleaning aspects. First, the folowing binary variables are introduced: Zij : 1 if product / precedes product); 0 otherwise. Ximr : 1 if r subcycles of product / take place at stage m; 0 otherwise. As the plant is a flowshop, every product j must be preceded by the same product / at all stages. Only one product succeeds and precedes the other: XZ,=1
Vy
and
^Z,.=l
V/
(2)
The number of subcycles required for product / at stage m is defined by: £X,,„=1
V/,m
(3)
The total amount of product / produced at stage ni during one subcycle is as follows: W^,,=G^,M„, V/,m (4) where RAim replaces the product between /?,>, (number of subcycles) and A,>„. The latter is determined by integrating the yield decay function over the duration of one subcycle: A», = c,,,?:,, +a,„, / b^ [l - exp {-b,J,„)] V/, m (5) or alternatively the following convex-region constraint is used: An, ^^/.7;„,+a,„,/Z7,.,,[l-exp(-Z7,„,7;.J ^ijn
(6)
The amount produced at stage m must be completely consumed at stage m+7 in order to avoid accumulation of material within cycles:
616 (7) ^•.=^,>.i/^7;.,.i V/,m = l...M-l where RTi^i, which represents summed duration of subcycles, replaces the product of Ri^ and Tim. The following constraints are required in order to define the RAi^ and /?r,>„ variables: ^'•-^A:^/.. Am=
V/,m,r
J.'AXirr. r
(8) (9)
\J i,m
r
TX^„r
^i,m,r
(11)
Vi>
(13)
r
/??;„= ^/-T^™. r
The total demand on final products must be satisfied at the cycle end: W.^>d.Tc V/ (14) The total time that product / is processed at stage m, TPim, is given by the summed duration of subcycles, /?r,>„, plus the time spent for cleaning: TPi.=RT,„+x,„^{r-\)X,„,^ yi,m (15) r
At any stage, the sum of the total occupation time plus the transition times for all products must not exceed the cycle time, Tc.
rc>gr/^„, + X^.>,z,
Vm
(16)
As the schedule is cyclic, product I is arbitrarily chosen as the first to enter the production line to decrease solution degeneracy:
7'5„=X7,nZn
(17)
Constraint (18) states that product 7 starts immediately after the processing of the preceding product / plus the correspondent transition time. -rc(l-Z,)
ra,,,
vi,,« = I...A/-1
(i9)
There is a limit on the time that stage m can process product / before a cleaning task. \fi,m (20) T
f /
1
^ m
r
j
m=l
(21) ^
Note that the last term in (21) represents a small penalty in order to prevent solution degeneracy as start times at the stages other than the first are unbounded. The above mathematical optimization model corresponds to an MINLP model with linear objective function over a convex solution space. The resulting MINLP model can
617 be solved by the OA/ER/AP method (Viswanathan and Grossmann, 1990) up to the global optimality using GAMS 2.5 (Brooke et al., 1998) with DICOPT-2 solver.
4, Case study Given a cycle time of 360 days, 5 different products are processed in 3 consecutive stages (see fig. 3) with decaying first and second stages and third stages with constant performance. At most 4 subcycles are allowed for every product and the maximum time between stops in the units with decay is 60 days. Table 1 presents data involved. 0«;i, l « l . o - v t ^ J
0«;s, kw«:t.y« r a ^ l
OSi'v
B
~ . -
E
X
086-
''\ ^ -X
20
f
oae-
•v.
•
"\
- \
'.: "--..
OK-
^--
X
•
•
.
»
.
'
"
•
•
•
"
•
^
' " ^ ^--^^
"^ "^^~~-^
40
• ^ ^
so
Fig 3. Performance decay in stages I and 2. Table 1. Plant data for Case Study. Pr-
m
Pi
Gil
Gi2
Gi3
Tu
Ti2
($/ton)
(ton/d)
(ton/d)
(ton/d)
(day)
(day)
53 38 33 41 55
51 35 33 42 49
1 1.5 1 1.5 1
1.5 1 1.5 1.5 1.5
56.2 53.6 52.6 50.8 45.0
A B C D E
100 95 90 105 102
Cm ($/day) 6000 5500 4000
Cf,=0A5P,\Cl,„=C^T^ Ctr,^=O.Sj^C^T,^
Transition times, Zijm (days) Pr. A B C D
1E
A 0 2.5 2 3 1.5
B 1.5 0 3 3 2.5
Stage 1 C D 3 1.5 0 3 3
2.5 2 1.5 0 3
E 2 3 3 1.5 0
A 0 2.5 2 1.5 3
B 3 0 1.5 3 2
Stage 2 C D 1.5 2.5 3 2 0 3 3 0 2 1.5
E 2 1.5 3 3 0
A 0 2.5 2 1.5 1.5
B 1.5 0 1.5 3 2
Stage 3 D C 3 1.5 2 1.5 1.5 0 0 2 1.5 3
E 3 1 1.5 3
1.5 1 0
To demonstrate the effectiveness of the proposed MINLP model, a hierarchical alternative approach is studied which comprises three steps. At the first step, the product sequence is obtained by minimizing the total transition cost. The second step determines subcycle duration for the first processing stage of each product by keeping the corresponding yield above 80% of the initial value. This step introduces new upper bounds for the maximum subcycle duration, that varies from 18 to 25 days, depending
618 on the product. Finally, at the third step, a reduced version of the original MINLP model is solved with fixed product sequence and tighter bounds on subcycle lengths. Different product sequences and timings are obtained from this hierarchical approach and proposed MINLP model ^ as shown in fig. 5.
•
Stg 3
09 L
S t g 2 08
Stg 3 0 9 L
Fig. 5. Comparison between hierarchical and optimization solutions Moreover, the optimal sequence, ACBED, has an objective function value of 570 $/day which represents a 11.5% improvement when compared with the hierarchical solution (511 $/day).
5. Conclusions In this work, the problem of production and cleaning scheduling for multistage multiproduct continuous plants with performance decay has been studied. The overall problem has been formulated as a convex MINLP model. Finally, the advantages of applying optimization-based approach have also been emphasized when compared the proposed model with a hierarchical approach.
Acknowledgment The authors would like to acknowledge support received from FAPESP (Fundacao de Amparo a Pesquisa do Estado de Sao Paulo) under grants 99/02657-8 and 98/14384-3.
6. References Alle, A. and J.M. Pinto, 2001, Proc. DYCOPS 6, Korea, 213-218. Brooke, A., D. Kendrick and A. Meeraus, 1998, GAMS - A Users' Guide. The Scientific Press, Redwood City. Georgiadis, M.C. and L.G. Papageorgiou, 2000, Chem. Eng. Res. Des., 78, 168-179. Georgiadis, M.C, L.G. Papageorgiou and S. Macchietto, 2000, Ind. Eng. Chem. Res., 39,441-454. Jain, V. and I.E. Grossmann, 1998, A/C/zE 7., 44, 1623-1636. Pinto, J.M. and. I.E. Grossmann, 1994, Comp. Chem. Engng., 18, 797-816. Viswanathan, J. and I.E. Grossmann, 1990, Comp. Chem. Engng., 14, 769-782. Optimal solution required 356 CPUs in a Compaq ML-570 Workstation.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
619
Improving the Efficiency of Batch Distillation by a New Operation Mode H. Arellano-Garcia, W. Martini, M. Wendt, P. Li, G. Wozny Institute of Process and Plant Technology, Technische Universitat Berlin 10623 Berlin, Germany
Abstract Batch distillation processes are well-known for their high degree of flexibility. A feature of batch distillation is that it produces not only the desired products but also off-cuts. Conventionally, off-cuts are recycled to the reboiler of the column for the next batch. In this work, we propose a new operation mode for batch distillation, namely, the off-cuts will be recycled in form of a continuous feed flow into the column. The separation effect is promoted in this way and thus economical benefits can be achieved. Simulation and optimization based on a rigorous model are carried out to study the properties of this operation mode and develop optimal operating policies. Results of applying this mode to two industrial batch columns show significant improvements of operation efficiency in comparison to the conventional recycle strategy.
1. Introduction Batch distillation is used in chemical industry for producing small amounts of products with high added value and for processes where flexibility is needed. For processes with a rather limited conversion or separation performance, off-cut recycles can be introduced to recover unused raw materials. Many authors studied circumstances under which slop recycling is worthwhile (Bonny, 1999; Wajge & Reklaitis, 1998). The main problem is to decide when and how much recycle can be profitable. In the conventional operation mode off-cuts are recharged to the column reboiler for the next batch. Several previous studies on treating off-cuts in this mode have been made (Bonny et al., 1994; Sorenson & Skogestad, 1994; Macchietto & Mujtaba, 1996; Wajge & Reklaitis, 1998), in which recycle strategies were developed through simulation and optimization. However, mixing the off-cut with fresh feed for the next charge reduces the separation effect and thus leads to a low operation efficiency. A new operation mode for batch distillation is proposed to facilitate the separation effect by recycling the off-cut in form of a continuous feed flow into the column. Since the composition of the off-cuts lies between the initial composition in the reboiler and that of the distillate, feeding the offcuts to the column shortens the way of separation. Thus the efficiency of batch distillation can be improved, e.g., less batch time and energy consumption and more product amount. The performance of this new operation mode is carried out through simulation and model-based optimization of two industrial batch distillation processes. As a result, using this operation mode significant improvement of the operation efficiency can be achieved in comparison to recycling them to the reboiler.
620
CoMptRtats: A: educi ester B: educt akohol C; product ester D: product alcohol
/^
^-
I
t=^
i
DisuUate
I \ . 1 iiBincut 1 (iff-cul >99.9 weight 9F
B
>99,9 vwaght*
>54
700 mm
Process Data Qarge 15920 Kg. 55,29 weight's A
160T(6bar) number of Iriys: 30 tr»> active area: 0.2 raliquid holdup: 0.1 m-
^='=f^^^
volume: 0.5 m ' volume: O.Si m'
Steam
/
D
2,5 weight^.
B
41,0weight'?^
C
1.21 weight^
D
> 95,7 wwght%
Fig. 1: Two industrial batch distillation processes with the proposed operation mode.
2. Modeling of two industrial processes As shown in Fig. 1, we consider two industrial batch distillation processes to study the properties of the new operation mode. The first process is a semi-batch distillation with a chemical reaction (transesterification) taking place in the reboiler (Fig. 1, left). During the batch, a limited amount of educt alcohol will be fed to the reboiler to accelerate the reaction rate. With the conventional operation mode the total batch time was about 26h. We use a detailed dynamic model which is validated with the measured data from the experiment. Included are component balance, energy balance and vapor-liquid equilibrium on each tray in the model. The holdup, the pressure drop of each tray, and non-equimolar flow in the column are taken into consideration. The reaction kinetics are used to depict the chemical reaction in the reboiler. The second process is a batch distillation with a packed column for separating a 4-component mixture (Fig. 1 right). In this system there is an azeotropic point and the column pressure is used as a control variable as well. To describe the packed column we use a detailed dynamic tray-by-tray model. The number of the theoretical trays is calculated corresponding to the height of the packing. The holdup of each theoretical tray is computed with the correlation model proposed by Engel et al. (1997). The vapor load from the reboiler to the column is restricted by the F-factor (vapor load term) of the column as well as the heating capacity of the plant. These two processes were studied for optimization with the purpose of minimization of the batch time and some improvement compared to the empirical operating strategy was achieved (Li et al., 1998; Wendt et al., 2000). However, they did not consider recycling of the off-cuts. In this work, we investigate the possibility to further improve the performance of these two processes by the new operating mode. The model of both processes is modified such that the off-cut of the previous batch will be fed to a tray of the column during the next batch.
621
3. Problem formulation and solution approach In industry practice, it is often desired to minimize the batch time of such processes. Thus we consider the time-optimal problem to find optimal policies for the new operation mode for the two industrial processes, which can be described as follows:
min /,(F,(4F,(4/?,(/)/„,/J s.t
the model equation system and
min r;(F,(r),P(r),/?,(4r„„/„3,?/) s.t. the model equation system and
t
X
r
(t
>
)>x''
F,{t)dt<M,
J to
F,'
r
F,{t)dt<M,
K ^F^t
)^F; I
The recycle stream F^, pure educt alcohol stream F^. and reflux ratio Ry are the control variables to be optimized for the first process. The switching time t^ from the main-cut to the off-cut should also be decided. The output constraints are the distillate purity during the main-cut period and the product ester purity in the reboiler at the end of the batch. The input constraints consist of the limited amount of both off-cut M, and fresh educt alcohol M, as well as their bound limitations. In the second problem, beside the reflux ratio and the recycle stream the column pressure is also used as a control variable due to the available compressor. The process is operated in a packed column to separate a 4-component-mixture, with A, B, C, D representing from the lightest to the heaviest component (see Fig. 1). Three main cuts (fractions A and C from the top of the column and the fraction D remaining in the reboiler) will be obtained during the batch. An off-cut mainly containing B will be also received from the distillate. The heaviest component D has no vapor phase and remains in the reboiler during the whole operation. The output constraints are the specifications of the average composition in the three product vessels xf, jc^', jc;^'' and the composition in the reboiler x j ^ ^ , respectively. Both processes possess strong nonlinear behaviors and the model leads to a large-scale DAE system. To efficiently simulate and optimize the processes, we use collocation on finite elements to discretize the dynamic model equations. Through this discretizafion the dynamic optimization problem is transformed to a nonlinear programming (NLP) problem. It should be noted that the choice of the feed tray for the recycle stream should also be optimized. But this will lead to a dynamic MINLP problem. To prevent the problem becoming too complicated, simulation has been made to decide the feed tray.
622 To solve the optimization problem, we applied the sequential optimization method by Li et al. (1998), i.e. the entire algorithm is divided into one layer for optimization and one layer for simulation. The model equations are integrated in the simulation layer, so that the state variables and their sensitivities can be computed by given controls. The control variables are computed in the optimization layer by SQP as the only decision variables. A detailed derivation of the optimization approach can be found in Li et al. (1998).
4. Computation results The two problems are solved by the sequential approach and the results compared with those of the conventional recycle mode. To have a base for comparison, the same amount of off-cut is either continuously fed to the column or charged to the reboiler at the beginning of the batch. Fig. 2, 3 and 4 show the optimal policies by the new operation mode for the first process. The recycle stream should be fed to the column in the main-cut period (Fig. 2) since it contains the two alcohols and thus it is favorable to separate it in this period. The educt alcohol feed flow has a similar profile (Fig. 3). 7 I "
6
I 'I
i<
I/I
I '
1"wo
nf
Oft-Cut
y
iJVlr^
I ^1 0
n
2
4
6
8
10
12
14
16
Timefh]
Fig. 2: Optimal feed of recycle stream.
OC
1
1
0,5
Main-Cut
- Oft-Cut ~ 8
10
12
14
16
Time (h]
Fig. 3: Optimal feed of educt alcohol.
4
6
8
*U10
Time [h]
Fig. 4: Optimal reflux ratio.
12
14
t|16
This is because the temperature of the reboiler will be considerably raised in the off-cut period. The reflux ratio (Fig. 4) should be small at the beginning and increased gradually in the main-cut period in order to guarantee the purity constraint of the product alcohol, while it will be decreased for a period of time after the switching to the off-cut so as to quickly pull out the product alcohol remained in the column and condenser. Then the reflux can be increased so that the evaporated educt alcohol will be brought to the reboiler. Since the transesterification is a reversible reaction, the effect of the recycle stream to the column can be seen from the composition of the two alcohols in the reboiler. Fig. 5 shows their optimal profiles of the amount in the reboiler by both the new mode and the conventional mode. During the main-cut period, the amount of product alcohol is significandy smaller by the new mode than that by the conventional mode. This means that the reaction velocity in the direction of products will be increased. The total batch time by the new mode is 15.8h and it is 17.8h by the conventional mode.
623
I
—educt alcohol pfodict alcohol
Fig. 5: Alcohol composition by the new mode (left) and conventional mode (right) 10
J
newmcxJe 8 —conventional mode
g
I 6 X
^
4
8 I - - -, - -
-
U
2
2
0
4 Time [h]
4 L
.
-
.'
-
-
3,6 Time [h]
Fig. 6: Optimal reflux ratio for the conventional and the new operation mode (left) and the optimal feed position of the recycle stream (right).
C
7
Time[h]
Fig. 7: Optimal recycle flow rate
For the second process, due to the amount of component A in the off-cut, the whole content of the off>cut should be pumped into the column by the end of the first main cut period. Thus only in this period, the differences between the conventional and the new operation mode can be seen. Fig. 6 (left) show the optimal trajectories of the reflux in this period for both the conventional and the new operation mode.
In Fig. 6 (right) the optimal feed position of the recycle stream corresponding to the theoretical tray number is illustrated. This is approximately corresponding to the position in the column where the composition is equal to the feed composition. Fig. 7 shows the optimal recycle flow rate during the first period. It has to be noted that the difference concerning the optimal policies of the column pressure are only marginal, since a higher pressure is favorable in case of a constant F-Factor during the first main cut period. But the pressure is restricted by an upper bound due to the temperature of the reboiler headng steam. Due to the fact that in the new operation mode the off-cut of the previous batch is kept separated from the liquid mixture in the reboiler from the beginning, it has only a little amount of component B compared to the composition in the conventional operation mode. This leads to the fact that at the beginning the VLB relation between component A and C is more dominating, which causes an increasing
624 volatility and thus a much lower reflux rate is required for fulfilling the purity constraint of the first fraction. However, due to the decreasing amount of component A in the column during the batch, the supply of the recycle flow has to begin at a certain time as a compensation of this decrease. On the other hand, the content of the feed tank has to be depleted early enough before the first switch to the next fraction, in order to provide enough time for separating the remaining amount of component A, which originally comes from the recycle flow. The computed trajectory of the recycle flow (Fig. 7) indicates two physical phenomenon. Actually with the proceeding time, with a decreasing amount of component A, a stronger compensation and thus a higher feed flow rate is desired. On the other hand high liquid flows in later time intervals also causes a longer period until the first switch to the next fraction can be done due to the higher amount of liquid in the column, which needs to be distilled. Thus the optimized curve indicates a trade-off between those two contradictory desires. However, it should be noted that the shape of the curve does not have a strong impact on the targets of this optimization problem.
5. Conclusions and Acknowledgement We propose a new operation mode to improve the efficiency of batch distillation processes. For the minimization of the batch time two industrial processes have been considered. Model-based optimization was used for developing time-optimal policies for the batch distillation processes. The computation results show that the new operation mode is effective and workable. The effect of the recycle stream will be further studied concerning multiple batch campaigns. We thank the financial support from Deutsche Forschungsgemeinschaft (DFG) under the contract WO 565/12-1.
References Betlem, B. H. L., H. C. Krijnsen and H. Huijnen, 1998, Chem. Eng. Journal, 71. Bonny, L., 1999, Ind. Eng. Chem. Res., 38, 4759. Bonny, L., S. Domenech, P. Floquet and L. Pibouleau, 1994, Chem. Eng. Process., 33. Engel, v., J. Stichlmair and W. Geipel, 1997, IchemE Symposium No. 142, 939-948. Kim, K. and U. M. Diwekar, 2000, AICHE Journal, Vol. 46, Nr. 12. Li, P., H. Arellano-Garcia, G. Wozny and E. Reuter, 1998, Ind. Eng. Chem. Res., 37. Mujtaba, I. M, and S. Macchietto, 1992, Comput. Chem. Eng., 16, 273-280. Reid, R. C , J. M. Prausnitz and T. K. Sherwood, 1977, The Properties of Gases and Liquids, McGraw-Hill, New York. S0rensen, E. and S. Skogestad, 1994, Journal of Process Control, 4, 205-217. Wajge, R. M. and G. V. Reklaitis, 1998, Ind. Eng. Chem. Res., 37, 1910-1916. Wendt, M.; P. Li and G. Wozny, 2000, ESCAPE-10, May 7-10, Florence.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
625
Planning and scheduling the value-added chain M. Badell\ M.A guer\ G.Santos^ and L. Puigjaner ^Dept.Enterp.Management, Esc.Univ.Ing.Tecn.Indust EUETIB., Urgell, 187, Barcelona ^Cimade S.L., Rambla Catalunya 17 3° T , Barcelona 08007, Spain •^Chem.Eng. Dept., Univ. Politecnica de Catalunya, Diagonal 647, Barcelona 08028 This paper contains details of a computer aided decision-making tool under development for manufacturing and process industry that integrates financial/production trade-off planning, scheduling and optimisation in the enterprise value added chain.
Abstract The enterprise-wide optimisation of its value-added chain is performed using as target key performance indicators, KPI, that consider its business operation influence in the value added. A value-added chain can be seen as the value view of the supply chain. Having this in mind, new schedule strategies can be devised reshaping current developments for the supply chain management. The new approach is to perform an integrated and simultaneous fmancial/supply chain trade-off planning and scheduling optimising its economical/shareholder value. Consequently the result of this work is a decision-making tool for the optimal management of a value-added chain in manufacturing and process industries. The systematic deployment of more qualified information and the substitution of intuitive sequential decision making by simultaneous optimisation of activities could be the key of new enterprise-wide optimisation systems (Badell, 2001). Managers could make better business and technology decisions if they can use accurate simulation tools and timely process data to evaluate plans. The challenge is to obtain trade off solutions with common maximum performance measures while satisfying customers. This system gives to the financial-managers (company economist side), a tailor-made version of the scheduling tool helping them do their work. The overall simulation allows to test different alternatives during planning through a supply chain schedule with all the operation and financial information online with absolute transparency of the limitations and interactions occurring at plant and business level within each alternative. With this help it is possible to keep the visibility of the cumbersome interactions between the plant floor in multi-site levels and to change the today slave/blind position of business level during the supply chain decisionmaking procedures. The benefits of the value added chain management system are shown through a case study.
!• Introduction The aim of this work is to change the current slave/blind position of the financial manager in the supply chain decision-making procedure. With the system proposed their claimed problems can disappear, being able to improve enterprise competitiveness by knowing where the money is and where will it be due to production scheduling decisions; by knowing projections of financial and/or production decisions; by looking
626 at a rough-cut activity/process/product costing; by knowing the company cumulative value added/profit graphically showing distance from the breakeven point; by looking at interactions of cash flow with production at the plant floor; by making simulations with business level scope using updated financial status of the company. During simultaneous financial-supply chain scheduling, financial operations are placed in dominant position. In the classical time variable is overridden also money supporting the well known ''time is money". This time-money variable is included in the trade-off solution with the multiobjective economic optimisation of the whole system. A friendly intuifive interface allows the financial decision selection from a set of possible actions, operadons and indicators as discounts, investment in marketable securities, credit and loans, factoring, pledging, advancing, etc. needed to simulate/evaluate the convenience of the monetary uses or reproductions proposed. The status of value, cash and profit and the effects of this operadons are visualized in the simulation tool environment.
2. Previous work In the past the clerical role of the enterprise finance organizadon was centered on oversight and control consuming even 80% of working time on fiduciary dudes. Dramadc changes in business scenario have driven finance organizations to reconfigure its role at a more prevalent position and active participation as value producer, strategist, high qualified adviser, annalist and business partner in order to improve the corporadve value, and hence, its competitive advantages. The working time percent reladon of the old financial profile has to be changed from a 80 to 20% reladon to its inverse 20 to 80%, leaving the most time to fulfill the new tasks. The information demanded by decision makers to measure and manage performance requires greater quality, interoperability and time precision, so consequently a more complex computeraided tool. The majority of finance applications are commercial off-the-shelf packages with individual analysis of items. This system modifies finance tools when uses the simultaneous analysis of actions to better assess the current role and organize finance to add value. The system is not only flexible in the type of enterprise and the type of compeddve advantage, but also in the type of procedure and which performance measures and objecdve functions to use. Due to the system modular approach the applicadons are easily set in house establishing the adequate KPI to control and measure the performance of acdvides included in the objecdve funcdons to opdmize. The traditional financial data, usually overaggregated and too late informed, is translated into a single set of meaningful informadon given on line in a partnership interoperable platform shared by management, finance and supply chain operadons, including the value market diversified operations. Thus two systems must be shared in parallel in financial dudes in the management space: the classical transacdonal accounting system and the simuladon tool capable of giving optimal alternadves with in live overall information of acdvities interoperating to add value to corporadons. Almost all aspects of the cash management deterministic modeling problems have been discussed in the literature several decades ago. A review reveals that the majority of cash management deterministic models deal with a combination of three decision types: cash posidon management, short term investing and short-term borrowing.
627 Was Keynes in 1936 who introduced the transactions precautionary and speculative motives for holding cash. Lutz and Lutz, 1951, pointed that cash inflows/outflows are not normally synchronized so a positive cash balance is required to operate. Several deterministic inventory models were suggested by Gregory, 1976, Baumol, 1952, and Miller and Orr,1966 who consider the fluctuations in net cash flows as completely stochastic. The intertemporal features of cash management attracted attention (Beranek, 1963) to the uncertain possibility of using dynamic programming to solve the problem. In contrast succesful results with linear programming were first applied to finance by Charnes et al., 1963. Most of these studies concentrate either on capital budgeting and the problem of financial planning. Robichek, Teichroew, and Jones, 1965, focused the cash management with the intertemporal aspect of the problem for short-term considering several decision variables, but excluding securities transactions and cash balance. Orgler in 1969 used an unequal multiperiod linear programming model capturing the day-to-day aspect of the cash management problem in which includes the four major types of decision variables: payments, short-term financing, cash balance, and securities transactions with the amount and maturity that are defined and also derived by the model. Srinivasan, 1986, reviews his formulation about the cash management problem as a transshipment problem. A simpler network optimization approach gives advantage over conventional linear programs in computer performance. Lerner (1968) uses simulation by using the standard deviation of the elements in financial planning. Klein, 1998, created a knowledge based decision support system coupling financial planning and production planning for the financial analysis giving as output the balance sheets and income statements. The research in cash management modeling focused more on the specific types of decisions paying less attention to a broader objective. Although the intertemporal aspects in the financial environment were rigorously studied, the whole sequence of interrelated problems in an enterprise were not considered, likely due to the lack of adequate software and computers. But thanks to the segmented focus now are available polished methodologies that can be applied in the adequate computer framework.
3. Key Performance Indicators, KPI The selection of the set of operative and financial performance measures appropriate for the objectives to optimise in the integrated system is one of the most difficult and polemic aspects to decide. Further, for each enterprise the selection must capture its strategic aim to hold the competitive race that is also dependent on the kind of business. If response to market is the key, or if the product innovation is the advantage, or if it is service, then KPIs must have the appropriate scope to explore the different areas having this point of view. If several KPIs determine the exact advantage, its interrelations must be balanced in order to avoid the overlapping in detriment between them. In addition accounfing systems are failing in the companies where the assets are increased via intangibles assets. The KPI problem in software development is that usually management tools use profit as the primary indicator of corporate performance, viewing all aspects on those terms.
628
Figure 1. Value-added chain activities in interoperation and enhanced interoperable framework to be covered by in house meaningful KPls. The profit policy frequently gives more weight to the short term objectives and long term viability becomes threatened. As a result of this disadvantage good decisions to add value could be affected in an effort to diminish costs. Executives have a strong preference for this single indicator of performance which is well tested and gives unambiguous signals. The task of selecting KPIs is cumbersome: the accounting system based on the transactional principles and stated five centuries ago is in crisis. The financial and non-financial types of performance measurements has to be reconsidered in an effort to fill the gap or deficit of information about the enterprise intangible assets in the actual accounting and value market systems. Using multiple indicators is hard because they are difficult to design and to relate one to another. But the new trends emphasize on multiple indicators that help to draw the complexity of corporate activity and give more weight to treasury. What comes in and out - cash inflows/outflows - is unmistakable. Nevertheless, taking into account that net cash flow could only be post determined, the expected future treasury balance is the most reliable performance measure today available to set an objective to achieve. But the expected cash flows are mainly dependent on forecasts and non-financial (not quantitative) performance measures. Consequently, enterprises now have not well tested neither defined the adequate tools to provide reliable information for internal/external decision making. Dramatic changes in enterprises are now taking place. Main benefits are obtained through intangibles assets' values. Enterprises are now supported not by profit but by development expectative. While a new accountancy version is under construction or retrofitting, all enterprise system must have a flexible framework for easy KPI adaptation during this changing period. The tool designed is prepared to include traditional or shareholder value-added-based KPI objectives.
4. Case Study One of the tasks of a financial officer involves the cash management and related financial instruments on a short-term basis so as to produce extra revenues for the corporation that otherwise would go to the banks. The goal is to determine the optimal cash management decisions for the firm. An example implemented via Excel illustrates the modeling framework of the system capable of making the simultaneous opUmization of the supply-chain and financial operations. By means of a web-based agent customers
629 orders are sequenced using the TicTacToe algorithm and a timing algorithm that optimizes the schedule giving as output a Gantt chart with the timing of supply chain operations and financial needs. A linear programming model with rolling horizon possibilities uses the set of cash inflows and outflows optimally scheduled taking into account customer satisfaction. A simplified objective function to maximize in time horizon T is the sum of payments X^j and marketable securities (y,j - z,j) revenues discounting the costs of a credit line w/,^ Technical coefficients C, D, E, and F adjust quantities depending on the timing of periods /, g, j , h (payment, maturity, sale periods, etc.) where actions incur. j = 2 j=\
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It is chosen a time horizon of a week (128 h), divided unequally into twelve periods (see figure 2). The producfion plan considers sixteen batches to fulfill the orders received of eight products. The information of the monetary needs is included in the financial model. Salaries, utility costs, and fixed costs will not be taken into account. Production expenses during the week will consider initial zero stock and raw material needs. The economic situafion of the case study is based on the information given in a financial balance sheet. Inifial capital considered will be 1000 monetary units (mu), two times the minimal net cash flow 500 mu beneath which a short term loan must be requested. This value was assumed taking into account the variability of cash outflow, a random variable with normal distribution and mean value 372 mu per day for a week producfion cycle. The portfolio of marketable securities held by the firm at the beginning of the first period includes several sets of securities with known face values and maturity periods, only one maturing beyond the horizon. All marketable securities can be sold prior to maturity at a discount or loss for the firm. A short term financing source is represented by an open line of credit. Under the agreement with the bank, loans can be obtained at the beginning of any period and are due after one year at a monthly interest rate of 0.5 percent. Early repayments are not permitted. The payment decisions to be considered correspond to accounts payable with 2 percent 10 days, net 30 days terms of credit (2-10/N-30). All payments of raw materials are fulfilled within the horizon. It is assumed that all bills are received in the first half of the periods and that payments, including sales of final products, are made beginning the periods.
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630 Any part of the bills can be paid either at the first ten days with a 2% discount or at face value after 30 days. It remains to be decided upon what part of the bills to pay in which period. The net fixed cash flows, costs and revenues associated with the transactions in marketable securities are given. The company has heuristic rules: take all discounts, if are all possible; use fully the line of credit; if it is necessary to get a discount, sell not mature marketable securities. Several possibilities to "balance" the cash budget in periods are obtained introducing subjective management constraints. Revenues of 6% were obtained besides the not measured opportunity cost of assuring liquidity all time.
5. Conclusions and future work Good manufacturing practices in financial management must begin with an appropriate logistic support to solve the present insuficiencies. The focus of this work has as target the design of adequate tools able to support in time optimal budgets. This work constitutes an advance since not only syncronizes cash inflows and outflows and assures a safety cash stock but solves the perishable property of money, making full use of the possibilities of the short term funding. The benefit of using the idle cash in financial operations gives revenues of more than 6%, enough to compensate the implicit monetary losses to its perishable character. This, together with other advantages already mentioned, leads to a competent, rigorous and consistent financial system prototype adequate for the competitive race. Financial support from European Community is gratefully acknowledged (VIPNET and GCO projects).
6. References Badell, M. and L. Puigjaner, 2001, Discover a powerful tool for scheduling in ERM systems, Hydrocarbon Processing, 80, 3, 160. Baumol, W. J., 1952, The Transactions Demand for Cash: An Inventory Theoretic Approach, Quarterly Journal of Economics, 66, 4 , 545. Beranek, W., 1963, Analysis for Financial Decisions, R.D. Irwin, Homewood, Illinois. Chames, A., Cooper, and W. Miller , 1963, Breakeven Budgeting and Programming to Goals, Joumal of Accounting Research, 1, 1, 16. Gregory, G, 1976, Cash Flow Models: A Review. Omega 4, 6, 643. Ijiri J., Levy, F. K., and Lyon, R. C, 1963, A Linear Programming Model for Budgeting and Financial Planning, Joumal of Accounting Research, 1, 2, 198. Keynes, J. M., 1936, The General Theory of Employment, Interest and Money, Harcourt Brace and Company, New York, 170, 194. Klein, M.R., 1998, Coupling financial planning and production planning models. Xth Intern. Work. Sem. on Production Economics Proceed., V.3., Innsbruk, Igls, Austria. Lemer, E., 1968, Simulating a cash budget. California Management Review. 11,2, 78. Lutz, F. A. and Lutz, V., 1951, Theory of Investment and the Firm, Princeton Univ., NJ. Miller, M. H., and Orr, R., 1966, A Model of the Demand for Money by Firms, The Quarterly Joumal of Economics, 80, 3, 413. Orgler, Y. E., 1969, Unequal-period Model for Cash Management Decisions, Manag.Science, 14. Robichek, A. A., Teichroew, D. and Jones, J. M., 1965, Optimal Short-term Financing Decision, Management Science, 12, 1, 1. Srinivasan, V., 1986, Deterministic Cash Flow Management, Omega, 14, 2, 145. Weingartner, H. M., 1963, Mathematical Programming and the Analysis of Capital Budgeting Problems, Prentice-Hall Inc., Englewood Cliffs, NJ.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Published by Elsevier Science B.V.
631
Optimisation of Oilfield Development Production Capacity Richard J. Barnes, Patrick Linke and Antonis Kokossis Centre for Process & Information Systems Engineering, School of Engineering, University of Surrey, Guildford, GU2 7XH, U.K.
Abstract The paper presents an optimisation method for the systematic evaluation of the economic production capacity of an offshore oil production platform. The problem is formulated as sequential mixed-integer linear programs to keep the mathematical comlexity at a low level. Continuous variables represent individual well, jacket and topsides costs and discrete variables are used to select or deselect individual wells within a defined field grid. The mathematical formulation is concise and efficient to enable future extensions to consider uncertainty in reservoir performance and actual development costs. The new method is illustrated with two hypothetical fields that are based on real-life examples in the North Sea.
Introduction The production capacity is the most critical design decisions in the development of a new oil field as it defines the overall size of the facility and the rate of revenue generation. Particularly for an offshore installation, it can be expensive and difficult to change the capacity of an installation after it has been installed. If the facility has been oversized, it is not economic or practical to replace existing processing equipment with smaller units. If the facility were undersized, it is equally difficult to install an extra train of processing equipment or to install a larger processing train. It is possible to debottleneck installations, but this can be very costly. It is current practise to design the facility to produce between about 10% and 20% of the reserves each year. Economic analysis should ideally be used to explore the effect of different production capacities, but there are no guidelines or support systems to determine the most economic capacity from basic field data. The objective of this study is to identify the factors affecting the most economic capacity for the facilities and to develop a method of calculating the optimum design capacity. Such a method could either make savings in avoiding building an oversized installation, or produce extra revenue by ensuring that the installation was built with adequate capacity. Previous work in oilfield infrastructure planning has produced large scale MILP (Nygreen et al., 1998; Iyer et al., 1998) and MINLP (van den Heever and Grossmann, 2000) formulations. This work will draw on problem domain knowledge to reduce the overall problem into sequential, easy-to-solve but realistic MILPs.
Problem representation and model developments This work aims at the development of a generic model for an offshore field that can be used to identify the criteria and capture the relationships between the different field
632 parameters. In view of the large number of possible design aspects in oil field exploration, the model needs to strike a fme balance between its size and the accuracy of its predictions so that efficient optimisation techniques can be deployed for decision making. For the model to reflect the major system trade-offs, it needs to capture the reservoir areal extent, the productivity of individual wells, drilling costs, platform cost, processing and topsides cost, export pipeline cost, operating and maintenance cost, export pipeline cost, and oil revenue. These variables directly or indirectly affect the platform design capacity. We develop a model that captures these interactions from which the configuration giving the greatest financial return can be determined by optimisation. The complete model that has been developed is an MILP and will be the subject of a future publication. The main modelling aspects and assumptions are discussed in the remains of this section. The reservoir is assumed to act as a single tank and no account of drawdown around a particular well is considered. However, in the fully developed model this will be an unrealistic assumption. The reservoir is divided up in a grid with each cell being selected to give a reasonable and typical well spacing assuming one downhole target in the centre of each cell. The peak production rate of each individual well is specified as an input parameter. The model is structured such that other depths and peak production rates can be incorporated. The field is assumed to be offshore. The model is capable of modelling fields in other locations and water depth with minimal alteration. Well and drilling costs are based on the total length of the well drilled to the top of the reservoir. Two types of wells are considered. The first is a vertical well in which there is no well deviation from the centre. The second type of well geometry is known as build and hold. In this type of well, the initial section is drilled vertically, and then a deviation angle is built until a straight projection from this point intersects the target location. The cost of each well is a combination of a fixed cost for the wellhead, design and project management and a variable costs such as casing and actual drilling costs that depends on the well length. The total cost of the production platform accounts for the costs of the jacket, topsides and pipeline. An in-house cost estimation package was used to estimate the cost of these three portions of the platform over a wide range of Design Production Rates (DPRs). Quadratic equations to describe the costs were fitted to enable cost calculations of platform depending on the production capacity. For each DPR a target production profile was generated: 25% of the DPR is produced in Year 1, 50% in Year 2, 75% in Year 3 and 100% in the fourth and subsequent years until 85% of the recoverable reserves have been produced. After this point, the field is assumed to be in decline resulting in an exponential decline in production. The actual production profile assumes that the full capacity of each selected well is produced and that there is sufficient capacity in the production facilities to process and export this flowrate. This assumption results in more production than the target profile, particularly when the field is in decline. This profile represents a similar practise to that which would be adopted in a real field. Each well production rate is reduced by a function of the cumulative oil produced at the end of the previous year and the recoverable reserves (Individual Well Profiles). This reduction represents the general decline in productivity with field life and is described by the production reduction factor
633
"^PR
RR - 0.5CP RR
Where RR are the recoverable reserves and CP is the cumulative production. For each field, an optimum location for the drilling centre for the development is initially calculated. This is determined from a simplified model of all possible downhole targets. For each surface location, represented by the grid locations, the lowest drilling cost wells are determined to meet or exceed the specified production rate. The optimum drilling centre was determined using a mean value of the DPRs to be investigated for that particular field. The minimum well cost was calculated for each grid location and the minimum cost of all locations was selected as the drilling centre. This optimisation problem gives rise to a MIL? formulation to explore all possible drilling centre locations. Initially, the cost of drilling to each downhole target from each location is calculated. These parameters are used in the optimisation model with two scalars representing the coordinates of the drilling centre. The existence of each well is represented by a binary variable. For each drilling centre location, a minimum total cost is calculated by solving the MILP problem. This minimum cost corresponds to the optimum drilling centre location. This was then used as a fixed location for subsequent calculations involved in planning for field development. The financial analysis is performed using a spreadsheet. Costs for drilling, platform and topsides, and export pipelines are annualised over a period of four years before the field comes into production. After this time, wells are assumed to be drilled in the year preceding the year they are required in operation. The total capital expenditure for the jacket and topsides is annualised over a period of four years in the proportions of 15%, 25%, 30% and 30%. The pipeline costs are assumed to be annualised over a two year period with expenditure being equal in both years whereas the drilling costs are not adjusted for inflation. The operating cost is assumed to be 4% of the capital expenditure and not adjusted for inflation. The revenue from the field is calculated for a range of oil prices. From this data the Net Revenue and Cash Flow is calculated for each year. The Net Present Value at a specified discount rate and the Internal Rate of Return, IRR, are then calculated for each case.
Implementation The basic model to determine the drilling centre locations for each year of the project has been implemented in the General Algebraic Modelling System (GAMS) and solved using the CPLEX solver. The field model is implemented using a mathematical modelling package and a customised program written in C++. This structure was chosen to reduce the complexity of the overall problem when the full field life is modelled and also to more easily manage data transfer between the optimisations performed in the different years of the project. The customised program generates the GAMS input file for each year, executes the optimisation of the basic model, extracts the required information from the GAMS output file and then generates the new GAMS input file for the subsequent year. The program also calculates the target production rate and the Production Reduction Factor, FpR after each year. At the end of each set of runs at a specified production rate, the data is transferred to a spreadsheet to calculate
634 the economic performance of the field at that production rate. Results are reported as an Internal Rate of Return (IRR). Once the optimum drilling centre has been determined, the development of the field is determined for each year of the exploration project in a subsequent optimisation stage. A DPR is specified for the particular case to be optimised. Annual production rates follow the profile described in the previous section. The objective is to minimise the cost of the wells. The binary variables are handled such that wells selected in the previous year are forced to exist in the current year and only those idle wells can be selected. Moreover, the Production Reduction Factor is incorporated into the production equation to capture general field and well productivity decline. The coordinates of the fixed, single drilling centre from the previous optimisation stage and the minimum production rate for the year are defined as input parameters. A set of downhole well targets are defined by coordinates and potential production rate. This data is held in a separate field specific data file to enable simple change of field characteristics. Parameters are also defined to describe well geometry and well cost. Well length and drilling costs are then calculated from the fixed drilling centre and the downhole target for each well. The well cost remains constant for any given drilling centre so that the original problem is reduced from a MINLP to an easy-to-solve MILP with the existence of wells being associated with a binary variable. The objective function is to minimise the total drilling cost from the single drilling centre. An inequality constraint on the specified production rate is also defined. For each year, the MILP is implemented in the General Algebraic Modelling System (GAMS) and solved using the CPLEX solver. A C++ program is created to manage the optimisations and the data transfer between the optimisation problems solved for the individual project years.
Examples Two hypothetical fields have been modelled to determine the effect of DPR. Each field has been based on a 500m by 500m grid. This grid is used to represent the downhole targets of each of the potential wells to be drilled from a single drilling centre. Two hypothetical reservoirs have been modelled, one with recoverable reserves of 500 MM bbl comparable to a large North sea field such as Nelson or Fulmar; the other with recoverable reserves of 50 MM bbl comparable to one of the smaller North sea fields such as Fife or Banff The grid representations of the two fields is shown in Figure 1. The West Field contains 224 potential well locations and has 500 MM bbl recoverable reserves. The model was run with DPRs of between 50,000 and 450,000BPD. IRRs were calculated using an oil price of between $5 and $30/bbl. These results are shown in Figure 2. Above a DPR of about 150,000 BPD, the IRR curve initially increases with DPR at a decreasing rate and is essentially flat over a wide range of DPR. A peak is not observed. The North Field contains 59 potential well locations and has 50 MM bbl recoverable reserves. The model was run with DPRs of between 5,000 and 70,000BPD. The field requires a minimum oil price of $15/bbl to be economic and generate a positive IRR. IRRs were calculated using an oil price of between $15 and $30/bbl. These results are shown in Figure 3. Above a DPR of about 25,000 BPD, the IRR curve is essentially
635 flat, indicating similar returns irrespective of the DPR. Again a distinct maximum did not occur.
Conclusions The present model does not indicate any clear optimum in the Design Production Rate for these fields for any oil price considered. The results indicate that, provided the fields are developed with a DPR in excess of a minimum rate, the IRR will be within 5% of the maximum. These results appear to indicate that the decision on DPR is less important than perceived by the practitioners. Future work will address the identification of the factors causing the stabilisation of IRR. It is also intended to enhance the model so that the Production Reduction Factor is applied to individual well production rather than to total field production. The assumption of the reservoir acting as a single tank is an over-simplification that could be contributing to the present results. -5.000 -4.500 •4,000
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636 The current assumption of build up to full production capacity is also perhaps somewhat conservative and the effect of a delay in reaching full production will also be investigated. This change will have the effect of significantly increasing the discounted revenue from the field. Another area to be investigated further is the effect of uncertainty. Uncertainty arises in a number of different areas, including the estimation of recoverable reserves, the actual cost of the installation compared with the original estimate, and the well drilling cost and well productivity. It is intended to investigate these effects as the continuing work in this study. Return on Investment, 500 MM bbl Reserves
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References Iyer RR, I.E. Grossmann, S. Vasantharajan, and A.S. Cullick (1998). Ind. Eng. Chem. Res. 37 (4): 1380. Nygreen, B., M. Christiansen, K. Haugen, T. Bjorkvoll and O. Kristiansen (1998). Ann. Oper. Res. 82,251. Van den Heever, S. and I.E. Grossmann (2000). Ind Eng. Chem. Res. 39, 1955
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
637
Optimal operation of fed-batch sugar crystallisation with recycle Ben H.L.Betlem, Sander van Rosmalen, Peter B.Thonus, Brian Roffel Dept. of Chemical Technology, University of Twente, The Netherlands
Abstract In sugar batch crystallisation, multiple sequential crystallisers (pans) are necessary to obtain the required exhaustion of the feed. In this simulation study, the production rate is optimised by distributing the exhaustion effort between the first two pans. To describe the crystal size distribution (CSD), a dispersion based population balance has been used. An appropriate CSD of the seeds and a nucleation equation fulfil the initial and boundary conditions. In the growth rate kinetics, supersaturation and activation energy are formulated as a function of the impurities. The pan simulations as well as the interactions between the first two pans have been fitted by industrial data. Optimisation of the production rate implies maximisation of the exhaustion of the first pan.
1. Introduction Many (fed-)batch processes can be optimised by partitioning the operation in a production step and one or more subsequent separate exhaustion steps. The intermediate products of the exhaustion steps are recycled to the feed of the first step (Figure 1). By this partitioning, the production step becomes easier as the exhaustion remains limited and the feed is enriched by recycle flow, which has a higher purity. Also, the exhaustion becomes easier since the product of the exhaustion steps satisfies a lower requirement. However, a larger recycle flow means a larger throughput in both steps. Therefore, a maximum in the production rate may arise for a certain distribution of the exhaustion between the production step and exhaustion step (Betlem and Roffel, 1997). In the case of sugar fed-batch crystallisation, it is not possible to produce pure sugar crystals with the required size from impure syrup in one process step, while ensuring that the syrup is sufficiently exhausted. The sugar purity of the feed dry mass contents is about 93%. The first crystallisation step produces 720 jiim crystals with a purity of nearly 100%. The subsequent exhaustion step produces 520 jim crystals, which are melted for recycling. The syrup is exhausted to a remainder with a purity of 75%. The
product
< •<±>*
recycle production step
exhaustion step
remainder (recovery)
Figure 1 Batch operation consisting of production and exhaustion phase.
638 distribution of the exhaustion between the two steps will strongly depend on the influence of the impurities on the growth rate. When the influence is higher, the exhaustion of the production phase will be more restricted. In this article, first, a suitable crystallisation model will be presented. Next, the model parameters will be fitted on three industrial pans with large differences in impurity levels. The fit is performed according to a sensitivity analysis. A number of simulations have been made to analyse the influence of the control variables on the production rate. Finally, conclusions will be drawn about the optimal recycle strategy. The simulations are performed with the program package gPROMS and the partial differential equations are solved numerically by the second order backwards finite difference method.
2. Sugar Crystallisation Model The process concerns a fed-batch evaporation crystallisation of sucrose in a mixed suspension vessel at low constant pressure. The heat supply is condensing steam. The major assumptions for the process are the following. • • • •
The vessel is sufficiently mixed to consider all variables to be lumped. The controllable heat supply can be set directly. The temperature of the boiling suspension is in equilibrium with the vapour. This means that the temperature directly depends on the pressure and composition. Only the net growth rate and nucleation are described.
One fed-batch step consists of several phases. First, a fixed part of the feed juice amount is initiated in a pan. This is thickened until a pre-defined supersaturation level has been reached. Then, a crystal seed slurry is introduced and the crystallisation phase starts. A constant feed juice flow and heat flow are added until all feed has been consumed. Next, the suspension is centrifuged to separate the crystals from the mother liquor. To wash away the impurities thoroughly, hot water is injected in the centrifuges. For the identification and simulation, only the crystallisation phase is considered. For the optimisation, also the thickening phase and the centrifuge separation are included. The driving force of crystallisation is the supersaturation of the solution. The supersaturation /SZs can be modelled according to Eq. 1. In this equation 5, /, and W are the weight of the sucrose, impurities and water contents of the mother liquor. The sucrose content at equilibrium, E^^^, can be calculated according to the Wiklund equation. Some impurities, such as salts, will increase the supersaturation, whereas sucrose like substances, such as raffinose, have an opposite effect. For low impurity-water ratios, (//W), the second term will result in a slightly decreased Seep whereas for higher ratios the third term is dominant and Seq will increase linearly with the ratio. AZ5=5/(5 + / + w ) - 5 , j ( 5 , , + / + l v )
(1)
S,, /W = mklund{HV^\ S,,,^„,, /W
(2)
Wi/t/«nd{//w}= a + (1 - a) • exp(-c • //W) +fo• //IV
(3)
639 It is generally accepted that the mechanism of crystal growth is determined by two steps: mass transfer by diffusion from the mother liquor bulk to the crystal surface followed by surface integration. Both rate constants are temperature dependent. The kinetic growth model of Eq. 4 is often used. In the presence of raffinose, sucrose crystals grow with approximately a second-order dependence on supersaturation (Liang et ai, 1989). Eq. 5 takes the increase of the activation energy due the presence of impurities into account. It contains a Langmuir term considering the influence of the surface coverage by impurities. G=A{//lv}.exp(-^^VAZ/
(4)
,with
EAl/w}=E,^^^,,(l^a,(e,)'')
Oj =-^~^
(5)
l-\-bi • I/W Theories to describe crystallisation are based on constant growth rate, size-dependent growth rate, or growth rate dispersion (GRD). GRD means that crystals of the same size grow under the same conditions at different rates (Ulrich, 1989). This phenomenon should only be observed when the surface integration step is rate controlling. Although, sucrose crystallisation is mass transfer rate controlled, crystals grown exhibit GRD (Liang, 1987). So, the population density for growth rate dispersion can be given by:
In Eq. 6, population density n* [m^] is the amount of crystals with characteristic length Lc at time t. G and Dg describe the growth rate and the diffusivity. The initial condition is the CSD of the seed slurry. Its properties can be effectively described by: Az*(4,0) = a - ( L , ) ^ . e x p ( - 7 . L , ) ,
a,p.y>0
(7)
The boundary condition is determined by the nucleation of new crystals. Crystal generation from solution is called primary nucleation and requires high levels of supersaturation. Secondary nucleation is the term used to describe the formation of new crystals from already present crystals. This nucleation is more likely to take place in industrial crystallisers, as it requires a supersaturation level which is considerably lower (Myerson, 1993). Empirical expressions are used to describe the nucleation. n (0,0 = —^^•^=
,where
B^c^G'-
(8)
From the population distribution the i^-moment can be derived according to Eq. 9. Subsequently, the mean aperture MA and the coefficient of variation CV of the CSD can be calculated.
640
<-jn-L-
with MA = ^
dU
L,=0
-yjni^/m^ -MA ,and C V = ^a^ = MA
(9)
In addition to the Eqs. 1 to 9, which model the crystal formation, mass balances and an energy balance are needed for the accumulation of crystal sugar, dissolved sugar, impurities (non-sugars) and water. Nearly all balances are simple differential equations. The amount of crystal sucrose is determined from the third momentum by considering the density psc. and volumetric shape factor ky, which is assumed to be constant. The physical properties are taken from the Sugar Technologist Manual (Bubzik et ai, 1995). This concerns properties such as: (1) solubility of sugar, (2) viscosities of the mother liquor and the massecuite (overall pan contents), (3) specific heat capacities of crystal sugar, feed, mother liquor, and the massecuite, (4) densities of crystal sugar and pure or impure, under-saturated, saturated or supersaturated mother liquor, and (5) boiling point and boiling point elevation of the mother liquor, which depends on pressure, solid content (brix) and impurity level.
3. Model IdentiHcation and Simulation of Industrial Pans For identification, limited information was available: the season average of the initial and final compositions of the pans, the batch times and some global indications about final MA, CV, viscosity, and supersaturation level. The average feed rate, average heat supply and seed slurry amount, which are the input variables, are determined by the
30% 29% 28% ^ 27% 26% -A-pan
-A-pan
25%
B2-pan
B2-pan
C-pan
C-pan 150
200
250
300
350
150
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i
A-pan ^ c JO
1.08 -
a
1.06 -
•a 3 m
B2-pan
1^
!
It
i
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1 02 -
; ! •
1.00 c
50
100
i
•
0 100
O o
0 010
•
./T
150
200
time [min]
300
350
250
300
1
350 1
.••••"'"
/^'^
\ 1
^.,
^
250
1 000 1
j
C-pan
3 S
200
time [min]
time [min]
-A-pan
^
>
1
B2-pan ''
H
C-pan
0 001 c
50
100
150
200
250
300
350
time [min]
Figure 2 Simulation results of MA, CV, supersaturation, and viscosity oftheA-, B-, Cpan during a batch.
641 final values of the massecuite, brix and MA. From a sensitivity analysis, it was found that neither the supersaturation exponent, /c, nor the frequency factor, A, exhibit any discernible effect on the key output variables. The average activation energy, E^, has been identified by the final crystal mass while the diffusivity factor, D^, can be derived from the final CV. The simulations of the A-, B-, and C-pan (Figure 2) show that after thickening, supersaturation is high. Therefore, initially the MA increases fast and consequently the CV decreases at the same pace. During the batch the supersaturation level remains low, consequently the MA and CV increase slowly. However, simultaneously the viscosity increases exponentially and determines the possibilities to control the exhaustion.
4. Production Curve and Optimal Operation The process requirements are the MA of the A-pan product and the exhaustion of the Bpan. The available manipulated variables are: the feed rate, the heat supply rate, the initial juice volume (relative to the feed amount), the initial supersaturation, and the seed slurry amount. The initial values are taken constant. The seed slurry amount is used to obtain the requirements. Therefore, the only remaining control is the feed to heat ratio. This determines directly the supersaturation. In sugar crystallisation, producing a narrow crystal size distribution (CSD) is required. To achieve this objective, the supersaturation level should be kept within the metastable zone without causing nucleation. To determine the influence of the feed to heat ratio, production curves are drawn. These curves reflect the production for different batch times, while maintaining the product requirements (Rippin, 1983). The different batch times are obtained by varying the feed, while the heat supply is constant. Figure 3a shows the normalised production curves of the A-, B- and C-pan. The normalisation is obtained by plotting the relative production against the batch time corrected for differences in heat supply. For feed without impurities, the production curves of the pans coincide. From these curves and additional simulations, the influence of the impurities can be determined. The decrease in relative growth rate can amount up to 25% for the C-pan (Figure 3b).
00% -, y/^
80°/ jT
70%
^ ^
"^
^^^ ..>•'..-'" X/ ^, .- * ' i A-pan ^ /' ' i B-pan V^
no.
20°/
^
j . - - -C-pan
10%
, 1
100% purity ,
0% 0 40
0.60
0 80
100 average Impurity-water ratio [•]
relative batch time [-]
(a)
(b)
Figure 3 A-, B-, C-pan. (a) normalised production curves, (b) influence of impurities on the growth rate, (arrows indicate the three industrial pans.)
642 80% •o £ 70%
L
1 1
^,„^91
.. ._ ___/!
/
87,/^
89/87 91
30%
/
yss
• A-pan, MA=700mm
81
4 B-pan, purity syrup 75% 20% 200
300
400
85%
(a)
90%
purity A'syrup
batch time [min]
(b)
Figure 4 Determination of the optimal operation . (a) production curves of the A and B-pan. (b) production rate inclusive thickening At the lower limit of the production curve (Figure 3), at high feed rate no supersaturation is obtained, whereas at the upper limit nearly all water is evaporated and the viscosity is limiting. The tangent from the origin to the curve indicates the maximum production rate. The A-pan appears to have a constraint optimum, whereas the B-, and C-pan have a free optimum. In Figure 4a the production curves of the A- and B-pan are shown. The requirement for the A-pan is a product MA of 720 ^im , whereas for the B-pan the requirement is 75% syrup purity. The obtained distribution of the exhaustion effort between the pans is indicated by the purity of the A-syrup. The production rate is optimal when the A-pan is exhausted to 81%. A decrease of the impurity level does not justify a larger recycle.
5. Conclusions A model for fed-batch evaporation crystallisation has been developed, which describes the dynamic behaviour and production of the different pans well. The influences of the impurities on the production are implemented through functions for the supersaturation and the activation energy. From the simulations it was found that the growth rate decrease due to impurities are approximately 5%, 15% and 25% for the A-, B-, and Cpan respectivily. As this influence of the impurities is restricted, the optimal recycle is a constraint optimum obtained by maximum exhaustion of the A-pan.
References Betlem, B.H.L., Roffel, B. (1997). Integrated manufacturing of cyclic/batch processes. ICSC-WMC 97, Auckland, New Zeeland, 428-432. Bubnik, Z., Kadkec, P. (1995). Sugar Technologist manual. 8* edition, Bartens. Liang, B., Hartel, R.W., Berglund, K.A. (1989). Effects of raffinose on sucrose crystal growth kinetics and rate dispersion. AlChE Journal, 35, 2053-2057. Myerson, A.S. (1993). Handbook of Crystallisation. Butterworth-Heinemann, Boston. Rippin, D.W.T. (1983). Simulation of single- and multi-product batch chemical plants for optimal design and operation. Comp. Chem. Engng, 30, 137-156. Ulrich, J. (1989). Growth rate dispersion - A review. Cryst. Res. Technol., 24, 249-257.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
643
Economically Optimal Grade Change Trajectories: Application on a Dow Polystyrene Process Model Wim Van Brempt\ Peter Van Overschee^ , Ton Backx^, Jobert Ludlage^, Philippe Hayot^, Louis Oostvogels\ Shamsur Rahman^ ^IPCOS, Technologielaan 11/0101, 3001 Heverlee, Belgium [email protected], [email protected] ^IPCOS, Bosscheweg 145a, 5282 WV Boxtel, The Netherlands [email protected], [email protected] ^DOW Benelux, P.O. Box 48, 4530 AA Terneuzen, The Netherlands [email protected], [email protected] ^The Dow Chemical Co., 1400 Building, Midland, Michigan 48640, USA, srahman @do w.com
Abstract A novel dynamic optimizer PathFinder has been applied to a dynamic model of a Dow polystyrene production facility at Tessenderlo, Belgium. PathFinder optimizes grade transitions subject to an economic cost function. Introduction of process constraints allows for a gradual migration from the currently used transition towards a more optimal transition. The results show a significant improvement in added value during a grade transition.
1. Introduction The chemical process industry is facing a huge problem to increase their capital productivity. A solution to this problem is demand driven process operation such that exactly these products can be produced that have market demand and take price advantage of a scarce market. Flexible operation of production is therefore required [Backx, et ai, 1998]. A new integrated process control and transition optimization technology is needed for this purpose. A very important requirement for this technology is to enable the calculations of feasible and economically attractive grade transitions. The idea of optimization of grade transitions in the polymer industry has been introduced by McAuley [McAuley and MacGregor, 1992]. Based on rigorous dynamic models optimal open-loop paths are calculated. The cost function has been improved into a more straightforward economical framework [Van der Schot et al, 1999]. In order to cope with the strong non-linear cost function, the PathFinder rigorous model based dynamic optimizer has been developed. An application on a Dow polystyrene production facility is discussed. The paper is organized as follows. In Section 2 the formulation of an economic optimization criterion is given. Subsequently, in Section 3 a framework for integration of trajectory control and trajectory optimization is explained. In Section 4 relevant aspects of the polystyrene solution process are described. Finally,
644 Section 5 describes the application of PathFinder on a Dow polystyrene production facility.
2. Economical optimization criterion Since the incentive for the elaboration of optimal grade changes is merely economical, it is reflected in an economically driven optimization criterion (Eq. 1). The goal is to maximize added value (AV) during a time horizon T while making the transition from one grade to another grade. AV{T) = j price{t)throughput(t)dt - J ^ feed^ (t) cos /, (t)dt
(1)
The first term accounts for the benefits gained during the trajectory by producing the desired end-product. It depends on the production throughput and is a highly non-linear function with regard to product price. The high non-linearity arises from the fact that mostly a good price is paid for on-spec material, while the market price for wide-spec or off-spec material is significandy lower. Although integration softens the non-linearity, it remains nevertheless very severe. The specifications are typically expressed in terms of product properties, which are themselves non-linear (dynamic) functions of the process conditions. The second term in (Eq.l) accounts for the economic costs of all the feed stock materials and cost of plant operation. The optimizer searches for the optimal process manipulations, such that the resulting trajectory is economically optimal. The relation between optimization parameters, the process and the economic cost is shown in Figure 1. It is clear that a dynamic process model is needed to enable the calculation of the Added Value given the applied process manipulations. Objective Function
t MV's= Optimization Parameters
PV's
Added Value
Figure 1. Relation between Manipulated Variables (MV's), Process Variables (PV's) (such as flows, product properties, holdups...) and Added Value End-point constraints on CV's, path constraints on CV's and input constraints (absolute boundaries and rate of change constraints) on MV's are added to the optimizer restricting the optimization freedom. These constraints guarantee a safe and feasible operation during the transition and also guarantee that the desired product properties and production level are achieved after the transition. They also constrain the optimizer freedom such that the new trajectory doesn't differ too much from the initial trajectory. The last reason is important when one has no blindfolded confidence in the process model. Adding constraints will allow one to migrate slowly from a well-known recipe to a new recipe. PathFinder is a robust and fast solution for the above optimization problem. Though the objective function is strongly non-linear, typically 5 up to 10
645 trajectory simulations and model linearizations are needed for the cases that have been analyzed (compared to 500 up to 1000 model evaluations with a SQP optimization scheme). These model evaluations are the bottleneck for a faster calculation time. In [Van Brempt, et ai, 2001] relevant implementation topics are discussed.
3. Integrated Trajectory Control and Optimization Technology A general framework has been set up in order to cope with the challenge to integrate trajectory optimization and trajectory control [Van Brempt, et al, 2000]. The key idea is explained in Figure 2.
l-%
Latest Process Model
PathFinder
^opi
JL
+
Optimal Trajectory Recioe
<—
Off-Line On-Line
yopt
MPC INCA®
Ay^
+Au —w
y
Process
Extended Kalman Filter
Figure 2. Integration of MPC control technology and optimization technology
PathFinder calculates off-line dynamic economically optimal grade change recipes. These manipulated and controlled variable trajectories are as such applied to the process. The controller only corrects for the deviations Au and Ay ('delta mode') from the process input-output setpoints Uopt and yopt that are given by the optimizer. The deltamode guarantees a best of both worlds operation [Van Brempt, et al, 2000]. Indeed, it would be a pity to have the trajectory, which has been carefully designed with nonlinear knowledge, overridden by a linear model controller. Therefore this trajectory is applied as such to the process and the controller is only allowed to shift the deviations of the input-output trajectory (Uopt, yopt) between the controller input and output. As explained in the previous section, the long trajectory simulation time determines the optimization calculation time. Therefore PathFinder is started several hours before the trajectory has to be initiated, with up-to-date market conditions and the latest instance of the rigorous model that is known, with the latest state updates in case an Extended
646 Kalman Filter is available. Once the optimal trajectory is calculated and acknowledged by a Product Engineer, it is sent to the controller environment. It will be graphically available to the panel operators, such that they can intervene depending on actual circumstances.
4. The Polystyrene Solution Process Polystyrene is mostly produced using a solution process. A simplified layout of a typical polystyrene solution process can be found in Figure 3. The process consists of a combination of several plug flow reactors. Styrene, a solvent and in some cases an initiator are fed to the first reactor. Reactors are usually operated at sequentially higher temperatures with a final conversion at 60-90%. Unreacted monomer and the solvent are separated from the polymer under vacuum. The hot melt is then pelletized while the monomer and solvent are condensed and recycled. Additives may be added at different places in the process. Various polystyrene grades can be produced on the same process using a carefully chosen set of flow, temperature and pressure setpoints (SPi) that we will refer to as a "recipe". A given polystyrene grade will be characterized by a set of properties (PROPj). When changing from one polymer grade to another polymer grade, setpoints must be moved from one recipe to the other, driving the process through a zone where off-specification product is made. Typically the transition path for recipe setpoints will be selected to minimize production of low value off-spec product. Dow developed a rigorous dynamic model for the specific process used for the production of polystyrene at Tessenderlo, Belgium. The model has been validated over the entire operation range. In order to minimize plant-model mismatch an on-line state estimator has been implemented. Styrene, Solvent, ^ r
>
1
¥
¥ V
1
1
1
PS
Figure 3. Schematic description of a typical solution polystyrene process
5. Application of PathFinder on the Dow Polystyrene Process PathFinder's optimization technology is applied on a model of the Dow polystyrene production facility at Tessenderlo, Belgium. Fourteen variables were used as MV's for PathFinder including feed flows and composition setpoints and temperature setpoints for the reaction as well as for the separation sections of the process. For each variable 13 fixed move times were defined, resulting in 182 degrees of freedom for the optimizer. Path constraints are applied on 8 process variables, rate of change constraints on 10 MV's and absolute boundaries on all 14 MV's. Optimal trajectories were
647 subsequently calculated for two different market situations depending on the price of off-grade product (Table 1). Indeed, off-grade product can be used in low-end applications and its price is therefore subject to offer and demand fluctuations. Both market situations are characterized by a considerable difference between the on-spec and off-spec material price. In the first case off-spec material represents a serious loss compared to the raw material, while in the second case off-spec material can be sold with a benefit compared to the monomer that is being used. Table 1: Two different Market Conditions Price Case 1 High Onspec - Offspec Offspec - Styrene Negative
Case 2 Low Positive
In Figure 4 optimized trajectories are shown for both situations. PathFinder started from a trajectory that was given by Dow. The initial trajectory shows first a production decrease, followed by a transition from one grade to another grade. Notice that in both situations the off-spec time has considerably been shortened to about one quarter of the original off-spec time. The optimizer fully exploits the dynamical behavior of the process within the freedom of the specification range to optimize the transition. In both cases production has been increased in the original grade. The production has been increased more in the second case, since the off-spec is not penalized as much as in the first case. In Figure 4 also some MV trajectories are shown. Observe that the original trajectory was a rather quasi steady state transition, while the new trajectories are fully dynamic. The reactor feed flow (MV2) is obviously manipulated and related to the production increase. The rapid change of MV3 is responsible for the fast clipping of the PR0P2 value. If this brusque change of the value of MV3 would be unacceptable for some reasons, it could be limited by introducing a proper rate of change constraint.
m Figure 4 Trajectories for quality variables PROPl and PR0P2 (together with specification boundaries) and for production rate, as well as trajectories for three selected MV's for three cases: initial trajectory (—), Case I (solid) and Case 2 (...)
648 In Figure 5 a path constraint (PCI) is shown that was imposed on a specific process variable known as a constraint: operating beyond that point would lead to undesirable effects. The optimizer pushes the process against this boundary to increase profit.
Figure 5. Trajectory for a constrained process variable for three cases: initial trajectory (—), Case 1 (solid) and Case 2 (...). The thick solid line corresponds to the upper boundary that is imposed on this variable.
6. Conclusion A robust optimization technology PathFinder for the calculation of economical optimal grade change trajectories has been presented. It is seamlessly integrated with a model predictive control technology such that controlled (optimal) grade transitions are straightforward. PathFinder derives dynamic grade change-over trajectories that optimize added value. Several types of constraints can be entered, such that a safe operation can always be guaranteed and such that a gradual migration from a known recipe to a renewed recipe is obtained. PathFinder has been successfully applied on a dynamic model of a Dow polystyrene production facility at Tessenderlo, Belgium. The results showed considerable shortening of the off-spec time as well as a reduction of the overall cost of a grade transition for two different market conditions.
7. References Backx, T., O. Bosgra and W. Marquardt, (1998), Towards intentional dynamics in supply chain conscious process operations, proc. FOCAPO 1998, 5-7 July 1998, Snowbird Resort, Utah, USA Backx, T., O. Bosgra and W. Marquardt (2000), Integration of Model Predictive Control and Optimization of Processes, proc. AdChem 2000, June 2000, Pisa, Italy McAuley, K.B. and MacGregor, J.F., 1992, Optimal Grade Change Transitions in a Gas Phase Polyethylene Reactor, AIChE Journal, October 1992, Vol. 38, No 10, pp 1564-1575 Van der Schot, J.J., R.L. Tousain, A.C.P.M. Backx and O.H. Bosgra (1999), SSQP for the solution of large scale dynamic-economic optimization problems, proc. ESCAPE 1999, June 1999, Budapest, Hungary Van Brempt W., Backx T., Ludlage J., Van Overschee P., De Moor B., Tousain R. (2000), A high performance model predictive controller: application on a polyethylene gas phase reactor, proc. AdChem 2000, June 2000, Pisa, Italy Van Brempt W., Backx T., Ludlage J., Van Overschee P.. 2001, Optimal Trajectories for Grade Change Control: application on a polyethylene gas phase reactor. Preprints DYC0PS6, June 2001, Cheju Island, South Korea
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
"^^
Short-Term Scheduling of a Polymer Compounding Plant p. Castro', A. P. F. D. Barbosa-Povoa*"^ and H. Matos' Departamento de Engenharia Quimica and ^Centro de Estudos de Gestae Institute Superior Tecnico, 1049-001, Lisboa, Portugal
Abstract This paper addresses the optimal short-term scheduling of a three parallel production line polymer compounding plant, whose equipments require cleaning between product changeovers. A very effective user-friendly software tool was developed, which consists of a general scheduling model coupled with the capabilities of Microsoft Excel for data handling and analysis. The scheduling model is based on a Resource Task Network discrete time formulation and leads to Mixed Integer Linear Programming problems. As outputs the user can access the optimal schedules for a number of different objectives.
1. Introduction The short-term schedule of a multiproduct plant is to be periodically made to satisfy a given set of production orders for different products within a fixed time horizon. The orders may come directly from customers or be generated to meet inventory requirements. In the multiproduct plant considered (a two-stage, three parallel line facility), the list of products to be manufactured changes every day. Hence, a cyclic production policy is inadequate to follow market demands and a non-regular production pattern is adopted. The scheduling of a given set of product orders involves important decisions regarding: /) the assignment of products to a given production line and //) product sequencing. The aim is to minimise changeover times and meet due dates. To help with this decision making process and to improve customer satisfaction an efficient scheduling tool was developed.
2, Problem Analysis The process can be viewed as consisting of two limiting stages: /) mixing and //) extruding. Three batch mixers (CPE-049, CPE-055 and CPE-075) are suitable for the first stage while three semi-continuous extruders (CPE-001, CPE-002 and CPE-003) can handle the second stage. Each mixer has its own maximum and minimum capacities. Whenever the amount to be produced exceeds the maximum capacity of the mixer more than one batch is required. The extruding task can start as soon as the first mixing batch is completed.
* Corresponding author. Tel: +351-218419014. Fax: +351-218417638. E-mail: [email protected]
650 The plant is capable of producing several products resulting from the incorporation and dispersion of different colours into polymers, usually low-density polyethylene. The number of colours that can be produced at the plant exceeds one thousand. These can be grouped into 15 different families (set Q according to their main tint, ranging from white to black. Every time a product is changed, the mixers and the extruders require cleaning. Changeover times are dependent on three entities: /) the equipment handling the product (an element of set E); ii) the colour of the product that has been processed and ///) the colour of the product that is going to be processed immediately after. For a given equipment there are at most three distinct cleaning times for every colour tone. Also, cleaning from a colour tone to a different one lasts the same time regardless of the fmal colour tone. These two characteristics clearly suggest that, in certain situations, two or more colour tones can be grouped into a new family. To reduce the number of colour tones to consider in each equipment and consequently the number of cleaning tasks required to model the process, a colour-grouping algorithm was developed. The algorithm identifies functional equivalences among subsets of the available resources, thereby allowing a more aggregate treatment of such resources (Dimitriadis et al.,1998). As a result, the RTN representation of the process becomes simpler and the size of the resulting formulation becomes smaller.
3. Mathematical Model The general Resource Task Network (RTN) representation (Pantelides, 1994) is used to model the scheduling problem. The mathematical formulation is based on a discrete representation of time, where the horizon is divided into a number of intervals of equal and fixed duration, and gives rise to a MILP problem. All system events are forced to coincide with one of the interval boundaries. 3.1. RTN Process Representation The RTN representation regards all processes as bipartite graphs comprising two types of nodes: resources and tasks. Each task is an operation that transforms a certain set of resources into another set. On the other hand, the relevant resources are classified into two different types: /) resources representing the relevant material states, set /?; //) resources representing the two possible equipment states, set S. The adopted superstructure, where the three production lines are clearly identified, is given in Figure 1. For each product, five different resources are required (Rl represents the raw materials, R2 through R4 represent the intermediate mixing states, one for each line, and R5 the final product. Clean (51) and dirty (S2) states are also considered. Besides the equipment index, these states need to be referred to a certain colour tone, omitted for simplicity. In Figure 1 it is assumed that the total amount of the mixed intermediate is produced at the end of the first mixing batch. These two situations are distinguished in the RTN: arrows denoting production of resources at the end of the first batch are connected to the block in an intermediate position while those denoting production at the end of the last batch are connected further to the right. Although the extruders operate semi-continuously, the extruding tasks are modelled as batch: the mixed product (/?2, /?3 or R4) is totally consumed at the beginning of the task, while the final product is totally produced at its end.
651
•O Mixing in CPE-075
I ~^
Extruding in CPE-001
-*@
Mixing in CPE-049
-*©
Mixing in CP&055
•*©
I "*^
H^© -© ®*" Extruding in CPE-003
Extruding in CPE-002
Figure 7- RTN process representation for each product. Part I: processing tasks. Between processing tasks the equipments require cleaning. Each cleaning task consumes a dirty state (52) at its beginning and produces a clean state (SI) at its end. However, it is not enough to refer to a piece of Cleaning (CO,CO) equipment as clean. It is also necessary to state Cleaning (C0,C14) |» which colour it has been cleaned for. As the process superstructure must include all possible colour combinations (15x15) for each Cleaning (01,00) equipment, its representation is quite complex Cleaning (01,014) (see Figure 2) The first cleaning task (CO,CO) consumes the white dirty state 52(0) and / ^ I I \r;;;—. ' ,^,, . • produces the white clean state, 51(0). As the Cleanmg (014,00) | \ ^ z p 0 4 | cleanmg times vary between equipments, even ^ Cleaning (014,014)| ' ^ ' - " ' - ' ' ^ 1 ^ ^ ^ ^ ^ ' — - ^ ^ ^ for the same two colours, the number of cleaning tasks to consider is usually not the same in different equipments (a certain colour group will Figure 2- RTN for each equipment. include more or less colours, see section 2). Part 2: cleaning tasks.
t^
3.2. Mathematical Formulation The short-term scheduling of the plant is achieved by the following constraints. Excess resource balances: the excess amount of a resource at a given time interval t is equal to that at the previous time interval adjusted by the amount consumed by all tasks starting at the beginning of / and by the amount produced by all tasks starting at a previous time interval and ending at t. (1) eeE e=0 ^_^
^e,s,c,t -^e,s,c,t-\
''e.p
'•e.cx
^ /^/^"^ e,p,s,c,6^ e,p4-e + ^ 2^^ e,c,s,c\e\s=2Ae=()^ e,c,c\t-e + 7e,5,r,/1 .v= pEPe=0 C'EC 6=0
- NB,^c,,l-2.e>3 + S C'EC
il^'rU-.A.Ue--ri,,, e=o
^e.cV,^
V^6 £ , . £ 5,C6 C,te
T
(2)
652 In the above equations, the parameters liie,r,p,e ^^^ ^e,p,s,c,d represent the amount of resource r (state s of colour c) of product p produced in equipment ^ at a time 0 relative to the start of the processing task. Parameters Ve,c,s,c',9 ^^^ ^e^,c,c\s,e ^^ ^^^ equivalents of v^p,.^Q for the cleaning tasks, while parameter r^'^^ represents the duration of the first mixing batch for mixers {e<3) and the total processing time (r^^ ) for extruders. Parameter T^^^' represents the required cleaning time in equipment e to change from colour c to colour c\ Finally, to account for equipment usage resulting from orders that began in the previous time horizon, parameter Y^^s,c,t i^ ^^^^- The excess variables Rr^j are positive continuous variables while the excess variables Se,s,c.t as well as the extent variables N^p^_Q (for processing tasks) and Ne,c^c\t-e (^^^ cleaning tasks) are of the binary type. Variables NB^ct are also binary and represent the consumption, at time r, of the dirty state of colour c in equipment e (if extruder). By using these variables, degenerate solutions near the end of the time horizon are avoided. Excess resource constraints: In the beginning of the time horizon all raw materials are available in an amount equal to the order size {M^). At that point, all intermediates and final products are unavailable and the state of all equipments is neutral: ^i,p,o=M^VpGP (3) ^r,/7,o =OVrE / ? , r > l , p e P
(4)
•^.,.,^,0 =OV^G £,5G 5,c6 C
(5)
To reduce the number of degenerate solutions, the beginning of the extruding tasks is made to coincide with the end of the corresponding mixing tasks. Also, the cleaning tasks are executed immediately after the end of the processing tasks. R,p, =0\freR,\
(7)
E.ceCjeT
Objective function: Two alternative objective functions will be used. The first, production maximisation, will be used to select from the set of total orders, those that can be produced in a given time horizon. Then, the second objective will be used to minimize product delays among schedules with minimum cleaning times. •"axXS^s,,,,
- - E S S E ^Lc:,rL.rUNIT teT ceCc'eCeeE e>3
(8)
.
^
^
pep\
'
(9) '
The tardiness of each order (TRp, a positive continuous variable) is given by:
™^ - S E ^'p-' (^' +
(10)
where dp represents the due date of product /?, T^ the absolute starting time of interval t and TUNIT ihQ length of each time interval.
653
4. A User Friendly Interface To handle the scheduling problem a user-friendly software tool was developed. By userfriendly we mean easy data input and generation of the problem's solution without going through its complex modelling aspects. The scheduling tool has two components: i) the general scheduling model (described in section 3.2), implemented in GAMS and //) an interface to the GAMS software. Microsoft Excel was chosen as the interface since most users are familiarised with spreadsheets and because it has a programming tool associated with it, Visual Basic. Any problem is solved in four steps. In the first step, the problem data is provided in a worksheet of the Excel spreadsheet. In the second step, the data is analysed in order to reduce the number of colour tones to consider and converted to the GAMS format. The problem data is then included in the general GAMS input file, which is solved in the third step. Once the optimal solution is reached, the results are exported to Excel and displayed in the form of Gantt charts.
5. Model Solving A real case study will be used to illustrate the behaviour of the model. It consists of nineteen products and a horizon of one week (the plant operates 5 days a week and 9 hours per day). Product data is provided in Table 1. Due to previous orders, the extruders only become available after 450 (CPE-001), 330 (CPE-002) and 90 minutes (CPE-003) and still require cleaning (from preto, branco and amarelo, respectively). The OSL solver solved the resulting models on a Pentium III-450 MHz machine. Table 1-Product data (demand in kg, due date in days) Product 1 2 3 4 5 6 7 8 9 10
Colour Verde Verde Verde Azul Azul Azul Azul Azul Amarelo Amarelo
Demand 500 900 150 250 150 450 100 150 3000 1250
Due Date 2 3 4 5 2 1 4 3 5 1
Product 11 12 13 14 15 16 17 18 19 Total
Colour Vermelho Preto Verde Vermelho Castanho Castanho Vermelho Castanho Cinzento
Demand 500 500 3000 400 250 150 100 1000 3000 15800
Due Date 3 4 3 2 2 5 4 4 3
Table 2- Computational statistics Obj(active function TVNIT {mm) Ti me Intervals Integer variables Continuous variables Constraints Obj. relaxed MILP (Obj. MILP CPU (s)
Eg. 8 15 180 65639 99356 31317 13462.5 13450 7670
Eg. 9 5 540 119249 185900 62770 258.49 264.78 16732
When solving the problem using the first objective function (equation 8) it is found that it is impossible to produce all 19 products. A maximum of 13450 kg was obtained, corresponding to the sum of the demands of products 1, 2, 3, 9, 10, 13, 14, 15, 18, 19.
654 These products are then selected for production. Next, in order to minimise changeover times, the problem is solved using the second objective function (equation 9). A lower discretisation of the time interval is chosen (5 instead of 15 minutes) in order to reduce the errors involved in rounding the cleaning and processing times to a multiple of the interval length. The optimal schedule is given in Figure 3. As expected, products with the same colour tone are adjacent in the schedule. Product delays are the following: 2.94, 0.43, 1.86, 1.99, 0.92 and 0.71 days for products 1, 10, 13, 15, 18 and 19, respectively. The problem statistics are given in Table 2.
1200
1400 Time (min)
Figure 3- Optimal schedule
6. Conclusions
This paper addresses the short-term scheduling problem of a polymer compounding plant. A model, based on a RTN discrete-time formulation was developed to handle the problem. The process superstructure consists of three production lines for each product and all the cleaning possibilities for each equipment item. In order to reduce the number of cleaning possibilities, an algorithm was developed which, by looking at the cleaning times between colours involved in a given set of products, groups, if possible, two or more colours into one new group. The scheduling model was coupled with Microsoft Excel so as to provide a user-friendly interface. Using the production maximisation criterion, the developed software tool selects, from a given set of orders, those most suitable to be produced in a given time horizon. Then, for the selected orders, the optimal schedule that minimises product delays among schedules with minimal changeover times is generated.
References Castro, P., 2001, Optimal Short-Term Scheduling of hidustrial Batch Processes. PhD Thesis. Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Portugal. Dimitriadis, A., Shah, N. and Pantelides, C , 1998, Comp. Chem. Eng., 22, S563. PanteUdes, C. C , 1994, Unified Frameworks for the Optimal Process Planning and Scheduling. In Proc. 2"^^ Conference on FOCAPO. CACHE Publications, New York.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
655
Short-Term Site-Wide Maintenance Scheduling Kwok-Yuen Cheung\ Chi-Wai Hui\ Haruo Sakamoto^, Kentaro Hirata^ and Lionel, O'Young"^ * Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. 2 , ' Safety Engineering and Environmental Integrity Lab., Process Systems Engineering and Production Technologies Field, MCC-Group Science & Technology Research Center, Mitsubishi Chemical Corporation, 1, Toho-cho,Yokkaichi Mie 510-8530, Japan. ^ MC Research & Innovation Center Inc., 444 Castro St., Mt. View, CA 94041, USA.
Abstract Preventive maintenance is essential for every chemical production site, as it can prevent equipment failure and accidents. In order to minimize the production loss caused by maintenance, the maintenance has to be carefully scheduled. To obtain an optimal maintenance strategy for a whole production site, the maintenance schedules of the production and utility plant have to be tackled simultaneously considering site-wide utility and material balances. However, the interconnections between production and utility system make the scheduling problem become very complex and difficult. In this paper, a multi-period mixed integer linear programming model, Site Model, is proposed as an aid to optimize short-term site-wide maintenance schedule. A special formulation is adopted to handle pre-set utility and material demand profiles during the shutdown, maintenance and start-up periods of plants.
1. Introduction A chemical production site consists of variety production and utility plants. All of them require regular maintenance to enhance their reliability and to reduce losses caused by emergency shutdowns. Plant maintenance requires huge capital and human resources and causes losses due to the suspension of production. It also affects the material and utility balances between the production and the utility units. A good maintenance schedule should carefully take into account all these factors providing a feasible and economical solution. However, these considerations make the maintenance scheduling be a very complicated task. In this paper, a multi-period mixed integer linear programming (MILP) model, called Site Model, is proposed for solving this type of maintenance scheduling problem. A Site Model includes all the major energy and material balances of the production and utility plants, and their interconnections and constraints. Optimizing maintenance requires tackling both long-term scheduling and short-term schedules (Cheung and Hui, 2001). On a chemical production site, not every plant has to be maintained every year. There are always some plants being operated while others are down for maintenance. Due to the limits on the availability of materials, utilities and skilled workers, production plants and utility facilities are normally divided into
656 groups and maintained separately in different maintenance periods. A long-term maintenance schedule provides the grouping and rough timing of plant maintenance over a two to five year time span. It takes into account the basic material and utility balances during each maintenance period to determine long-term production and inventory strategies, and policies on selling and buying material and utility services. Superimposed on the long-term maintenance schedule, short-term scheduling determines the details, such as the exact timing of the main operations (e.g. shutdown, overhaul, inspection and startup) of all the maintained plants during a maintenance period. These operations are normally completed in a time span of four to ten weeks in which utility and labor demands could be significantly fluctuated. The main objective of short-term scheduling is to guarantee the feasibility during the main operations, smooth utility and labor demands, and minimize production losses. Only small literature addresses scheduling maintenance of large-scale chemical production sites. Some of it focused on the short-term scheduling of multipurpose batch plants with integration of maintenance and production (Dedopoulos and Shah, 1995 and Pistikopoulos et al., 2001). Kim and Han (2001) have discussed maintenance scheduling of utility systems by assuming a fixed configuration of utility and production systems and utility demands. Hui and Natori (1996) included maintenance feasibility as a consideration in a case study of utility system expansion. Tan and Kramer (1997) scheduled maintenance for chemical processes. So far, maintenance schedules for a chemical production complex including the interaction between production and utility systems have not been investigated in any of this work. The idea of applying a site model for long-term site-wide maintenance scheduling has previously been discussed by Cheung and Hui (2001). This paper will focus on the short-term maintenance scheduling of a chemical production complex that contains both production and utility plants.
2. Problem deflnitions A simple example has been adopted from Hui (2000) to illustrate short-term scheduling in this paper (Figure 1). The example site consists of eight production plants (ETY, VCM, PVC, PE, PP, A, B and C) and a utility plant that consists of two boilers (Bl and B2) and three turbines (Tl, T2 and T3). Not all the processes and utility facilities need to be maintained in every maintenance period. In the following examples, only the ETY, VCM, PP, PE and PVC plants are maintained together with B2 and T2. Other units are operated during the maintenance period. The whole maintenance period lasts for a maximum of 31 days. Each day has three shifts (day, night and midnight). At each shift, prices and importation limits of electricity are different. Before a unit undergoes maintenance, a few days are required to shutdown all the process equipment. Overhaul is then carried in the coming days of weeks requiring intensive labor and heavy machinery involvements. After the overhaul, the plant requires a few days to resume (or startup) to normal operating
657 conditions. The lengths of the shutdown and startup periods are fixed. The length of maintenance could either be fixed or given a lowest limit to allow the optimizer to determine the optimum length. Utility, material and manpower demands of the maintained plants vary during their shutdown, maintenance and startup. These demand profiles are given as an input for the optimization. The objective of the site model is to maximize the overall profitability (or minimize operating cost) of the chemical production site during the shutdown maintenance period by simultaneously taking into account raw-material costs, product prices, inventory holding costs, fuel costs, electricity variable and fixed costs, and maintenance demand profiles.
i Bl
^^^
ETY PLANT
B2 I
K)-
jdti.
Aii. - • -It.
Ccxidensate
oiler Feed Water
Aii.
K^_^rH
•i-'*-» VCM pmN
Aii Aii
PLANT
o
K>i -- K^' K)r^^^^
Storage Tank VHP Steam HP Steam LP Steam Eleciricity
Figure I: Sample site-model.
3. Deflnition of a Site Model A site-model is a general mathematical programming model that contains the major material and energy balances of both production and utility system of a chemical production site. Some site-wide constraints, such as, seasonal production and utility demands, fuel and electricity supply contracts, are also included. To enable a sitemodel for production planning, the model has to be formulated in multi-period time frame such as days or months. Since the main applications of a site-model targets for large-scale planning or scheduling problems therefore using a linear model is the most suitable and sufficient. For problem like shutdown maintenance scheduling, integer variables are imposed into the model for determining equipment on/off and/or selection. The number of parameters, equations and variables used in a site model increases rapidly with the number of plant units and time periods. Managing the model becomes
658 an important issue for industrial scale problems. In the proposed site-model, variables and equations are carefully defined to make the model manageable using a database system. Details of the model such as data, constraints, equations and GAMS (Brooke et al., 1992) input files are given at totalsite.ust.hk.
4. Case Studies 4,1 Base Case: Currendy, plant maintenance is scheduled by experience and with some simple heuristics. The main concern of the scheduling is to guarantee feasible material and utility balances. Payment for skilled labor is one major concern in scheduling process. To maintain steam balance, the boilers are normally the last one to shutdown and the first to be started up. With these considerations, a B J ^ ^ •Shutdown [oOverhaul maintenance schedule can be glnspectionj manually created as in Figure 2. I Startup This schedule is then verified using the site model to make sure of its I Holiday/ Weekend feasibility. 1 3
5
7
9 11
13 15 17 19 21 23 25 27 29 31
Figure 2: Schedule of Base Case. B2 1 ETYBl
1 i1 1 M• i i i
PEllj
iShutdown •Overhaul
i
PP PVC
EL^ffliM ii t1' \ ti i l l '
T2 1 1
VCM
'
11 H _L^M
ginspectioni r
1 Startup
B Holiday/ 1 Weekend |
i M 1 1 ^ H 1 3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
Day
Figure 3: Schedule of Case 1.
Figure 4: PE production rates oj Base Case and Case L
4.2 Case 1: Figure 3 shows an optimum maintenance schedule generated using the site model direcdy. The model optimized the overall profit taking into account of the variation of electricity prices given in current electricity contract and other sitewide constraints. Compared with the base case, this schedule increases the total labor cost. Since most of the plants are maintained in the early part of the maintenance period, increased the requirement of external skilled workers. Electricity cost has also increased due to increased production. In return, the overall profit increases, mainly due to the increase of PE production. The PE plant is shut down together with the ETY plant that reduces the need from ETY inventory allowing the downstream processes of ETY to
659 increase production during and after the ETY plant shutdown. The PE production rates in the Base Case and Case 1 are shown in Figure 4. 4.3 Case 2:
Figure 5: Schedule of Case 2.
Figure 6: Electricity importation of Case 1 nnH 2.
Assuming that variation of current contract is allowed electricity an additional degree of providing in the optimization. freedom Electricity cost in a long-term purchasing contract is normally divided into two parts; fixed cost and variable cost. The fixed cost of electricity depends mainly on the amount of peak importation that is normally happen during the main utility facilities shutdown. The variable cost is calculated based upon the overall electricity consumption. In this study, the current electricity contract allows maximum importation of 45MW during the shutdown period. The optimum schedule with consideration of renewing the electricity contract is shown in Figure 5. Peak consumption of electricity is reduced to 28MW resulting a 43 million Yen decrease in electricity cost. Since the production rate remains high compared with the Base Case, the overall electricity price is still 2 million Yen higher than the Base Case. Electricity importation levels for Case 1 & 2 are shown in Figure 6.
4.4 Case 3: In the previous cases, the length of Day the overhauling interval is fixed. In Case 3, the length of the overhaul is Figure 7: Schedule of Case 3. allowed to change providing an additional degree of freedom for increasing profit. The resulted schedule is shown in Figure 7. The PP plant has extended its overhauling interval from 11 days to 14 days. PP is the least profitable product. Extending the PP plant maintenance allows utilities and materials to be better utilized for other productions resulting a slightly better profit. A comparison of the cases is given in Table 1. 11 13 15 17 19 21 23 25 27 29 31
660
5, Conclusions A MILP model, Site-Model, is evaluated in this paper for tackling a short-term sitewide maintenance scheduling problem. The case studies demonstrated the importance of considering utilities, materials and inventories, as well as variation of utility contracts simultaneously when scheduling the maintenance. A good maintenance schedule should not only guarantee feasibility, but also maximize overall profitability. Table I: Results
Cost & Profit (Million Yen) Product Revenue Labor Cost Production Cost Total Elect. Cost Total Profit
Comparison.
Base Case
Case 1
Case 2
Case 3
2331 -18 -1175 -65 1137
2707 -19 -1504 -110 1183
2630 -19 -1411 -67 1200
2600 -21 -1377 -64 1202
6. References Brooke, A., D. Kendrick, and A. Meeraus, 1992, GAMS - A User's Guide (Release 2.25); The Scientific Press: San Francisco, CA. Cheung, K.Y. and C.W. Hui, 2001, Total-Site Maintenance Scheduling. Proceedings of 4th Conference on Process Integration, Modeling, and Optimization for Energy Saving and Pollution Reduction (PRES'Ol), Florence, Italy, 20-23 May 2001. Dedopoulos, I.T. and N. Shah, 1995, Optimal Short-term Scheduling of Maintenance and Production for Multipurpose Plants. Industrial and Engineering Chemistry Research, 34, 192-201. Hui, C.W., 2000, Determining Marginal Values of Intermediate Materials and Utilities Using a Site Model, Computers and Chemical Engineering, 24, 1023-1029. Hui, C.W. and Y. Natori, 1996, An Industrial Application Using Mixed-Integer Programming Technique: A Multi-Period Utility System Model, Computers and Chemical Engineering, 20, S1577-S1582. Kim, J.H. and C. Han, 2001, Short-term Multiperiod Optimal Planning of Utility Systems Using Heuristics and Dynamic Programming. Industrial and Engineering Chemistry Research, 40, 1928-1938. Pistikopoulos, E.N., CG. Vassiliadis, J. Arvela, and L.G. Papageorgiou, 2001, Interactions of Maintenance and Production Planning for Multipurpose Process Plants — A System Effectiveness Approach. Industrial and Engineering Chemistry Research, 40, 31953207. Tan, J.S. and M.A. Kramer, 1997, A General Framework for Preventive Maintenance Optimization in Chemical Process Operations, Computers and Chemical Engineering, 21 (12), 1451-1469.
7. Acknowledgments The authors would like to acknowledge financial support from the Research Grant Council of Hong Kong (Grant No. 6014/99P), and financial and technical supports from Mitsubishi Chemical Corporafion, Yokkaichi Plant.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Published by Elsevier Science B.V.
661
Modelling Multi-site Production to Optimise Customer Orders D. Feord\ C. Jakeman^ N. Shah^ ^Dow Deutschland GmbH & Co. OHG, D-77836 Rheinmunster, Germany ^Process Systems Enterprise Ltd. London W6 9DA, U.K. ^Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine,London SW7 2BY, U.K.
Abstract Short term planning of production networks across geographically linked sites is complex, especially during periods of severe or catastrophic system failure. This paper presents a method of modelling the production network, incorporating actual operational, demand and inventory data and predicting, based on the commercial strategy, which orders should be met, which delayed and which will not be delivered. The development and operation of this model will be discussed.
Introduction The internal restructuring of large multinational chemical companies into smaller business units has meant that the emphasis of optimising production & supply has shifted away from single large integrated sites to the management of business production networks across several regional sites. Each of these sites may in turn be a stand-alone plant or part of an integrated site. Correct management of this geographical integrated production network is critical for business success. Enterprise management tools such as SAP allow for basic production planning but are not suitable for addressing problems that occur when the production network under-performs. These tools often have difficulty in representing the complexity of a chemical plant in sufficient detail to make them useful for optimisation. Examples of under-performance can include unplanned production outages, late arrival of raw materials, variable shift patterns etc. A model has been developed to predict the performance of a specific business geographical production network. Its primary role is to identify, depending on the optimisation criteria, which orders can be fulfilled on time, those that will have to be delayed and those that cannot be fulfilled. The predictions have a time horizon of one to two weeks, depending on the sparseness of the available demand data. Optimisation criteria used are based upon heuristic customer priority levels set by the commercial organisation, which represent the business strategy and therefore optimal long-term returns.
662
Production Problem The problem is composed of a network of three regional sites, which employ both continuous and batch processing. Figure 1 shows a simplified schematic of this problem.
Figure 1 Simplified Production Network Description
From initial to final product there can be up to four different product formations, including combinations with different products also made in the chain. Each product produced in the chain may either be consumed downstream or sold direcdy to a customer. Only one product in the whole chain is made in more than one plant. Optimal profitability implies that production plants are run at 100% and inventories are held to a minimum. There is very little slack in the system to compensate for errors. Problems occur primarily because of, • • • •
Unplanned production outage. Reduced production manpower levels. Delayed internal deliveries. Backlog of late orders that delay other orders in the system.
In order to maintain committed deliveries, on time, to strategic customers a tool is needed that represents the network mathematically, optimises existing orders against customer priority levels and indicates to the planners what actions should be taken.
Production Network Model The production network has been represented and solved using the gBSS software package, N. Shah et al (1995) from PSE, an MILP programme. gBSS uses the State Task Network representation to model a process. A state corresponds to any material during the processing, e.g. raw material, intermediate or products. A task corresponds to any processing step, which can include standard physical and chemical processing but
663 may also include packaging, loading, transportation etc. The flexible gBSS modelling language allows the production units in the network to be broken down into smaller units that represent the critical steps. Continuous units may be represented simply e.g. one task with a given hourly capacity. Batch units require a more detailed representation to determine each potential rate-limiting step, either volume e.g. reactor contents or time-wise e.g. a drumming line, Rickard (2000). The tasks are then modelled individually to form a network representing the total process. The flexibility to represent the chemical process close to its actual operation was the major reason for choosing the gBSS package. The total multi-site network is modelled as one single network. Each individual site is modelled as one State Task Network. The sites are then linked together by modelling the transportation as tasks that hold the transported material making it unavailable for the given transportation time after which it becomes available in the new location. In this way one large network is formed. The complete production model is large, comprising about 100 equipment items, 180 materials and over 200 recipes. gBSS also allows for utilities to be defined optionally with a given capacity profile. For this problem classical utilifies such as steam and process water availability are not important. However the ability to include labour levels in gBSS enables some of the more important constraints in the batch processing to be accounted for. The labour required for tasks that have labour as a constraint are defined as part of the recipe. A matrix of recipes and resources representing the products made on the network is maintained. In this way the equipment on which each individual product may be made, process rates and utility requirements, as well as raw materials, unit ratios and material storage are defined.
Data Conditioning The representation of the production system with the network model enables the customer order production sequence to be optimised over a given time horizon, which is normally 1-2 weeks. Meaningful optimisation is only possible with adequate and representative input data. Data must also be easily obtainable i.e. in an electronic form, to enable the manipulation and conditioning necessary to create the complete problem file to be solved by gBSS. This data necessary is shown in figure 2. Process capability data represents the actual processing capability of the plant. In times of unplanned outages a database enables production personnel to enter the estimated outage time and the reduced production rate of the plant during this period. This results in a fast response to the supply chain on the potential impact of processing problems. This information is included in the simulation, thus a closer representation of the true state of the production network capacity is obtained. If no reduced capacity is entered the process capability is assumed to be maximum production rate. Demand data. Obtained on a daily basis. Data for 1-2 weeks is taken, depending on the problem time horizon. In general, sparseness of data beyond 10 days can give
664 misleading results. Included in this data is, order volume, delivery date, material ID, customer, and priority level.
Figure 2 Data requirements to complete problem description
Inventory data. This data includes inventory volume and storage capacity. It is obtained as for the demand data. Filtered process control data, collected directly from the plant is used to obtain the intermediate and final product inventory held in plant tanks. In this way the best system data is made available. Rav*' materials data is obtained initially as part of the inventory data. It is assumed that during the time horizon sufficient raw materials are available. A potential raw material shortage during the optimisation time horizon may be entered manually. Labour availability via shift planning is obtained on a weekly basis, and can vary due to illness, holiday's etc. As labour is a constraint for some batch operating equipment this information is important to obtaining good results. Data entry is manual.
Optimisation Criteria The primary objective of this work is to develop a tool which optimises orders against customer priority levels for unplanned production circumstances. This is a business decision and is made today by the commercial organisation, working with the supply chain and manufacturing. It is not intended at the moment that this tool should automate the process but rather to aid and improve the timing and accuracy of the decisions that need to be made. In order to do this the model optimisation must reflect the decisionmaking process, which is strategic rather than on a direct cost basis. The network model may be run in two optimisation modes The first is "make in full" where all orders must be produced .An extended deadline is created and those that are made in this extended deadline are penalised. The cost function for this mode is given in equation 1. The optimisation attempts to minimise the value of the cost function. This
665 method requires that the problem solution is feasible, which is not always the case. The second mode is "make on time" where all the products that are not made on time are neglected from the schedule. The cost function for this mode is given in equation 2. The optimisation attempts to maximise the objective function and therefore ensure that maximum number of high priority customers is served. Make in Full
If (delivery date < Nominal date) Cost function = 0 If (nominal date < delivery date < Hard extended deadline) Cost function = 2^(Amount delayed (MT)* Weighting * (Delivery date - Nominal date))
Make on Time
(1)
Cost function = 2-^(Weighting * Amount made (MT) + Inventory Created (MT) * Value)
(2)
The Weightings for each order drive the cost function. Each customer is assigned a priority level by the business. The customer priorities are maintained in a matrix within the programme and are not changed unless a strategic change occurs By scaling the priority levels correctly the model will be biased towards always delivering the highest priority customers. Weighting too biased towards high priority customers will tend to lead the model to ignore lower priority customers even if there is spare capacity for production. A weighting biased too evenly could mean that high volume low priority customers take preference over lower volume higher priority customers. Therefore care and attention are needed in finding the correct balance for the cost function weightings.
Results The model is run at most on a daily basis. Running time is typically 10 - 30 minutes depending on the problem size and complexity as well as the method of optimisation. All operation is controlled via an Excel interface. It has been found that it is useful to run the model often (once a day) as even under ''ideal" operating conditions the model will pinpoint errors in planning that had been overlooked. Figure 3 shows a typical results format. Each order is available in a spreadsheet, including order information such as amount, deadline, product ID, customer etc. Orders where the optimisation has indicated that the order should be delayed or cancelled are highlighted for easy identification. A Gantt chart is also available. This is particularly useful in identifying bottlenecks and reasons behind the delay of products if it is not obvious.
Benefits The model benefits the business and planners as it enables a quick and accurate assessment of customer order fulfilment at any particular time, over a given working horizon. This foresight has often allowed corrective action to be taken to enable order
666 fulfilment. Scenarios where this approach has already proved useful include planned and unplanned shutdowns, unplanned labour shortages and over demand (allocation). Other benefits have included the identification of processing bottlenecks and the early recognition of problem orders due to sub-optimal planning.
I ^ ^ . ^ E i 3 4—]^-cz:-::
HE Figure 3 Typical results indication product availability
Conclusions Through this approach it has been shown that it is possible to prioritise customer orders for geographically separate but linked production sites. Under abnormal operating conditions or catastrophic failures, the supply chain and commercial organisation can swiftly decide which orders will be met, delayed or not fulfilled in accordance with the strategy of the business. Also consistent use of the model has enabled system problems to be identified such as inter-site transportation, manpower levels and over demand. In this way long term solutions for these problems are sought. gBSS has proven capable of modelling the continuous and batch processing by breaking down these units into small units representing the critical steps. In addition the ability to represent transportation and utilities such as process manning levels has enabled an accurate representation of the total production network and its constraints.
References J.Rickard, 2000, Combining Optimal Off-line Scheduling with On-line Supervisory Control, World Batch Forum - 2000 European Conference N. Shah, C.C. Pantelides, L.G. Papageorgiou, L. Liberis, P. Riminucci and K. Kuriyan, 1995, gBSS: An Integrated Software System for Multipurpose Plant Optimisation, 46-48, (Proc. IChemE Res. Event, Edinburgh, 1995)
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
667
A New Event-Driven MILP Formulation for Short-Term Scheduling of Continuous Production Facilities Nikolaos F. Giannelos* and Michael C. Georgiadis Chemical Process Engineering Research Institute (CPERI) PO Box 361, Thessaloniki 57001, OR, *Email: [email protected]
Abstract A new mathematical formulation for scheduling multi-purpose continuous processes is presented, based on an event-driven representation of time, and resulting into a mixedinteger linear programming (MILP) model. The formulation is applicable to arbitrary process structures, variable unit processing rates, sequence-dependent changeovers, and flexible storage requirements. A medium-to-large scale manufacturing process is examined to illustrate the applicability and efficiency of the method. The formulation is shown to compare favorably with existing continuous-time models.
1. Introduction A novel event-driven formulation is developed and applied to the short-term scheduling of continuous processes. Relevant existing approaches may be grossly divided into models applicable to mixed (batch/continuous) production facilities (Schilling and Pantelides, 1996; Zhang and Sargent, 1998; Mockus and Reklaitis, 1999) or specialized formulations for continuous processes of restricted topologies (Karimi and McDonald, 1997; Mendez and Cerda, 2001). To manage problem sizes, schemes for treating continuous time rely on uniform (a single time scale for all process resources) or nonuniform (multiple time scales) representations. The proposed approach utilizes a nonuniform continuous time construct outlined in section 2 below. The resulting MILP model is presented in section 3. Applications are shown in section 4 and the relative strengths of the proposed approach conclude the discussion.
2. Methods The formulation is based on the state-task network (Kondili et al., 1993). To avoid usage of tri-indexed variables (task/unit/time), any task that can be performed in k units (k>l) is treated as k distinct tasks of identical properties (states consumed/produced, stoichiometric coefficients of task recipe, etc). The time grid construction relies on the definition of n event points, (ti,t2,...,tn.i,tn), for every task in the process. Event points are determined by the end of task execution. Actual event times corresponding to the same event point are generally non-identical for different tasks in the process. The nonuniform time representation can therefore be thought of as |I| distinct time axes/grids, where |I| is the number of distinct tasks in the process. Coordination of the multiple time scales is driven by unit utilization considerations (for tasks performed in the same unit).
668 as well as material balance considerations (for tasks processing the same material state). To ensure material balance feasibility, all continuous tasks processing the same state and terminating at the same event point (if any), are forced to start and end at the same real time, so that a rate-based balance is posed unambiguously.
3. Mathematical Formulation 3.1 Nomenclature Sets: I (index i) is the set of all tasks. S (index s) is the set of material states. T (index t) is the set of event points. U (index u) is the set of units. lu is the set of tasks performed in unit u, !„ c I. T is the set of continuous tasks, f c I. f s is the set of continuous tasks consuming or producing state s, f s c f. I^^ is the set of flexible storage tasks, I^^ c I. V\ is the set of storage tasks suitable for state s, V\ c l^\ f ^n' is the set of continuous tasks i, I with changeover requirements. S^'^ is the set of states with finite intermediate storage requirements, S^'' c S. S^ is the set of final product states, S^ c S. The HEAD(*) operator indexes the first element of set •. Parameters: D™"s (D'"'^) is the min (max) demand for state s. e™"i (0"^"^) is the min (max) duration of task ie f. R"""i (R'^'^i) is the min (max) processing rate of task ie f. ST'^^^'s is the maximum dedicated storage limit for state s. ST^s is the initial amount of state s. Vs is the value of state s. k^i is the fraction of state s engaged in task recipe i. 6ii' is the changeover time from task i to i'. C'"'^'' is the time horizon, and C^^u the minimum cumulative changeover time in unit u. Variables: 9it is the duration of task le l^ for event t. 0st is the duration of all continuous tasks processing state s {ie Ts) at event t. ijt is the ending time of task ie I for event t. Tst is the ending time of all tasks engaging state s (iG Is) at event t. ^it is the total extent (rate X duration) of task IGT for event t. x^ is 1 if task iGl terminates at t, 0 otherwise. STst is the amount of state s at the end of t. yst is 1 if STst ^ 0, 0 otherwise. 3.2 Basic Model x,0"^". < 0, < x,0"^"i
Vi6f,teT
(1)
0uR"^"i < ^it < 0itR""i
VieT.teT
(2)
T,-Tst < C"^^(l-Xu)
VseS, iefs.teT
(3)
T,-Tst > - C " ^ ^ ^ ( l - X j
VSGS,
iel's, teT
(4)
0u-0st < C"^"^(l-xu)
VseS, ieTs, teT
(5)
0u-0st > - ^ " - ( 1 - X u )
VseS, i e I \ , t e T
(6)
Tit - 0it > Ti,t-i
ViGr,teT
(7)
'I^st ~ 0st ^
VseS,teT
(8)
'^s,t-l
669 teT
(9)
VueU,teT
(10)
V ueU, i,i'elu, i=HEAD(IJ, iVi, teT
(11)
VSGS,
STst - STs,t-l + S^si Sit S Xu < 1 16 lu '^it — '^i't
T,, - e,, - Ti, > e,i. - C"''" ( 2 - Xi, - X,, + I
X x,., •)
V ue U, i,i'e lj%,
t
(12)
i"elu t
VUGU
(13)
VsGS^teT
(14)
VsGSP,tGT
(15)
tGT ielMu
STst < ST'^^s D™"s < STstn < D"^^s
Constraint (1) ensures that continuous task durations lie within allowable processing times. Constraint (2) limits the total extent of any continuous task based on allowable rates. Constraints (3)-(6) coordinate the multiple time scales for continuous tasks engaging the same state s at the same event index t. When Xit = 1, constraints (3)-(4) enforce 6it = Gs^; applied to all continuous tasks processing s, the ending times of these tasks are commutatively forced to equality. Similarly, constraints (5)-(6) equate the durations of all continuous tasks performed at the same event point and engaging the same state. Constraints (7)-(8) impose the required monotonicity on event times for the same task/state. Note that the upper bound of all ij m, is.m variables is C"'''''. Constraint (9) is the typical multi-period material balance expression; in view of the multiple time scales, (10) is only meaningful by virtue of constraints (3)-(6). Constraint (10) ensures that no unit is assigned to multiple tasks concurrently. Because of the non-uniform time representation, (10) is sensible by virtue of constraint (11), which replaces multiple independent time grids with a single one for tasks executed in the same unit. Constraint (12) ensures that if task i is performed in unit u at event t, task i' is performed in unit u at event t' > t, and no other task is performed in u between i and i', then the starting time of task i' is sufficiently greater than the ending time of task i for the required changeover. Changeovers are used to provide improved linear relaxations of the model in constraint (13), where C'^^u is the minimum cumulative changeover time required in unit u. Dedicated intermediate storage requirements are expressed in (14), and product demands are deterministic as in (15). 3.3 Flexible Storage Extensions STst < I XuV"-i
VseS'^teT
(16)r
VseS^'.teT
(17)
VseS'"'Mer's,tGT,t
(18)
VseS"', iel"s,teT
(19)
iel i
Xi,,+i ^
Xi, + y s t - l
Ta > X s . - C ' ^ d - x , )
670 Ti,-i < Tst - 0st + C " " ( 1-Xi,)
V sG S''\ ie 1% te T, t^ti
(20)
Ti,-, < Ts,t-, + C"^^ ( 2-Xi-Xi,_i) + C"^^ ( l-ys,-i)
V SG S''\ ie r\. ie T, t^t^
(21)
Tit > Ti,t-i
ViGr\tGT,t7^ti
(22)
Constraint (16) activates suitable storage tasks when the amount of state s at the end of all t events processing the state is positive. Constraint (17) identifies all events t where there is a positive amount of state s by setting y^^ = 1. By virtue of (16), there will be one or more suitable storage tasks active (x^ = 1). Equation (18) then ensures that all these storage tasks become active for the next event point, t+1, as well. The timings of storage tasks are determined in (19)-(21). Equation (19) ensures that the ending time of a storage task for state s at t is at least equal to the ending times of all continuous task events involving the state. Constraint (20), combined with (19), ensures that the duration of a storage task for state s at t is at least equal to the duration of all continuous task events involving the state. In fact, it can easily be verified that the combined effect of (19)-(20) is the condition Tit-Ti,t-i > Ost- For storage tasks active at consecutive events t-1 and t with STst_i:?!: 0, a stronger condition is required as posed by (21). In this case, the storage task duration as determined by combined application of (19) and (21) becomes iit-Tit-i >Tst-Ts.t-i. Constraint (22) simply imposes the time monotonicity on ending times of storage events. 3.4 Objective max z = X Vs STs ^^ seSP
(23)
subject to constraints (1)-(15) for dedicated intermediate storage, or, (1)-(13) and (15)(22) in the flexible intermediate storage case.
4, Application The continuous processing facility under investigation is shown in Figure 1. Process specifications are compiled in Table 1. Feed-stocks are mixed in three units to produce seven intermediates, which are packed into fifteen final products ('D' denoting drummed products, ' C canned ones). There are three storage tanks for the intermediate states. Packing is the rate limiting step, and yield is affected by storage utilization. Two cases are investigated: (a) unlimited intermediate storage with min demands from Table 1, and, (b) flexible intermediate storage with the same demands. Analytical results for (b) are presented in Table 2 and Figure 2. The instance is solved to absolute optimality. The modest running time is due to the number of decision variables and the linear relaxation value. The schedule features maximum utilization of all packing lines in terms of rates and changeovers. Process yield is not critically limited by storage. In fact, additional runs indicate that only two storage tanks are sufficient for obtaining practically optimal solutions within 0.22% of the reported one in 2 cpu mins. The unlimited intermediate storage instance is also solved easily to optimality. Random
671 instances are generally solvable within \% of their relaxation in a few cpu mins. All results were obtained on a single 400 MHz processor via a GAMS/Cplex interface.
Pack-Dll
Dill
Pack-Dl2
DI2)
Packmen
D13)
J PackXl2
g)
Pack.D21
D2?)
Pack.C21
C2l)
Pack.D22
D22)
Pack.C22
C22)
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D3l)
Pack.C23
C23)
Pack.D41
D4l)
Pack,C31
Pack.D42
MakeJiitS
-(iiiie)-
Make-Int?
D42)
Pack.C32
Figure I. Simplified STN representation for case study.
Table L Summary of specifications for case study. State ST'' 00 BasA, BasB, BasC Intl-Int? 0 DII 0 D12 0 D13 0 D21 0 D22 0 D31 0 D41 0 Unit Rate/ Capacity Mixer 1 17 Mixers 2-3 12.24 17 Tanks 1-3 60 Line 1 5.8333 Line 2 2.7083 Line 3 5.5714 Line 4 2.2410 3.3333 Line 5 5.3571
|-|min
-
220 251 116 15 7 47 8.5 Suitability
State D42 Cll C12 C21 C22 C23 C31 C32
ST* 0 0 0 0 0 0 0 0
j^nun
144 42.5 114.5 53 2.5 16.5 13.5 17.5
Changeovers
lntl-lnt2 Int5-lnt7 Int3-Int4 Intl-Int7 D12, D2LD42 D13, D22,C11,C31 D11,D31 C23, D41 C22, C32 C12,C21
-
-
-
{D12,D21}^ D42 {D13,D22}<-^ {C11,C13} PDU <->PD31
Ih 4h Ih
{C23,D41}f^ {C22,C32}
2h
-
-
Table 2. Comparison of results for case study. This work Schilling (1997) Zhang and Sargent (1998)
Events 4 30 30
Cll)
Pack.D13
MakeJnt5
Int/Cont Vars 220/673 1042/2746 1318/3237
Relaxation 2695.32 2724 2724
Solution 2695.32 2604 2556
CPU 397 3407 1085
C3l) C32)
672 1 Unos
I
114.5 C12
16.5 Une4 C23
I
—\
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I
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,328 8
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.90 5
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II
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11
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,16.2
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1
l|7 ,8.5, 43.6 114 1 1 17
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470
, 144 0
'
l|4
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1
1
1
1
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,90 5
I|2
111
1
-L
Il3
1 1 1
Figure 2. Finite intermediate storage schedule for case study.
5. Conclusions This work introduced an event-based formulation to the short-term continuous process scheduling problem. The model was shown to produce optimal schedules on several variants of a benchmark case study. Due to the utilization of a non-uniform time grid, the proposed method reduces the number of event points by taking advantage of any independent sub-structures existent in the state-task incidence matrix. Depending upon the specific problem at hand, the handling of events in this work may be functionally equivalent to a uniform time scale employing an order of magnitude more points/slots. Work is in progress towards a unified formulation for mixed production facilities.
6. References Karimi, LA. and CM. McDonald, 1997, Ind. Eng. Chem. Res. 36, 2701. Kondili, E., C.C. Pantelides and R.W.H. Sargent, 1993, Comput. Chem. Eng. 17, 211. Mendez, C.A. and J. Cerda, 2001, Proceedings of the European Symposium on CAPE 11, Eds. R. Gani and S.B. J0rgensen, Elsevier, New York, p. 693. Mockus, L. and G.V. Reklaitis, 1999, Ind. Eng. Chem. Res. 38, 197. Schilling, G., 1997, Algorithms for Short-Term and Periodic Process Scheduling and Rescheduling, PhD Thesis, Imperial College, London, UK. Schilling, G. and C.C. Pantelides, 1996, Comput. Chem. Eng. 20, S1221. Zhang, X. and R.W.H. Sargent, 1998, Comput. Chem. Eng. 22, 1287.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
673
A supporting tool for improvement of batch manufacturing Petra Heijnen and Zofia Verwater-Lukszo Delft University of Technology The Netherlands
Abstract Integration of the activities in a complex system as a batch plant will need extra support. The Management Decision Tool, as described in this paper, will give batch plant managers the opportunity to perform a quick scan of the plant focusing on objectives and activities, and their interactions. As a result the most promising activities for improvements are determined, an optimal way for performing product innovations is stipulated and new ideas for process innovations are generated.
1. Introduction and objectives At the Delft University of Technology an interfaculty research project aimed at improving the efficiency of batch process operations was started in 1998. The industrial partners in this research endeavour represent the chemical and metallurgical process industry, the food industry and a number of agro-based companies. The close cooperation with the industrial partners and the identification of their needs for support on integrated plant management result among others in the development of a management decision tool. The objective of this tool is to support management in taking the following decisions: Which improvements at which activities in the plant will contribute to the overall objective of the enterprise? Which interactions between the activities should be improved to increase their effectiveness as well as their efficiency? How to organise product innovations in the existing plant in an optimal way? To answer these questions a thorough analysis is needed of the company's structure, the goals, the desired performance, the way the activities are performed and the expected improvements. System engineering will be a suitable approach for such an analysis. An industrial site can be seen as a large integrated system, which is built from objects (subsystems: departments, key activities etc) linked together by material and information flows in a complex structure. In this paper the implemented ideas for the Management Decision Tool (MDT) will be discussed. The execution of the proposed method will give the plant manager the opportunity to get a quick scan of promising improvements in the batch plant and to manage his plant in an integrated way. Moreover, it will support the plant manager in the creative search for process innovations.
674
2. The Management Decision Tool The MDT is applicable to each batch plant. The key model of MDT uses a so-called standard plant containing activities, which take place in almost every batch manufacturing plants, as well as activities that take place in all kind of production environments. To use the MDT in a dedicated plant, the present models need only to be adapted to the specific situation and do not have to be built from the ground. The MDT consists of four different steps through which the user is guided. After performing all the steps the user has gained more insight in the organisation of the plant with respect to the plant objectives and their relative weights compared to the main objective of the plant, the activities in the plant and their mutual interactions and the most important possibilities for improvement.
3. Objectives of the plant The first step in the MDT covers a thorough analysis of the objectives of the batch plant. For the MDT an objective tree is developed that will be applicable to most batch plants. In an objective tree the objectives are arranged in a hierarchical way starting from the main objective, which characterizes and defines the area of interest, i.e. the system involved (Keeney, 1992 & 1993). The main objective of the standard batch plant is formulated with respect to the long-term existence of the plant. This strategic objective is made operational by dividing it into more concrete subobjectives. The objective tree is split up in this way into three horizontal levels (Keuning, 2000) 1. The strategic objectives as regards the organization of the relations between the plant and its environment. 2. The tactical objectives as regards the choice of a good structure for the organization and management of the resources in the plant in such a way that the strategic objectives are achieved. 3. The operational objectives as regards the use of the resources that are available in the plant in such a way that the tactical objectives are achieved. Figure 1 shows part of the objective tree as available in the MDT. The user of the MDT could adapt the objective tree to the specific situation in the plant considered. The adaptations will in general be carried out at the level of operational objectives.
4. Activities and their interactions In the second step of the MDT the activities, which guarantee that the operational objectives of the plant are achieved in practice, are formulated. For the MDT an activity model is developed in which the activities of the standard batch plant have been modelled by using the so-called IDEFO technique. IDEFO is a method to model activities and their mutual relations in a hierarchical way (FIPS PUBS 183, 1994). In the MDT firstly all activities are defined that may influence the measure in which the operational objectives are achieved. These activities have a high degree of detail and they can be combined into domed activities. After a number of these compositions the
675 so-called context activity will be achieved. In the same way the interactions between the activities will be modelled and combined into interactions with less degree of detail.
W^ Figure J: Part of the objective tree of the standard batch plant The interactions that are distinguished in an IDEFO model are 1. Input; what is transformed by the activity into the output, 2. Output; what is produced by the activity, 3. Control; what controls the activity, i control Im Ic. 4. Mechanism; what performs the objective activity.
11
input
AcnvrrY mechanism
output
676 In the activity model as used in the MDT a fifth flow is being introduced for the objectives to which an activity may contribute. An arrow coming from the activity and pointing upwards will denote this flow. As an example the objective high product quality will be considered. This objective is decomposed into five sub-objectives. For every operational objective several activities are performed to guarantee that the objective is achieved. These activities are denoted by the grey boxes.
good production process
high quahty of materials
high quahty of installation
sr-i-
's~sr
"XX"
IVfANAGE MAINTENANCE
PURCHASE INSTALLATION
good production personnel
CONTROL PRODUCTION
MANAGE RECIPES
Figure 2: Activities added to the operational objectives
In Figure 3 a small part of the activity model for the standard batch plant is shown, where the interactions between the activities "manage maintenance" and "control production" are central.
i
iL
1
CONTROL
¥
PRODUCTION iV control tools
high quality of mstallation
mamtenance standards and methods
maintenance requests
T
^r
MANAGE MAINTEt^ANC:E
responses & technical feedback
Figure 3: IDEFO diagram of ''Control Production" and "Manage Maintenance"
677
5. Contribution of the objectives to the overall objective In the third step of the MDT the user will be asked to order the objectives in the objective tree to the degree in which they contribute to the main objective. The pairwise comparison method (Saaty, 1980) will be used for the ranking of the objectives. In every group of objectives in the objective tree, i.e. all child objectives of one parent, every pair of two objectives is compared. The user decides for every two objectives which one will yield the highest contribution to the main objective. Doing this for every pair in the group a unique preference list will result. With the formula m 1 where /:,• = S ~ r=i r
kwI
m
j=\
(^)
}
the weight w, of the /th objective in the preference list is calculated. The relative weight of an operational objective can now be calculated by multiplying its weight with the weights of all its parents. As an example we look again at the five child objectives of high product quality. a. Good production process b. High quality of installation c. High quality of materials d. Good insight in quality e. Good production personnel The user is asked to decide for randomly chosen pairs, which of the two is more important to achieve a high product quality. Assume that the user decides to the following preference structure: dycya>-bye. With Equation (1) the relative weights of the objectives are: 1. Good insight in quality, weight: 0.46 2. High quality of materials, weight: 0.26 3. Good production process, weight: 0.16 4. High quality of installation, weight: 0.09 5. Good production personnel, weight: 0.04 When all weights of the operational objectives are known by multiplying the relative group weights with all the weights of their parents, the objectives with the highest contribution to the main objective of the plant are determined. With the results of the previous step the activities that are responsible for these objectives can be found. For improving the degree in which the main objective is being achieved, the user should focus on the improvement of these activities.
678
6. Efficiency of activities and possible improvements In the last step of the MDT the possible improvements of the activities are being mapped. Here a difference is made between the effectiveness of the activity and the efficiency of the activity. The effectiveness of the activity is defined as the degree in which the activity contributes to its objectives. Most operational objectives can be influenced by more than one activity, as shown in the activity model. To be able to measure the degree in which an objective is achieved, a performance indicator will be linked to the objective. To measure the effectiveness by activity the performance indicator of an objective should be split into several sub indicators that each can be influenced by just one activity. The efficiency of the activity is defined as the costs or time needed to transform the input of the activity into its output. As mentioned before the objective of the MDT is to find activities that are good candidates for improvements, i.e. which will yield a high contribution to the main objective of the plant. Measuring the effectiveness of the activity by its performance indicators can determine the actual contribution of the activity to the main objective. Improvement of the effectiveness of the activity may not be performed at the expense of the efficiency of the activity. With the assistance of the activity model the user can determine which input the activity requires to deliver the right output. The right input on the right moment will improve in general the efficiency as well as the effectiveness of the activity. With the results of the MDT the user may 1. determine which activities are good candidates for improvement, 2. determine the actual performance of the activities, 3. improve the activities by better tuning of their interactions.
7. Results and final remarks The MDT is still under development. The modelling phases for objectives and activities are already evaluated and applied in the plants of the industrial partners. At the moment research concentrates on the definition of the contribution of the chosen operational objectives to the overall plant goal. This will be realised by discussions with a selected team of process people (plant managers, quality managers, schedulers and operators) from an industrial plant in the food and chemical industry. Next, several case studies will be performed for further evaluation.
References IDEFO: FIPS PUBS 183, 1994, Standard for Integration Definition for Function Modelling. Keeney, Ralph L., 1993, Decisions with multiple objectives; preferences and value tradeoffs. New York Cambridge University Press. Keeney, Ralph L., 1992, Value-focused thinking; a path to creative decisionmaking. Cambridge, Mass. Harvard University Press. Keuning, D., D.J. Eppink, 2000, Management en Organisatie, theorie en toepassing, Educatieve partners Nederland BV, (in Dutch). Saaty, T.L., 1980, The Analytic Hierarchy Process, McGraw-Hill Book Co, New York.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
679
Comparison of three scheduling methods for a batch process with a recycle J.Heinonen and F.Pettersson [email protected], [email protected] Heat Engineering Laboratory, Department of Chemical Engineering Abo Akademi University, Biskopsgatan 8, FIN-20500 Abo, Finland
Abstract A short term scheduling problem from an industrial case is modelled in three different ways. A discretized MILP-model, a heuristic approach and a genetic algorithm are tested. The scheduling task is complicated by a recycle connection. Storage capacity for raw materials and intermediates are restricted. Some productions steps are operated continuously and some in batch mode. A one-week schedule and a two-week schedule are constructed with the different models and compared with industrial results. The genetic algorithm outperforms the other approaches, though the computational efforts are larger.
1. Introduction The significance of effective methods for production scheduling and planning has increased due to a general effort to improve profitability. The difficulties in scheduling tasks arise primarily from sharing of resources and equipment and the desire to operate the process cost-effectively. Most commonly the need for such methods occur in production operating in batch mode. Typically batch processes are used in production of pharmaceuticals, food and specialty chemicals. One of the most important advantages of batch processes is their flexibility. To be able to exploit this advantage, an effective production planning is of great importance. Some process steps (e.g. crystallization) are also better controlled in batch mode compared to their continuous counterparts. In spite of a common trend towards continuous processes, the batch process continues to be the best alternative in several cases. Methods for short term scheduling using mathematical programming have been studied since the 1970s. The problems are described with either discretized time intervals or continuous time formulations. Although, the computational capacity increases continuously, one problem using mixed integer programming is the combinatorial character of scheduling problems, resulting in large problems with long execution times when problems of industrial relevance are considered. The planning horizon obtained in this work varies from 10 to 15 hours using a state task network formulation, presented by Kondili et al. (1993), when modest computational times are required. In order to speed up the computation, a heuristic solution method was developed based on an exhaustive search of the most promising steps for each new decision to be made.
680 The drawback of this kind of methods is the need for major changes in the strategy when changes in the process setup occur. Evolutionary programming, and especially genetic algorithms, has been quite successfully applied to i.e. job shop and travelling salesman problems thus encouraging the development of a genetic algorithm based planning tool for the case problem.
2. Problem description The process consists of ten pairs of pre- and main reactors operating in batch mode. Two different feeds exist. Processing times in the reactors are considerably long, varying from 30 hours to 40 hours, depending on the selected raw materials. The reactors are fed through four parallel concentration units, operating continuously. After the reactors the product is separated. The remaining part is recycled through a continuously operating reactivation process step back to the concentration units. The task is to find an optimal sequence of unit operations when the planning horizon is a week or more. The process is operated in campaigns, thus all available resources are dedicated to the current product. The objective is to maximize the product stream, while fulfilling a number of constraints imposed upon the process equipment, e.g. available units capacity limits, storage limits for all intermediates and amount of raw material available. Concentrators should also, if possible, be operated without interruptions. The cistern profiles must also be within certain safety limits. Unexpected changes or disturbances leads to situations where replanning has to be performed, setting conditions on a fast planning method. Feed A ^^^
Intermediate storage
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Reactivation step I
Concentrators Reactors
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ALm
Separators
Intermediate storage
Figure L An overview of the studied process.
681
3. Model development The concentrators are modelled with four distinct properties in mind. First, they should (if possible) be operated continuously, since interruptions in their performance are costly. Second, since there are a number of different concentrators to choose from, the number of changes in the concentration procedure, once started, should be restricted to maximum one. This is beneficial from an operators point of view since many changes take much of an operators time. Thirdly, the first property should not be allowed to compromise in any way the overall optimum of the solution, meaning that the concentration should still be efficiently sequenced but not by cutting down the number of starting reactors just to ensure continuous concentrator operation. And as fourth, the concentration event into a reactor should be continuous, and not contain any gaps in the resulting sequencing. The separation step can only separate a single batch at any given time. The MILP-model does not necessary use a FIFO-queue for the separation, since a batch can be put on hold in favour of a batch that might prove to be more advanteguous. The heuristic method and the genetic algorithm use a strict FIFO-queue. The reactivation step is modelled as a linear input/output flow with the reactivation yield determining the difference in their values. 3.1 The discretized MILP-model The scheduling of the multistage process was formulated as a state-task network for the MILP-model inspired by Sahinidis and Grossmann (1991) as well as Voudoris and Grossmann (1996). The time interval for the discretization was set at 1 hour. This means that the resulting Gantt-charts contain some crudeness, but it is no real problem since the concentrator flow rates can be regulated accordingly. For detailed information of how the model works see Heinonen and Pettersson (2001). The solution times for different subproblem horizon lengths can be seen in table 1. It is clear that large problems must be divided into smaller subproblems and solved sequentially in order to make schedules with reasonable computational efforts. Schedules with longer timespans than 15 hours are required since the cycle times for the reactors are very long. It can be mentioned here that a continuous MILP-model was tried but due to the recycle step the problem was not easily translated into the formulation proposed by Pinto and Grossmann (1995).
Table 1. Solution times for different subproblem horizon lengths (discretized MILP-model). horizon (h)
solution time (min)
# binary variables
10 20 30
0.18 10.48 128.87
1000 2000 3000
Solution procedure The user inputs the current state of the process through a graphical user interface (GUI) (including flow rates, reactor stages and cistern profiles etc.) and a MILP-model is generated on basis of the input state. The model is sent to Cplex, which is a
682 commercially available solver for MILP-problems. Cplex solves the subproblem and returns the results to the GUI where the next subproblem is automatically generated. The system loops until the required time horizon for the entire schedule is reached and the entire schedule is presented to the user. Gantt-charts for the reactors, concentrators and the separation step are created as well as cistern profiles. 3.2 The heuristic approach This model utilizes a decision based search-tree in determining the best way to use the concentrators. The unit operation times are floating point variables and the resulting charts are more accurate since no time discretization is needed. Solution procedure The basic input task is the same as described above, and one gets similar Gantt-charts upon completion of the run. No separate solver is needed since search trees are standard data structures and can be implemented with ease. The operation alternatives for the concentrator at time t are added to the search tree and the cistern profiles and such are simulated for those decisions. After the simulation step we are at time t + tconc » where tconc is the time it takes for the actual concentrator to fill the reactor, and new decisions have to be made. The alternatives are again inserted into the tree and the system loops until the desired schedule length is reached. At average there are three decisions at every decision point in the schedule for the concentrators. Each level in the search tree thus contains 3" decisions, where n is the tree level, which have to be simulated. This effectively puts an upper bound on the subproblem size. 100 hours has been found as a reasonable length. 3.2 The genetic algorithm The basic input procedure and Gantt-charts are similar as above. The genetic algorithm (GA) is capable of solving the entire time horizon of one month in one single system though it requires seemingly lengthy calculations. Shorter systems might save some valuable calculation time but may not give as good schedules. All unit operation times are floating point variables and result in a greater accuracy than the discretized MILPmodel. No separate solver is used. A single gene is a decision of which reactor and which feed a concentrator should start with. The chromosomes thus use a 'permutation encoding'-type of representation. A chromosome consists of a sequence of genes (unit operations for a concentrator) and all the concentrators and their operations formed together allow the system to be simulated. A fitness value is calculated for each chromosome, which is based upon the amount of resulting products as well as how the cistern profiles behave in the simulated system. Solution procedure The algorithm is initialized by creating a set of chromosomes with random genes. Then the schedule implied by the chromosomes is simulated and fitness values calculated. The next generation of chromosomes is determined by means of roulette-wheel selection and single- or two-point crossover. A small mutation rate is kept, as well as elitism. Since the elitist chromosome has a tendency to often be a selected parent, a diversity check is made at regular intervals. If deemed necessary, the mutation rate can
683 be increased by a user chosen amount during a single generation. Thus the GA can effectively break free from local optimums. The system can be set to loop for a specific amount of generations or the user can stop it at will and later restart it from the same point.
4. Illustrative problem A common starting situation for the three approaches has been taken from an already implemented industrial schedule. A one-week schedule and a two-week schedule are constructed with the different models and compared to the industrial results. The amount of products manufactured while fulfilling all cistern profile- and other constraints is taken as a measure of the schedule's relative goodness. Table 2 shows the results for the scheduled horizon. A resulting two-week Gantt-chart for the process can be seen in figure 2. Table 2. The amount of manufactured products for the different schedules and models. 1 week schedule 48.41 49.98 47.83 50.44
Implemented MILP Heuristic GA
2 week schedule 98.50 97.35 95.37 97.73
The GA outperforms both the discretized MILP and the heuristic approach. The relatively good results with the MILP-model, despite the time discretization, can be partially explained by a different approach to the separation step as explained before. The good implemented result for the two-week schedule is possible since in real life feeds are sometimes mixed if a cisternprofile is near its upper extreme. Mixing should be avoided, and the models thus operate under strict rules not to mix the feeds.
£J
MSSIH
;j
SE;ssa.j i 'EISi^SO S U S I E ] IJ
^ssa. KSISIBL
M^
»^i2_l liai^LLk.. i l l
'^:"l;>k.^i
lls^
Xl
13
J E ^ S 0 { Ii;;:^^^y:;-i1l I 1:^;:^!^^^:.^ [ i {e-:^:Tr;\j1 Q1 i&: ^'py:^W n r n
Figure 2. A resulting two-week schedule with full sequencing made with the GA. The letters stand for the different feeds, the sequence in front is the concentration part after which the reaction starts, and the sequence at the end is the cleaning part.
684
5. Computational experience Table 2. The average solution times for the different approaches, on a 600MHz PC MILP Heuristic GA
1 week schedule 2.45 min 2.26 min 6.77 min
2 week schedule 4.60 min 6.82 min 38.25 min
__
_
The discretized MILP-model with problem subtasking performs best while the heuristic comes as close second. The GA requires the lengthiest calculations. Since the GA is stochastic by nature, the time required until convergence varies. A two-week schedule could take 20 minutes or it could last for as long as an hour. Two-point crossover was used during the runs for the GA. The times are average from three different runs.
6. Conclusion The GA not only gives the best schedules, it also allows us to construct schedules with a large enough time horizon in a single step. When speed is an issue (e.g. during a mechanical breakdown when a modified schedule is needed fast) the MILP model in its present form could be preferable. Then again a shorter time horizon could be run with the GA. The heuristic approach did not meet the expectations on either the computational speed or the obtained solutions. It does, however, cope with reasonable planning horizons without the separation into subproblems as in the MILP approach. It can be argued that longer horizons give better schedules since you can be relatively sure that the process is not driven into unwanted states. This was seen as a problem in the MILP approach. Acknowledgment - The financial support from TEKES, the Finnish Technology Agency, is gratefully acknowledged.
7. References Goldberg D.E. 1999, Genetic algorithms in search, optimization & machine learning. Addison Wesley, California. Heinonen J. and Pettersson F. 2001, Scheduling a process with a recycle and units operating continuously and in batch mode, ICHEAP 5, Florence, Italy. Kondili E., Pantelides, C.C, and R.W.H. Sargent. 1993, A general algorithm for scheduling batch operations. Comput. Chem. Eng. 17,211. Pinto J.M. and Grossmann I.E. 1995, A Continuous Time Mixed Integer Linear Programming Model for Short-Term Scheduling of Multistage Batch Plants. Ind. Eng. Chem. Res. 34, 3037. Sahinidis N. V. and Grossmann I. E. 1991, Reformulation of Multiperiod MILP Models for Planning and Scheduling of Chem. Processes, Computers chem. Eng. 15, 4, 255. Voudoris V. T. and Grossmann I. E. 1996, MILP Model for Scheduling and Design of a Special Class of Multipurpose Batch Plants, Computers chem. Eng. 20, 11, 1335.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
685
Computer-Aided Design of Redundant Sensor Networks Georges Heyen^ , Marie-Noelle Dumont\ Boris Kalitventzeff 1 : Laboratoire d'Analyse et Synthese des Systemes Chimiques, Universite de Liege, Sart Tilman B6A, B-4000 Liege (Belgium), email [email protected] 2 : BELSIM s.a., rue Georges Berotte 29A, B 4470 Saint-Georges-sur-Meuse, Belgium
Abstract A systematic method to design sensor networks able to identify key process parameters with a required precision at a minimal cost is presented. The procedure is based on a linearised model, derived automatically from a rigorous non-linear data reconciliation model. A genetic algorithm is used to select the sensor types and locations.
1, Problem position The application of data reconciliation to plant monitoring in now considered as standard practice. Redundant measurements allow reducing the uncertainty due to random errors. Unmeasured parameters can be estimated safely from reconciled state variables. However little has been published on the design of measurement systems allowing to achieve a prescribed accuracy for key process parameters, and to secure enough redundancy to ensure resilience with respect to sensor failures. Madron (1972) solves the linear mass balance case using a graph-oriented method. Bagajewicz (1997) analyses the problem for mass balance networks, where constraint equations are linear. In this study, we propose a general mathematical formulation of the sensor selection and location problem, in order to reduce the cost of the measurement system while providing estimates of all specified key process parameters within a prescribed accuracy. The goal is to extend the capability of previously published algorithms, and to address a broader problem, not being restricted to flow measurements and linear constraints. For sure minimising the cost is not the only possible objective function: designing a sensor network should better be treated as a multi-objective optimisation problem, and address other features, such as redundancy and resiliency to equipment failure or capability to detect gros§ errors. This study is a first step toward that goal, and we plan to later refine the technique by selecting a more general objective function. In the optimisation problem formulation, the major contribution to the objective function is the annualised operating cost of the measurement system. The set of constraint equations is obtained by linearising the process model at the nominal operating conditions, assuming steady state. The process model is complemented with link equations that relate the state variables to any accepted measurements, or to key process parameters whose values should be estimated from the set of measurements. In our case, the set of state variables for process streams comprises all stream temperatures, pressures and partial molar flow rates. In order to handle total flow rate measurements, the link equation describing the mass flow rate as the sum of all partial molar flow rates weighted by the component's molar mass has to be defined. Similarly,
686 link equations relating the molar or mass fractions to the partial molar flow rates have also to be added for any stream where an analytical sensor can be located. Link equations have also to be added to express key process parameters, such as heat transfer coefficients, reaction extents or compressor efficiencies. In the present study, we assume that all variables are measured; those that are actually unmeasured will be handled as measured variables with a large standard deviation. Thus data reconciliation, taking linearised constraints into account, requires the solution of : min(X-X'/w(X-X') (1) s.t. A X + D = 0 where X is the array of process variables (size m), X' is the set of measured values and W=diag(l/a,') is the weight matrix (the diagonal terms of the inverse of the measurement covariance matrix). The linear approximation of the constraints is easily obtained from the Jacobian matrix A of the non-linear model evaluated at the solution. Constraints are handled using Lagrange formulation: min L = (X-X')T W (X-X') + 2 X^ (A X + D) (2)
x,x
Solving for stationarity conditions:
fx'
^
w
A^"
-1
"WX'"
"WX'"
(3) _ -D _ -D A 0 Matrix M can easily be built, knowing the variance of measured variables appearing in sub matrix W, and the model Jacobian matrix A (which is constant). This matrix will be modified when assigning sensors to variables. Any diagonal element of matrix W will remain zero as long as a sensor is not assigned to the corresponding process variable; it will be computed from the sensor precision and the variable value when a sensor is assigned, as shown later in section 2.3. In fact we are not interested in the solution of system (3), since measured values X' are not known. The variance of reconciled values X is related to the variance of measurements X' as shown in Heyen et al. (1996):
_x_
= M'^
The elements of M'are obtained by calculating a LU factorisation of matrix M. In case matrix M is singular, we can conclude that the measurement set has to be rejected, since it does not allow observing all the variables. Row i of M^ is obtained by back substitution using the LU factors, using a right hand side vector whose components are 5ij (Kronecker factor: 6, = 1 when i=j, 5,=0 otherwise). In the summation of equation (4), we only take into account variables X'^ that have been assigned a sensor, the variance of unmeasured variables being set to infinity.
2. Algorithm description Solution of the sensor network problem is carried out in 7 steps: 1. Process model formulation and definition of link equations. 2. Model solution for the nominal operating conditions and model linearisation. 3. Specification of the sensor database and related costs. 4. Specification of the precision requirements for observed variables. 5. Verifications of problem feasibility. 6. Optimisation of the sensor network.
687 7. Report generation. Each of the steps is described in details before presenting a test case. 2.1 Process model formulation and definition of link equations. This is easily done in the Vali 3 data reconciliation software, which is used as the basis for this work (Belsim 2001). The model is formulated by drawing a flowsheet using icons representing the common unit operations, and linking them with material and energy streams. Any acceptable measurement of a quantity that is not a state variable (T, P, partial molar flow rate) requires the definition of an extra variable and the associated link equation, what is done automatically for standard measurement types (e.g. mass or volume flow rate, density, dew point, molar or mass fractions, etc). Similarly, extra variables and link equations must be defined for any process parameter to be assessed from the plant measurements. A proper choice of extra variables is important, since many state variables cannot be measured in practice (e.g. no device exists to directly measure a partial molar flow rate or an enthalpy flow). To solve the model, enough variables need to be set by assigning them values corresponding to the nominal operating conditions. The set of specified variables must match at least the degrees of freedom of the model, but overspecifications are allowed, since a least square solution will be obtained by the data reconciliation algorithm. 2.2 Model solution for the nominal operating conditions and model linearisation. The data reconciliation problem will be solved, either using a large-scale SQP solver, or the Lagrange multiplier approach. When the solution is found, the value of all state variables and extra variables is available, and the sensitivity analysis is carried out (Heyen et al, 1996). A dump file is generated, containing all variable values, and the non-zero coefficients of the Jacobian matrix of the model and link equations. All variables are identified by a unique name indicating its type (e.g. S32.T is the temperature of stream S32, E102.K is the overall heat transfer coefficient of heat exchanger E102, S32.MFH20 is the molar fraction of component H20 in stream S32). 2.3 Specification of the sensor database and related costs. A data file must be prepared that defines for each sensor type the following parameters: • the sensor name • the annualised cost of operating such a sensor • parameters ai and bj of the equation allowing to estimate the sensor accuracy G- from the measured value Xj', according to the relation: a^ = aj-H bj Xj' • a character string pattern to match the name of any process variable that can be measured by the given sensor (e.g. a chromatograph will match any mole fraction, thus will have the pattern MP*, while an oxygen analyser will be characterized by the pattern MF02) 2.4 Specification of the precision requirements for observed variables. A data file must be prepared that defines the following information for the selected key performance indicators or for any process variable to be assessed: • the composite variable name (stream or unit name -h parameter name) •
the required standard deviation a^, either as an absolute value, or as a percentage of the measured value.
688 2.5 Verification of problem feasibility. Before attempting to optimise the sensor network, the programme first checks the existence of a solution. It solves the linearised data reconciliation problem assuming all possible sensors have been implemented. This provides also an upper limit Cmax for the cost of the sensor network. A feasible solution is found when two conditions are met: • the problem matrix M is not singular; • the standard deviation a, of all selected reconciled variables is lower than the specified value cr- . When the second condition is not met, several options can be examined: • add more precise instruments in the sensor definition file; • extend the choice of sensors by allowing measurement of other variable types; • modify the process definition by adding extra variables and link equations, allowing more variables besides state variables to be measured. 2.6 Optimisation of the sensor network. Knowing that a feasible solution exists, we can start a search for a lower cost configuration. The optimisation problem as posed involves a large number of binary variables (in the order of number of streams * number of sensor types). The objective function is multimodal for most problems. However identifying sets of suboptimal solutions is of interest, since other criteria besides cost might influence the selection process. Since the problem is highly combinatorial and not differentiable, we attempted to solve it using a genetic algorithm (Goldberg, 1986). The implementation we adopted is based on the freeware code developed by Carroll (1998). The selection scheme used involves tournament selection with a shuffling technique for choosing random pairs for mating. The evolution algorithm includes jump mutation, creep mutation, and the option for single-point or uniform crossover. The sensor selection is represented by a long string (gene) of binary decision variables (chromosomes); in the problem analysis phase, all possible sensor allocations are identified by finding matches between variable names (see section 2.2) and sensor definition strings (see section 2.3). A decision variable is added each time that a match is found. Multiple sensors with different performance and cost can be assigned to the same process variable. The initial gene population is generated randomly. Since we know from the number of variables and the number of constraint equations what is the number of degrees of freedom of the problem, we can bias the initial sensor population by fixing a rather high probability of selection (typically 80%) for each sensor. We found however that this parameter is not critical. The initial population count does not appear to be critical either. Problems with a few hundred binary variables were solved by following the evolution of populations of 10 to 40 genes, 20 being our most frequent choice. Each time a population has been generated, the fitness of its members has to be evaluated. For each gene representing a sensor assignment, we can estimate the cost C of the network, by summing the individual costs of all selected sensors. We also have to build the corresponding matrix M (equation 3) and factorise it. In an initial version of the code, we used SUBROUTINE MA29AD from Harwell library (1984), aimed to
689 factorise a general symmetric matrix. However later experience with larger test problems indicated that taking advantage of the sparsity of M matrix could save significant computer time. Indeed using Belsim's sparse matrix code reduced the computer time by a factor of 25 for a problem involving 312 variables. The standard deviation Gi of all process variables is then estimated using equation 4. This allows calculating a penalty function P that takes into account the uncertainty affecting all observed variables: P =^P. ,=1
where
P, = - 7 cr,
when a. < al
P =0.01min
•»-f^"
(5) when a. > G\
(6)
The fitness function F of the population is then evaluated as follows: • if matrix M is singular, return F = - C^ax • otherwise return F = -(C + P) Penalty function (5) increases (slightly) the merit of sensor networks that perform better than specified. Penalty function (6) penalises genes that do not meet the specified accuracy, but it does not reject them totally, since some of their chromosomes might code interesting sensor sub networks. The population is submitted to evolution according to the mating, crossover and mutation strategy. Care is taken that the current best gene is always kept in the population, and is duplicated in case it should be submitted to mutation. After a specified number of generations, the value of the best member of the population is monitored. When no improvement is detected for a number of generations, the current best gene is accepted as a solution. There is no guarantee that this solution is an optimal one, but it is feasible and (much) better than the initial one. 2.7 Report generation. The program reports the best obtained configurations, as a list of sensors assigned to process variables to be measured. The predicted standard deviation for all process variables is also reported, as well as a comparison between the achieved and target accuracies for all key process parameters.
3. An example As an example we tried to design a sensor network for an ammonia synthesis loop. The process involves a 5-component mixture (N„ H„ NH3, CH^, Ar), 13 units, 19 process streams, 10 utility streams (cooling water, refrigerant, boiler feed water and steam). The units are: 2-stage compressor with 2 intercoolers, recycle mixer, recycle compressor, reactor preheater, ammonia converter, waste heat boiler, water cooled condenser, ammonia cooled condenser, vapour-liquid separator, purge divider and flash drum for expanded ammonia condensate. The model involves 181 variables and 131 constraint equations. Accuracy targets are specified for 45 variables. It includes extra measurable variables (molar fractions) and some unit parameters to be monitored (e.g. heat exchange transfer coefficient, compressor efficiency, and extent of reaction, departure from equilibrium).
690 The sensor database allows choosing between sensors of different accuracy and cost, namely 2 temperature sensors, 3 flow meters, 2 composition analysers and 2 types of pressure gauges. The program detects that up to 166 sensors could be installed. Thus the solution space involves 2^^^ =9.3 10^^ solutions (most of them being unfeasible). We let the search algorithm generate 4000 populations, which required 80000 function evaluations and 776 CPU seconds (PC with 1.33 GHz AMD Athlon processor, program compiled with Compaq Fortran compiler, only local optimisation). The initial cost function was Cmax= 17340 cost units with all sensors selected. Optimisation brought it down to 1110 cost units. A solution within 60 cost units of the final one was attained after 5700 evaluations (in less then one minute).
4. Conclusions and future work The available software could be improved by allowing more flexibility in the sensor definition (e.g. defining acceptable application ranges for each sensor type), or by implementing different objective functions besides the cost. Possible objectives could address the resiliency of the sensor network to equipment failures, or the capability to detect gross errors, in the line proposed by Bagajewicz (2001). There is no guarantee that this solution found with the proposed method is an optimal one, but it is feasible and (much) better than the initial one. Thus we claim that this algorithm contributes to the rational design of sensor networks.
References Bagajewicz M.J., 1997, chapter 6 in "Process Plant Instrumentation: Design and Upgrade", Technomic Publishing Company. Bagajewicz M.J., 2001, Design and Retrofit of Sensor Networks in Process Plants, AIChE J., 43(9), 2300-2306 Belsim., 2001, VALI 3 User's Guide, Belsim s.a, B 4470 Saint-Georges-sur-Meuse, Belgium Carroll D.L., 1998, FORTRAN Genetic Algorithm Driver version 1.7, download from
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
691
Improving Gelatin Plant Productivity By Modelling The Demineralization Process Of Animal Bones D.A. Horneman\ M. Ottens\ M. Tesson^, L.A.M. van der Wielen^ ^Kluyver Laboratory for Biotechnology, Delft University of Technology, Julianalaan 67, 2628 EC, Delft, The Netherlands ^Delft Gelatin BY, P.O. Box 3, 2600 AA Delft, The Netherlands
Abstract A dynamic model is developed that simulates a large size, full scale animal bone demineralization process, which is operated as a simulated moving bed system. The model at the particle level, is based on the unreacted shrinking core model. To solve the system of couples PDE's and ODE's the simulating package gPROMS is used. The model described is succesfuUy applied to find the optimal process conditions and to increase the productivity of the existing significantly.
1. Introduction Gelatin is derived from collagen, the primary protein component of animal connective tissues like bone, skin and tendons (Croome and Clegg, 1965, Mark et al. 1987) Outside the United States, cattle bones are the most important source for collagen (Babel et al., 2000). Current industrial practice involves a first bone extraction step with hot water to reduce the fat content to approximately 1 %. After this step the bone particles contain on average 20-25 % of collagen. The principal non-collagen component of bone is the mineral salt tri-calcium phosphate Ca3(P04)2, which is extracted by a dilute acid stream (usually 5 % hydrochloric acid HCl (Makarewicz et al., 1980)). This extraction step is called the demineralization and is described by the following stoichiometric equations: Ca3(P04)2^ -h 4HC1^ -> Ca(H2P04)2^ + 2 CaCh^ CaCOs^ + 2 HCl^ -> C02^ + H20^ + CaC^^
(1) (2)
HCl diffuses into the bone particles and reacts with Ca3(P04)2 to form mono-calcium phosphate Ca(H2P04)2 and calcium chloride CaCl2 that are soluble in the acid stream. This demineralization of bone is considered to be diffusion limited, with the reactions considered being almost instantaneous and complete (Croome and Clegg, 1965, Makarewicz et al., 1980). Delft Gelatin BY, a Dutch gelatin producer, uses serial reactors for the demineralization of shredded animal bone "particles" (Figure 1). It is possible to have real counter current contact between the solid bone particles and the fluid by appropriately connecting and scheduling the reactors to the acid stream, like in a simulated moving bed system (Schulte and Strube, 2001). In these systems, the solid phase is not actually moving but the countercurrent motion is simulated. The acid stream flows through the reactors starting in the reactor with the most highly demineralized bone particles and ending in the reactor filled with fresh, non-demineralized bone particles. Upon
692 completion of the demineralization in the first reactor, this reactor is emptied and refilled with fresh bone particles and placed at the end of the series.
demineralized bones
5 % HCl
' ^
^
non-demineralized bones
Figure 1. Schematic layout of the reactors system The demineralization process has not changed much over the last 100 years. It is described in this study with the use of a dynamic mathematical model using the shrinking core model at the bone particle level. The purpose of this study is to show that this model is in good agreement with the real process and to use it for process optimization.
2. Theory To describe the demineralization process, some general assumptions are made. Firstly, the demineralization is limited by internal diffusion with the reactions (1) and (2) being instantaneous and complete (Makarewicz et aL, 1980), (Croome and Clegg, 1965)). Secondly, the demineralization process does not influence the size and the shape of the bone particles. In this way, a growing demineralized layer is formed around an unreacted core separated by the sharp reaction front (Makarewicz et ai, 1980). This is described by the unreacted shrinking core model ((Levenspiel, 1972). Figure 2 gives a schematic representation of the unreacted shrinking core model. The flux of HCl, Jnch in (moles/m^s) diffusing into the bone particle is described by Fick's first law:
'HCl
-'
dN
li£L^4nr'D,^£^^
dt
(3)
dr
where A^//ais the amount of HCl in moles, De is the effective diffusion coefficient (mVs) and r is radial position in the bone particle (m) (Figure 2). The change in mineral amount can be described in terms of radial position of the core boundary: -dN,,,ir, ^-PdNfici
=-Cnund(inr,^
) = -4^C„,inr,
dr,
(4)
693 where /? is the stoichiometric coefficient, C^,>, is the concentration of Ca3(P04)2 in moles per m^ and r^ the radius of the non-demineralized core (Figure 2).
Demineralized layer Non-demineralized core
'HCl
R r r ^
r_ R
0
-> r
Figure 2. The unreacted shrinking core model If the fraction of demineralized bone equals X than:
]-X
=
volume of non reacted core
j ^^c
total volume of particle
—nR^
( r,, ^
(5)
The two balances (eq. 4 and 5) can now be written in terms of fraction demineralized bone: dNHCl _,^^ ^ n dt - 4nUg C HClMlk '^ -
{l-^y
(6)
i-{i-xi^
dX _ dt
3b C
R
(}-X)'
^ „
(7)
l-(l-X)^
2.1 Demineralization in packed bed reactors The reactors used for demineralization are packed with the bone particles, which have a relatively broad size distribution, and also may differ in water and mineral content. To increase the accuracy of the calculations, the particles are divided into N fractions with different properties of the bone particles. The balances for these packed reactors now become: ^CHCl(t,z) _ dt
0,dCHci(t,z) 8 Adz
A^
^4nR^a,,D^ -X"""''"""^c„c/r^o (^-^''('''^'^\ 1-(1-X„(t.z))'
(8)
694
dXnit,z)
313 =
^^
{1-X,{t,z}y jDe<^Ha(^'Z)
Cmin^n
T I-(1 - X,,(t, Z))'
ze(0,L]
(9)
where e is the fraction of liquid in the reactor, Un the number of bone particles of fraction n in the reactor and L the length of the reactor bed.
3. Modeling in gPROMS For the calculations we have used the flexible scheduling and simulation package gPROMS. The first part of the model is a description of the demineralization in one reactor. This is done by the N partial differential equations (PDE's) of equation 7 and A^ ordinary differential equations (ODE's) of equation 8 where A^ is de number of bone fractions. The reactors are coupled by:
Q a i = CHa,rJ
ie[2,...m]
(12)
where NR is the number of reactors. The reactors are switched mathematically, moving the content of reactors /+1 to /:
Qi=Q[^Jid
/G[i,2,...,yv/?-i]
(13)
where Q.\ gives the value of variable or parameter Q in reactor / at length z after switching and Q^oiJ gives the value of variable or parameter Q in reactor / at length z before switching. After the switching procedure, new values are given to the parameters and variables in the new reactor (reactor NR). The PDE's are solved with the methods of lines. In this method, the equations are rewritten into ODE's by using a discretization method. In this case, a first order forward discretization method is used.
4. Process characteristics The dots in figure 3 show the fraction of undemineralized bones, K, in top of the reactor as function of time. From this profile, it is possible to determine the demineralization time of the bone particles in the top of the reactor. The demineralization starts when Y decreases and is finished when K = 0. In the real process, the demineralization is followed by monitoring the density of the liquid flows in and out each reactor. Figure 3 shows an example for one of the reactors. The dashed line gives the density of the incoming flow and the solid line of the outcoming flow. The reactor is first placed at the end of the series of reactors. The incoming flow has a high density due to the high concentration of Ca(H2P04)2, which has been produced in the former reactors. During the process the reactor moves forward into the series. At a certain time the incoming flow of the reactor contains less
695 Ca(H2P04)2 resulting in a lower density. At this point, the demineralization starts in this reactor. The density of the flow out of the reactor is still high due to the production of Ca(H2P04)2 in the reactor, but decreases when the production of Ca(H2P04)2 decreases. At this point, the demineralization in the next reactor will start. In the end, the density of both flows will be the same, this is the end of the demineralization. The reactor is now the first one in the series, it will be emptied and refilled again and placed back at the end of the series.
50
100
time(h)
150
200
250
Figure 3. Density profiles of incoming (dashed line) and outcoming (solid line) flow and the ash content of the bone particles in top of the reactor (diamonds) as function of time.
The model calculates the fraction of undemineralized bones in the reactor but it is difficult to measure this continuously during the real process. Therefore Y is compared with the measured density profiles.
5. Results modeling To simulate the real process the incoming flow and HCl concentration in this flow were measured as function of time and used as input variables in the model. All properties of the bone particles were given by Delft Gelatin. The particle diameter is between the 4 and 14 mm, the water content is about 9.6 wt % and the ash content (fraction non minerals) is about 34.5 wt %. The effective diffusivity was experimentally determined. Figure 4 shows the result of the model (line) with the measured density profile (squares and diamonds) of the real process. The left Y-axes gives a value of the fraction of undemineralized bone:
r = 7-X^^•^^
(14)
/i=7
Figure 4 shows that the simulated profiles are in good agreement with the measured density. Therefore the model could be used for the optimization of the demineralization
696 process. On the basis of the developed model it proved to be possible to increase the plant productivity substantially.
1 1,
0.8 0.6 0.4 0.2 0
0
25
50
75
100
125
150
175
200
225
250
time [hr]
Figure 4. The measured density (squares and diamonds) of the flows in and out a reactor together with the model results (line).
6. Conclusion The developed dynamical model, based on the unreacted shrinking core model, accurately describes the real large scale demineralization process. It has been applied succesfully for determination of the optimal process conditions to increase the productivity.
References Babel, W., Schulz, D., Giesen-Wiese, M., Seybold, U., Gareis, H., Dick, E., Schrieber, R., Schott, A., Stein, W., 2000. Ullmann's Encyclopedia of Industrial Chemistry (6* ed.). 2000 Electronic Release. Croome R.J., Clegg E.G., 1965, Photographic gelatin. The Focal Press, New York. Levenspiel O., 1972, Chemical Reaction Engineering. John Wiley & Sons Inc., New York. Makarewicz P.J., Harasta P.J., Webb S.L., 1980, The journal of Photographic Science. 28, 177. Mark, H.E., Gaylord, N.G., Bikales, N.M., 1987, Encyclopedia of polymer Science and Engineering, vol 7, New York. Schulte, M. and Strube J., 2001, J. Chromatogr. A. 906, 399. Westerterp, K.R., Swaaij, W.P.M., van, Beenackers, A.A.C.M., 1987, Chemical Reactor Design and Operation. John Wiley & Sons, New York.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
697
A Mixed Integer Optimization Strategy for Integrated Gas/Oil Production V. D. Kosmidis, J. D. Perkins and E. N. Pistikopoulos^ Centre for Process System Engineering, Department of Chemical Engineering , Imperial College, London SW7 2BY, U.K.
Abstract The paper describes a mixed integer optimization formulation for the well allocation/operation of integrated gas/oil production systems, which takes into account the interactions between the reservoir, the pipeline network and the surface separation facility. To address the complexity and high dimensionality of the resulting optimization model, an efficient approximation solution strategy is proposed, which is based on outer approximation principles and illustrated with an example problem.
1. Introduction-problem statement An integrated oil and gas production system comprises (i) the reservoir, which is defined as an accumulation of oil and gas in porous permeable rock, (ii) the wells, (iii) the headers, where the well streams are mixed, (iv) the flow lines which connect the headers to the separators, and (v) the separator facilities where the fluid is separated in oil, gas and water. Each well has a valve, the choke, which is used to control its flow rate; the region around the well inside the reservoir is called the well bore; finally, the wells, headers and flow lines define the pipeline network, as shown in Figure 1. The well allocafion/operation problem of such an integrated oil and gas production system, as the one shown in figure 1 can be stated as follows: given (i) a set of wells, which could be either connected to headers or to separators, and (ii) a set of headers which could be connected to separators, the goal is to determine (i) the interconnections of wells to headers and flow lines to separators, and (ii) the corresponding well flow rates, which maximize oil producdon, at a particular time instant, while satisfying the underlying governing equations, such as mass, energy and momentum balances, and operational constraints, such as separator capacity constraints, well flow rate (upper and lower) bounds, maximum number of interconnection changes, etc. Previous attempts to address this problem include: (i) heuristic-based decomposition strategies which typically employ rules of thumb, such as 'shut in a well if it's water production violate an upper bound', 'allocate high gas producing wells to high pressure separator' etc (Fentor, 1984, Litvak et al, 1997), and (ii) mathematical programming approaches which only deal with simplified or special cases of the problem at hand (Fuji and Home, 1994, Fang and Lo, 1996, Handley-Schachler et al, 2000). In this paper, we present a novel mixed integer optimization formulation and an efficient decomposition strategy for the solution of well allocation/optimization problems. 'Corresponding author. Tel: +44 20 75946620 Fax:+44 20 75946606. Email: [email protected]
698
Reservoir
\
>/ ^ , ^ \—' \ . ^ ^ \ / / V / ^...y^^''^ ^^^'^A^^"^^'^^ ^^^f""^ Headers j : ;p: ^ells
Surface Facilities
\ ^ ^ ^ - J l ^ ' Ni^^ay
Reservoir
Wells
Figure I. Components of an integrated oil and gas production system
2. Modelling issues In order to develop a mathematical model to describe the discrete (infrastrucutre) and continuous (flow rate requirements) characteristics of the well allocation/operation problem, we firsrt introduce the following set of 0-1 binary variables:
yk =
yk,n
1 if there is flow through well k( k = 1,...,K) 0 otherwise 1 if well k( k = l,...,k ) is connected to header n (n=^ ],...,N ) 0 otherwise 1 if header n( n = 1,...,N ) is connected to separator i (i = ],...,! )
yn,i
0 otherwise
The proposed mathematical model (see Kosmidis et al, for details) then includes (i) mass balances around the wells, headers and separators (ii) relationships regarding the flow rate of gas, water and liquid exiting the reservoir into each well (the well bore model, Litvak et al, 1997), (iii) relationships relating the flow through a well as a function of pressure and its design diameter (the choke model, Sachdeva, 1986), and (iv) logical, mixed-integer continuous constraints, such as for example, the requirement that each header be connected to one separator, etc. The mathematical model for the small pipeline network shown in figure 2 is given in Appendix I. Note that it involves (i) both algebraic and differential equations, and (ii) continuous and 0-1 binary variables, i.e. it corresponds to a mixed-integer dynamic optimization (MIDO) problem (Bansal et al 2000).
3. Solution Procedure As the number of wells and headers increase the dimensionality of both continuous and integer space becomes prohibitly large for the application of recently proposed state-of the-art MIDO algorithms, especially for realistic oil fields with tens to hundreds of wells. Therefore, we propose instead an approximation solution strategy, which is based on the following ideas:
699
r
U
0
I—
Weill
Well 3
Figure 2. Three well network (i)
Projection of the well momentum balance and the choke valve equations to the space of header pressure and well flow rate (P,Qo ) - The projection is achieved by setting the choke fully open and discretizing the header pressure. For each value of the header pressure, P, the well bore, well momentum balance and choke models comprise a square system of equations with its solution corresponding to the maximum well oil rate Q^^ (fully open choke).
(ii)
These pairs (P, Q^l ) of discrete values are then approximated by a polynomial expression, which represents the well hydraulic constraints in the network and replaces the well momentum balance and choke models in the optimization model. Note that the well pressure constraints will be satisfied for any well flowrate below the maximum by reducing the choke diameter, Solution of the momentum balance equations of the header pressures for
different values of the separator pressure, liquid flowrate, gas oil ratio and water cut, (see notation in Appendix) from which hydraulic look up tables can be developed (Litvak et al, 1997). Based on these tables, function and derivative evaluations can be performed simply by interpolation. Considering the above approximations, the MIDO formulation of Appendix (i) can be recast as a mixed integer nonlinear optimization (MINLP) problem, as shown in Appendix (ii), which can be solved with an outer approximation algorithm (Grossmann and coworkers, 1990, 1996).
4. Illustrative example The well characteristic of the three well pipeline network of Figure 2 is given in Table 1. The MINLP model of Appendix (ii) is then solved resulting in an optimal configuration for maximum oil production, which involves only two wells with the third one shut in, as shown in Table 2. It is interesting to compare the optimal solution with the one generated by the application of the heuristic rule, which states that for
700 maximizing oil production the wells chokes ought to be fully open. As also shown in Table 2, the comparison clearly demonstrates that (i) heuristic rules may lead to suboptimal strategies, and (ii) an increase in oil production of 175 barrels/day is observed by applying the proposed optimization strategy. Furthermore, it must be noticed that (i) the optimal solution of the proposed strategy is always feasible, when it is applied to the exact system due to certain concavity properties (see Kosmidis et al for details) and (ii) the optimal solution of the proposed strategy compared to that of the exact problem is extremely closed and mainly depends on the discretization of the header pressure (see Kosmidis et al for details). Table L Well characteristics of the three well network. Reservoir / pipe parameters Reservoir Pressure ipsia) Productivity Index {stb/psia day) GOR (scf/stb) WC Vertical length (ft) Horizontal length (ft) Diameter (in) Roughness Flow rate upper bound (stb/day) Flow rate lower bound (stb/day)
Well 1 2370 3.0 5100 0.93 8000 6000 3 in 0.0001 1600 200
Well 2 4650 9.0 1900 0.165 6000 4000 3 in 0.0001 10000 530
Well3 4250 3.3 1600 0.15 7000 3000 3 in 0.001 5300 470
Flowhne
22000 ft 0 6 in 0.0001
Table 2. Maximum oil production. Structure (yhy2^y3)
=
(lin
(yi.y2>y3)=(0,iJ)
Objective function (barrels/day) 11929.2 (Heuristic) 12104.2 (optimization)
5. References Bansal, V., Perkins, J. D., Pistikopoulos, E.N., Ross, R., and van Schijndel, J. M.G. Simulatneous Design and Control Optimization under Uncertainty. Comp. Chem. Eng. 2000, 24, 261. Dutta-Roy, K. and Kattapuram, J. A New Approach to Gas-Lift Allocation Optimization. SPE 38333, 1997. Fang, W. Y. and Lo, K. K. A Generalized Well-Management Scheme for Reservoir Simulation. SPERE. 1996, 14, 116. Fentor, D.J. A Multi-Level Well Management Program for Modeling Offshore Oil Facilities. SPE 12964, 1984. Fuji, H. and Home, R. N. Multivariate Optimization of Networked Production Systems. SPE 27617, 1994. Handley-Schachler, S., McKie, and Quintero, N. New mathematical Techniques for Optimisation of Oil and Gas Production Systems. SPE 65161, 2000. Kosmidis, V., Perkins, J., and Pistikopoulos, E. N. IRC technical report (manuscript in preparation), 2001. Litvak, M., Clark, B., Farichild, J., Fossume, M., MacDonald, C. and Wood, A. Integration of Prudhoe Bay Surface Pipeline Network and Full Field Reservoir Models. SPE 38895, 1997. Sachedeva C. Two Phase Flow Through Chokes. SPE 15657, 1986.
701 Turkay, M., and Grossmann, I. E. Logic -based MINLP algorithms for the optimal synthesis of process networks. Comp. Chem. Eng. 1996, 20, 959. Viswanathan, J., and Grossmann, I. E. A combined penalty function and outer-approximation method for MINLP optimization. Comp. Chem. Eng. 1990, 14, 769.
Appendix. Mixed integer optimization formulation (i) Original mixed integer dynamic optimization model max q^j
GOR,=f^(q,,) well bore ^..k=^^k^L.k ^L,k^^o,k^^^:k
f(P;\GOR,.WC,,q,^^).
dL
chjn
och.out
dPl dL
_ pf
LelOXri
Bx = 0
k = 1.2.3
pch.out
well momentum & choke
/ Tch+ I
= f( pl .GOR, ,WQ ,9,,^ ). Le[ Z.f*,L„ ]
P/(L„)=P„(L„)
dP„
f(P„,GOR
dL
„,WC„,qi_),
\fLe
Pi = P( L, ) GOR „q„„ = q^ "^C
nQL.n
=
^H.
k
\ header k
k
mass balance
[L,,L,]
flowline
momentum
702 (ii) Approximate reduced mixed integer optimization model (MINLP). max q^ s.t yk
l8,k=G0^k%.k Qw,k =
^^kQL,k
Bx = 0
k = 1,2,3
^L,k = ^o.k + ^w,k
q^
)
%,k^Qo
P,=f(P,,GOR,,WC,,qJ GOR„q„„ = q^
k
k
^i..n=Y^q..k k ^L.n=^(^,.k+qw,k)
(iii) GOR L Qp.k
Notation = Gas to oil ratio. = Length of the pipe. = Flow rate in standard conditions of phase p from well k = Flow rate of phase p at header. = Oil flow rate upper bound of k well. = Pressure of k well along the well tubing.
P/
= Pressure of pipe which connects the choke of k of well to header n .
P^
= Pressure of header n .
^k "^' ^k "^ ~ Pressure upstream and downstream of choke of k well. Pi
= Pressure of separator / .
WC p
= Water cut, namely the ratio of water to liquid flow rate. = Phase (oil, gas, water, liquid).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
703
Towards the optimisation of logistic decisions and process parameters of multipurpose batch plants Thomas Lohl and Sebastian Engell Process Control Laboratory, Dept. of Chemical Engineering, University of Dortmund, D-44221 Dortmund, Germany. Phone: +49 (0)231/755-5127, Fax: +49 (0)231/755-5129 e-mail: {t.loehl | s.engell}@ct.uni-dortmund.de
Abstract This contribution presents a simulation and scheduling environment which enables the simultaneous optimisation of both scheduling decisions and process parameters based on the reference model described in (lEC, 1997). The modelling effort is reduced since the structure of a batch plant is defined by generic model elements. The scheduling problem is solved by a genetic algorithm based on an imprecise model, the logistic optimisation model. During the genetic search the simulation of a model with more accurate elements - the process model - is utilised to refme the parameters of the logistic optimisation model. Despite the use of models of different accuracy and detail, no redundant modelling effort is required.
1. Introduction The profitability of multipurpose batch plants depends on the choice of the parameters of each processing step and on good logistic decisions. The development of life-cycle models is gaining increasing importance, since they reduce the effort to build and maintain different models for every stage of a life-cycle (Vankatasubramanian et al., 2001). Since a proper scheduling can reveal equipment savings in the design stage, reduce inventories in the operational stage and help to identify bottlenecks during plant retrofit of existing plants, we regard the optimal scheduling as a key problem to be solved. The remainder of this contribution explains the major elements of the optimisation model and their relation to the structural elements of the core model (section 2), a brief description of the solution approach including aspects concerning the overall architecture of the realised software environment (section 3) and the discussion of the results obtained for a benchmark example (section 4). The conclusion (section 5) closes this contribution.
2. Modelling We propose to use the reference models for batch control in multipurpose plants (lEC, 1997) as the basis of both the simulation models and the logistic optimisation model.
704 This standard defines the elements of batch processes, their basic properties and their dependencies. In the paper this object oriented model is called the core model for short. 2.1 The core model The key aspect of the core model is the division of the process into the plant and the recipes. A recipe describes the procedural steps which are required to produce substances or to execute services. The elements of the master recipe are formulated independendy of the specific equipment items of the plant. The plant model represents the physical structure of the plant and, in particular, the basic technical functions which the elements of the plant can perform. The assignment of the steps of the master recipe to the equipment items yields the control recipe. The control recipe determines the flow of material through the plant unambiguously and can be used by a supervisory control program without major adjustments. For simulation purposes, no additional structural information beyond the core model is necessary, only the interpretation of the dynamic behaviour of the elements (e.g. filling) according to the chosen level of detail (e.g. constant flow or hydrostatically driven flow) must be available. This concept has been realised in the Batch Simulation Package (BASIP), for which a thorough description of the modelling, the event detection and the numerical performance can be found in (Fritz et al., 1999). However, for the logistic optimisation the core model cannot be used directly for two major reasons: 1. The degrees of freedom, which the optimisation algorithm must be able to exploit are not stated explicitly. A search for alternatives would necessitate time consuming search through the entire object hierarchy. 2. The information about the current state of the plant, the arbitration strategy and the scheduling objectives, to name but a few, are not contained in the core model. 2.2 The optimisation model structure The logistic optimisation model is structured as three sets of graphs, where each set is assigned to one of the layers of abstractions represented by the type and the elements of the recipes (see figure 1). The substance graph is used to map alternative master recipes for the production of the desired substances or services on the master recipe procedure level. The activity graph refines the substances described in the substance graph to the states of material (e.g. location of material). The activity graph replaces each master recipe procedure of the substance graph by a set of control recipes to consider alternative resource utilisation. Both graphs are directed bipartite graphs consisting of circle nodes, representing the involved materials/states and rectangle nodes, representing the master or control recipe procedure respectively. The representation of these graphs is similar to the STN framework for mathematical programming (Kondili et al., 1992). ThQ phase-graph specifies the control recipe procedure by the sequence of the recipe phases, where the edges define precedence relations. Every state of the activity graph contains a reference to the associated unit. The recipe phases reference equipment items of units which can be used exclusively or shared. Each equipment model (unit) administers the inventory history and the list of allocation intervals (see for example unit B31 in figure 1).
705 2.3 The solution procedure The large number of combinatorial decisions suggests to apply genetic algorithms (GA) to determine the sequence and resource allocation of the batches (Corne and Ross, 1997). Therefore, a linear, indirect coding of the batch objects as decision variables is used. A batch object contains the interval where the batch run can be performed and a reference to the task of the activity graph, i.e. the control recipe. The interpretation of the sequence of batch objects to be scheduled is carried out I ( B11 ) ( B 1 2 ) ^^ R21 y^ ( B31 ) for each batch in the following two step Figure 1: A part of the optimisation model which describes the production recipe of the example (see procedure. In the first step, figure 4). The dashed lines are object references, the the material balances are solid lines represents precedence relations. The dashed evaluated to determine the time intervals where the lines from the states to the units are omitted for clarity. capacities of the states of the raw material and the products are within the feasible bounds. In the second step the starting time of the batch is adjusted to resolve all resource conflicts on the recipe phase level, using the concept of feasible time windows as proposed for example in (Rodrigues et al., 2000). The quality of the schedule is assessed using the logistic optimisation model for a specified objective. At predetermined number of generations the best schedule obtained so far is simulated by the BASIP Simulation module using the process model. The process model refines the elements of the core model. In this model, the dynamics are described by differential equations.
3. Realisation In the realised software environment, the processes in a multipurpose batch plant are modelled graphically by specifying the plant topology as a flowsheet extended by the equipment phases, the master recipes using sequential-function-chart-like semantics (see figure 4) and the assignment of the recipe phases to the equipment phases for the generation of control recipes. The logistic optimisation model is generated automatically. The GA creates feasible schedules which are simulated using the process model. The results of the asynchronously running simulator are imported and the
706 parameters of the optimisation model are updated. The figure 2 depicts the overall architecture focussing on the most important data flows. The local search strategy, the intermediate storage policy, the time representation and the configuration of the genetic algorithm (i.e. choice of the types of genetic operators, selection and mating strategy, constraint handling techniques, etc.) are determined by external configuration dialogs. Thus no modification on source code level is necessary to adjust performance critical
Figure 2: Architecture of the scheduling environment consisting of simulation, modelling and scheduling parameters. The complete software is implemented in C++ and uses the GALib (Wall, 1995) source code as the genetic algorithm class library.
4. Example and numerical results The example illustrates the modelling approach, the scheduling performance and the influence of the detailed simulation on the course of the optimisation.
Raw material: A B ,
JL B11 LJ
B12
1
c
JL B13L-J
r
4.1 Problem description JL 1 1 Jk The process under consideration is a batch process R22 R23 R21 in which two liquid products {D and E) are produced from three liquid substances {A,B and C). ^ Basically, the plant consists of three stages (see 832 B3 1 figure 3). —-~ 1 Product D 1 ——=\ • Raw material buffer: The vessels B11, B12 and 1 B13 buffer the raw materials A,B and C, respectively. Each tank is used exclusively for Figure 3: Topology of the one raw material and may contain at most two benchmark plant batches of substance.
707 •
Reaction: The three reactors R21, R22 and R23 may produce both products D or E. The master recipe of this stage consists of three parallel operations (see figure 4) which have to be synchronised. • Product buffer: The tanks B31 and B32 buffer the products and are exclusively used for either D or E. Both of the tanks may contain at most three batches of product. All equipment items of each stage are fully connected to the units of the next stage. The charge/ uncharge steps cannot be neglected and the flowrates are different for each control recipe and are only imprecisely known a-priori. The scheduling task is to produce an amount of 6 batches of each product. The algorithm has to determine the resource allocation on the recipe phase level, the sequence of batches and the starting times of every batch such that the makespan is minimised. The problem comprises 28 batches with 140 operations in total to be scheduled on a scheduling horizon of 600 equally spaced time units. 4.2 Numerical results A GA was used for this example with a linear [ s j Filling scaling, a roulette wheel selection and 20 % of the population are replaced by the offspring. The mutation Raact operator changes the Oascr i p t i o -Est. start t i n s : 0 Est. ion: 0.0 position and resource Raact °^'\ "'*"J''^°'^"_ assignment of a batch Figure 4: Screenshot of the production master recipe as a with a probability of 0.3. sequential function chart. The double bars indicate the The standard order parallel execution of the steps 3,4 and 5 until the mass in the crossover is applied with a probability of 0.9. The vessel called Reaktor exceeds 1.7 kg. population comprises 50 individuals and a constant duration of 100 generations was used as the termination criterion. This standard configuration is used without problem specific tailoring methods. The convergence of the genetic algorithm and the influence of the detailed hybrid simulation is depicted in figure 5. The figure shows the rapid convergence of the best and the worst individuals towards better solutions. The duration of all process operations are updated after the 15^ and then every 10 generations with the simulation results obtained by the hybrid simulation. The new parameter set had the greatest impact at the second time it occurred (see 25* generation). It is interesting to note that at this point the change of the durations of the operations has led to a broadening of the objective function value distribution of the population increasing the chance to find better solutions. The best solution of 319 time units is within the confidence interval obtained by a statistical t-test. An optimal solution for the problem without the more detailed simulation is 345 time units. The complete optimisation run took approximately 285 CPU seconds. Thus, 0.11 CPU seconds are required to create and evaluate one 1
n^acrinhini
-Filling Acid"
"E»t. duration: 13.0 ' From: "Vorlagabehaaltar Sauara' To: "Raaktor"
-LH(Raak.tor) -0.85 [kg]
Roaction __ Daicription: "Nautralisatii 3-Est. start tina: 13.0 — E s t . duration: 13.0
-LH(Raaktor) s
0
Stirring Honoganiza Dascription: "Honogi r-- Eat. start ti«a: 13 lt,~E«t duration 13 0 Vassal 'Raaktor" rpa 20C 0
^i
'Faed Base ma: 13.0 i; 13.0
708 schedule on the average. The algorithm was executed on a SUN UltraSparc-II (300 MHz) using 11 MB of memory. For a more constraint problem (the starting times of the batches of two raw materials were fixed) of this process, the results were compared against a standard discrete time MILP/STN model using the G A M S / C P L E X solver.
^^ convergence
The proposed approach found an optimal solution after 11 seconds, whereas the mathematical programming approach required 5.1 minutes.
5. Conclusions The validation of simulation models and the 40 60 Generation No effective representation of the optimisation model Figure 5: Convergence plot of the best, average and worst with little additional makesoan. modelling effort enables easy and consistent modelling as well as quick results. The proposed approach was able to improve the solution quality by 7 % compared to scheduling without simulation. Apart from the reduction of the modelling effort, the numerical experiments also demonstrate the efficiency of the optimisation method and the advantages of the combination of a detailed simulation with optimisation.
6. References Come, D. and P. Ross, 1997, Practical Issues and Recent Advances in Job- and Open-Shop Scheduling, in: Evolutionary Algorithms in Engineering Applications, Eds. D. Dasgupta and Z. Michalewicz, 531-546. Fritz, M., A. Liefeldt and S. Engell, 1999, Recipe-Driven Batch Processes: Event Handling in Hybrid System Simulation, in: Proc. of 1999 IEEE int. Symposium on Computer Aided Control System Design (CACSD '99), Hawaii, 138-143. lEC 61512-1, 1997, Batch Control, Part 1: Models and Terminology, Intemational Electrotechnical Commission (lEC). Kondili, E., C. C. Pantelides and R. W. H. Sargent, 1992, A general short algorithm for short-term scheduling of batch operations. Part I- MILP formulation. Computers and Chemical Engineering, 17 (2), 211-227. Rodrigues, L. A., M. Graells, J. Canton, L. Gimeno, M. T. M Rodigues, A. Espuna and L. Puigjaner, 2000, Utilization of processing time windows to enhance planning and scheduling in short term multipurpose batch plants. Computers and Chemical Engineering, 24, 353-359. Venkatasubramanian, V., J. Zhao, S. Viswanathan, C. Zhao, F. Mu, P. Harper and B. Russel, 2001, in: European Symposium on Computer Aided Process Engineering (ESCAPE) 11, Eds. R. Gani and S. B. J0rgenson, 925-930. Wall, M. B., 1996, GALib: A C++ Library of genetic algorithm components, Manual, Mechanical Engineering Department, MIT, http://lancet.mit.edu/ga/, (online: 22.10.01).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
709
Trilinear Models for Batch MSPC: Application to an Industrial Batch Pharmaceutical Process Lopes J.A.* and Menezes J.C.^ Cemter for Biological & Chemical Engineering, Technical University of Lisbon Av. Rovisco Pais, P-1049-001, Lisbon, Portugal ^[email protected]; ^[email protected] Phone: (+351) 218 417 347; Fax: (+351) 218 419 062
Abstract In this paper, PARAFAC and Tucker3 models were compared with the commonly used multiway principal components analysis approach (MPCA) for multivariate process control of an industrial batch antibiotic production process. Two different approaches for on-line monitoring were used: sliding window (multiple models) and global window (single model) monitoring strategies. The later approach requires orthogonality for the time dimension scores. In this context, a modification of the Parafac algorithm was proposed. The Tucker3 and Parafac models as proposed here share an identical structure. Scores (D) and residuals (Q) statistics were used to on-line identify faults. We concluded that Parafac and Tucker3 models outperformed MPCA in terms of detection of faults specially when the statistic for scores is used. All models performed equally well in the residuals statistics. The sliding window strategy proved to be more appropriate to identify faults than the global window strategy. This is, to our best knowledge, the first time such study was performed for an industrial batch antibiotic process.
1. Introduction Principal components analysis (PCA) has been extensively used in multivariate statistical process control (MSPC) in the past ten years (Martin and Morris, 1996; Chen and McAvoy, 1998; Kassidas et al., 1998). Multiway PCA (MPCA) is traditionally used to extend MSPC to batch processes (Nomikos and MacGregor, 1995; Albert and Kinley, 1996; Chen and McAvoy, 1998; Lopes and Menezes, 1998). Batch multivariate statistical process control (BMSPC) needs to account for three primary directions: the batch number, the monitored variables and the time dimension for each batch. Trilinear methods, such as Parafac or Tucker3 models, are more appropriate to model batch processes since they take into account the three-dimensional structure of the data (Bro, 1998). The purpose of this paper is to study the advantages of using trilinear models (Parafac and Tucker3 models) over standard MPCA models in terms of fault detection (Louwerse and Smilde, 2000). The Tucker3 and Parafac models are usually described with two different equations. In this paper, the same structure is used to describe both models. Two on-line fault detection strategies are compared. Equations to compute projected scores for each monitoring strategy are also presented here. A pharmaceutical process (antibiotic production by fermentation) was used as a case-study.
710
2. Methods 2.1 MPCA, Parafac and Tucker3 models The MPCA model is equivalent to the PC A model. It uses a data decomposition commonly named unfolding to transform a data three-way array X ( I X J X K ) in a twoway array X ( I X J K ) where in general the batch dimension (I) is preserved (Kiers, 2000). Using SVD it is possible to extract an array A ( I X R ) of scores (variability among batches) and a loading matrix C ( J K X R ) containing information on variables and time dimension. R is the model number of components. E(lxJK)is the residuals array that depends on the chosen R.. X = AC^+E (1) Parafac and TuckerS are trilinear models because they preserve the trilinear structure of the data ( B ( J X R ) and C ( K X R ) are the loadings for variables and time modes). They share the structure given by equation 2 (the operator ® denotes the Kronecker product) (Bro, 1998). X = AG(C(x)Bf+E
(2)
In a Parafac model, the core array G ( R X R X R ) is as shown in equation 3.
0 0
p 0 p 1
p 0 p 0
0 0
b 0
0 0
1 0 — parafac
(3)
The Parafac model can also be written as X = A ( C O B ) ^ - H E , where o denotes the Khatri-Rao operator (Bro, 1996). The Parafac model expressions are equivalent since for any two arrays B and C (with the same number of columns), it holds (C (S) B)Gparafac = C o B . The modcl algorithms can be found elsewhere (Bro, 1998). 2.2 Monitoring Strategies Two on-line monitoring strategies are presented here. 2.2.7 Sliding window strategy The sliding window strategy consists on the model projection of a segment of the batch data as the process evolves. At each new batch time interval [k-Ak/2 , k+Ak/2] (where Ak is the window size), new B and C loading matrices are needed. The number of necessary loading matrices depends on the batch size (sampling frequency) and window size (Louwerse and Smilde, 2000). Equation 4 shows how to obtain the new scores. a„ew=x„ewZG^(GZ^ZG^)-^
(4)
In this equation, Z = C 0 B . For a MPCA model, a ^ew = ^ new ^ • 2.2.2 Global strategy The global window approach involves using the full B and C loading matrices obtained from the collection of history nominal batches. At each new time interval the scores for
711 the entire batch are estimated (Nomikos and MacGregor, 1995). The point here is to restrict the available batch information (say at time point k) to be consistent with the correlation structure up to that time point k. The scores are obtained by projecting the known data at time k onto the model. Equation 5 is used to obtain the scores for a new batch at time k for Tucker3 and Parafac models (Lopes, 2001). ^new,k = Xnew,kZkG
\fjZ^Z^.G
(5)
)
In equation 5, Xnew,k is the vector containing the batch data up to time point k and the array Z^ is given by equation 6. Z,=|'cJc;^cJ'l(8>B
(6)
The loadings matrix, Ck, contains the first k lines of C. Equation 6 holds if C C=I. This is only true for the Tucker3 model. The Parafac algorithm must be changed in order to account for this constraint (Lopes, 2001). In the iterative Parafac algorithm, we need to impose
C = XZ(z^X^XZj
Z = (BOA)[(BOA)^(BOA)J
each time the C matrix is estimated
(where
). For the MPCA model, the scores are obatined with
equation 7. ^new ~ ^new,k^k V^k^k /
^^
2.3 Statistics Multivariate statistical process control is based on two statistics: one for the scores (statistic D or Hotelling T^) and one for the residuals (statistic Q). The D statistic measures the variability explained by the model, while the Q statistic measures the residuals. For each new batch i the statistic D can be obtained with equation 8 (Wise and Gallagher, 1996). D,^(a,-AK-^{a,-Xf
(8)
The residual statistic for batch i is obtained with equation 9.
The confidence limits for these statistics were computed as explained in Nomikos and MacGregor (1995). Westerhuis et al. (1999) explain how to compute robust limits for these statistics. Because the Qa limit is variable in time it is better to plot relative Q values in control charts (Qk/Q95% )• Table I. Non-nominal batches used to test models and monitoring strategies. Case A B C
Fault(s) description Abrupt change of the air inlet flow at 49 hours Low pH batch (since batch beginning) Step tests applied periodically on substrates feeds
712
3. Case-Study/Experimental An industrial pharmaceutical process (clavulanic acid production by fermentation) is presented here as a case-study (Neves et al., 2001). Fed-batch cultivation of a streptomycete strain was carried out using non-defmed (complex) medium. The conditions used were typical of those employed routinely in industry for aerobic microbial growth. A fully instrumented bioreactor with an operating volume of 200 dm^ was employed throughout this study. A total of twenty variables were measured. Some were obtained on-line while others were measured off-line with a frequency of about 4 hours except for the viscosity which was measured only once per day. Data preprocessing included outliers detection and noise reduction. Interpolation for missing/unavailable values was used due to the process slow dynamics. A 1 hour interval was chosen to synchronize the fermentation data. 3.1 Nominal Data 20 batches were operated under normal operation conditions (NOC). The 20 nominal batches were arranged as a three-way array with dimensions X(20x20xl40). 3.2 Non-Nominal Data Three non-nominal batches (with known faults) were used to test each model and each monitoring strategy, thereafter named cases A to C. Table 1 indicates the faults occurred in each non-nominal batch.
4. Results and Discussion A major difference between the models is the captured variance for the same number of components. Because the number of parameters of the MPCA model is greater it also captures more variance for the same number of components. Note that a two-component MPCA model captures 47.2% of variance while a four-component Parafac model captures 42.9% and a four-component Tucker3 model captures 45.8%. The captured variances for Parafac models considering orthogonal C scores (as required for the global window strategy) are slightly lower due to the imposed constraint. For a fourcomponent Parafac model with imposed orthogonality of C loadings, only 40.5% of variance is captured. To compare the error detection performance for the three models, two-component MPCA and four-component Tucker3 and Parafac models were selected. For the sliding window monitoring strategy 129 models (At=12 hours) were built (note that the first model can only be built after 12 hours). One global model was built for the global window strategy. D and Q statistics were computed and probability values (pvalues) were determined. It was considered that a fault is detected when the p-value is lower than 0.05 (95% confidence limit). Table 2 summarizes the times were an error was detected for the first time in the D statistic for each non-nominal case. From Table 2 it is possible to conclude that Parafac and Tucker3 models performs better in terms of statistic D than MPCA models. This is specially true for the global monitoring strategy where a single model is used. However, errors are detected earlier when the sliding window strategy is used. The sliding window size must be selected in order to avoid false alarms. In this case (12 hours size) no false alarms were detected.
713 Table 2. Fault detection times for the three tested cases (a fault is detected when pvalue<0.05). MPCA Sliding Global Window 53 h 75 h 40 h 82 h 59 h 125 h
Case A CaseB CaseC
Tucker3 Global Sliding Window 57 h 50 h 71h 37 h 74 h 51h
Parafac Global Sliding Window 63 h 52 h 78 h 38 h 82 h 52 h
With respect to the residuals statistic (Q) small differences were observed between the three models tested. In general, errors are detected earlier in the Q statistic. In case A, all models detected an error in the Q statistic at 49 hours (where the abrupt change in the inlet air feed ocurred). In case B, a fault (approximately at 40 hours) is detected by all models in both statistics. However, after a short period of time the control chart for statistic D returns to an incontrol region. This is not verified in the Q statistic where an out-of-control signal persisted even after the 40 hours region (see Figure 1). If an alarm is started, a comparison (difference) between the projected batch and the average of nominal batches variables contributions provides an indication of which variable(s) are causing the fault (Louwerse et al., 1999). These charts are called contribution plots. Figure 2 shows the contribution plots for residuals and scores, for case A, at the time the fault actually ocurred (49 hours). At 49 hours the fault was observed only on the residuals statistic (p-value<0.05). Clearly, variables 13, 14 and 17 (oxygen and carbon dioxide outlet-gas fractions and inlet-air flow) are causing the deviation.
5. Conclusions It was found that trilinear models perform better than MPCA in terms of the statistic for scores. In the three tested cases the faults were detected earlier in the D statistic when trilinear models are used. A marginal gain was observed for the Tucker3 model.
0
20
40
60
80
Time (hours)
100
120
140
20
40
&0
80
100
120
140
Time (hours)
Figure 1. Sliding window monitoring strategy (12 hours window size) MSPC charts based on MPCA models (charts for case B).
714
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Variables
ILiL^
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Variables
Figure 2. Fault identification contribution plots for scores and residuals, for case A at 49 hours (MPCA model/sliding window^ monitoring strategy). All three models performed equally well when residuals statistic was used. In these cases the faults were primarly detected in the residuals statistic. This happens when the models are unable to explain the variation observed in the projected data. The sliding window strategy is more appropriate than the global window strategy for fault detection. However, the later has the advantage to use only one set of loadings, thus, being less memory demanding in computer processing. Nevertheless, we found that the sliding window strategy is more sensitive to detect process deviations from normal operating conditions. The choice of the correct time window for fault detection would depend upon the process. Batches where faults occurred can be used to adjust this parameter. Future work is being directed, as new industrial data become available, to address these problems.
6. References Albert, S. and R. Kinley, 2001, Trends Biotechnol. 19(2), 53-62. Bro, R., 1996, IEEE T. Sig. Proces. 6795. Bro, R., 1998, PhD Thesis, 286, University of Amsterdam. Chen, G. and J. McAvoy , 1998, J. Proc. Cont. 8(5), 409-420. Kassidas, A., P. Taylor and J. MacGregor, 1998 J. Proc. Cont., 8(5), 381-393. Kiers, H., 2000, J. Chemometrics 14, 105-122. Lopes, J., 2001, PhD thesis, 296, Technical University of Lisbon (in Portuguese). Lopes, J. and J. Menezes, 1998, AIChE Symp. Series 94(320), 391-396. Louwerse, D. and A. Smilde, 2000, Chem. Eng. Sci. 55, 1225-1235. Louwerse, D., A. Tates, A.Smilde, G. Koot and H. Berndt, 1999, Chemometrics Intell. Lab. Syst. 46, 197-206. Martin, E. and J. Morris, 1996, Trans. Inst. MC 18(1), 51-60. Neves, A., L. Vieira and J. Menezes, 2001, Biotechnol. Bioeng. 72(6), 628-633. Nomikos, P. and J. MacGregor, 1995, Technometrics 37(1), 41-59. Westerhuis, J., S. Gurden and A. Smilde, 1999, J. Chemometrics 14, 335-349. Wise, B. and N. Gallagher, 1996, J. Proc. Cont. 8(6), 329-348.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
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A Mixed Integer Programming Approach for Scheduling Commodities in a Pipeline L. Magatao*, L.V.R. Arruda^, F. Neves-Jr.* CEFET-PR, CPGEI Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, Brazil Tel.: +55 41 310-4707 - Fax: +55 41 310-4683 {*magatao, Wuda, *neves}@cpgei.cefetpr.br
Abstract This paper addresses the problem of developing an optimisation model to aid the operational decision-making process on pipeline systems. The model is applied on a real world pipeline, which connects an inland refinery to a harbour, conveying different types of commodities. The optimisation model was developed based on mixed integer linear programming (MILP) with uniform time discretisation. The MILP well-known computational burden was avoided by the problem domain decomposition. Simulation examples have demonstrated that the optimisation model was able to define new operating points to the pipeline system, providing significant cost saving.
1. Introduction The oil industry has a strong influence upon the economic market. Research in this area may provide highly profit solutions and also avoid environmental damages. The oil distribution-planning problem is within this context. A wide net with trains, tankers, and pipelines are used to link harbours, refineries and consumers. According to Kennedy (1993), pipelines provide an efficient way to transport oil and gas. The maximum utilisation efficiency of this transportation medium becomes interesting to the oil industry. However, the operational decision-making on pipeline systems is still based on experience, with aid of manual calculation. According to Lee et al (1996), mathematical programming techniques for long-term planning have been extensively studied and implemented, but much less work has been devoted to short-term scheduling, which in fact reproduces the operational decision-making process. The short-term scheduling requires the explicit modelling of discrete decisions. The approach to solve this problem is manifold. A general one is to use a mixed integer linear programming formulation. A complete survey in mixed integer programming and techniques for several application problems is presented in (Wolsey, 1998). The great concern of a real-word MILP formulation is related to the difficulty of finding solutions in a reasonable computational time. According to Applequist et al. (1997), a MILP feature of a practical problem requires a large number of integer variables, thus the computational expense has to be concerned. Subrahmanyam et al (1995) demonstrate that decomposition strategies are a valid approach to avoid the combinatorial explosion introduced by integer variables.
716
2. Problem Deflnition This work focuses on the short-term scheduling of activities in a specific pipeline system. It connects a harbour to an inland refinery. The pipeline is 93.5 km length, it can store a total volume of 7,314 m^ and it connects a refinery tank farm to a harbour tank farm going along regions with 900-meter-altitude difference (Ah). The pipe conveys multiple types of commodities. It is possible to pump products either from the refinery to the harbour (this is called y?(9w operation) or from the harbour to the refinery (this is called reflow operation). There is no physical separation between different products as they move in the pipe. Consequently, there is a contamination area between products: the interface. Some interfaces are operationally not recommended, and a plug (small volume of product) can be used to avoid a specific interface. However, plug inclusions increase the operating cost. The tank farm infrastructure, an up-to-date storage scenario, the pipeline flow rate details, and the demand requirements are known a priori. The scheduling process must take into account product availability, tankage constraints, pumping sequencing, flow rate determination, and a wide variety of operational requirements. The task is to specify the pipeline operation during a limited time horizon (7), providing low cost operational procedures. Figure 1 illustrates the pipeline physical structure overview.
3. Methodology The methodology applied on this work is the mixed integer linear programming with uniform time discretisation. The computational complexity is concerned, and an optimisation structure is proposed to decompose the problem in blocks, providing a framework that aims to reduce the computational expense (Figure 2). The optimisation structure is based upon a MILP main model {Main Model), one auxiliary MILP model {Tank Bound), a time computation procedure {Auxiliary Routine), and a Data Base. The tank bound task involves the appropriate selection of some resources (tanks) for a given activity (pumping the demanded product). Its main inputs are demand requirements, product availability, and tankage constraints. As an output, it specifies the tanks to be used on operational procedures. The auxiliary routine takes into account the available time horizon, the product flow rate range, and demand requirements. It specifies temporal constraints, which must be respected by the main model. The main model determines the product pumping sequence and the flow rate details. It establishes the initial and the final time of each pumping activity. The final scheduling is attained by first solving the tank bound and the auxiliary routine, and, at last, the main model. The modelling and optimisation tool Extended LINGO/PC Release 6.0 (LINDO, 1999) was used to implement and solve the optimisation structure. LINGO is a commercial tool, which allows formulating linear and non-linear large problems, solving them, and analysing the solution. It has four internal solvers: a direct solver, a linear solver, a nonlinear solver, and a branch and bound manager. The LINGO's solvers are all part of the same program, which is directly linked to its modelling language. This allows the data exchange directly through memory, rather than through intermediate files.
717 Tank Bound
Auxiliary Routine
Figure 1. Pipeline physical structure overview.
\ \^
r
Data Base
Main Model
Figure 2. Optimisation structure.
4. Model Framework Basically, the modelling process takes into account the following conditions: (i) pipeline can fill or empty only one tank at a time; (ii) tanks being emptied can not be filled, and tanks being filled cannot be emptied; (iii) a tank always stores the same product; (iv) the tank farm infrastructure limits must be respected; (v) the product flow rate range must be respected; (vi) the product demand has to be within an operational range; (vii) every product must be pumped continually; (viii) it is possible to use a plug between incompatible products, but plug inclusion increases the operating cost; (ix) the plug volume is significantly smaller than any demanded batch, so that its pumping time is neglected; (x) changeover times are neglected; (xi) use of plugs should be minimised; (xii) it is required a minimum time horizon (r„^^) to pump the entire demand. In case r = r^„, every product is pumped at its maximum flow rate; (xiii) the system starts pumping at the initial time (/ = l). In case r > r ^ „ , the pumping procedure can be finished before T, but the pipeline must remain pressurised. There is also a cost to maintain the pipe pressurised. The mathematical approach, as stated, is based on MILP with uniform time discretisation. Space restrictions preclude a detailed problem formulation. Such information can be obtained in (Magatao, 2001). It is presented the main model objective function (1), exploiting its characteristics. minimize COST = = CR^.m^-^(TFB; -TIB;)^CP^^„,^Y.^TFB;
-TIBp^^[Cer(PP,+PRM-^
+ C,plug
(1)
-\-CSTS p
pa t=TlR + \
p
pa
i=TIP+\
where p and pa are different products; t is the discretised time (h); T is the available time horizon (h); TIR, TSR, TIP, and TSP are temporal constraints determined by the auxiliary routine (h); Cp,^g is the average cost to pump a plug ($); Ce, is the average electric cost per flow rate unit at a time t ($h/m^); CPpu„,p is the average cost to pump a product from the harbour to the refinery ($); CRp^„p is the average cost to pump a product from the refinery to the harbour ($); CS is the average cost to maintain the pipe pressurised ($/h); /^^^ is a dimensionless parameter that assumes one if pumping p
718 followed by pa requires a plug between then, zero otherwise; PP, is the flow rate (m^/h) at a time t (reflow procedure); PR, is the flow rate (mVh) at a time t (flow procedure); TFB^p is the end pumping time (h) of/? (reflow procedure); TFB'p is the end pumping time (h) of p (flow procedure); TlB^p is the start pumping time (h) of p (reflow procedure); TIB'p is the start pumping time (h) of p (flow procedure); TPp^p^, is a dimensionless variable that assumes one if the transition between p and pa occurs at a time r, zero otherwise (reflow procedure); TS is the time period that the pipe remains pressurised (h); TRpp^, is a dimensionless variable that assumes one if the transition between p and/?a occurs at a time r, zero otherwise (flow procedure). Expression (1) demonstrates that plug inclusions (/^^^-TO^^^,, Ip,pa-TPp^pa,.) increase the operating cost. So that, the optimisation solution method seeks scheduling solutions that minimise the plug usage. The product pumping time (TFBp, TIBp , TFB^, TIB^) increases the operating cost. This time is related to the flow rate {PR,, PP,) by an inverse ratio: if the flow rate increases, the product pumping time decreases. On the other hand, the operating cost is also directly influenced by flow rate variations - see the factor Ce, (PP, +PR,). Thus, the optimisation structure must determine the ideal flow rate policy during a limited time horizon (T). Maintaining the pipe pressurised also influences the operating cost (CSTS).
5. Results This section considers an example involving the pumping of four products from the harbour to the refinery followed by another four pumped from the refinery to the harbour. Each product has two tanks enabled to sending operations. For simplicity, units were standardised and omitted. The normalisation is based on the pipeline volume. The entire pipe has 7,314 ml It is admitted a NF (normalisation factor) that equally divides the pipe volume. The product demand is expressed based upon NF. As an example, NF = 4 determines batches of 1,828.5 m^ (7,314-^4). A normalised demand of two units represents a total demanded volume of 3,657 m^ (1,828.5x2). The system pumps, at most, one normalised volume per time unit. A normalised flow rate of one at a time t indicates that a volume of 1,828.5 m^ is pumped between times t and t+l. The time length selection of each discretised time span involves a trade-off between accurate operation and computational effort. The problem data was rounded, so that the time quantum could be increased and, thus, the number of decision variables decreased. It was adopted a uniform time discretisation of six hours, and NF = 4. Simulation covers since the minimum normalised time horizon (r„^„ = 20) up to twenty-five normalised time units (7 = 25). The pumping process starts from the harbour to the refinery; CPp^mp . CRpump ^ Cpi,g, and CS were considered unitary. Table 1 demonstrates the normalised electric cost value (Ce,) at each time unit (t). The cost variation is due to on-peak demand hours. Pumping start time is at 6 a.m. (r = I).
719 Tabli e 1. Electric cost variation. t
1
2
3
4
5
Ce,
1
1
5
1
1
6 1
7 5
8 1
9 1
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1
5
1
1
1
5
1
1
1
5
1
1
1
5
1
1
Table 2 is a system information sketch for the problem main features. It presents a priori information about demand requirements {Demanded Amount), flow rate range, and plug necessity. As an example, the sequence PI followed by P2 demands the use of a plug. Table 2. System information - main features. Operation
Reflow
Flow
Product PI P2 P3 P4 P5 P6 P7 P8
Demanded Amount 1 1 2 1 1 2 1 1
Flow Rate Range 0,5-1 0,5-1 0,5- 1 0,25 - 0,5 0,5- 1 0,5- 1 0,5- 1 0,5- 1
Plug Necessity P2, P4, P6, P8 P1,P5 P4, P8 P1,P3, P5,P8 P2, P4, P6. P8 P1,P5 P4, P8 P1,P3, P5,P8
Table 3 provides information about the optimisation structure simulation on a Pentium III, 933MHz, 256 MB RAM. For each time horizon (7), the optimisation structure is run, and a specific normalised cost is attained - expression (1) value. It was not applied any optimality margin (Shah et ai, 1993). The auxiliary routine and the tank bound simulation data were neglected. These structures required a computational time lower than one second, for all simulation instances (20 < 7 < 25). Table 3. Main model simulation data. Time Horizon (7) 20 21 22 23 24 25
Total Number of Variables 468 559 650 741 832 923
Total Number of Binary Variables 112 134 156 178 200 222
Total Number of Constraints 1,371 1,559 1,747 1,935 2,123 2,311
Computational Time (s) 4 24 189 1,346 2,633 19,140
Normalised Cost ($) 63 60 59 58 59 60
In order to pump the entire demand, it is required a minimum time horizon (T„^^ = 20). In such a horizon, every product is pumped at its maximum flow rate. However, in case T > r^„ the optimisation structure determines the ideal flow rate policy. This flow rate is established based on both the available time horizon (T) and the electric cost variations (Table 1). Considering r = 23, Figure 3 shows the normalised flow rate determined by the optimisation structure. Figure 4 shows the normalised cost expression (1) - as a time horizon function. It demonstrates the existence of a specific T that yields the minimum operating cost ( r = 23). The cost versus time horizon function clearly demonstrates that a correct pipeline timing policy provides significant cost saving. For 7 = 23 the pumping sequencing determined by the optimisation structure is P4, P2, P3, PI, P5, P7, P6, and P8, which implies no use of plugs (Table 2).
720 |Flow Rate] x Discretised Time (t) "57 ^
^ rS
1
0.75
0.5 0.25 0
(0 1 3 5 7 9 11 13 15 17 19 21 23
Figure 3. Flow rate versus discretised time.
Figure 4. Cost versus time horizon.
6. Conclusions It was presented a mathematical programming approach to the economically important problem of oil distribution through pipelines. The task was to predict the pipeline operation during a limited time horizon, providing low cost operational procedures. It was applied the scheduling approach based on mixed integer linear programming with uniform time discretisation. The computational expense was concerned and an optimisation structure was proposed (Figure 2). The large-scale mixed integer linear problem was implemented and solved by using the commercial tool Extended LINGO/PC Release 6.0. Currently pipeline operation is based on experience, and no computer algorithm is used; plug product usage and energy consumption are not rigorously taken on account. Simulation examples indicate that economic improvements of 8% are feasible (Figure 4).
References Applequist, G., O. Samikoglu, J.F. Pekny and G.V. Reklaitis, 1997, Issues in the use design and evolution of process scheduling and planning systems, ISA Transactions, 36, 2, 81-121. Kennedy, J.L., 1993, Oil and Gas Pipeline Fundamentals, Penn Well Publishing Company. Lee, H., J.M. Pinto, I.E. Grossmann and P. Sunwon, 1996, Mixed-Integer Linear Programming Model for Refinery Short-Term Scheduling of Crude Oil Unloading with Inventory Management, Ind. & Eng. Chem. Res., 35, 1630-1641. LINDO, 1999, LINGO: The Modelling Language and Optimizer - User's Guide, LINDO Systems Inc, Chicago, Illinois. Magatao, L., 2001, A Methodology for Sequencing Commodities in a Multi-Product Pipeline, Master Thesis, CPGEI/CEFET-PR, 170 pages (in Portuguese). Shah, N., C.C. Pantelides and R.W.K. Sargent, 1993, A General Algorithm for Short-Term Scheduling of Batch Operations - II. Computational Issues, Comp. Chem. Engng., 17, 229-244. Subrahmanyam, S., M.H. BasseU, J.F. Pekny and G.V. Reklaitis, 1995. Issues in Solving Large Scale Planning, Design and Scheduling Problems in Batch Chemical Plants, Comp. Chem. Engng., 19, suppl., S577-S582. Wolsey A.L., 1998, Integer Programming, John Wiley & Sons Inc.
Acknowledgements The authors acknowledge financial support from the Brazilian National Agency of Petroleum (PRH-ANP/MCT PRHIO CEFET-PR) and the CNPq (under grant 467311/00-5).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
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An MILP Framework for Short-Term Scheduling of Single-Stage Batch Plants with Limited Discrete Resources Carlos A. Mendez and Jaime Cerda* INTEC (UNL - CONICET) Guemes 3450 - 3000 Santa Fe - ARGENTINA E-mail: [email protected]
Abstract Dealing with limited discrete resources in batch scheduling problems usually produce a sharp increase in model size and computational requirements. This work introduces a novel MILP formulation where all discrete resources including processing units are treated uniformly. Moreover, the ordering of batches at any resource item is handled by a common set of sequencing variables so as to achieve an important saving in 0-1 variables. Pre-ordering rules significantly reducing the problem size can be easily embedded in the MILP framework. In addition, discrete resources could even be sequentially assigned when real world resource-constrained scheduling problems are tackled. Two examples involving the scheduling of up to 29 batches in a single-stage batch plant under severe manpower restraints were successfully solved. Comparison with prior work shows a notable reduction in CPU time of at least two orders of magnitude.
1. Introduction In multiproduct batch plants, the processing tasks to be accomplished generally share manufacturing resources. Plant resources are usually classified as renewable and nonrenewable ones. Renewable resources like processing units, manpower and utilities become again available for use after ending the task to which is currently assigned. A renewable resource is said to be discrete if it is consumed at a constant level throughout the entire processing task. Such resources are usually available by limited amounts that cannot be exceeded at any time of the scheduling period. To meet such constraints, it is necessary to monitor the resource usage level over the scheduling horizon. Schedules involving simultaneous tasks with a total resource requirement larger than the maximum supply are to be excluded from the problem feasible region. Discrete resource constraints are computationally costly when continuous-time domain representations are used. Reklaitis (1992) presented a comprehensive review of resource-constrained scheduling problems. Continuous time formulations based on both the resource-task network (RTN) and the partitioning of the time horizon into intervals of unknown duration usually account for resource constraints but at the expense of a sizable increase in the number of 0-1 variables (Pantelides, 1994; Schilling et al., 1996). In turn, Pinto and Grossmann (1997) introduced a logic-based approach treating the resource constraints as disjunctions to reduce the number of enumerated nodes by orders of magnitudes.
722 This paper introduces a novel continuous-time MILP framework for short-term scheduling of single-stage multiproduct batch plants under severe limitations in required discrete resources. Similarly to the RTN notion, all the discrete resources (processing units, manpower, utilities, etc.) are equally treated. However, the use of a common set of 0-1 variables to sequence the batches at any available resource item significantly bounds the model size growth when resource constraints must be considered. Compared with previous formulations, a notable saving in binary variables and CPU time is achieved and larger resource-constrained batch scheduling problems can be tackled.
2. Problem Statement The problem of short-term batch scheduling under severe limitations in discrete resource supplies can be stated as follows. Given: (a) a single-stage multiproduct batch plant with several units y'G 7 working in parallel; (b) a set of single-batch orders /G/ with specified unit-dependent resource requirements, release-times and due-dates; (c) a set of discrete resources reR with known limited supplies; (d) sequence-dependent changeover times at any resource item and (e) a specified time horizon. The problem goal is to determine a production schedule that, in addition to meeting resource allocation and batch sequencing constraints and optimising a particular problem objective like a weighted combination of batch earliness and tardiness, also satisfies the limitations on the total supplies of plant resources at any point in time.
3. The Mathematical Model 3.1 Timing constraints. The starting time of batch / (S,) can be computed from its completion time by subtracting the processing time at the assigned unity. Obviously, S, must never be lower than the release-time of batch /. Si = Ci-Y,ptiJyiJ
^iel
(1)
jeJ,
3.2 Discrete resource allocation constraints. Let RRr be the available discrete resources items of type reR and Virj be the fixed amount of resource r required to process a batch iel in unit ye 7,. Resource type r may be referred to processing units, manpower, electricity, etc. As stated by Eqn (2.1), every batch / e / must be allocated to just a single unit ye//. In turn, constraint (2.2) ensures that enough amount of each resource re /?,' be allocated to meet the requirement of batch /, ^Yij
=l
(2.1)
yiel
jeJi
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re R\ ,\/ieI
(2.2)
jeJi
where /?', is the set of resources required by batch / other than processing units. Moreover, q^. is the amount of resource r available at the resource item ze RRr. 3.3 Sequencing constraints. Assuming that a pair of batches {/,/'} requires a common resource re(Ri n Rr) and both have been assigned to the same resource item zeRRr (Yiz= yi'z= 1)» then the completion time of batch / acts as a lower bound on the starting
723 time of r only if batch / is processed before. In such a case, the sequencing variable X,,will be equal to one. Consequently, the sequencing constraint (3) is enforced and eqn. (4) becomes redundant. If instead the assignment variables F,, and Yf. are both still equal to 1 but batch /' is first processed, then X,,- = 0 and constraint (4) is now enforced. Otherwise, one or both assignment variables are zero and the value of X/,- is meaningless. Therefore, a single variable X,,- is required to control the relative ordering of any pair of batches {/,/'} sharing a resource item. Sequencing constraints (3) and (4) explicitly account for sequence-dependent setup times at each common resource item z. Ci + dirz<Si'+ M{1-Xn)+ M{2-Yiz-Yiz) Cr+di'iz < Si+ M Xii' + M{2- Yiz - Yrz)
\/i,i'e I, i
RRr
(3)
\/i,i' E I, i < i\re {Ri n /?/•), z e RRr
(4)
Constraints (3) and (4) can be applied to sequence any pair of batches {/,/'} at every resource item ze RRr of any type re (Ri n /?,) allocated to both (K,, = K,- = 1). Moreover, the same binary variable X,/- can be used to denote the sequencing of batches {/, /'} at any common resource item z. This implies an important saving of sequencing variables 0-1. 3.4 Sequencing constraints at any already allocated resource item z. Let us assume that resources of type reR^ d R have already been assigned. Suppose that I. stand for the set of batches to which the resource item z has been allocated over the time horizon. For any pair of consecutive batches {/, i'}e /, on the processing line of resource item z, the constraints (3)-(4) reduce to eqn. (5). V/,/'€ Iz(niz + \ = nizhre R^.ze RRr
Ci-\-Tii'z<Si'
(5)
Therefore, sequencing constraints (3)-(4) apply to non-allocated resources while constraint (5) controls the batch timing at already assigned resources. In this way, it is possible to sequentially allocate, for instance, first the processing units and then the manpower to batches to be processed. Batch sequencing at already assigned resources are assumed to remain unchanged during the allocation of other resource types. In a next paper, such a batch sequencing will be allowed to change too. 3.5 Order tardiness and earliness. The earliness £", or tardiness T, of batch / takes a positive value only if it is completed either earlier or later than its specified due date di. Ti>Ci-di
\/ieI
(6)
Ei>di-Ci
\fiel
(7)
3.6 Problem objective function. The problem goal is to complete the batches just in time by minimizing the overall weighted earliness and tardiness. Min Yi^aiEi^piTi)
(8)
iel
In this way, the batch scheduling problem with limited discrete resources has been modelled as an MILP involving the set of constraints (l)-(7) and the objective function (8). 3.7 Using preordering rules to get a near-optimal schedule. Preordering rules arranging the batches at any resource item by decreasing due dates or slack times can be easily embedded in the proposed formulation to attain a further reduction in the number
724 of sequencing variables and constraints. For example, if the batches are to be sequenced by the HDD rule and J, < J,-, then X,,- can be eliminated from the model. Moreover, constraint (4) for the pair of batches {/,/'} can also be removed while the corresponding constraint (3) should be rewritten without the term involving X,,-. Frequently, preordering rules allow one to discover a very good solution to real world scheduling problems that otherwise it would never be found.
4. Results and Discussion The proposed MILP approach to the batch scheduling problem under severe discrete resource constraints will be illustrated by tackling a couple of examples. Example 1 previously studied by Pinto and Grossmann (1997) involves the scheduling of 12 batches in a single-stage batch plant over a one-month period. Though four parallel units (Ui,112,1)3,114) can be run in parallel, limited manpower supplies (two or three operators crews) prevents from running all the units at the same time. A single operator crew per unit is required. Data for Example 1 are those related to the first twelve orders in Table 1. Three cases were studied: (a) non-limited manpower; (b) three operator crews available; (c) two operator groups available. Similarly to prior work, the problem objective was to minimize batch earliness, assuming that due dates are imposed as hard constraints on the completion times (7, = 0, for any batch /)• Since processing units and manpower are allocated to batches at the same time, constraints (5) are ignored. Gantt charts describing the optimal solutions to the three cases are depicted in Figures la, lb and Ic, respectively. Model sizes and computational requirements for both (i) the logicbased approach of Pinto and Grossmann (1997) with preordering constraints and (ii) the proposed MILP formulation with/without applying the minimum-slack-time rule, are shown in Table 2. It can be observed that our approach discovers a better solution to any of the three case studies. For Examples (lb) and (Ic), the CPU time is reduced by a factor of 450 and 1100, respectively, when compared with the highly efficient logicbased approach. Example 2 deals with the scheduling of the twenty-nine orders in the same single-stage batch plant (see Table 1). This example was studied before by Mendez & Cerda (2001) who reported, for the unconstrained case, the production schedule included in Figure Id. Assuming that the equipment items have already been Table 1. Order data due unit date processing time (day;) order (day) ' U4 U3 u. U2 15 0. 1.194 1.538 0. 30 1.500 0.789 22 O3 1.607 0.818 25 O4 1.564 2.143 20 1.017 O5 0.736 30 06 3.200 5.263 21 O7 3.214 4.865 3.025 26 Os 1.440 1.500 30 O9 2.459 1.869 29 1.282 0,0 30 3.750 3.000 On 21 0,2 6.796 7.000 5.600 30 6.716 11.25 0,3 25 2.632 1.527 0,4 24 5.000 2.985 0,5 setup time 0^180 0.175 "C):'237
order 0,6 0,7 0,8 0,9 O20 O21 O22 O23 O24 O25 O26 O27 O28 O29
due date (day) 30 30 30 13 19 30 20 12 30 17 20 11 30 25
unit processing time (day)
u.
U2
U3
1.250 4.474 1.429 3.130 1.074 3.614 0.864 3.624 2.667 3.448
2.424 7.317
5.952 3.824 6.410 5.500 0.180
U4
0.783 3.036
0.175
2.687 1.600
4.000 4.902 1.757 3.937 3.235 4.286 a237
725 allocated as shown in Figure Id, the proposed MILP formulation, including constraints (5) for the units, was applied to also allocate the required manpower. It is supposed that three operator crews are available to run a similar number of units. In this manner, the schedule depicted in Figure le has been found. Gantt charts describing the assignment of units and manpower over the time horizon are shown. Model sizes and CPU requirements are also included in Table 2. EZZ]
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(e) Fig. I. Optimal schedules for Example 1 (a) without resource constraints (b) with three operator crews (c) with two operator crews. Schedules for Example 2 (d) without resource constraints (e) with three operator crews. (d)
Table 2. Model sizes and computational requirements Example
CPU nodes binary vars, objective time cont. vars, rows function #1. Logic-based approach with preordering constraints (Pinto and Grossmann, 1997) 283 63.56" 1.581 No resource constraints 673 125.42'^ 2.424 Resource constrained (at most 3 units) 7341 927.16^^ 8.323 Resource constrained (at most 2 units) #1. This approach without preordering rules 64 O.ll' 1.026 82 12, 202 No resource constraints 3071 7.91^ 127 12 610 1.895 Resource constrained (at most 3 units) 35.87' 19853 7.334 Resource constrained (at most 2 units) 115 , 12 ,478 -This approach with preordering rules 12 0.05' 1.026 25 12 119 No resource constraints 127 0.28' 1.895 61 12 329 Resource constrained (at most 3 units) 708 0.82' 7.334 49 12, 263 Resource constrained (at most 2 units) # 1 MILPfbrnniuiatio^^ No resource constraints 62.479 #2. This approach with preordering rules 89.97' 21907 8 7 , 2 9 , 1352 110.57 Resource constrained (at most 3 units) ' Seconds on IBM 6000-530 with GAMS/OSL." Seconds on Pentium III PC (933 MHz) with ILOG/CPLEX
726
5. Conclusions A highly efficient MILP formulation for the resource-constrained batch scheduling problem equally treating different types of discrete resources including equipment items has been developed. When compared with previous approaches, the proposed model shows a remarkable saving in 0-1 variables and a two-order-of-magnitude reduction in CPU time.
Nomenclature (a) Sets / orders to be scheduled /, orders assigned to the resource item z at the current schedule (/. c /) available processing units to process order /e 7 ( 7 , = /?/?/,) J, R renewable resources (processing unit, manpower, utility, etc.) /?* renewable resources that have akeady been assigned (/?* c R) Ri renewable resources required by order / (/?, c R) R'i renewable resources required by order / except processing units (/?', Q R , ) RRr available resource items of type re R (b) Parameters di due date of order ie I M a very large number riir position of the order /G / on the current sequence of resource item r pUj processing time of order /G / in unit; qn amount of resource r assigned to resource item z sli slack time of order /, sh ~ di - Min {pt,j. je J,} suij setup time of order ie I in unity T„ > sequence-dependent setup time between orders ie I and /'G / in resource item r v,rj amount of renewable resource re R required to process order i when order / is allocated to unity a, weighting coefficient for earliness of order ie I Pi weighting coefficient for tardiness of order ie I (c) Variables Ci completion time for order / Ei earliness for order / S, starting time for order / T, tardiness for order / Xir binary variable denoting that order ielis allocated before (X, = 1) or after (X„ = 0) order i'el in some common resource item r Yir binary variable denoting the allocation of order i to resource item r
References ILOG OPL Studio 2.1 User's Manual, 1999, ILOG S.A. France. Mendez, C.A., Cerda, J., 2001, Dynamic Scheduling in Multiproduct Batch Plants. Proceedings of 2 - Pan American Workshop on Process Systems Engineering, Guaruja, Sao Paulo, Brazil. Pantelides, C.C., 1994, Unified Frameworks for Optimal Process Planning and Scheduling. In Foundations of Computer Aided Process Operations, Austin, TX, 253. Pinto, J. M., Grossmann, I. E., 1997, A Logic-based Approach to Scheduling Problems with Resource Constraints. Comput. Chem. Eng., 21, 801. Reklaitis, G.V., 1992, Overview of Scheduling and Planning of Batch Process Operations. NATO Advanced Study Institute-Batch Process Systems Engineering, Antalya, Turkey. Schilling G., Pantelides C. C , 1996, A Simple Continuous-Time Process Scheduling Formulation and a Novel Solution Algorithm. Comput. Chem. Eng., 20, S1221.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
727
State-Space Residual Based Monitoring E. Mercer, E. B. Martin and A. J. Morris Centre for Process Analytics and Control Technology University of Newcastle, Newcastle upon Tyne, NEl 7RU, England
Abstract Although the process performance monitoring tools of dynamic Principal Component Analysis (PCA) and Canonical Variate Analysis (CVA) take into account process dynamics, the monitoring statistics still contain serial correlation. Consequently the traditional statistical basis for the calculation of control limits will be invalid resulting in either missed out-of-control signals or an excess of false alarms. A methodology is proposed whereby a CVA state-space model is first developed and then a PCA based monitoring scheme is formed using the model mismatch. In this case, the residuals will be independent and identically distributed and the standard control limits will be valid. The methodology is demonstrated on the benchmark Tennessee Eastman problem.
1. Introduction The traditional approach to developing process representations for the monitoring of continuous processes have been the multivariate statistical projection techniques of Principal Component Analysis (PCA) and Partial Least Squares (PLS). These approaches do not take into account the serial correlation in the data. This has led to the development of dynamic monitoring tools including dynamic PCA (Ku et a/. 1995) and Canonical Variate Analysis (CVA) (Larimore, 1997). As the latent variables/states do not necessarily represent a specific physical measurement on the plant, the control limits are determined statistically. Calculation of the limits is based on the assumption that the data underlying the metrics is independent and identically distributed (i.i.d). This assumption tends to be invalid for the majority of chemical processes. If serial correlation is present in the data, then calculating the limits based on the assumption that the data is i.i.d. will result in an increase in the number of false alarms or in the fault detection time. One approach to removing the serial correlation is to increase the sampling period. However adopting this approach can delay the detection of process changes. An alternative approach that has been proposed to address the problem of serial correlation in the resulting metrics has been to base the monitoring statistics on the CVA state space equation residuals, w and e (Simoglou et al, 1999), calculated from: x(/ +1) = Cx(0 + Gu(0 -H w(r)
(1)
y(r) - Hx(0 + Au(0 + Bw(0 + e(0
(2)
728 where x is a state vector, and u and y are the process inputs and outputs respectively. The noise terms, w and e, are assumed to be i.i.d. with covariance matrices Q and R. Simoglou et al. (1999) proposed two metrics based on the state-space residuals, 7^^^ and T^e- If the state space equations adequately define the process, the residuals will be i.i.d. and the control limits can be calculated using the appropriate statistical distribution. Simoglou et al also compared the statistical distributional approach to calculating the limits with the empirical reference distribution (Willemain and Runger, 1996) in terms of the false alarm rate and the time to fault detection. A number of monitoring statistics have also been developed based on the CVA states. The first of these was the 7^ Statistic based on the k significant states, 7^„ (Negiz and Cinar, 1997). This idea was further developed by using those states that are not considered significant as the basis of another metric, 7^,^, (Russell et al 2000). Both statistics, 7^, and 7^,,, have control limits based on the F-distribution. However the data is not i.i.d. and thus the calculated limits are not valid and will result in spurious alarms. This paper proposes an extension to the work of Simoglou et al where PCA is performed on the state space output residuals, e. This allows the use of the PCA 7^ and SPE metrics to monitor the residuals, i.e. T^PCA and SPEpcA' The limits can be calculated using an F-distribution as the residuals will be independent identically distributed.
2 The Tennessee Eastman Process The Tennessee Eastman process simulation forms the basis of the subsequent study. The process consists of a reactor/separator/recycle arrangement and includes two gas/liquid reactions and two-side reactions that produce the same by-product. The simulation comprises 12 manipulated variables and 41 measured variables (see Downs and Vogel, 1993 for details). In this study, one manipulated variable, agitator speed, is constant so is excluded from the subsequent analysis. The Lyman control scheme for the 50/50 product ratio was used (Lyman and Georgakis, 1995). Twenty-three data sets were generated of which three were associated with nominal data and twenty were based on pre-programmed faults (Table 1). Each run lasted 48 hours with samples taken every three minutes, with faults occurring after 8 hours (160 observations). A CVA state-space model was built from nominal data set one. This is in contrast to the approach of Russell et al (2000) where a separate model was developed for each fault. A lag of three was selected for all variables and 29 states were included in the CVA model. These values were selected using the small sample corrected Akaike Information Criterion (AIC) (Hurvich et al 1990). All data was scaled to zero mean, unit variance based on the first nominal data set. The CVA based state space model built on the first nominal data set, resulted in residuals that were artificially low in magnitude. Thus a second nominal data set was used to calculate the residuals. These formed the basis of the PCA model and the control limits. The residuals, e, were obtained by subtracting the output model estimates from the process simulation outputs, whilst subtracting the state step-ahead predictions from the CVA calculated states gave the state errors, w. These two sets of residuals
729 were then used to calculate the respective covariance matrices and the T,,, and T^ statistics. The output prediction residuals, e, were normalised to mean zero and unit variance and PCA was applied. The number of principal components retained were those with an eigenvalue greater than one. The false alarm rate for a 99% action limit was then investigated using the third nominal data set. From the results in Table 2, it can be concluded that T^PCA gave the value closest to that expected, i.e. 0.005. The poorest performance was exhibited by 7^,^ and T^vv which had false alarm rates in excess of 10%. The next step was to examine the serial correlation retained by each metric. This was done using the autocorrelation function (ACF) (Fig. 1) and the partial autocorrelation function (PACF) (Fig. 2). It can be seen that the false alarm rate is typically proportional to the level of serial correlation, i.e. the higher the level of serial correlation retained in the metric, the greater the false alarm rate. The first measure used to compare the statistics' ability to detect faults was their sensitivity to the presence of faults, illustrated by the proportion of in-control signals once an initial fault had occurred. Faults 3, 9, and 15 were excluded due to them having no apparent effect on process operation. Table 3 shows the proportion of observations that showed an in-statistical-control reading once the prescribed fault had occurred. These results indicate that 7^^ gives the lowest missed detection rate for the majority of the faults with SPEpcA giving marginally poorer results. However from the ACF and PACF, Figs. 1 and 2, SPEpcA appears to contain less serial correlation. This suggests that the 7^^ limits may be artificially low, giving a high false missed detection rate. Table 1. Tennessee Eastman Process Faults Fault Number
mvd) rov(2) IDV(3)
rov(4) IDV(5) IDV(6) IDV(7)
rov(8)
IDV(9) EDVdO) IDV(ll) IDV(12) IDV(13) IDV(14) IDV(15)
rov(i6) rov(i7) IDV(18)
rov(i9) IDV(20)
Fault Description A/C feed ratio, B composition constant (stream 4) B composition, A/C ratio constant (stream 4) D feed temperature (stream2) Reactor cooling inlet temperature Condenser cooling inlet temperature A feed loss (stream 1) C header pressure loss- reduced availability (stream 4) A, B, C feed composition (stream4) D feed temperature (stream 2) C feed temperature (stream 4) Reactor cooling water inlet temperature Condenser cooling water inlet temperature Reaction kinetics Reactor cooling water valve Condenser cooling water valve Unknown Unknown Unknown Unknown Unknown
Type Step Step Step Step Step Step Step Random Variation Random Variation Random Variation Random Variation Random Variation Slow Drift Sticking Sticking Unknown Unknown Unknown Unknown Unknown
730 Table 2. False Alarm Rate Statistic
PT n ^
False Alarm Rate 0.1076 0.0549 0.1014
" "
W
Statistic ~~T'e ipCA
SPEpcA
Table 3. Missed Fault Detection Rates (bold denotes Fault F ^ 9 IDV(l) 0.0038 0.0025 0.0000 IDV(2) 0.0126 0.0100 0.0088 IDV(4) 0.8952 0.0013 0.7215 IDV(5) 0.6604 0.6625 0.4811 rov(6) 0.0013 0.0013 0.0000 IDV(7) 0.3586 0.3864 0.0013 rov(8) 0.0189 0.0164 0.0151 0.1604 rov(io) 0.1581 0.0821 IDV(ll) 0.7854 0.5069 0.0909 rov(i2) 0.0076 0.0289 0.0063 0.0467 0.0442 0.0351 rov(i3) rov(i4) 0.0278 0.0013 0.0000 0.2551 rov(i6) 0.1280 0.0543 0.1048 rov(i7) 0.0402 0.0240 0.0922 rov(i8) 0.0896 0.0853 0.6995 IDV(19) 0.0025 0.2120 0.3333 IDV(20) 0.1982 0.3526
^v^
False Alarm Rate 0.0327 0.0052 0.0378
best result) ^. 0.0013 0.0125 0.0000 0.0000 0.0000 0.0000 0.0088 0.0828 0.2271 0.0013 0.0402 0.0013 0.0703 0.0221 0.0878 0.0088 0.0765
T^PCA
SPEpcA
0.0025 0.0138 0.6939 0.0013 0.0000 0.0088 0.0176 0.0979 0.4881 0.0025 0.0439 0.1343 0.1205 0.0263 0.0954 0.1882 0.2509
0.0013 0.0176 0.0188 0.0000 0.0000 0.0000 0.0113 0.1380 0.3099 0.0038 0.0402 0.0100 0.1092 0.0226 0.0966 0.0452 0.0790
The next metric examined was the time to the initial out-of-control signal once a fault had occurred. Examining Table 4, the time between the fault occurring and an out-ofcontrol value being registered, the best performers were 7^^, T^^, and SPEpcA- This is because although the statistics containing serial correlation, 7^^ and 7^,^, have a higher false detection rate (see Table 2), their ability to detect a true out-of-control signal more quickly is compromised by the effect of the previous in-control measurements. In terms of sensitivity to faults and speed of detection once a fault has occurred, the above results show that the monitoring metrics based on w and e, the residuals from the state space equations, gave better results compared to those statistics based on the calculated states. The residual based statistics also gave fewer false out-of-control signals. These results are due to the reduced level of serial correlation in the residuals compared to the states, resulting in more appropriate statistical limits as the i.i.d. assumptions are upheld. The statistic with the lowest level of serial correlation, T^PCA^ did not give the best results but were close to those of the T^e and SPEpcA results, although the level of serial correlation and false alarm rate in T^PCA suggests that it provided the best statistical control limits. The performance of the residual based techniques are comparable to those presented by Russell et al. (2000) who used a library of faults and limits based on the 99* percentile of a fault free validation data set.
731
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SPEPCA
3 Conclusions The Tennessee Eastman simulation was used to compare dynamic monitoring statistics based on process states and their residuals. The metrics examined were T^ based on the used CVA states, 7^,^ based on the unused states, 7^^ and f^e based on the state space equation residuals, and two proposed statistics, fpcA and SPEpcA calculated by performing PCA on the output prediction residuals, e.
732 Table 4. Time to Fault Detection (mins) (bold denotes best result) Fault T", K I^^ l^e T'pcA 9 6 3 6 0 rov(i) 30 IDV(2) 21 15 18 21 IDV(4) 6 0 3 0 0 3 3 IDV(5) 6 0 0 0 IDV(6) 3 3 0 0 0 3 3 0 0 rov(7) 33 18 IDV(8) 15 30 18 63 IDV(IO) 63 63 60 63 21 IDV(ll) 33 18 18 18 6 9 12 3 0 rov(i2) 108 IDV(13) 111 102 105 102 3 IDV(14) 3 3 3 0 21 33 24 IDV(16) 18 18 57 54 63 IDV(17) 54 57 225 237 IDV(18) 222 228 210 6 45 6 rov(i9) 0 0 219 222 IDV(20) 225 195 186
SPEp 3 12 0 0 0 0 18 69 18 3 102 3 18 51 240 0 186
The metrics based on the state space equation residuals, w and e, were shown to contain less serial correlation than those using the system states themselves, as the majority of the structure is contained within the states. The ability to detect faults both accurately and quickly was best shown by the 7^^ and SPEpcA statistics, followed J^PCAJ T^W^ and T^,j. The validity of the performance of the 7^vv, and T^^ statistics was questionable due to the inherent level of serial correlation. The serial correlation contained within, and the false alarm rate of, 7^^ and SPEpcA raises questions about the results. Based on these initial studies using monitoring statistics formulated from the output prediction residuals from the state space equations seem to give improved fault detection without the need to modify the statistically calculated control limits.
Acknowledgements Mr Mercer acknowledges the EPSRC, BP and CPACT for financial support of his PhD.
References Downs, J. J. and E. F. Vogel, 1993, Computers Chem. Engng., 17, 245. Hurvich, C. M., R. Shumway and C. L. Tsai, 1990, Biometrika, 77, 709. Ku, W.F., R.H. Storer and C. Georgakis, 1995, Chemo. & Intell. Sys. 30, 179 Larimore, W. E., 1997, In Statistical Methods in Control and Signal Processing (Eds, Katayama, T. and Sugimoto, S.) Marcel Dekker, New York, 83. Lyman, P. R. and C. Georgakis, 1995, Computers Chem. Engng., 19, 321. Negiz, A. and A. Cinar, 1997, AIChE Journal, 43, 2002-2020. Russell, E. L., L. H. Chiangand, R. D. Braatz, 2000, Chemo. & Intell. Sys. 51,81. Simoglou, A. E. B. Martin and A. J. Morris, 1999, Computers Chem. Engng., S277. Willemain, T. R. and G. C. Runger, 1996, Journal of Quality Technology, 28(1), 31.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Published by Elsevier Science B.V.
733
Performance Monitoring for Process Control and Optimisation Arnoud Nougues, Pierre Vadnais, Rob Snoeren Shell Global Solutions The Netherlands
Abstract Over the last two decades many oil and petrochemical companies have installed Advanced Process Control (APC) and Closed Loop Optimisers in their plants. Within Shell for example there have been about 550 APC projects and 30 Closed Loop Optimisers installed that add 430 million Euro's per annum to the bottom line. Potentially this can grow to 700 million Euro's per annum. For the coming years the challenge will be to maintain the optimum performance of the existing applications while at the same time implementing new projects. With skilled resources remaining limited, it means that more innovative steps have to be taken. With this background, Shell has developed a number of methodologies to monitor the performance of controllers and optimisers. The key objective is to benchmark the performance against 'best in class' performance, identify non-compliances, and diagnose possible problems (tuning, modelling, etc.) so that the appropriate corrective actions can be taken.
Economic Incentive of Application Performance Monitoring APC applications essentially consist of multivariable controllers, varying in size from small local applications, e.g. with 2 or 3 Manipulated Variables (MV's), to unit-wide optimising multivariable applications where an economic variable is explicitly included in the control strategy for LP type optimisation. In a refinery where all major APC applications have been implemented, reported benefits are typically in the range of 10-15 US cents/BBL overall. For an average site processing 150 KBBL/D, this translates into benefits in the order of 5 to 8 MMUSSAf. Similar numbers apply to petrochemical plants. For refinery optimisers the tangible benefits range from 5 to 10 US cents/BBL throughput. Achieved benefits come from three main sources: • Feed increase from more stable operation, closer to the limiting operating constraints and products specifications, • Yield improvement of the more valuable products, • Energy savings. Besides the tangible benefits, extensive lists of non-tangible benefits have been acknowledged as important spin-offs from APC /optimisation projects. The benefits at stake are therefore considerable. If the applications performance is not sufficient, then only a fraction of the expected benefit will be achieved, implying an
734 economic loss from non-compliance of the applications, and a lower than expected return on investment of the APC and optimisation implementation projects.
The importance of Performance Monitoring for Applications Maintenance Effective applications maintenance nowadays implies putting in place a performance monitoring system with the following general characteristics: • Provision of the right amount of information (%uptime, performance index, information to allow diagnosing the root cause of potential problems) in a systematic and concise way. • The information should be readily available, at the engineer's desktop, and available in real time with no effort, preferably using web-enabled technology. • Remote maintenance option: as more and more applications are put on-line, and under the general competitive pressure to cut costs, control and optimisation manpower at sites is generally scarce and overloaded. Remote maintenance then becomes an attractive option, to allow support from specialised staff in a central location (technical head office, or contractor's office). • Application performance should be measured against pre-defined targets, and corrective action taken as required when non-compliances occur.
APC and Base Layer Control Monitoring Shell has developed and is in the process of further developing a complete suite of software packages, called MD (Monitoring and Diagnosis) for monitoring the performance of control loops and to assist in troubleshooting loops that fail to meet their performance target. The tools apply to multivariable controls, of Shell technology (SMOC) as well as multivariable controls from any APC vendor, and they apply to traditional Single Input-Single Output loops (e.g. PID controller). The central element of MD is a client-server information system for control loop performance tracking. MD is linked to various commercial plant data historians (e.g. Yokogawa's Exaquantum, OSI PI), where the basic real-time control loop status and performance information resides. Each day performance statistics are automatically calculated and stored in a dedicated Relational Data Base. Control engineers are notified if control loops are performing below predefined targets by daily email summary reports. Next to this, the user can enter the report mode where the statistical information can be browsed. For every control loop and Controlled Variable (CV), MD provides the following statistical information: • % in Service: optional controller and unit availability tags are monitored to determine whether the controller is in service. • % Uptime: loop uptime is determined from the controller mode status. • % in Compliance: this statistic indicates if a CV deviates significantly from either a setpoint or Min./Max. limits. The bound to indicate a significant deviation is determined from a user specified tolerance (CL) for each CV. If the CV is within the ± CL bound about the control limits (set range), the CV is considered to be in
735 compliance. The information is reported as daily and monthly averages based on calculations carried out using typically one-minute data. %in Service, %Uptime, %in Compliance together with a user defined cost factor are used to derive a cost incentive which is reported on a daily and monthly basis to the user (PONC : Price of Non-Conformance). Monitoring loop performance is not sufficient. Additional tools are required to help analyse loop performance related problems and troubleshoot under-performing loops efficiently. A number of innovative proprietary loop performance diagnosis techniques have been developed by Shell and are part of the MD suite of packages: • Average closed loop response curves: both CV error and MV response curves are calculated and plotted. Average response curves provide a visual summary of the shape and response fime of SISO as well as multivariable control loops, in response to the actual disturbances affecting the process and in response to setpoint changes. The average response curves are derived from fitting an ARMA (Auto-Regressive Moving Average) model to the loop time series data, typically over several hours or days of normal closed loop operation. • Plot of sliding window CV error standard deviation; comparison with best achievable performance from Minimum Variance Controller (MVC): the CV error standard deviation is calculated over a representative time span, and then the calculation is repeated by sliding the window from the start to the end of the data time range. The standard deviation that would have been achieved by the fastest possible feedback controller (MVC) is shown on a parallel plot. The CV error standard deviation plots are useful in assessing the loop performance in relative terms (achieved CV error standard deviation, and how it evolves in time) as well as in absolute terms (comparison with reference MVC controller). • Degrees of freedom and constraint analysis: this technique applies to multivariable control applications. The idea is to track and report which controlled variables in a closed loop mukivariable system are active (i.e. driven to their upper or lower specification limit) and how often these are active. Correspondingly, the activity status of the manipulated variables is reported, i.e. which MV's are limited or unavailable and how often. This information, presented in the form of bar plots and trends, provides insight into the activity and performance of a complex muUivariable controller, and helps diagnose control structure problems (e.g. insufficient degrees of freedom to achieve the required control objectives).
Real Time Optimisation Monitoring and diagnosis techniques Recent monitoring and diagnosis developments have been focussed on the business objective of an optimiser. An optimiser is totally and strictly driven by an economic objective funcUon. This objective function is based on the best calculation available for the unit's margin using current market values for feeds, products and utilities. With such an objective function, the optimiser is clearly focused on the bottom line, on maximising profit.
736 In an attempt to help the understanding of the optimised solutions and to facilitate the communication between all involved, a few graphs were developed to show how the optimiser is improving the unit's margin and to pinpoint all the relevant constraints. Product upgrading The first graph, in Figure 1, shows the cumulative relative contribution of each element of the objective function to the increase in the unit's margin from the base case to the optimised case. A positive contribution is shown in green (light) and a negative contribution is shown in red (dark). The cost elements (feeds, imports, utilities) have a positive contribution when they are decreased. The revenue elements (products) have a positive contribution when they are increased. 100.0%
Ul
ill
3=
Q
c -50.0% -
^
a:
8
5
i
5
9L
g
Figure 1 - Product upgrading The graph gives a quick overview of the changes to the mass balance. For someone who knows the unit and has an idea of the current economic drivers, it is easy to evaluate the appropriateness of the changes. In some cases, further analysis is required to fully understand how the changes are actually implemented in the unit. Independent moves The second graph, in Figure 2, displays the cumulative relative contribution of each independent variable to the objective function. A positive contribution is shown in green (light) and a negative contribution is shown in red (dark). The individual contributions are an approximate estimation of the true contribution of each independent variable based on the actual move and the average effect of the independent on the objective function (dO/dX). The purpose is mainly to single out the independents with the largest contributions.
737 First, it is important to know the main positive contributors represented by the longest green (light) bars on the right hand side. They can generally explain the changes to the mass balance that generated the profit seen in the first graph. Second, it is important to understand the main negative contributors represented by the red (dark) bars on the left hand side. Why would an independent move ''knowingly" in the direction that reduces its contribution to the objective function, unless it is to help some other independent(s) to provide an even greater contribution.
Figure 2 - Contribution of independent variables Would you expect operators to predict that a move in the "wrong" direction for one variable will be favourably compensated by relaxing a constraint on another variable? Would you consider using a slogan going against the known profitable direction? Would you recommend to use a linear multi-input constraint pusher to reach the tenuous balance of a crude preheat train integration with the column pressure? The fact that an online optimiser can move some independents in the ''wrong" direction to make more money with the others largely explain why the results of online optimisers are sometimes difficult to understand for all involved. It should also prove beyond any doubts the superiority of the technique for continuous optimisation of any operation. Finally, it is possible to identify the independents with little or no effect on the objective function and investigate if they are over constrained, redundant or temporarily ineffective. Constraints Having seen what is happening in the first graph and how it is produced in the second graph, it is normal to wonder "Why not more?". The third graph. Figure 3, shows how far each independent has moved relative to its range of freedom. The range of freedom is shown by a green (light) frame in the direction (up or down) of a positive contribution to the objective function and by a red (dark) frame in the direction of a negative contribution to the objective function. If the independent moved in the "right" direction, the green frame is partially or totally filled by a green (light) bar proportional to the
738 extent of the move relative to the potential move. If the independent moved in the ''wrong" direction, a similar red (dark) bar fills the red frame. To maintain the business focus, the frames are relatively sized on their potential contribution to the objective function with 100% assigned to the largest potential contribution.
a
Figure 3 - Freedom of independent variables But why are some frames only partially filled? Because some independents are limited by constraints on resulting variables. The independents can be self limited by their own direct constraints. They can also be limited by a number of dependent variables, which have reached the limit in the direction where the specific independent is trying to push. Optimiser maintenance and performance monitoring services Online optimisation is highly technical, but it is also a collaborative tool, a synergistic tool, which brings together the sum of the knowledge of all the team members about the process technology, the unit configuration, the economic incentives, the operating constraints, and the mathematical algorithms. It appears that the commitment of the entire team (optimisation engineer, process engineer, operations, and economist) with full support of management is the most significant constant behind a successful optimisation project. Too often, we have seen a successful optimiser loose momentum after the transfer of the champion who could keep the spirit alive by stimulating the communication of the right information within and around the team. Based on this experience, the continues pressure on operating costs, the lack of sufficient skilled supporting staff. Shell Global Solutions International has extended their services with APC / optimiser maintenance and performance monitoring making use of developed monitoring tools and highly skilled professionals. Via web enabled technology day-to-day, worldwide support can be given from our head-offices in Amsterdam, Singapore, Kuala Lumpur and Houston.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
739
Optimization of Naphtha Feedstock Blending for Integrated Olefins-Aromatics Plant Production Scheduling Y. Ota, K. Namatame, H. Hamataka, K. Nakagawa and H. Abe Mitsubishi Chemical Corp. Kashima Plant, Ibaraki, Japan A. Cervantes, LB. Tjoa^ and F. Valli, MC Research and Innovation Center, Mountain View, CA 94041, USA.
Abstract An optimization system for the planning and scheduling of a petrochemical complex is presented in this work. The complex uses naphtha, with a wide range of properties and costs, as its main feedstock. Depending on the plants production constraints and market conditions, the optimization of the feedstock blending has a significant economical impact. We describe the production planning and scheduling system for the integrated petrochemical complex. This application covers scheduling decisions in the naphtha and gasoline tank yards, two olefins plants and two aromatics plants. The optimization problem poses challenges due to the nonlinearities of the process model and the combinatorial part of the naphtha delivery and tank yard operation. A model in the form of a mixed integer nonlinear programming (MINLP) problem is used. In this work, we discuss the practical project implementation. The benefits of using the scheduling system are reflected on the plant operation. Two case studies are presented. The optimal naphtha scheduling on the tank yard section gives more flexibility to the naphtha blends that are fed to the plants. This flexibility is reflected on a better plant utilization that makes the whole operation more profitable.
1. Introduction Production scheduling is an important operational activity in a chemical industry. The main focus of scheduling has been on batch processes. A comprehensive review on scheduling and planning of batch plant operations can be found in Reklaitis (1991, 1992), and a summary of basic scheduling techniques can be found in Pekny and Zentner (1994). Although there are many reported linear programming approaches for the planning problem of continuous processes, there are few studies reported on the optimization of the scheduling problem because it requires a large nonlinear model with many degrees of freedom. A reliable solution method for this type of problem is not widely available. The first reported MINLP based ethylene production scheduling model was presented by Tjoa et. al. (1997). Pinto et. al. (2000) addressed a nonlinear model for the a refinery plannning and scheduling problem where the blending problem is also a very important issue. In this paper, we extend the methodology of Tjoa et. al. (1997) to solve a large petrochemical complex model that includes a tank yard, two ethylene plants and two aromatic plants. ^ Corresponding author: [email protected].
740
2. Problem Description The production scheduling covers operational decisions for a tank yard section, two ethylene and two aromatics plants. Given a monthly production target of key products from the planning system, the main task for the scheduling system is to decide daily operational decisions such as transfer rates of feedstocks in the tank yard section, production target of key products, etc. The main challenge in this scheduling problem is to allocate the right composition of naphtha in the charging tank that maximizes the utilization of the main resources: furnaces and separation units. Since naphtha prices vary with its composition, the optimal allocation of the naphtha feedstock has a significant financial impact. In the following subsections, we describe the problem and constraints without presenting the mathematical formulation. The main purpose is to understand the economic opportunity and complexity of the model. 2.1 Tank Yard Section Naphtha feedstock poses a fairly complex problem for the ethylene producer due to the wide variety of supplied naphtha qualities. The complexity of the tank yard management is related to naphtha delivery, storage, transfer, and mixing. In this problem, most of the naphtha is delivered by ship, and the rest is delivered through pipes on fixed dates. On a typical month, we need to schedule about 10 vessels that carry naphtha of various grades. There are few tanks at several stages before the feedstock reaches the charging tanks. Availability of extreme naphtha compositions at the charging tanks is very crucial for providing flexible online blending for meeting the targeted compositions and amounts required for each ethylene plant. Thus the objective of the tank yard section is to satisfy naphtha demand, maintain inventory volume and provide the ethylene plants with the optimal naphtha mixture. In the model formulation, besides total and component mass balances on the naphtha tanks, the following constraints are formulated as a mixed integer nonlinear model: Naphtha storage tanks are divided into 3 stages that consist of 2 tanks each. A limited number of berths are available at stage 1 and 2. Other tank yard operational constraints due to piping network. 2.2 Ethylene plants The main purpose of the ethylene plants model is to predict production rates for various feedstocks under an attainable plant operating condition. Here is the list of constraints for the ethylene plant: Nonlinear yield model for furnaces to represent 9 major products, from hydrogen to heavy ends. There are several furnace types for each plant. Furnace operational variables such as severity are considered in the model. Material flow between the two plants is allowed for certain products. Separation constraints that reflect the capacity of the separation columns. Utilities consumption and production in each plant. A major economic trade off is the production of gasoline that is fed to the aromatic plants. Light feedstock produces insignificant amount of gasoline; on the other hand, heavy naphthas produce a significant amount of gasoline.
741 2.3 Aromatic Plants The heavier products coming out from the separation units in the ethylene plants (gasoline and others) are the main input of the aromatic plants. We also have additional supplies of crude gasoline. Here, the compositions also vary. The objective of the aromatic plants is to transform the heavier products coming from upstream into aromatics. The most valuable product from the aromatics plants is benzene.
3. Modeling and Solution Approach An accurate representation of the production scheduling requires a fairly detail model. The resulting model is often very large and complex. In order to reduce the modeling effort, we have developed an in house process planning and scheduling system that allows us to implement a project efficiently. Here we describe briefly the basic approach of our system. 3.1. Modeling Approach. Our process planning and scheduling system uses a unit operation modeling approach, a similar approach as in the familiar process simulator systems. Here, we can handle both, process as well as operation constraints. Depending on the production operation flow diagram, the modeling environment will generate the appropriate mathematical programming formulation in GAMS modeling language, Brooke et. al. (1988). It can generate from a simple linear programming (LP) model to a more complex MINLP model. In this application, the system generates a MINLP model. The main advantage of this modeling approach is that we can modify the model representation efficiently. 3.2. Solution Approach. The main advantage of our modeling approach is that the entire problem can be solved and optimized simultaneously using a mathematical programming method. Thus we can avoid a 'sequential' solution approach which can be very expensive for finding just a feasible solution given so many degrees of freedom in this model. The main challenge, however, is to develop a reliable solution methodology for solving a large scale MINLP model that has over a thousand binary variables and several thousands of continuous variables with over a thousand degrees of freedom. To illustrate the size of this application, this model has approximately 1,500 binary variables and 60,000 continuous variables. Here we developed a proprietary solution strategy for solving a large scale MINLP model. Without disclosing the details, the basic idea of this strategy lies in decomposition of the full problem into two subproblems, a Mixed Integer Linear Programming (MILP) subproblem and a Nonlinear Programming (NLP) subproblems. The MILP model contains reliable linearized constraints based on process knowledge as well as mathematical programming techniques. The binary variables are decided from the MILP subproblem and the continuous variables are obtained from the NLP subproblem. Since we use GAMS modeling environment, there are several choices of
742 solvers for solving each subproblem. Here, we use OSL (IBM, 1991) and CONOPT (Drud, 1992) for solving the MILP and NLP subproblems respectively. Due to complexity of the process model, we use a similar decomposition strategy as described in Tjoa et. al. (1997). First, we solve the ethylene and aromatics sections based on two average naphtha compositions (light and heavy) in order to get an approximation to the amount of required feedstocks. This is an NLP problem, the nonlinearities appear in the yield models and the non-sharp split separation columns. Once we have an approximation of feedstock requirement, we solve the tank yard section as an MINLP model, trying to maintain the classification on the feedstocks into light and heavy. The objective is to keep one charging tank with heavy naphtha and the other one with light naphtha. The solution of the tank yard section satisfies the approximation of the feedstock requirements and also achieves a good classification of the naphtha. After solving these problems we resolved the ethylene and aromatics plants model with the fixed input to the charging tanks. This is again an NLP model. It takes approximately 20 min to solve the whole problem in a Pentium III, 800 MHz computer.
4. Case studies. In order to demonstrate the importance of having an automated system for the plant scheduling we present two case studies. Both represent a one month schedule where the arrival of the naphtha shipments has been fixed. For a comparison purpose, the initial composition of the charging tanks is set the same, in this way the composition profiles only reflect the current schedule and not previous scheduling policies. In the first case we solved the problem with an open demand for ethylene and propylene, for the second case we set the demand equal to 85% of the production obtained for the open demand. The benzene demand for both cases is open.
%f^.jL:::'' ' ^
-T1_0pt T1 Base
-T2_0pt T2 Base
Figure 1. Charging Tanks Composition The solutions of the optimization problem (Opt) are compared with those obtained by just getting a feasible solution (Base) for the tank yard section and an optimal solution for the ethylene and aromatics sections. The charging tanks (Tl and T2) compositions for these two solutions can be seen in Figure 1. For the optimized solutions (opt) there is a clear classification of the naphthas fed to the plant, the light naphthas go to Tl and the heavy naphthas go to T2. The naphthas are stored and transferred in the tank yard in
743 an optimal way. In the base solution, there is no classification at all, the naphthas are stored and transferred just to satisfy the piping and inventory constraints. Table 7. Cases Comparison
Fix Demand Base
Naphtha AvgSG 0.679
Ethylene Units 85%
Propylene Units 85%
Benzene Units 92%
Hydrogen Units 92%
Fix Demand Opt Open Demand Base Open Demand Opt
0.690 0.689 0.686
85% 97% 100%
85% 100% 99%
95% 100% 100%
92% 97% 100%
Case
The effect of the classification can be seen in Table 1 where the production value has been normalized based on the maximum value of each product for all cases. For the fix demand case the ethylene and propylene productions are obviously the same, but the production of benzene is larger for the optimal solution. Furthermore, the average SG of the naphtha feed in the optimized case is higher; this is reflected as a decrease in the operating costs because heavier naphtha is cheaper than lighter naphtha. The proper classification of the naphtha allows a wider range of naphtha compositions that can be fed to the plants, increasing the plant utilization and efficiency. In the second case the wider range of compositions is reflected as an increase in the production of ethylene, which is the most valuable product. The propylene and benzene productions are lower in the optimal case because their prices are lower than that for the ethylene. The efficiency can also be seen on the average SG as the plants are producing more ethylene with almost the same grade of naphtha. For both cases, the production of other byproducts is very similar for both solutions, and the economical impact is smaller than that of ethylene, propylene and benzene. We also looked at the naphtha consumption by analyzing the naphtha final inventory at the end of the schedule. This can be done because the naphtha feed is fixed for all cases to the same value. Figure 2 and 3 show the results. Here we classified the naphthas in inventory according to their SG into 3 groups: light (< 0.675), heavy (>0.69) and medium (in between).
0 80
1 60
c Z 40
o 20 Base Opt i Light D Medium • Heavy
Figure 2. Final Naphtha Inventory Fixed Demand
Base Opt S Light D Medium • Heavy
Figure 3. Final Naphtha Inventory Open Demand
744 The decrease of medium naphtha in the optimal solutions is obviously the effect of the better classification of the naphtha feedstock. More importantly, the increase of light naphtha in the final inventory for both cases is larger than the increase of heavy naphtha for the optimal solutions. For example, for the fixed demand case the increment is 34 % for the light naphtha, but only 18 % for the heavy.
5. Actual Implementation The current system was implemented under a Microsoft Windows environment. The general user interface (GUI) provides a modular, plug and play framework for connecting to the solution engine as well as accessing information on a database. The information is displayed in excel kind cells for easy editing. It also includes graph capabilities, which provides a better image of the process. The GUI allows easy browsing and editing of the information in the database. It can easily be reconfigured to the changing needs of the user, by simply creating, modifying or deleting queries in SQL language.
6. Conclusions We have developed a system that allows us to improve our plant operation. The entire model that consists of a complex feedstock management, ethylene and aromatic plants is solved simultaneously using the proposed decomposition strategy. The robustness and utility of the system have been demonstrated on a real industrial application. The optimized solutions consistently demonstrate the benefits of the system. The economic impact can easily be identified, less light naphtha utilization and improve capacity utilization.
7. References Brooke, A.; Kendrick, D. and Meeraus, A, GAMS - A user's guide (release 2.25). The Scientific Press, San Francisco (CA), 1988. Drud, A.S., CONOPT - A GRG code for large scale nonlinear optimization - Reference manual. ARKI Consulting and Development A/S, Bagsvaerd, Denmark, 1992. IBM, OSL (Optimization Subroutine Library) Guide and reference, release 2, Kingston, NY, 1991. Pekny, J.F. and Zentner, M.G., 1994, Learning to solve process scheduling problems: the role of rigorous knowledge acquisition frameworks. In Foundation of Computer Aided Process Operations; (Rippin, D.W.T., Hale J.C. and Davis, J.F. eds.). CACHE, Austin, pp. 275-309. Reklaitis, G.V., 1991, Perspectives on scheduling and planning of process operations. Presented at the Fourth International Symposium on Process Systems Engineering, Montebello (Canada). Reklaitis, G.V., 1992, Overview of scheduling and planning of batch process operations. Presented at NATO Advanced Study Institute- Batch Process Systems Engineering, Antalya (Turkey). LB. Tjoa, Y. Ota, H. Matsuo and Y. Natori, 'Ethylene Plant Scheduling System Based on a MINLP Formulation,' a special edition of Computers Chem. Eng., 1997.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
745
A Systematic Approach for the Optimal Operation and Maintenance of Heat Exchanger Networks Hernan Rodera and Hiren K. Shethna Hyprotech Ltd, Suite 800, 707 - 8"^ Avenue SW, Calgary, Alberta, T2P 1H5, Canada e-mail: [email protected]; [email protected].
Abstract An automatic approach to study the behaviour of an existing heat exchanger network design when the operating conditions vary is presented. The purpose of this methodology is to facilitate the involvement of the engineer in the daily operation and maintenance of the network and its optimal performance prediction. The approach also assists in proposing improvements in the design that could lead to retrofit implementations. The fmal aim of the proposed methodology is to provide a tool capable of monitoring the performance of the individual heat exchanger units. Specification of the fouling factor is the first step in this direction. An example showing the application of this approach to a crude distillation column preheat train is presented.
1. Introduction Synthesis of industrial heat exchanger networks is carried out usually by considering all possible set of operating conditions and by solving the problem to a final stage in which a practical design is obtained. The conceptual design phase evaluates the capital vs. operating cost trade-off and sets targets that are used to construct and optimise the network. Then, the final design is the result of a compromise between the optimal design and the designs capable of performing under different operating conditions. Therefore, limitations in the flexibility of the final design (when changes in these operating conditions occur) and in the capacity of the network to cope with fouling of heat exchanger units are usually encountered. Moreover, opportunities for retrofit improvements and their impact in the existing network are very difficult to evaluate. By these reasons, the operating engineer is faced with the challenge of adjusting the given design to the current operations. Because of the lack of tools to assess the effects that these changes have in the design, keeping the optimal conditions becomes a trial-anderror procedure. As a result the optimal operation of the network is compromised, the savings in energy consumption are reduced, and additional capital investment often is required. The need for a tool that allows the performance monitoring of the heat exchanger network is evident. In this paper, an automatic approach to study the behaviour of heat exchanger network designs when the operating conditions vary is proposed. Its purpose is to facilitate the involvement of the engineer in the daily operation and maintenance of the network. Results showing the impact in the design are automatically provided when changes in
746 the operating variables are observed. The tool is therefore available for online optimisation and the simultaneous evaluation of retrofit opportunities. Fouling of heat exchanger units (that produces a reduction in the area of transfer) and the evaluation of its impact in the whole network are easily conducted. An example of the preheat train of a crude distillation column shows the different aspects and advantages of the approach are presented.
2. The Heat Exchanger Network Calculator Approach The basic equations for a single shell and tube heat exchanger unit that consider process and utility stream segmentation are:
Q^=m,,{Cp':u;;;~cpTT:r^c:,)
(1)
Q,^m^,{CpTTr-Cp^:^T:^^C:,)
(2) (3)
Q,=F,{UA),^T,^
where C]^ IC*^ is a constant that accounts for the segmentation of the temperature vs. enthalpy curve of the corresponding hot/cold process stream in order to approximate the specific heat and consider a linear dependency of the enthalpy with temperature within the segments. In equation (3), F^, refers to the Ft correction factor for the tube and shell exchanger unit / to account for noncountercurrent flow. The relations used in calculating this factor that requires values for all four extreme temperatures are summarised in Shenoy (1995). These temperatures also are used in calculating the logarithmic mean temperature difference M^i^. Once the mass balances are solved for the entire heat exchanger network, the set of variables for the single unit is comprise by the heat load Qi, the heat exchanger area A, and the four extreme temperatures. 2.1 Sequential vs. Simultaneous approach Consider the simplest possible heat exchanger network shown in Figure 1. It comprises two heat exchanger forming a loop between the same hot and cold process streams. A set of equations (1) to (3) is written for each unit and two additional equations assure that the outlet temperatures of the first unit are the same as the inlet temperatures of the
uncc
TQCC
Figure 1. The Simplest Possible Heat Exchanger Network second unit. The heat balances form a system of eight equations with twelve unknowns.
747 Four variables should be specified based on the degrees of freedom available. An iterative sequential approach that considers a single unit at a time can solve the system of equations provided that the two heat exchanger areas are not specified simultaneously. For the purpose of monitoring the performance of an existing heat exchanger network, it is required that the areas are kept constant. This is not possible with a sequential approach. Solving the whole set of heat balances simultaneously for the entire heat exchanger network has the advantage of allowing the specification of the both heat exchanger areas. The heat exchanger network calculator approach presented in this paper solves the entire system of equations for the heat exchanger network when this system is completely defined. In the case of partially defined systems, the approach determines and solves the largest completely define system. To assure global optimality for the solution, the nonlinear equation that has to be solved when area is specified is linearised as explained in the next section. 2.2 Linear Approximation when the Area Is Specified When the area of a heat exchanger unit is specified, it becomes fixed and the system of equations (1) to (3) can be solved by iteration if the value of two of the extreme temperatures or one of these temperatures and the duty are know. The system is, however, nonlinear and difficult to solve. The partition of the entire area of the unit into equal area portions is proposed. The mean logarithmic temperature difference within each area section is then approximated by the corresponding linear arithmetic mean temperature difference. 2.3 Calculation of Utility Mass Flowrates In the particular case of utility streams, loads are calculated using the Grand Composite Curve (GCC) based allocation method. For a single utility exchanger extreme temperatures are fixed. Therefore, the utility mass flowrate is calculated using equation (1) or (2) depending on utility side. In the case of multiple heat exchangers for a utility an iterative procedure is employed to calculate simultaneously the intermediate temperatures and the utility mass flowrate.
3. Heat Exchanger Fouling Analysis In order to consider heat exchanger fouling, specification of the fouling factor for each particular unit is allowed. The new heat transfer coefficient for the heat exchanger is calculated by adding the additional resistance due to fouling. The capability of the Heat Exchanger Network Calculator approach to solve the system simultaneously makes possible the consideration of the different operating conditions through which an existing heat exchanger network undergoes when the fouling increases. Calculation of fouling factors based on measured extreme temperatures also is possible by using the Heat Exchanger Network Calculator approach. An step further is the analysis of alternative cleaning schedules and their influence in the total annual cost applying techniques existing in the literature (O'Donnell et al, 2001). Moreover, designs that avoid fouling or consider fouling mitigation can be investigated (Policy et al., 2000). All
748 these new features are currently under development, and they will be part of future implementations.
4. Example In this example, the preheat train of an atmospheric crude distillation column is considered. Process and utility stream data is presented in Table 1. The low-temperature crude stream is preheated prior to enter the pre-flash drum. The liquid from the pre-flash (high-temperature crude) is preheated before entering the furnace and mixed with the vapour stream coming from the pre-flash. The resulting stream enters the column that produces four products (naphtha, kerosene, diesel, and AGO) and bottoms residue. The condenser stream, waste H20, and three pumparounds are also available for integration. The heat exchanger network corresponding to the preheat train of the atmospheric crude distillation column is shown in the grid diagram of Figure 2. The low temperature crude stream is heated up in five heat exchanger units prior to enter the pre-flash. The last unit uses high-pressure steam in order to reach the pre-flash temperature. The liquid exit Table 1. Stream and Utility Data for Example Stream Low Temperature Crude
Segment
i
2 3 High Temperature Crude Naphtha Waste H20 Condenser
Pumparound 1 Kerosene Diesel Pumparound 2 AGO Pumparound 3 Residue Cooling Water LP Steam Generation MP Steam Generation HP Steam Fired Heat
1 2 3 4 1 2 1 2 1 2
1 2
1 2
-
Tin (°C) 30.0 108.1 211.3 232.2 73.2 73.2 146.7 133.3 120.0 99.9 167.1 116.1 231.8 176.0 248.0 147.3 263.5 297.4 203.2 319.4 347.3 202.7 20.00 124.0 174.0 250.0 1000
Tout (°C) 108.1 211.3 232.2 343.3 40.0 30.0 133.3 120.0 99.9 73.2 116.1 69.6 176.0 120.0 147.3 50.0 180.2 203.2 110.0 244.1 202.7 75.0 25.00 125.0 175.0 249.0 400.0
MCP(kW/°C) 333.6 381.3 481.5 488.2 57.69 6.842 233.9 202.2 170.1 338.5 172.4 157.5 50.73 46.31 67.75 58.44 123.1 21.99 19.44 136.2 217.1 179.6 10800 10258 10258 33177 63.00
749
MP StetmGtacntiM
Figure 2. Crude Distillation Column Preheat Train of the pre-flash is preheated in two units prior to enter the furnace. A series of coolers are used to reach the target temperatures for the products. Medium and low-pressure steam are generated using the two top pumparound streams. Notice the presence of loops in the network. With all process heat exchanger areas specified, a sequential iterative approach fails to calculate the entire network. By using the simultaneous Heat Exchanger Network Calculator approach, solution of the entire system of heat balance equations is possible. Process to process heat exchanger data is presented in Table 2. In order to consider fouling of the process heat exchangers during the period of a year, a series of designs is obtained in which the fouling factor is specified. A first order degradation in the clean overall heat exchanger heat transfer coefficient (Uo) is used (O'Donnell et al., 2001) with k = 0.35 year'^ Therefore, the fouling factor (R^) is calculated by the equation:
«.=f(^'-')
(4)
Table 3 shows the results obtained considering a design for each month starting from the clean-based design. As expected, all utility loads increase due to the degradation of the process to process heat exchanger performance. An increase in the Residue target temperature is also observed. Finally, the furnace inlet temperature degrades at a rate of Table 2. Process Heat Exchanger Data for Example Exchanger EI E2 E3 E4 E5 E6
Load (MW) 15.22 0.67 0.12 0.73 39.11 0.18
Area (m^) 685.7 20.03 82.25 21.45 2029 39.98
Shells 2 1 1 1 6 1
HTC (kJ/h.m^°C) 1505 1604 1765 1566 1098 1268
FtPactor 0.8858 0.9993 0.9718 0.9983 0.8037 0.9948
750 Table 3. Designs Considering Increasing Fouling Factor Tr (°C) 75.0
Time (months) 0
Furnace Load (MW) 37.84
HP Steam Lx)ad (MW)
CWl Load(kW)
CW2 Load (kW)
33Ti
018
175
1
38.03
33.20
0.23
4.77
75.8
2
38.21
33.30
0.27
4.78
77.1
3
38.41
33.40
0.32
4.80
78.4
4
38.60
33.52
0.36
4.81
79.8
5
38.80
33.64
0.41
4.83
81.2
6
39.00
33.77
0.46
4.84
82.7
7
39.20
33.90
0.51
4.86
84.2
8
39.40
33.95
0.56
4.87
85.2
9
39.60
34.11
0.60
4.88
86.9
10
39.81
34.26
0.65
4.89
88.6
11
40.02
34.45
0.69
4.91
90.5
34.63
0.74
4.92
92.3
12
40.22
near 5°C per year due to fouling. This represents an increase in furnace duty of 2.4 MW or 6% of the total heat load. Considering a crude oil priced at 21$/barrel, the cost of heat exchanger fouling would be of around $250,000 per year.
5. Conclusions An approach to simultaneously calculate a given heat exchanger network has been developed. The Heat Exchanger Network Calculator approach solves the entire set of equations for the network and allows area specification. The system is linearise by partitioning the total area in portions where the logarithmic mean temperature difference can be approximated by the corresponding arithmetic mean. Fouling factors are included to consider the degradation in the heat transfer capacity of the heat exchanger units. The approach is the base for the development of a performance-monitoring tool. Acknowledgements Dr. H. Rodera and Dr. H.K. Shethna thank Hyprotech Ltd. for allowing the use of HXNet software to produce the results of this paper. References HX-Net, Development Version, Hyprotech Ltd, Calgary, Alberta, Canada. O'Donnell, B.R., B.A. Barna and CD. Gosling, 2001, Optimize Heat Exchanger Cleaning Schedules, Chem. Eng. Prog. Vol. 97, No. 11, 56. Policy, G.T., D.L Wilson and S.J. Pugh, 2000, Designing Crude Oil Pre-Heat Trains with Fouling Mitigation, AIChE Spring Meeting, Atlanta. Shenoy, U.V., 1995, Heat Exchanger Network Synthesis. Gulf Publishing Company, Houston, Texas.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
751
Scheduling of Continuous Processes Using Constraint-Based Search: An Application to Branch and Bound L. C. A. Rodrigues, R. Carnieri, F. Neves Jr. Centro Federal de Educa^ao Tecnologica do Parana (CEFET-PR) Programa de Pos Gradua^ao em Engenharia Eletrica e Informatica Industrial (CPGEI) Rua Sete de Setembro, 3165, 80230-901, Curitiba PR, Brasil (Icar, carnieri, neves)@cpgei.cefetpr.br
Abstract In this work a Branch and Bound approach based on constraint-based search (CBS) is proposed to the scheduling of continuous processes. The purpose of this work is to extend to continuous processes a CBS approach proposed previously to batch production (Rodrigues et al, 2000). Tasks' time-windows are submitted to a constraint propagation procedure (CBS) that identifies existing orderings among tasks. Linear programming is used to determine the optimal flow rate for each bucket whenever all buckets are ordered in the branch and bound.
1. Introduction Several MILP procedures have been recently proposed to the scheduling of continuous processes (Pinto et al, 2000; Pinto and Moro, 2000; Stebel,2001; Magatao, 2001). But these authors are unanimous about the difficulties in solving real world problems due to its dimension. In this work a Branch and Bound approach based on CBS is proposed to the scheduling of continuous processes with the minimization of earliness. Production is identified in terms of buckets of tasks. The size of product buckets is made equal to the highest common factor of the storage capacity among available tanks. Therefore it takes an integer number of buckets to fill a tank. This assumption is necessary if the scheduling is posed as an ordering of tasks, as when CBS is used. As in previous works (Rodrigues et a/., 2000; Gimeno et ai, 2000; Rodrigues, 2000), the proposed approach uses a time-window to impose a time interval for the execution of each bucket. This approach has two phases: planning and scheduling. Time-windows are initially determined during planning based on the availability of raw materials and on the demand of final products. These time-windows are submitted to a constraint propagation procedure that identifies existing orderings among buckets of different tasks. Constraint propagation also enables the identification of disjunctions (undefined orderings) among buckets of different tasks. During scheduling the branch and bound procedure is used to set an ordering to these disjunctions. When all the orderings are set during branch and bound, linear programming is used to determine the flow rates of the continuous processes.
752
2. Problem deHnition A problem proposed by Pinto and Moro (2000) is used to test the proposed branch and bound approach. The proposed problem is the optimization of production and inventory management of a liquefied petroleum gas (LPG) plant. The STN representation (Kondili et a/., 1993) of this plant is presented in figure 1. A petroleum refinery yields a mixture of propane and butane (C3C4) to the LPG plant. C3C4 is sent to the depropanizer where two different tasks may be performed: /) production of separate yields of propane (C3) and butane (C4); //) production of separate yields of propane for petrochemical purposes, also known as intermediate propane or propint (C3i), and butane (C4). An amount of C3C4 is bypassed whenever propint is produced. In this problem, any mix of C3 and C4 may be delivered to consumers as LPG, provided that the amount of C3 must be greater or equal to that of C4. It is also possible to deliver C3 as LPG. C4 may also be marketed as bottled gas or used to produce methyl-tert-butyl-ether (MTBE). The production of MTBE also results in the production of rajfinate (Raff) as a byproduct. Raffinate is also used to the production of LPG. The storage farm comprises 8 spheres suitable for LPG, C3, and C3i. There are also 4 spheres suitable for C4 and raffinate. All buckets in this problem have the size of the storage sphere FEED
I
C3
Raff
ny "^ ^^ 2 0 h . | 0.5 ^ -^M. Ref Pron Umtbe n s Wm ^ /^ 6h. (My X ns 1f y X ^ ^ ^ p;sLPG Pint J W 0 5 1r ^ C3C4 MTBE Xo? s 6h^
6h.
Y
r
C3i
Table 1. Assignment and storage. Task
Units
State
Storage
Ref Prop
Prod Dep
Pint
Dep
Umtbe
Umt
Cons
Pipe
C3C4 C3 C4 C3i C4 LPG MTBE Raff Clients
ZW FIS(LPG) FIS(C4) FIS(LPG) FIS(C4) FIS(LPG) UIS FIS(C4) UIS
Con
I
Clients Fig. 1. STN representation
3. Production planning The amount of buckets of each product that will be produced is defined from final product demands. Latest finishing times (LET) of the time-windows are obtained from a backward propagation of final products demands and its due dates. Earliest beginning times (EBT) are obtained from a forward propagation of raw materials availability. Time-windows are calculated based on the optimistic assumption that production will be accomplished with maximum flow rates. This is done because any time-window obtained from smaller flow rates fits within the time intervals defined by the optimistic time-windows. These initial time-windows are then submitted to constraint propagation that identifies exisfing orderings among buckets of tasks (Caseau and Laburthe, 1995; Bapfiste and Le Pape, 1995). The problem becomes infeasible if these orderings are not respected. Therefore planning is in fact a pruning procedure that reduces the dimension
753 of scheduling. The identification of orderings may impose time-windows reductions. A time-window is reduced during constraint propagation if it is identified that it is infeasible to process a task during a certain interval of time that belongs to the timewindow of this task.
4. Constraint-based search Constraint propagation takes into account equipment contention, storage restrictions, and pegging considerations due to mass balance. Analysis of equipment contention is performed by two procedures known as explosion and edge-finding (Caseau and Laburthe, 1995; Baptiste and Le Pape, 1995). Explosion and edge-finding are based on the analysis of time-windows feasibility. These two procedures are responsible for the identification of existing orderings between buckets of tasks that are processed in the same processing unit. Mass balance propagation is performed whenever a time-window is reduced. An example of pegging is that if the EBT of a bucket a is postponed, the EBT of the buckets that depend of a to be produced will also have its EBT postponed. Storage restrictions (as ZW, NIS, and PIS policies)* are also considered during constraint propagation (Rodrigues, 2000). ZW and NIS restrictions impose timewindows reductions in a similar way. Consider a task a that produces a product consumed by task j3, and that this product has ZW storage restrictions. Any timewindow reduction in task a will result in a similar change in task /3. If storage demand is greater than the storage capacity, PIS restriction may impose orderings or result in disjunctions between tasks that share the same set of tanks. When scheduling is performed it is necessary to set an ordering for all disjunctions (imposed by processing and storage units) identified during planning. Therefore equipment contention and PIS storage must not be neglected, especially in continuous processes where PIS storage may be the main restriction to production.
5. Branch and bound The branch and bound procedure is based on the definition of orderings to the disjunctions (among pairs of buckets) identified during planning. The proposed branch and bound uses constraint propagation to identify feasible orderings among buckets with disjunctions. New orderings may be identified by constraint propagation whenever an ordering is imposed during branch and bound execution. When an ordering is defined to all planning phase disjunctions at a solution node of the branch and bound, it can be stated that the solution is feasible. But the optimal flow rates must be determined since the optimistic assumption of maximum flow rates only leads to the lower bound of the analyzed solution. Linear programming is used to determine the optimal flow rate for each bucket of this solution. The branch and bound procedure minimizing earliness that has been implemented is presented bellow.
ZW means "zero wait"; NIS means "no intermediate storage"; PIS means "finite intermediate storage".
754 Branch and bound procedure minimizing earliness: /.
//. Hi.
iv. V.
vi.
vii. via.
Create a search tree G consisting solely of the start node s. Put 5 on a list called OPEN (indicating the nodes to be expanded). Set LOWERBOUND (the lower bound of the search procedure) to infinite (+ ©o). The search tree G contains all the nodes identified during branch and bound. Node s is the output of the planning phase. If OPEN is empty, end of search. In this case, if LOWERBOUND value is infinite (+ oo), there is no feasible solution. Otherwise, optimal solution has been identified with an earliness value equal to LOWERBOUND. Select the first node on OPEN and remove it from this list. Call this node n. Expand node n, generating the set of its successors. Each expansion imposes an ordering to a pair of disjunctive buckets. Therefore each expansion generates two successor nodes. Constraint propagation is performed to all expanded nodes. Include successor nodes are that feasible but aren't solution nodes (where an ordering has been set to all planning disjunctions) in OPEN. Use linear programming to define start time, processing time, and flow rates to all buckets of continuous processes to each successor node that is a feasible solution node. If there is a solution node with earliness value smaller than LOWERBOUND, set this earliness value as the new value of LOWERBOUND. Eliminate from OPEN all nodes with an earliness bound value greater than LOWERBOUND. Latest finishing times (LPT) of the time-windows impose an earliness bound to every node of the search tree. In a minimization problem, the monotone condition of the branch and bound search guaranties that a successor node always has an earliness bound value greater or equal to its ancestors earliness bound. Reorder OPEN according to the earliness bound value of its nodes. The smallest earliness bound is placed first in the list. Return to step //.
The linear programming (LP) model used within the proposed branch and bound is presented bellow. Equation 1 presents the objective function, minimizing earliness. DD,;^ represents the due date of bucket b of task /. F//, is a variable representing the end of processing of bucket b of task /. B, identifies the set of buckets of task / that are produced. Earliness = 5 ^ 5 ^ /
(DD,,-F,,)
(1)
be B,
Although not presented in this paper, the variables representing the start and end of processing, S,, ^ and F, /,, are constrained by its time-windows. The most important restrictions in the LP model are those concerning precedence relations (equations 2 and 3). Equation 2 is used whenever there is a ZW restriction among two buckets. This equafion is used for tasks linked by states with ZW storage and also to avoid processing gaps, assuring continuous processing among buckets of the same task. Equation 2 is generated if bucket b of task / is preceded by another task with zero wait connection (ZW, IJ = True). The task that precedes bucket b of task / with zero wait connection is identified by i_zwij, and its bucket is identified by b_zwib. Equation 3 only imposes that when there is a precedence relation among tasks (Prec,, b = True) the processing end of
755 bucket b_pru, of task i_priij is smaller or equal to the processing start of bucket b of task /. There are no flow rate variables in this LP model. Since buckets have a fixed size, flow rates are indirectly obtained from its processing times (TP/^), as shown in equation 4. Maximal processing times are taken from minimal flow rates and minimal processing times are obtained from maximal flow rates. S/, /; = F/(LzH<J, b), b_zw{i,
b))
V/,^G B,/ZW,-/, = True
S/, t ^ ^(i_prii, b), b_pr{i, b)) TPmin, < TP,, ^ < TPmax,
\f i, b E Bi / Prcc,, b = True
(2) (3) (4)
V /, Z? 6 B,
6. Results Part of the proposed approach had been implemented previously to batch production (Rodrigues 2000). The most important differences between the two implementations are in the branch and bound, that had to be included. The constraint propagation mechanism was only changed to perform a deeper analysis of storage constraints. The LP procedure was implemented using LINGO (1999). A DLL (Dynamic Link Library) was developed to allow the solution of the LP within Visual Basic. Figure 2 is the result of preliminary tests with the proposed approach. It presents the time windows obtained from planning. : P
SdWMtulmq M»tJ»
Midv/Snow
Fig 2. Time windows.
7. Conclusions In this work we present an approach to the scheduling of continuous processes. The branch and bound that is proposed relies heavily on the pruning capacity of constraintbased search techniques. We claim that the definition of an ordering among tasks may reduce the dimension of the problem when compared to the existing MILP approaches. Since MILP procedures involve the definition of orderings and allocation of tasks at the same time. We believe that the smaller amount of decisions to be made in the branch and bound will allow us to solve problems of higher complexity than those solved by MILP.
756
8. Acknowledgements The authors acknowledge the financial support of the Brazilian National Petroleum Agency (PRH-ANP/10 CEFET-PR) and of CNPq (Kit Enxoval Recem-Doutor).
9. References Baptiste P. and Le Pape C , 1995, A Theoretical and Experimental Comparison of Constraint Propagation Techniques for Disjunctive Scheduling, Proceedings of 14th International Joint Conference on Artificial Intelligence, Montreal, Canada. Caseau Y. And Laburthe P., 1995, Improving Branch and Bound for Job Shop Scheduling with Constraint Propagation, Proceedings of the 8* FrancoJapanese 4* Franco-Chinese Conference. Gimeno, L., Rodrigues, M.T.M., Rodrigues, L.A., 2000, Constraint Propagation Tools in Multipurpose Batch Plants Short Term Planning, Proceedings of 2"^ Conference on Management and Control and Production and Logistics, Grenoble, France. Kondili, E., Pantelides, C.C, Sargent, R.W.H., 1993, A General Algorithm for Short Term Scheduling of Batch Operations - I. MILP Formulation, Computers and Chemical Engineering, 17, Pp. 211-227. Elsevier, Amsterdam. UNDO Systems Inc., 1999, LINGO: The Modeling Language and Optimizer - User's Guide, Chicago, USA. Magatao, L., 2001, Mathematical Programming Applied to the Optimization of Pipeline Operation, M.Sc. thesis, CPGEI, CEFET-PR, Brazil (in portuguese). Pinto, J. M., Joly, M., Moro, L. F. L., 2000, Planning and Scheduling Models for Refinery Operations, Computers and Chemical Engineering, 24, Pp. 22592276. Elsevier, Amsterdam. Pinto, J.M., Moro, L.F.L., 2000, A Mixed Integer Model for LPG Scheduling, In: European Symposium on Computer Aided Process Engineering-10, Comp. Aided Chemical Engineering (8) (S. Pierucci(Ed)), 1141-1146, Elsevier, Amsterdam. Rodrigues, L.C.A., 2000, Planning and Scheduling of Multipurpose Batch Plants: A Decomposition Strategy Using Time-Windows, Phd Thesis, UNICAMP, Campinas, Brazil (in portuguese). Rodrigues, L.C.A, Graells, M., Canton, J., Gimeno, L., Rodrigues, M.T.M., Espufia, A., Puigjaner, L., 2000, Utilization of Processing Time Windows to Enhance Planning and Scheduling in Short Term Multipurpose Batch Plants, Computers and Chemical Engineering, 24, pp. 353-359. Elsevier, Amsterdam. Stebel, S.L., 2001, Modeling of Liquefied Petroleum Gas Storage and Distribution in an Oil Refinery, M.Sc. thesis, CPGEI, CEFET-PR, Brazil (in portuguese).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
757
Real Time Batch Process Optimization within the environment of the Flexible Recipe. Javier Romero, Antonio Espuna and Luis Puigjaner Chemical Engineering Department, U. P. C. E.T.S.E.I.B., Diagonal 647, E-08028 Barcelona, Spain. e-mails: [romero, aec, lpc]@eq.upc.es
Abstract Batch processes are normally thought to operate using a traditional fixed recipe. However, the fixed recipe has to be approximately adapted in a rather unsystematic way depending on experience and intuition of operators. Therefore, the concept of flexible recipe seems to be the adequate way to rationalize and systematize the adjustment procedure. In the flexible recipe context, the term recipe is used in a more abstract way by referring to a selected set of adjustable recipe items that control the process output. In the present work, a framework for batch process real time optimisation considering the flexible recipe concept is presented. As soon as a deviation is detected, the control recipe can be readjusted. Deviations are only detected at the process state assessment. The process state assessment compares the operating conditions at the sample time with the expected operating conditions that will depend on the initial conditions. For this reason and in order to perform this comparison, a predictive model is necessary. Then, a corrective model is required to adjust the control recipe. This second model describes the ultimate effect of the values measured at the time of the process state assessment as well as any run-time corrections applied during the remainder of the processing time.
1. Introduction The traditional fixed recipe does not allow for adjustment to plant resource availability or to variations in both quality of raw materials and the actual process conditions. However, the industrial process is often subjected to various disturbances. For this reason, the fixed recipe is in fact approximately adapted but in a rather unsystematic way. Therefore, in order to systematize these adaptations the concept of flexible recipe might be appropriate, as the flexible recipe is systematically adapted at the moment (Rijnsdorp, 1991). According to the standard ISA-SP88, the flexible recipe might be derived from a master recipe and subsequently used for generating and updating a control recipe. Verwater et al (1995) introduced the concept of different levels between these two stages defined at ISA-SP88. With these levels, a better description of the different possible functionality of the flexible recipe is obtained. These levels are: a) Master control recipe: a master recipe valid for a number of batches, but adjusted to the actual conditions, from which the individual control recipes per batch are derived.
758 b) Initialized control recipe: the adjustment of the still-adjustable process conditions of a master control recipe to the actual process conditions at the beginning of the batch. c) Corrected control recipe: the result of adjusting the initialized control recipe to process deviations during the batch. d) Finally, for monitoring and archiving purposes, the accomplished control recipe. Figure 1, describes the interaction of these four flexible recipes in a real-plant environment. Production and operations planning (Market reauirements)
Master Recipe improvement & nxxiel development
Switching program regulatory control: Corrected Control Recipes
Switching program regulatory control: Initialised Control Recipe
Actual process conditions
I
Operational database inanagement of Accomplished control recipes
Figure 7. Real time flow data among different possible flexible recipes. These models may be developed in laboratory experiments, during normal production by a systematic introduction of small changes in certain inputs, or by adjusting white models and simulating them under different conditions. In this work, a new framework for the recipe correction is introduced. The aim of this approach is to optimize the entire batch process in front of disturbances. It is proposed a framework for generating the corrected control recipe from the initialized (master) control recipe. In Romero et al. (2001) is presented a framework for recipe initialization based on the same basic framework presented here.
759
2. Recipe correction interacting witii a Rescheduling tool. The recipe initialization is performed at the beginning of the batch phase only taking into account known initial deviations. But other run-time deviations may arise. However, under certain circumstances, it is possible to compensate the effects of these unknown disturbances during the batch run, provided that continuous or discrete measurements are available. 2.1 Flexible Recipe model for recipe correction. Within the flexible recipe context, the term recipe is used in a more abstract way by referring to a selected set of adjustable items that control the process output. The Flexible Recipe Model is the relationship that correlates a batch process output as a function of the selected input items of the recipe. This model is regarded as a set of constraints on quality requirements and on production cost (Romero et aU 2001). Recipe items are classified into; i) The vector of process operation conditions, poc„ of stage / of a recipe. It includes items like temperature, pressure, type of catalyst, etc. ii) The product specification vector, psi, at the end of each process stage / of a recipe. It may consider items like conversion of a reactant, quality aspects, etc. iii) Processing time, TOP,, at each stage / of a recipe, iv) And, waiting time, 7W„ the time between a stage finish and the next stage start time. From this, the vector of product variables of one batch stage may depend, according to a function • , on the processing time, on waiting time, on process set-points, and on product specifications at different stages /*, where the different inputs to stage / are produced. Then, within this model, product specifications, ps, and process operation condition, poc, are subject to be corrected within a flexibility region, • and • respectively. While a batch process takes place, different on-line continuous process variables and discrete variables sampled at different moments are taken (see Figure 1). From this information, di process state assessment is performed. From this assessment, information about how the batch-process is being carried out is given to Model III (Flexible recipe model for recipe correction). The time at which process state assessment is performed, and so, at which Model III takes different actions might be different from the moment at which a deviation is detected. Finally, Model III will interact or will have integrated some scheduling algorithms. In order to perform the process state assessment comparison a prediction model is necessary. In order to adjust the control recipe, a correction model is necessary. Finally, in order to adjust the actual schedule, a rescheduling strategy may be necessary. Prediction Model This model, Eq.l, estimates the continuous and discrete sampled product variables in function of the actual control recipe, established in Model II, at the sample moment. rTPA; ^ 1 f™' (1) poc ,dt TPA ; , p s , . , TPA TPA ^ J where TPA\ is the wth moment at which stage / of a batch is assessed, pvs'^v is the expected vector of product specifications at TP/Ci moment.
pvs r = "I"
pred
f
\
760 Process State Assessment The Process State Assessment basically consists on the evaluation of the batch-process run. This assessment consists on comparing the predicted product specification /, pvs^^i, (expected by Model II) with the real variable observed at the \\;th process statement, ps^^i. If this comparison is greater than a fixed permitted error, • , some actions will be taken in order to compensate this effect. Correction Model Describes the ultimate effect of the values measured at the time of the process state assessment as well as any run-time corrections applied during the remainder of the processing time. This model adjusts process operation conditions to set the values, established at the recipe initialization (Model II), of product specifications. This model adjusts these values in function of the processing time, waiting time, product variables at the beginning of the batch stage and of the deviations detected at the wth process assessment moment, Eq.2. There must be one correction model for each process state assessment moment. 1
T0P,-TPA7
rTOPi
JTPA:
f
poc.^r =H^7"'M
1
rTPA^
TOP,,7W,,ps,,ps..pvsr,
poc^dt (2)
Rescheduling strategy The output of the flexible recipe model for recipe correction might give variations in processing fime or resource consumption, which would make the existing schedule suboptimal or even infeasible. Therefore, a rescheduling strategy will have to be used. There are two basic alternatives to update a schedule when it becomes obsolete: generating a new schedule or altering the initial schedule to adapt it to the new conditions. The first alternative might in principle be better for maintaining optimal solutions, but these solutions are rarely achievable in practice and require prohibitive computation times. Here, it is proposed a retiming strategy to be integrated into the flexible recipe correction framework. At each deviation detected, optimization will be necessary to find the best corrected control recipe of the process. From this, it is proposed to solve the LP in Eq.3 to adjust the plant schedule to each recipe correction. When dealing with multipurpose plants, this strategy might not be able to make feasible some infeasible schedules. If this would happen, further actions should be taken. However, this aspect is out of the scope of this work. mm(Performance _ Criterion) subject to, TI. > 0 Vi TR =TL-{-TOP-^TW:
Vi
TIj =TF, Vi,i73sG{S, n S , }
(3)
TWj < TW^^ Vi and subject to the correction flexible recipe model constraints where TI, and TF, are the initial and ending time of each stage / of the batch. 2.2 Batch correction procedure Within each batch-run, the algorithm of Figure 2 will be applied. This algorithm first predicts the expected deviations in process variables from the nominal values. Then,
761 verifies if there exist significant discrepancies between the observed variables and the predicted. If so, fixes process variables of all batch-stages already performed and of the batch-stages that are currently being performed and are not the actual batch stage being assessed and re-optimizes the actual recipe taking into account the effect on the schedule timing.
Solve: 'Flexible Recijje Correction Model
&
'Schedule Timing'
Figure 3. Batch correction procedure algorithm.
3. Case Study To evaluate the potential use of this framework, a case study proposed by Cott and Macchietto (1989) is discussed. The plant produces three products (D, E and F). The production recipes of these four products are presented in Figure 3. The batches were scheduled into the plant using a minimum makespan scheduler with the nominal processing times and zero waiting time poUcy. The resuhing proportion of batches scheduled were 6 batches of product E, 2 of D and 3 of F. During actual operation, the processing times of all phases for all products were assumed to be subject to independent and normally distributed variations of a mean equal to zero and a standard deviation equal to 10 % of the nominal processing time. In Figure 3 is presented the % equipment utilization for the nominal schedule and the resultant equipment utilization after introducing disturbances. It can be seen how
762 disturbances introduce a waiting time in equipment unit 4 of 1.56 h., when readjusting the schedule with the retiming algorithm POMA (Projected Operation Modification Algorithm) proposed by Cott and Macchietto, 1989. When applying the batch correction procedure proposed, just without considering the possible flexible recipe, this waiting time is already reduced to 0.42 h. The nominal makespan of the case study is of 143 h which is increased to 145.6 h when applying the POMA but only to 145.1 h when applying the optimization procedure of the proposed batch correction procedure to react in front of the disturbances. In order to consider flexibility in the recipe, black-box-linear models are assumed to be available for the different products. Process state assessments are performed only at the end of each recipe stage. Therefore, the prediction model itself is not necessary, as the expected product specifications at each process state assessment (pvs\ ) are the ones given at the initialized control recipe. The correction model will be as, dpoCi^i=di 5ps,>i - Ci 5pvs"-fl, 5 T 0 P , - b^ 5TW, where 5 means the deviation from the nominal and expected values. It has been assumed an ideal process where all the parameters of the flexible recipe model are equal to 1. Finally, it has been assumed a performance criterion given different weights to production makespan, set-point modification and waiting time. With this, it is shown in Figure 3 how waiting time is removed from equipment unit 4 as well as production makespan is reduced. Notice that this improvement will probably imply a higher production cost due to some required set-point modifications. Prod. Equip. D
E
F
1 4 5 2 3 4 5 1 2 3 4 5
TOP (h) 3 5 10 8 4 3 10 6 1 6 6 13
n° lots 3
6
Equipment unit 1 2 3 4 5
Nominal Recipe 27 51 42 51 135
1 2 3 4 5 Makespan
0 0 0 0 0
3
1
(h)
143
% Equipment utilization No Recipe POMA Flexibility 27.5 27.5 51.0 51.0 42.2 42.2 51.6 51.6 134.3 134.3 Waiting time (h) 0 0 0 0 0 0 0.42 1.56 0 0 145.6
145.1
Flexible Recipe 27.5
51.0
1
42.2 51.6 133.8 0 0 0 0 0 144.8
Figure 3. Case study results. The proposed batch correction procedure has been solved using Lingo^^.
4. References Rijnsdorp, 1991, Integrated Process Control and Automafion, Elsevier, Amsterdam. Romero et aL, 2001, A new framework for batch process optimization using the flexible recipe. Ind. Engng. Chem. Res. Submitted. Z. Verwater and K.J. Keesman, 1995, Computer-aided development of flexible batch production recipes. Production Planning and Control, 6, 320-330. Cott and Macchietto, 1991, Minimizing the effects of batch process variability using online schedule modification. Comp. Chem. Engng, 13, 105-113.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
763
Smart Enterprise for Pulp & Paper Mills: Data Processing and Reconciliation M. Sanchez\ D. Leung^, H. Konecsny^, C. Bigaran^, J.A. Romagnoli^* ^PLAPIQUI (UNS-CONICET), C. C. 717, 8000 - Bahia Blanca, ARGENTINA ^ Visy Pulp & Paper Pty. Ltd. Tumut, NSW 2720, AUSTRALIA Chemical Engineering Department, University of Sydney, Sydney, NSW, 2006
Abstract An ad-hoc data reconciliation procedure developed for the recausticizing section of a new pulp and paper industry is presented in this work. A comprehensive model was formulated to take into account different unit operation modes. It was also extended to incorporate specific knowledge of some pieces of equipment to increase redundancy, and consequently enhance estimate precision and gross error detectability.
1. Introduction Visy Industries, one of a few major players in the Australian paper business, has recently started up a state of the art modern pulp and paper making facility. The mill produces 240 000 tonnes of Kraft pulp and brown packaging paper each year; 800 000 tonnes per annum of raw materials is sourced from local softwood plantations, that are supplemented by 50000 tonnes per annum of waste paper from domestic and commercial sources. Smart Enterprise refers to a division at Visy Pulp and Paper Tumut that encompasses several areas of responsibility relating to data reconciliation, process modelling and simulation, process control and optimisation. Smart Enterprise's goal is to improve plant performance by allowing operators to make more informed decisions in a shorter amount of time. With this objective in mind, data processing and reconciliation arises as a key component of Smart Enterprise, as raw data are processed by this technique to produce a consistent set of data that constitutes a reliable input for other procedures. Furthermore, it will also allow the detection of faulty sensors, bias in measurements or other anomalies in the process operating data. A comprehensive analysis of data reconciliation strategies can be found in Romagnoli and Sanchez ( 1999). In this work we present the distinctive features of the data reconciliation procedure especially developed for the recausticizing section of the plant. To our knowledge, it is the first strategy reported for a pulp and paper industry. The methodology succeeds in considering changes in operating conditions associated with global plant production strategies or simply with equipment changes for maintenance cleanings. Furthermore extra knowledge is incorporated in the process model, by using pump characteristic curves, to overcome the lack of redundancy of the system. The procedure is completed To whom correspondence should be addressed
764 with a gross error detection technique based on the individual residuals test and a graphical interface in which input may be entered and results may be displayed. The graphical interface was generated using Excel with the data imported/exported directly between the two programs.
2. Procedure development Data Reconciliation is the process of adjusting or reconciling the process measurements to obtain more accurate estimates of flow rates, temperatures, compositions, etc... that are consistent with the material and energy balances. The classical procedure is based on the minimisation of the sum of the weighted difference between measured and estimated variables subject to a set of process constraints. It is valid under the assumption that no gross errors are present in the set of measurements. If it is not the case, gross error identification strategies are initially applied to determine and eliminate the set of suspect measurements. Data Reconciliation strategies are supported by the location of a set of sensors in the plant, that allows the existence of redundant measurements. To know what type of information is available for a given process and set of instruments, an instrumentation analysis is performed. It allows classifying variables according to their feasibility of calculation. Measurements can be classified into redundant and non-redundant variables. The redundant ones are those whose value can be computed from the mathematical model that represents the plant and other measured variables. In turn, the unmeasured variables are called observable when they can be evaluated from the available measurements using model equations. Otherwise they are unobservable variables. A data reconciliation procedure has been developed, following the classical approach, for the recausticizing area of the mill. This section plays two important roles in the production of white liquor. Firstly, it removes process impurities from the system in the green liquor filter. Secondly, the causticizing area increases the hydroxide (an important cooking chemical in the Kraft delignification process) content of the cooking liquor before it is fed into the digester area. The main features and motivations for each development stage of the procedure are presented below. 2.1 Model formulation The model of the process is incorporated into the optimization problem as the set of constraints that should be satisfied at the solution point. But for the plant section under analysis, the model structure changes with time because the flow of some streams can be zero depending on the positions of 43 switches operated through the Distributed Control System. These streams are related to the green liquor feed (controlled from the recovery boiler) and to the cleaning cycles of the filters. Furthermore the presence of manual valves and streams connected to water locks can modify the model. To tackle the problem of model variability with time, the broadest possible incidence matrix is first formulated. It has 146 rows and 306 variables associated with units and streams respectively. The model contains all streams interconnecting units (with the exception of those streams utilised only during plant maintenance) and fictitious streams
765 that account for the net accumulation in the filters and tanks. It is assumed that only these units have an unsteady state operation mode. A particular operation state is associated with specific positions for the switches, valves, and tank heights. Starting from this information and the all-inclusive model, a procedure was developed that determine the set of units and streams that participate in an operation state. It is based on row and column elimination of the general incidence matrix. 2.2 Instrumentation analysis Considering only mass balance calculations, an instrumentation analysis is performed to know what information is available from the process with the existing instrumentation, that consists of 17 magnetic flowmeters and 15 level sensors. The analysis is based on the variable classification procedure that uses the Q-R matrix decomposition (Sanchez and Romagnoli, 1996). Different case studies are conducted whose results are included in Table 1. It contains the number of units and streams involved in the corresponding model, and the amount of redundant (R), non-redundant (NR), observable (O) and unobservable (UO) variables. Normal operation condition is first considered (Case 1), i.e. all switch positions, height of tanks and valves are as indicated by the P&I diagram as normal. In this case the number of measurements whose values can be adjusted through reconciliation, i.e. the redundant measurements, is zero. This is a serious drawback not only to enhance measurement's precision but to detect and identify gross errors. Then a model that excludes vent tanks from the previous one is analysed (Case 2). Other vapour streams are also eliminated in Case 3, but the increment in system redundancy was not important. The incorporation of temperature measurements and energy balances in the model was discarded beforehand because its benefit was obviously worthless, due to the low number of temperature measurements that are present. As any implementation of a data reconciliation procedure is settled on the existence of redundant variables, the next goal was to incorporate extra knowledge to the process model to increase system redundancy. Table 1: Instrumentation Analysis Results Case 1 2 3
Units 120 120 106
Streams 262 255 218
R 0 3 3
NR 32 29 29
0 31 35 46
UO 199 188 140
2.3 Extended model formulation A study was undertaken to determine if measured variables related to pump operation could be used to formulate equations for flowrate adjustment, that is, relationships containing measured flowrates, constants and other measurements. In the recausticizing area there are five centrifugal pumps of measured variable-speed. They are important pieces of equipment because they act as final elements of flowrate control loops that manage the input flow to filters and the slaker from tanks. The sensor
766 configuration around each pump is shown in Figure 1. It consists of a magnetic flowmeter and two pressure gages installed on the pump output stream and a level sensor located on the tank, but there is no pressure sensors at pump inlet. First a correlation was obtained using information from pump characteristic curves. It allows estimating the flow rate of liquor/slurry being processed by the pump in terms of the head and speed of the pump. This correlation is not useful for flowrate adjustment by itself because, the lack of pressure sensors at the pump suction avoids measuring directly the pressure rise of the liquid over the pump. Instead the energy balance of the fluid, from the liquid maximum height in the tank to the pressure sensor downstream of the pump, is used as an extra model equation because it can be formulated in terms of measured variables and parameters. It is expressed as follows W. - + ai 77- + hi +- - + h f + h . pg ' 2g g
2
(1) - + a' - ^ + h^ Pg 2g ^ where: P = Pressure (kPa), p = Density (kg/m3), a = Velocity profile factor, g = Acceleration due to gravity (m/s^), h = Height (m), Wg/g = Shaft Work (m), hf = Head loss due to friction (m), he = Head loss due to contractions (m). In equation (1), the velocity in the tank is assumed to be zero, the pressure at point 1 is given by the tank pressure (usually atmospheric), the pressure at point 2 is read from the pressure indicator, the height terms are derived from the level indicator on the tank and the geometry of the pipe work obtained from isometric drawings, the head loss due to friction and contractions is formulated in terms of pipe geometry and flow rate. The process model is enlarged with seven non-linear semi-empirical equations, corresponding to balances indicated by formula (1). Also redundant equations that consider the equality of pressure due to hardware redundancy are included in the model. For the normal operation state, the results of variable classification are: R=25, NR=22, 0=32, UO=197. Redundant measurements include flowrates, but also pressures and pump speeds that has not been considered in Case 1.
" Energy Figure 1: Tank and Pump Configuration 2.4 Data reconciliation (DR) A current set of measurements and switch positions is provided by the DCS at intervals of 2.5minutes. The current operating status is compared with the previous one. If the
767 operational mode had changed the model is modified accordingly and the DR is performed on the measured redundant variables. Its inputs are the vector of average measurements, obtained from sensor outputs corresponding to the same operational state, and the current sector model is made up of the mass balances associated to nonzero flow streams and the redundant equations obtained for pump operation. If there is no change in the operational state of the process, the DR procedure is performed after a prefixed interval of ten minutes. Two data reconciliation procedures were derived, linear and non-linear. The linear one, uses the linearized version of the redundant model, consequently estimates are calculated straightforward. The solution of the nonlinear problem was achieved using a SQP technique. Measurement errors are estimated using the information provided by the vendors. Parameter values are updated using joint parameter and measured variable estimation at sensible fixed interval. A graphical interface was developed in Excel to allow the model's results to be presented in a recognizable and easy to read format. Figure 2 below shows a sample of the on-line results for reconciled and measured values in the case of the dregs filter pressures. These graphs are used as a monitoring tool for operators, to decide if process is running under normal conditions. Their judgement is based on confirmed information.
^^S€|^fe^Sl^^^^^^^fe^?lS??'ft,^ -^^
;\ , rr-r^' ' f 5 ^ V ^^ ,'v^V'S't/"\^%Y'fx!'t ^ i^vT'^'fc^ ^-;r"fF't'^'~i ^^ " j!!S4,',*-S^'Ci, W ; 4 i ^ * * - ^ ^ - i ' ^ ^ - S - ™
(a)
(b)
Figure 2: DR results for dregs filter pressures 2.5 Gross error detection Data reconciliation deals with the problem of random errors. If gross errors are also present in the process data, they must be identified and removed before reconciliation. In our approach we have used the test on the estimates for gross error identification. Following Mah and Tamhane (1982), measurement residuals are defined as e= y-y
(2)
where y and y are the vector of measurements and reconciled estimates respectively. It is easy to show that ^ -- N ( 0 , V) if there are no gross errors in the measurements, where V is the covariance of the residuals. We conclude that there is a gross error present in the i'^ observation, if
768
\E\ = \e\>E.all
(3)
where E^,^ is the upper a/2 point of the standard normal distribution and a is the level of significance. In our case, E^/^ was obtained from the actual distribution of the plant data by approximating the distribution function of the residuals, based on historical information for normal operating conditions. Figure 3 illustrates the bar plot of the residuals (critical value based on statistics
£'^/2 and actual value
6- after
reconciliation) for two operating conditions. GiMnU^uourfittr D3
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Figure 3: Residuals plot (a) Normal condition (b) Faulty condition
3. Conclusions In this work, the study conducted for implementing a data reconciliation procedure at the recausticizing sector of the Visy Pulp and Paper Mill, Tumut is presented. The procedure points towards solving two main aspects. One of them is the variability with time of the model representing sector operation, a point not considered in previous data reconciliation procedures that work for only one operative mode. The second aspect involves the necessity of increasing system redundancy in order to detect and identify gross errors. In this sense semi-empirical relationships among measured variables are obtained using information from pump characteristic curves. This constitutes also an innovative way to overcome the difficulties associated with the lack of instrumentation.
4. References Romagnoli, J.A. and M. Sanchez, 1999, Data Processing and Reconciliation for Chemical Process Operations. Academic Press, San Diego. Sanchez, M. C. and J. Romagnoli, 1996, Use of Orthogonal Transformations in Classification/Data Reconciliation. Computers and Chemical Engineering, 20, 483493. Mah R.S.H., Tamhane A.C., Detection of gross erros in Process Data, AIChE Houmal, Vol. 28, No. 5 pp. 828-830(1982).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
769
A Framework for Consistency Analysis of Safety Properties and its Use in the Synthesis of Discrete-Event Controllers for Processes Systems A. Sanchez\ R. Gonzalez, A. Michel Dept. of Elec. Eng. and Comp. Centro de Investigacion y Estudios Avanzados (CINVESTAV) Apdo. Postal 31-438, Guadalajara 45091, Jalisco, Mexico
Abstract This paper proposes an "ad-hoc" formal framework for the analysis of three types of safety specifications and its use in the synthesis of a class of discrete event controllers. The notion of a specification set free of errors and redundancies is introduced as a minimal set of consistent specifications as well as procedures to establish it. The satisfiability verification of the specifications by the closed-loop behaviour model is also discussed. The use and advantages of the framework are illustrated with the synthesis of a class of discrete-event controller, termed procedural controller, for the operation of a batch reactor. Conflicts on the specification set were easily identified and corrected, reducing the synthesis effort. Satisfiability verification of the specifications by the closed-loop behaviour established to what extent the controller fulfils the specifications.
1. Introduction Formal synthesis of controllers for event-driven operations using automata-based methods is frequently carried out employing a discrete event model of the process and a set of closed-loop behaviour specifications established by the designer. Both the model and the specification set are frequently assumed to be initially free of errors, which in few occasions is the case. Thus, the synthesis task becomes an iterative procedure in which the process model or specifications are modified in each iteration, until obtaining a satisfactory result. In most cases, not to mention large systems, it is very difficult to identify errors or inaccuracies either with incremental or monolithic specifications. This paper proposes an "ad-hoc" formal framework to facilitate the synthesis of a class of discrete-event controllers, termed procedural controllers, for three types of safety specifications describing the conditional execution of sequences of controlled events. The framework helps the designer i) to capture this particular class of specifications, ii) to analyse the specification set and initiate the controller synthesis with a specification set free of inconsistencies and redundancies and iii) to determine what specifications are satisfiable by the obtained closed-loop behaviour. The paper first describes the modelling of a discrete-event process using standard automata as well as the notion of Corresponding author, e-mail address: [email protected]. Fax: +52 (33) 3134 5579
770 semitrajectory. The definition of procedural controller is also presented together with an explanation of the type of process in which these controllers can be used. Section 3 introduces the definition of the safety properties realised as semitrajectories with specific semantics. Then, the notions of a minimal set of consistent specifications and satisfiability by the closed-loop behaviour are given. Computation procedures were implemented to carry out all required calculations. By using a minimal set of consistent specifications, the synthesis effort can be reduced substantially because this set is free of internal inconsistent behaviour and does not contain repeated specifications. Thus, if the resultant closed-loop behaviour does not satisfy some of the specifications, it is due only to controllability restrictions. Moreover, by verifying the satisfiability of the specifications by the closed-loop behaviour, it is established to what extent the controller fulfils the specifications. The paper then describes, using an example, how to incorporate these ideas in the synthesis of a procedural controller for the filling operation of a batch reactor. A closing section discusses what are the benefits of the proposed framework.
2. The modelling framework A discrete-event process is modelled in the standard fashion by a finite state machine (FSM) P = {Q, V, 2 5, qo, Q^}, where Q is the state set; V = {Vj, V2,..., Vn) is the statevariable set with n, state variable number; a state-variable Vj takes values from a finite domain Dj plus the distinguished symbol ^ meaning "any value". I is the transition set, divided into two disjoint sets: !„ (uncontrollable transitions) and Zc (controllable transitions); 8 :Z X Q ^ Q, is the state-transition partial function; qo is the initial state; Qm is the marked states set. A transition i is enabled in state q if 5(q,T) is defined. Definition. Assignment. An assignment over the variable set is a function s:V->D defined by the rule \\ ->s(Vi) for i=l,...,n, where D is the union of the state-variable domains including the distinguished symbol c». The assignment s is represented by an ntuple (s(vi), s(v2),..., s(Vn)) for i=l,...,n. For each state q E Q, an assignment can be associated by a function P:Q -^A defined by P(q)=Sq, where A is the set of all possible assignments. Notice that P(q) is not injective, thus it is possible to have two states with the same assignment. Definition. Non-executable transition set. For an assignment s such that P(q)=s for some qeQ, the set of non-executable transitions of s is given by net(s)={TGS | x is an enabled transition in any q with assignment s which, by design, is not permitted to occur}. DeHnition. Semitrajectory. A semitrajectory of length m+1 is a finite mixed sequence of assignments and transitions 7i= SQ^^ Si^-...^"'Sm. The relationship between assignments and transitions is established by y: A X S -^A with the following rule: for Sj+i = Y(Si,x), there exists exactly one j G {l,2,...,n}such that Si+i(Vj) ^ s,(Vj ) and for all k=l,...,n such thatJT^k, Si+i(Vk) = s,(Vk).
771
DeHnition. Covering (refinement). The assignment s covers assignment s* (equivalently, s' refines s) if and only if there exist at least one j G {l,2,...,n} such that s(vj) = oo, s'(Vj) ?t oo and for all k=: l,2,...,n such that j^k, s(Vk) = s'(Vk). The control device used in this work is termed procedural controller (Sanchez et al. 99). This device is capable of forcing the execution of controllable transitions by preempting uncontrollable events. The controller is modelled as well as an FSM C = {X, I Y, Xo, Xm}, where X is the state set; S is the same transition set as in the process model; Ti : Z X X -> X, is the state-transition partial function; XQ is the initial state and X^ is the marked states set. In particular, for each xe X, and GE S such that r|(a, x) is defined, either, 1) ae Eu and for all acG Ic. "HC^C, X) is undefined, or 2) GG ZC and for all a' ^ a G S, ri(a', x) is undefined. That is, a procedural controller can either be in: 1) a state in which one of a set of uncontrollable transitions occurs or 2) a state in which the execution of the only controllable transition defined is enforced. Sanchez et al. (99) presented conditions of existence of a procedural controller, closed-loop invariant properties as well as procedures to calculate the controller.
3. Speciflcation of safety properties A safety property informally states that a "bad thing must not occur". When dealing with batch plants, we have found that most of the specifications for designing discreteevent controllers can be described in terms of three following types of safety properties. These specifications are modelled as semitrajectories. Definition. Type 1 Safety Specification. Given an operation state, a set of control commands is executed sequentially. Any semitrajectory TI^SQ^^SI"^... "^S^ of length m+1, with m>0, models a type 1 specification. SQ is called the initial assignment ofn. Definition. Type 2 Safety Specification. Given the occurrence of a process event, a set of control commands is executed sequentially. A semitrajectory 7i=So^^Si^... ^s^ of length m+1, with m>0, models a type 2 specification if and only if SQ covers Si. Y(SO,X) is used as the initial assignment Si from where the rest of n is executed. This semitrajectory is a compact way of representing the triggering of sequence by a transition. This type of semitrajectory will be written as 7i = SQ^^^SI ^^...Sm, where expression AT, indicates the emphassis over the exectuion of the event. Definition. Type 3 Safety Specification. Given an operation state, a set of control commands is not permitted to occur. A specification of type 3 is modelled as a set of two-assignment semitrajectories of the form 7i=So^Si. SQ and S\ are fixed assignments whereas x can be any transition such that Y(SO,X)=SI and x does not belong to set net(so). So is the initial assignment. This type of semitrajectory will be written as 7i=So''^ Si.
772
3. The minimal set of consistent semitrajectories Semitrajectories TT, and n, are consistent if and only if one of the following is true: 1. Initial assignments of 7C, and n, are not equal and do not cover each other. 2. If condition 1 is not true, and semitrajectories are not of type 3, then they must coincide after the initial assignments. 3. If condition 1 is not true and only one semitrajectory declares the execution of transition x from its initial assignment, then the other semitrajectory must not forbid the execution of such transition. Semitrajectories 7i, and n, are duplicated if and only if their initial assignments are 1. Either equal or one assignment covers the other and 2. Either coincide after the initial assignments or, for specifications of type 3, at least one transition of net(so) of Tii belongs to net(so) of 7I2. A set of consistent semitrajectories is a set in which all semitrajectories are mutually consistent and a minimal set of consistent semitrajectories is a set of consistent semitrajectories without duplications. Computational procedures were implemented to obtain a minimal set of consistent semitrajectories.
4. Semitrajectory satisflability in the closed-loop behaviour. A semitrajectory 7i=So^^Si^^... "^Sr of length r+1 is accepted by an FSM M^ ={X,V, E, p, Xo, Xm}. X is the state set; V is the state variable set of the process model with state assignments given by P(Xi)=Si for i=l,2,...,r; S is the transition set of the process model and Xm = {Xr}. The transition function p: X X I —> X, such that if Y( Si,T) is defined in 71, then p( Xi,T) is defined in M;^. The initial state of M^ for specifications of type 1 and 3 is XQ. Otherwise, the initial state of M^ is Xi. A semitrajectory n accepted by an FSM M;t is satisfiable in a closed-loop FSM model McL = {C, I , e, CQ, C^}, if and only if 1) each assignment (3(Xi) of state Xj in M^ is equal to or covers at least one assignment P(c) of state c in of MCL and 2) if p(Xi,T) is defined, then e(q, x) is defined. That is, the FSM M^ *'fits at least once into" the closed-loop FSM MCL- Thus, semitrajectory satisfiability by the closed-loop model can be established by using a standard search strategy. The current computer procedure implementing the verification of specification satisfiability outputs whether the controller model satisfies a given specification. Otherwise, those specification states not existing in the closed-loop model are displayed as well as the closed-loop states in which e(q, x) is not defined for a given p(x,x).
5. Example The use of the framework is illustrated with the synthesis of the PLC control logic for the filling up phase of a batch reactor (shown in Figure 1) currently installed in a special lubricants factory. Table 1 includes the process and software components involved in the operation, together with FSMs models. Uncontrollable transitions are marked with an *. The rest of the transitions are commands that the PLC can use to control the
773
POsmoN INOCATOR FV1
POSITION INOCATOR FV2
Drocess. All initial and marked states are labelled with
(0). The objective of the phase is to feed into the reactor a measured amount of a component from a warehouse location. During the filling, the operator can stop/restart the operation by pressing button Bl and the PLC must be able to stop and resume safely the filling. If the procedure does not succeed or the operator presses the emergency button, the controller blocks any possible action of the operator, issues the command to stop the pump and shuts any valve still open. The designer captured this phase using 12 semitrajectories shown in Table 2. Specification 1 and 7 are of type 2. The rest is of type 1. They are Figure 1. Diagram of batch classified as normal, emergency and recovery procedures, which were exhaustively checked by hand. For each specification, a natural language statement is given first, followed by its associated semitrajectory. In order to present these semitrajectories in a concise manner, only the first assignment is shown. For subsequent assignments, it is indicated only what position and with what value it differs from the previous one. The process model was built using the method discussed in Sanchez (96) and the synthesis of the procedural controller was carried out using available tools (Sanchez et al. 99). The sizes of the process model and resultant procedural controller were 1024 and 124 respectively. Satisfaction of each semitrajectory by the closed-loop behaviour was then manually verified to the designer's satisfaction. Before translating the resulting FSM into programming code, a second verification round was carried out using the framework proposed here. It was found that the specification set was inconsistent and not minimal. Duplication was Component 1 Transitions | spotted in semitrajectories 2 Lbl Description From St. To St. 1 and 11. Both specifications 1 Bl 11* switchOn on(l) off (0) command to start the pump button 12 switchOff on(l) off(O) 2 emerg. 21* switchOn on (1) off(O) when the process is ready for stop on (1) off (0) the filling. The duplicate button 3 vol. 31 FeedSP Ijiic(l) Notlnic (0) behaviour was attributed in set point 32 Notlnic (0) NoSP Inic(l) this case to an oversight of the 4 vol. tot 41* voIOK OK (1) clear (0) designer. More importantly, it flag 42* clear clear (0) OK (1) 5 FV2 51 openFV2 open(l) closed (0) was found that semitrajectories 52 valve close FV2 closed (0) open (1) 8 and 9 were inconsistent. 6 FV2 61* FV2opens open (1) closed (0) Both specifications used pos. ind. 62* FV2closes closed (0) open(l) 7 FVl 71 openFVl open (1) closed (0) coverings in their respective valve 72 closeFVl closed (0) open(l) initial assignments and were 8 FVl 81* open (1) FVl opens closed (0) pos. ind 82* closed (0) FVlcloses open(l) not comparable. That is, state91 strtng (1) startPmp off (0) variable 3 was covered in 9 pump on (3) pmpStarts 92* strtng (1) specification 8 while it was status stppng (2) stopPmp 93 on (3) 1 94* pmpStops stppng (2) off (0) refined in specification 9, and Table 1. Elementary components of reactor and state-variables 5 and 7 were their associated FSM models (* = uncontrollable transition).
774 covered in specification 9 while they were refined in specification 8. This caused that there were 8 state assignments being shared by both states, which lead to the inconsistent behaviour. By verifying satisfiability of the specification set by the closedloop model, it was found that the controller did not satisfy specifications 9 and 12. In the case of specification 9, this was because specification 8 was applied first in the synthesis procedure. For specification 12, it was found that it was not possible to reach in a controllable manner the state declared as initial in the specification. In a second exercise, specifications 11 and 12 were eliminated from the specification set. Specification 9 was modified, as shown in table 3, to avoid shared assignments. Using the new specification set (minimal and consistent), the newly synthesised controller was the same as the previous controller. All specifications were satisfied by the Normal Operation closed-loop model. 1. If button B120 is toggled, then PLC issues commands to open FV2 and F^l (0, 0, 0, 0, 0, 0, 0, 0, 0) A " (1:1)^' (5:1)^' (7:1) 2. Once valves are open, then PLC issues command to start pump (1,0,0,0, 1,1,1, 1,0) ^'(9:1) 3. If pump is successfully started, then the volume setpoint is feed to the totalizer (1,0,0,0, 1, I, 1, 1,3)^'(3:1) 4. Once volume amount is reached, B120 is toggled, setpoint flag returns to not initialized and the PLC issues commands to close valves and stop pump (1,0, 1, 1, 1, 1, 1, 1,3)'^ (1:0)^^ (3:0)" (5:0)'^ a:Of (9:2) 5.
6. Conclusions Obtaining the same results in both exercises indicates that, although the controller guarantees a safe (i.e. controllable) closed-loop behaviour, it was not known what the controller was actually controlling. From a practical point of view, this is not acceptable for any real process. Although it is not essential for the controller synthesis to debug the specification set to a minimal and consistent one, it was very useful for the better understanding of the controller role.
Emergency Operation If FV2 shuts while filling up, then the PLC issues command to stop pump ( 1 , 0 , 0 0 , 0 , 1,0,00, oo, 3)^^^(9:2)
6. If FVl shuts while fining up, then the PLC issues command to stop pump (1,0,00, 0, oo, oo, I, 0, 3)^^ (9:2) 7. If emergency stop is activated, then button Bl is freed and operation cannot start again ( 1 , 1, ° ° , °°, oo, oo, oo, oo, oo)/i^
(1:0)
Bibliography
8. Once the emergency stop has been activated and the filling was under way, then the PLC must issue commands to close valves FV2 and P / 1 (0, 1, oo, 0, 1, oo, 1, oo, 0 ) " (5:0) ^^ (6:0) 9. Once the pump is off, clear the volume set point (0, 1, 1,0, oo, oo, oo, oo, 0)^^ (3:0) 10. If emergency stop is activated and pump is on, then the PLC must issue a command to stop pump (0, l,oo, 0,00,00, oo,oo,3)^\9: 2)
Sanchez
Recovery procedures 11. After restarting normal operation, valves were opened successfully and pump is off, then the PLC must issue a command to start pump ( 1 , 0 , 0 0 , 0 , 1 , 1, 1,1,0)'^'(9:1)
12. After restarting normal operation, if valves were closed, then the PLC issues commands to open them (1,0, 1, 0,0, 0,0, 0,0)^'(5:1)'^6:1) Table 2. Semitrajectories for filling pfiase 9. Once the pump is off, clear the volume set point (0,1,
1,0,0,00,0,00,0)^2(3:0)
Table 3. Modified semitrajectoryfor
specification
A.,
G.
Rotstein,
N.
Alsop
and
S.
Macchietto. Synthesis and implementation of procedural controllers for event-driven operations. AIChE J., 45,8,1753-1775,1999 Sanchez A. Formal Specification and Synthesis of Procedural Controllers for Process Systems. LNCIS, V. 212, Springer-Verlag, 1996.
9
Acknowledgements. Partial financial support from CONACYT (grant 31108U) and the use of the application example from Interlub S.A. are kindly acknowledged.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
775
Aggregated Batch Scheduling in a Feedback Structure Guido Sand and Sebastian Engell Process Control Laboratory, Department of Chemical Engineering, University of Dortmund, Germany
Abstract In this contribution a two-layer feedback structure to schedule batch chemical processes is specified. For a real-world benchmark the master scheduling problem is stated as a two-stage stochastic program and solved by a decomposition algorithm. Evaluation results are presented which prove the applicability of the approach.
1. Introduction Scheduling operations in batch chemical plants constitutes large-scale combinatorial optimisation problems. Their modelling and solution has been addressed by a number of approaches from the mathematical programming domain, see e.g. Reklaitis (1996). In the majority of the publications the models are stated under the assumption of complete information, but in recent years researchers increasingly address the integration of uncertainties. Representative examples are the papers by Balasubramanian and Grossmann (2000), Honkomp et al. (1997), lerapetritou et al. (1995), Petkov and Maranas (1997), and Sanmarti et al. (1997). However, almost all papers which consider the rescheduling of activities aim to maximise reliability, flexibility or robustness of a nominal schedule in the sense that the probability of having to perform extensive reactive modifications is minimized. Our approach is different because we assume that reactive scheduling can be performed by solving detailed scheduling problems on an appropriate horizon online. Schulz (2001) showed for a real-world polymer production process that an MINLP-model for a 6 days horizon can be solved nearly optimally in one minute. Stimulated by this result, we propose to generate aggregated master schedules which explicitly reflect this considerable recourse capability. To this end, two-stage stochastic integer programs (2SSIP) provide an appropriate modelling framework, see e.g. Birge and Louveaux (1997). We choose a formulation with a scenario based representation of uncertainty, which is, for instance, efficiently computational tractable by the algorithm of Car0e and Schultz (1999). Motivated by the mentioned process (chapter 2), we specify a two-layer feedback structure for real-time scheduling with recourse (chapter 3). The model on the upper layer is presented and evaluated (chapters 4 and 5), and an outlook on the future work concludes the contribution (chapter 6).
2. Problem Statement Figure 1 schematically shows a multiproduct plant, which is used for the production of 2 types of polymer (expandable polystyrene - EPS) in 5 grain size fractions each. The
776 preparation stage is less important for the problem and will not be considered here. Each polymerisation is performed batchwise according to a certain recipe (out of a set of 10 possible recipes), which specifies the EPS-type and the desired grain size distribution. Due to safety restrictions, a minimal offset between two batch starts has to be kept. The produced suspension is transferred into buffering mixers and continuously fed into separation units. The mixer levels have to be kept between certain thresholds while the corresponding finishing line is running; moreover, it may temporarily be shut down. The mixing process is described by a nonlinear equation. Degrees of freedom are: 1. the starting times of the polymerisations, 2. the recipes for the polymerisations, 3. the operation modes of the finishing lines (on/ off), and 4. the feed rates into the separation units (within certain bounds). The process is influenced by significant operational and demand uncertainties, namely: 1. equipment breakdowns (polymerisation reactors), 2. not precisely reproducible grain size distribudons, 3. variable processing times, and 4. vague demand forecasts. The production goal is to maximise expected profit, which is mainly determined by: l.the revenue, which decreases with exceeding the due date, 2. fixed costs for each polymerisation, 3. fixed costs for each change in the operation modes of the finishing lines, 4. variable inventory costs, and 5. variable penalties for production shortfalls.
3. Solution Approach Since the process is running continuously and the demand profiles reveal no periodical pattern, we propose a cascade-like feedback structure to optimise the process operations online (see figure 2). The aggregated layer works as a regularly master scheduler (MS), whereas the detailed layer takes the role of a quickly reacting scheduler (RS). Both parts are supposed to be based on stochastic integer programs with recourse. Their precise specifications are derived by applying these generic ideas to the specific process. Considering the process dynamics, it appears reasonable to perform master scheduling updates in intervals of 2 days and to look 10 intervals ahead. This establishes a
Figure I: Flowsheet EPS-process
777
disturbances
MS i i
guidelines
RS
operations
t
1
w
EPS
observation Figure 2: Cascade-like feedback structure multiperiod time representation and an aggregation of decisions according to the intervals. The alternating sequence of observation and decision would naturally lead to a 10-stage stochastic program. However, to keep the problem computationally tractable it is approximated by a 2-stage stochastic program. In accordance with the specification of the detailed scheduler, the first stage covers the intervals 1 to 3. Decisions which are relevant to the indicated horizon and granularity are: 1. the number of polymerisation starts in each interval classified according to the recipes, and 2. the operation modes of the finishing lines. Relevant constraints are capacity restrictions imposed by the polymerisation stage and the finishing lines, whereas mixing effects are of less importance and do not affect feasibility. Since uncertainties are supposed to be modelled by scenarios with a priori known probabilities, it is favourable to consider the exogenuous disturbances based on fixed time intervals. The probability of the occurrence of an exogenuous event during an interval is only a function of its length, and for a point of time it is zero. Concerning the EPS process, the exogenuous uncertainties, namely equipment breakdowns and demand forecasts, are exactly those which have long-term effects, such that they should be considered on the aggregated layer. The generated production profiles and the operation modes of the first stage intervals serve as guidelines for the detailed scheduler. Since the focus of this contribution is on the aggregated layer, the detailed scheduler is only sketched briefly. The model is intended to be based on an event driven time grid with 20 points. This corresponds to a horizon of 4 to 6 days, which is identical with the aggregated first stage. Those decisions which are associated with the first grid point form the first stage of a 2-SSIP. The process is modelled by detailed constraints, except for the nonlinear mixing effects: Simulations studies show, that the inaccuracy caused by a linear approximation is in the order of the inaccuracy caused by inexact data. The scenarios cover the endogenous (short-term) uncertainties, namely grain size distributions and processing times.
4. Aggregated Scheduling Model The deterministic equivalent of a 2-SSIP with discrete scenarios is defined as follows (cf Birge and Louveaux (1997)):
778
max
x,yi,...,yQ
1
S ^ w F ^0) +qwya))s-t. T(^X(^ + W ^ y ^ = h^^, x
(1)
(0=1
X(j^3GX,y^G Y,co = l , . . . , Q
The first- and second-stage variable-vectors x and y belong to polyhedral sets X and Y with integer requirements. The parameter Q denotes the number of scenarios with corresponding probabilities n. The constraints are formulated by means of the matrices T und W and the right hand side-vector h of suitable dimensions. The objective is to maximise a weighted sum of variables subject to the weighting-vectors c and q. For the aggregated scheduling model of the EPS-process we define the following indices, parameters and variables: 1. Indices: intervals i,j,kG [1,...J], delays dG [1,...,D], products pG [1,...,P], grain size fractions f G [1,...,F], and recipes r G [1,...,R]. 2. Parameters: ratio p of a certain grain size fraction according to a certain recipe, customer demands B, bounds for polymerisation starts N'"^, thresholds of mixer levels ^min^ (.max^ ^ ^ ^ ^^^^^^
^^ ^^^^
^^^^^ pmin^ pmax
3. Variables: number of batches N G IN, operation modes of finishing lines z G {0,1}, mode change indicators w G IR+, amounts of delivered produkts M G IR+, amounts of inventories M"^ G IR+, and amounts of production shortfalls B' G IR+. The capacity constraints of the polymerisation stage (2) (see below) are stated dynamically, i.e. they represent the couplings between the intervals correctly, by bounding all interval sequences. The dynamic capacity constraints of the finishing lines (3) bound the polymerisation starts such that a) if the finishing line is operating, feed rates exist which keep the mixer level feasible or b) if the finishing line is not operating, the mixer is empty. The so-called non-anticipativity constraints (4), which say that the first stage variables do not depend on a particular uncertainty realisation, are straightforward.
SISNj,p^^N,1J^
Vi,k,co|i
j=ip=lrp=l
^n
^kpwX(k+i)pcoCp(o - j
[C*^ if i = 1
(2) ]
k
min
^^^ [ + Z^jpa)Fp [^(i-l)pcD^ipo)'-pU) ^'S^J j=i
2XkpcoXM)pa,C^-
|C^fi = l
I
k
' ^^„ , +ExjpcoFp"[^(i-l)pco^ipa3^pca ^*^^J j=i
Nirp,(o=l=--- = Nirp,a)=Q,
^ip,ii^\=---
= ^ip,io=Q
Vi,p,fp
*" '^''
-ZZ'^jrpO) j=irp=l
(3)
Vi,k,p,(0|i
To maximise the expected profit, demand and production profiles have to be balanced as well as possible. The demand constraints (5) (see below) offset the customer demands against the production shortfalls and (possibly delayed) product deliveries. The dynamic supply constraints (6) bound the amount of product which is delivered
779 until a certain interval by the produced amount. The inventory is computed by the oversupply constraint (7), and finally, the mode changes of the finishing lines are detected by equation (8). The objective is to maximise a weighted sum of M, M"^, N, B" and w subject to (1) - (8). This results in an MINLP which is tranformed into a large but structured MILP by common linearisation techniques (see e.g. Williams (1994)).
Mt ^^ = max|
i
^p
Vi,fp,a)|i
(7)
ifpO)
Wip(o=|^(i-l)pa)-Xipa)|
Vi,p,co
(8)
5. Model Evaluation The 2-SSIP reveals a block-angular matrix structure, which is exploited by the dual decomposition algorithm of Car0e and Schultz (1999). It is based on the Lagrangian relaxation of the non-anticipativity constraints (4), without which the formulation decomposes into Q independent subproblems, which differ in the right hand sides h only. CPLEX (2000) is used as a subproblem-solver and imbedded in a branch and bound-algorithm, which branches on the first-stage variables and generates bounds by solving the dual problems. The model was implemented with the parameter values I = 10, D = 3, P = 2, F = R = 5 and randomly generated values for B and N"""^, representing uncertain demand forecasts and equipment breakdowns, respectively. For Q = 1 the problem degenerates into a purely deterministic MILP with 725 variables (122 discrete) and 680 constraints; it is solved by CPLEX 7.0 within 20 s CPU-time on a SUN Ultra 2 300 with average optimality gaps of 2,8 % (all < 6 %). The problem size is nearly proportional to Q. so that 1000 scenarios yield much more than half a million variables and constraints. It is reasonable to limit the CPU-time to 8 h, which corresponds to up to 2 polymerisation starts in the running process. On a SUN Ultra Enterprise 450, average optimality gaps of 7,6 % (all < 10 %) could be achieved for the stochastic program. By approximating the dynamic constraints statically the problem size decreases by 285 constraints (^ = 1) and the average optimality gap is approximately cut by half at the expense of reduced modelling precision. In fact, the static model leads to solution vectors which are structurally different from the above and of inferior quality. Furthermore, we substituted the model property that demands can be satisfied partially by the restriction that each demand must be met entirely, possibly later than the due date. The overall model size shrinks to 284 variables and 479 constraints, but the number of discrete variables rises. CPLEX is not capable to efficiently process this additional complexity such that the deterministic problems yield average optimality gaps of 63,8%.
780
6. Conclusion And Perspectives The results of this work demonstrate that for a real-world production process aggregated batch scheduling with recourse can be performed under real-time conditions. The explicit modelling of extensive recourse actions anticipates different reactions to feedback information and thereby expands the optimisation potential. The modelling of various sources of uncertainties is straightforward and allows for immediate utilization of empirical probability distributions. The large problem scale is successfully tackled by a well-posed model structure in combination with an efficient decomposition algorithm. Encouraged by the results for the master layer we will also apply the above methodology to the detailed layer based on the model from Schulz et al. (1998). Special challenges arise from the fact that the model will have stochastic matrices and continuous first stage variables.
7. Acknowledgements We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft within the priority programme "Real-Time Optimisation of Large-Scale Systems" under grant EN 152/17-1,2,3, and the very fruitful cooperation with the Chair of Discrete Mathematics, Gerhard-Mercator-University of Duisburg, Germany.
8. References Balasubramanian, J. and I.E. Grossmann, 2000, Scheduling to minimize expected completion time in flowshop plants with uncertain processing times, Proc. ESCAPE-10, Ed. S. Pierucci, 79. Birge, J.R. and F. Louveaux, 1997, Introduction to Stochastic Programming, Springer, New York. Car0e, C.C. and R. Schultz, 1999, Dual decomposition in stochastic integer programming, Oper. Res. Lett. 24, 37. CPLEX Optimisation Inc., 1989-2000, Using the CPLEX Callable Library, http://www.ilog.com. Honkomp, S.J., L. Mockus and G.V. Reklaitis, 1997, Robust scheduling with processing time uncertainty, Comp. Chem. Engg. 21, 1055. lerapetritou, M.G., E.N. Pistikopoulos and C.A. Floudas, 1995, Operational planning under uncertainty, Comp. Chem. Engg. 20, 1499. Petkov, S.B. and CD. Maranas, 1997, Multiperiod planning and scheduling of multiproduct batch plants under demand uncertainty, Ind. Eng. Chem. Res. 36, 4864. Reklaitis, G.V., 1996, Overview of scheduling and planning of batch operations. Batch Processing Systems Engineering, Eds. G.V. Reklaitis, A.K. Sunol and D.W.T. Rippin, Springer, Berlin, 660. Sanmarti, E., A. Espunia and L. Puigjaner, 1997, Batch production and preventive maintenance scheduling under equipment failure uncertainty, Comp. Chem. Engg. 21, 1157. Schulz, C, R. Rudolf and S. Engell, 1998, Scheduling of a Multiproduct Polymer Batch Plant, Proc. FOCAP098 (Snowbird/ USA), 224. Schulz, C, 2001, Modelling and Optimisation of a Multiproduct Batch Plant, PhD thesis. University of Dortmund, Germany (in German). Williams, H.P., 1994, Model Building in Mathematical Programming, John Wiley & Sons, New York.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
781
Analysis of Parametric Sensibility of the Process of Production of Cyclohexanol Santos, M.M. and Maciel Filho, R. School of Chemical Engineering - State University of Campinas (UNICAMP) CP 6066 - Campinas - SP - Brazil - 13081-970 e-mail: [email protected]
Abstract The main objective of this paper is to carry out a parametric sensibility analysis for a cyclohexanol production multiphase reactor. Firstly, the mathematical steady-state model validated with industrial data, based on mass, energy and momentum balances, was submitted to a fractional factorial design. This method allows the understanding of interactions between variables with advantages over conventional methods, which involve changing one variable while fixing the others at certain levels. From this, some variables could be neglected and a complete factorial design could be developed. Using this procedure was possible to identify not only the main effects of the variables, but also the interaction effects among these variables. The behavior of the cyclohexanol reactor was extensively studied bearing in mind the determination of control strategies and optimization.
1. Introduction The more economic operation form of a great scale production plant is the continuous one. Moreover the plant should satisfy demands of equipment operation, product quality, more rigorous environmental restrictions and also follow some economic order. A typical system of continuous processing is the cyclohexanol production plant, raw material for obtaining of several industrial products of high commercial value as, for example, nylon. Due to the interest of this substance, it is desirable to obtain models that reproduce qualitative and quantitatively the behavior of the system, beyond of the phenomenological understanding of the process. In this work a deterministic model for the steady state of the cyclohexanol multiphase reactor is considered and used as tool for process variable interactions. The reactor is composed by tubular modules immersed in a boiler. Some of the modules are constituted by concentric tubes through where the reagents and the refrigerant flow. Some modules, through where the only reagents flow, are used concentric tubes. The reagent flow from one module to another and the refrigerant is added to each of the modules. The refrigerant stream, that comes from the boiler and eventually of a new feeding of condensed (make-up), is divided among the tubular modules, with different flow rates. This arrangement allows the operator to play with individual coolant flow rates, which are added to the conventional manipulated variables (as reactant pressure, temperature and concentrations). This makes the problem to be highly multivariable so that the usual techniques to perform parametric
782 sensibility tend to fail. Bearing this in mind, in this work is proposed the implementation of fractional and complete factorial design procedure in a hierarchical approach to solve the parametric sensibility analysis of a large scale multivariable process. The proposed is to apply the fractional factorial design to reduce the process dimensionality and then use the factorial design to the remaining variables. Although the factorial design be a known tool extensively used in experimental planning considering real data very few attention has been given to its use as sensibility analysis tool coupled with process modelling. In fact the application of statistical process control procedure do not cover the whole potential as factorial design coupled to deterministic model does.
2. Mathematical modeling and statistical analysis of the process 2.1 Mathematical Modeling The deterministic mathematical model used to describe the reactor is based on the work by Santana (1999) and Toledo et al. (2001). This model considers the peculiarities of tubular modules that compose the reactor. The process model is constituted of conservation laws of mass, energy and momentum. The mass and energy balance are in the form, respectively:
^ = /(c,r,p,rr„,...)
(1)
^ = f{CJ,PJr„r") dz
(2)
dz
where C is concentration, T is temperature, P is pressure and Tr^ is refrigerant temperature in the ^-tubular module. Moreover, equations for predicting the heat coefficients must be present as well as a way to describe evaporation that may occur, depending upon the operating conditions. Each of these equations must be applied to each module for both regions, namely, the tubular and annular. Since the reactor is essentially a tubular one, axial dispersion is considered. Thus, the steady-state process model presents a set of ordinary differential equations if radial dispersion is neglected, which is, together with the hypothesis that the solid-liquid phase is a single pseudohomogenized fluid, a reasonable simplification that can be made in order to reduce the complexity of the process model. The mathematical model was validated with industrial data. 2.2 Fractional factorial and complete factorial design The univariable analysis involves changing one variable while fixing the others at constant levels and usually it does not provide good information on the problem since interactions among variables are not taken into account. To overcome such limitations the fractional factorial design (Box et al., 1978) appears to be a suitable procedure. To apply the method, it is necessary to plane the trials through of factorial design. This method is based on selection of a fixed level number to each variable and executes all
783 possible combinations. When many variables or parameters are involved in the process, one can choose a fractional factorial design. The design is reduced and it is possible to evaluate the variable importance in the responses. In this proposed hierarchical approach the fractional factorial design is useful in the initial stages of process development, so, that technique can be used to identify the more important independent variables and select them to realize a complete factorial design. Once the statistically significant variables were selected by means of fractional factorial design, the complete factorial design experiments were planned to obtain the principal and interaction effects of such variables.
3. Results and discussion 3.1 Fractional factorial design Firstly, the influence of eight variables on mole fraction of phenol in the liquid phase, reactor temperature and pressure will be analyzed. The independent variables are: initial reagent temperature (To) and pressure (Po), flow rate of refrigerant introduced into some tubular modules (Vrefl, Vref2, Vref3, Vref4, Vref5 and Vref6). Each variable is tested at two level, a superior (+) and an inferior (-), as shown in Table 1. The range of values is in agreement with real data and the temperature is normalized. Table 2 presents the simulations as well as the way they are to be conducted for the 1^'^ fractional factorial design with 16 trials. The statistically significant effects of the variables on each response for a 95% confidence level are presented in Figures 1 and 2 for mole fraction of phenol in the liquid phase and reactor temperature, respectively. For pressure only the initial reagent pressure have statistically significant effect, so the graphic is not be presented. In these figures, the flow rate of refrigerant is not statistically significant in any tubular module. Thus, they can be excluded of the complete factorial design. These results make possible to define the variables, which have an effect on the responses of interest so that complete factorial design can be elaborated to study the reactor behavior. Taking into consideration the present results, the selected variable for the responses are initial reagent temperature and pressure. Table 1. Superior and inferior Level To Po (kg/cm') -1 0.817 23.97 +1 1.105 32.43
levels to independent variables. Vrefl Vref2 Vrefi Vref4 (kg/h) (kg/h) (kg/h) (kg/h) 306.0 323.0 2329.0 561.0 414.0 437.0 3151.0 759.0
VrefS (kg/h) 1011.5 1368.5
Vref6 (kg/h) 1190.0 1610.0
3.2 Factorial design In addition to the selected variable from fractional factorial design, the initial concentration of phenol (X«)) and the quantity of catalyst (Wcat) will be used in the complete factorial design, since they are important manipulated variables. Table 3 presents the superior and inferior levels and Table 4 outlines the complete factorial design, where the variables used are To, Po, X^ and Wcat- The data are not experimental, but produced by simulation, as it consists in the Table 4 (trial 17), it is
784 only possible to present a trial in the central point and the answer models are free from pure error.
h
rv-Toi
[-•—Po|
^ o
J
—^—*-
^ziXziXi:
& °-t— 1^'
..^•^~.-\ ^^.^' ^ . •--.-^ 2
3
4
cv.^
5
6
7
8
reactor lenght reactor lenght
Figure 1. Perceptual effects in the mole fraction of phenol.
Figure 2. Perceptual effects in the temperature.
Table 2. Trials used in fractional factorial design. Trial Vref2 Vref3 Po Vrefl To +1 +1 -1 -1 1 -1 2 +1 -1 -1 -1 -1 3 + 1 -1 -1 -1 +1 4 +1 -1 +1 +1 -1 5 -1 +1 -1 +1 -1 6 +1 +1 -1 +1 -1 -1 7 -1 +1 -1 +1 8 +1 +1 +1 +1 +1 9 -1 -1 -1 -1 -1 +1 10 +1 +1 -1 -1 11 -1 +1 +1 -1 -1 12 +1 +1 -1 -1 +1 13 -1 +1 -1 -1 +1 14 -1 -1 +1 +1 +1 +1 -1 15 +1 +1 +1 +1 16 -1 +1 -1 +1
Vref4 +1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 +1 +1 +1 -1 -1
Vref5 -1 +1 +1 -1 +1 -1 -1 +1 +1 -1 -1 +1 -1 +1 +1 -1
Vref6 +1 +1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 -1
Figures 3 and 4 depict the effects for a 95% confidence level for mole fraction of phenol in the liquid phase and reactor temperature, respectively, and for pressure, only the initial reagent pressure have statistically significant effect, and the graphic is not presented. For the mole fraction of phenol in the liquid phase, seen in Figure 3, only principal effects of the amount of catalyst are not statistically significant in any region of reactor. The increase in initial reagent temperature increases the mole fraction of phenol at the reactor exit that prejudices the reaction. The increase in initial reagent pressure and initial concentration of phenol decreases the mole fraction of phenol, and this is suitable
785 to reaction, since phenol must present lower concentrations as possible because the rigorous environmental restrictions.
reactor lenght
Figure 3. Perceptual effects in the mole fraction of phenol.
Figure 4. Perceptual effects in the temperature.
Table 3. Superior and inferior levels for independent variables for factorial design. Level To Po (kg/cm^) Xfo Wcat -1 0.0 40.0 0.817 23.97 0 0.1 60.0 0.961 28.2 +1 0.2 80.0 1.105 32.43 Table 4. Trials used in complete factorial design. Trial To Po -1 1 -1 2 +1 -1 3 +1 -1 4 +1 +1 5 -1 -1 6 -1 +1 7 -1 +1 8 -fl +1 9 -1 -1 10 +1 -1 11 +1 -1 12 +1 +1 13 -1 -1 14 +1 -1 15 -1 +1 16 +1 +1 17 0 0
Xfo -1 -1 -1 -1 4-1 +1 +1 +1 -1 -1 -1 -1 +1 +1 +1 +1 0
Wcat -1 -1 -1 -1 -1 -1 -1 -1 +1 4-1 +1 +1 4-1 4-1 +1 4-1 0
As Figure 4, where the principal effects in the reactor temperature are shown, again only principal effect of the quantity of catalyst are not statistically significant in any region of reactor. The increase in the other three independent variables generates decrease in the temperature.
786 The all interaction effects in the temperature are not statistically significant. In the mole fraction of phenol, shown in Figure 5, the effects that contain the amount of catalyst are not significant. The interaction among initial reagent pressure and initial concentration of phenol presents positive values in the last tubular modules of reactor and assumes null value at the reactor exit. The simultaneous increase of initial reagent temperature and initial concentration of phenol worsens the reaction, since greater concentration of phenol appears. On the other hand, the increase in the initial reagent temperature and pressure simultaneously is suitable to reaction (negative effects).
£ •E
"1
ta> -co.
20 J
J 10 J
1 1i
^^T-^
^\
^
td^L_A-A^—A—A—A' ••-•--•-=J^ • • • '•«—1 1—|-q[_-|-' • ^ • ^ t ^ . . ^ ^
.1 ° -'°] o o 1 5 - ^ -204 2 « H
— ^ - To-Po —•— To-X^
Q. t -40 J (D I O -50-J Q. 1
—•— Po-W^^
0
,^\
yy-^V /
30 J
'
•
\
—*~ x„-w[|[ 1
2
3
4
5
6
—T 7
'
\ \ 8
reactor lenght
Figure 5. Perceptual interaction effects in the mole fraction of phenol.
4. Conclusions The proposed hierarchical procedure based on fractional and complete factorial design appears to have a large potential to deal with large scale non-linear multivariable process. The methodology of fractional factorial design was shown to be very useful for determination of relevant variables. This makes possible to consider initially a large number of variables and then to obtain a smaller set of variables that are the most significant ones. For the case study, the use of this method reduced the number of variables from eight to two. With this, it was possible apply the complete factorial design, including more two variables. Using the technique of factorial design it was possible study the reactor behavior through the principal and interaction effects, which are difficult to be identified in conventional parametric analysis procedures.
References Box, G.E.P., Hunter, W.G., Hunter, J.S., 1978, Statistics for experimenters. An introduction to design, data analysis and model building. New York, Wiley. Santana, P.L., 1999, Mathematical Modeling for three phase reactor: deterministic, neural and hybrid models, PhD Thesis, School of Chemical Engineering, Unicamp, Sao Paulo, Brazil (in Portuguese). Toledo, E.C.V, Santana, P.L., Maciel, M.R.W., Maciel Filho, R., 2001, Dynamic Modelling of a Three-Phase Catalytic Slurry Reactor, presented at 5th International Conference on Gas-Liquid and Gas-Liquid-Solid Reactor Engineering - GLS 5, 6th World Congress of Chemical Engineering, Melbourne, Australia - To appear in Chemical Engineering Science, 2002.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
787
On line optimisation of maintenance tasks management using RTE approach Sebastian Eloy Sequeira\ Moises Graells^, Luis Puigjaner^ Chemical Engineering Department. Universitat Politenica de Catalunya ^ETSEB, Av. Diagonal 647 Pav. G, 2° P Barcelona (08028) Spain ^EUETIB, Comte d'Urgell, 187, Barcelona (08036) Spain
Abstract The efficiency of equipment units decreases depending on the time they have been operating due to fouling, formation of byproducts, catalysts deactivation etc. Therefore, periodical maintenance and/or cleaning is required to restore original conditions. This imposes a trade>off between shutdown and productivity improvement. When considering different production lines, the problem is usually addressed using mathematical programming (MINLP- Jain, and Grossmann, 1998, Georgiadis et al. 1999). The implementation of the decisions obtained in this way is unlikely to result in an optimal operation because of plant-model mismatch. Therefore, this work introduces both: simpler formulations (NLP and LP) and a procedure for the subsequent implementation of the discrete decisions involved in a real time environment. This methodology is an extension of the Real Time Evolution (RTE) approach for continuous processes (Sequeira et al., 2001b). The main advantages of the proposed approach are the robustness resulting from the use of on-line information and a scarce model dependency. An example illustrates the methodology from the planning stage to the on-line optimization including model mismatch and model uncertainty.
1. Introduction The simplest case of maintenance is a single process whose performance (Instantaneous Objective Function, lOF) decreases with time illustrates. This will be the case for common semi-continuous (cyclic) operations as filtration, ionic exchange, catalyst with decreasing activity, etc. At some time r, the operation must finish in order to perform a maintenance task, which re-establishes the process initial conditions. Suppose that the maintenance task consumes a known time r^ and has a cost Cm- Thus, the lower the time r, the higher the performance but at the expense of a cost and time occurrence because of the maintenance itself. Therefore, there is a trade off determining the time topt that maximizes the overall performance of the whole operating cycle. A possible formulation for this optimization problem consists of maximizing a Mean Objective Function (Buzzi et al., 1984; O'Donnel et al. 2001): MOF (r) = {Benefits - Costs ) I Cycle _ Time
]lOF{T)dT-C^
Thus, the optimal maintenance time (topt) must satisfy:
(t^rj
1
(1)
788
*»^ lOF
*»»,^
^S^
•Ill
/ MOF
1 'op.
Figure!: lOF and MOF as functions of operation cycle duration
Figure!: Motivating example scheme
lOF(t^^,)it^^, ^TJ-^C,„ - jlOFiT)dT jlOF(T)dT-Q„ . ^ =0 hencq IOFit^^,)=^ r- • = MOF{t,J (2) Kt-^rJ [top,+rJ Therefore, using an appropriate model of 10F the optimum policy can be found (Fig. 1).
2. Off -Line Approaches Industrial practice poses more complex problems. In order to introduce such complexity, this section illustrates the effect of multiple feed presence trough a motivating example (Jain and Grossman, 1998). Three different raw materials (A, B, Q are available for arriving continuously to the corresponding storage tanks at a constant rate. These feed stocks are then processed sequentially in a reactor (the furnace), where because of cocking, the conversion decreases with time. The rate of arrival of different feed stocks is a decision variable {Fm^ bounded by Floi and Fupi. Every feedstock is processed in the furnace at a rate D,. Whenever there is a product changeover, the furnace is cleaned, and the operating parameters are set so that the furnace operates at the best possible conversion for that particular feed. The changeover time for feed / is known and given by rnii (sequence independent). The set up and cleaning cost for every feed is given by the constants Cm,. Raw material at different grades (A, B, C) is processed in the furnace to obtain a final product S (Fig. 2) at a conversion depending, for a given set of conditions, on the grade / and that decreases with time: (3)
Xi(t)= Ci + Ui ' e-^i ^
where «,, bi, c, are experimental parameters. Revenue is directly proportional to production through the price parameter P,. The problem is to determine the policy that maximizes the profit. Specific data for the example considered are found in Table 1. Table 1: Data for the motivating example
A B C
2 3 3
1300 1000 1100
0.20 0.18 0.19
0.10 0.13 0.09
0.18 0.10 0.12
160 90 120
100 90 80
350 300 300
650 600 600
789 2.1 MINLP Formulation (Fl) Jain and Grossmann (1998) propose the following formulation. The objective function to maximize is the average profit during the cycle Z. There are seven decision variables: total processing time of feed / to the furnace (r,j, number of sub-cycles during the total cycle (rii) and the total cycle duration (T^vr/J. The total time devoted to feed / in furnace, including processing and cleanup time is given by Ati.
_ ?K'''-
c , . - i , + ^ - n , ( l - e - - ' " . ) -Cm,.-n,. (4)
Tcycle ST:
A?, = « , T , + r ,
FmrT^,,, = Drt, ,
V/
(5) (6)
l^ti^T^yCe
(7) The solution reported corresponds to that in Table 2 (Fl): Table 2: Results applying formulation Fl, F2, F3 and F4.
fli
4
1
2
ti(d)
42.44
41.74
37.94
tSi(d)
10.61
41.74
18.97
\Pmi($/d)
9.76
115
21.11
5.95
9.19
11.25
9.76
9.19
11.25
53109
9827
23694
51476
13905
24950
53109
13905
24950
Tfi
0.363
0.322
0.316
0.380
0.308
0.311
0.668
0
0.332
0.600
0
0.390
Fnii (t/d)
397
300
300
411
300
300
650
0
288
650.0
0
345.3
53065
11168
Tcycle (d)
139.1
Z($/d)
30430
24057
30640
42670
41913
This formulation has some disadvantages. The solution is strongly dependent of arbitrary upper bounds used for Tcyde and n,. The reported solution corresponds to a maximum value for n, of 4 and 140 for Tcyde- For higher values of Tcyde and /z, bounds, the dependence still remains (i. e. for n, < 6 and Tcyde ^ 200, Z = 30507). Such fact suggests the possibility of another formulation, which does not consider the Tcyde concept. In the following, this formulation is introduced. 2.2 NLP formulation (F2 and F3) Assume that the furnace is operated with the feed / during a time tsi. The resulting mean conversion (Xm,) during the sub-cycle time r5/, can be easily computed, according to: i-b^.is,)
Xn\ (tSi) = ts;^ • f c,. + a. e dt = ts;^ \ c. -ts, +—• (1 - ^"^'"') i L ^i Therefore, for that feed, the contribution to profit Pm, will be (Eq. 2): Pm. =(ts. +Tm.y^iP.-D.'Xm.-tSi -Cm,)
(8)
(9)
790 Additionally, the average consumption of feed / (Fnii) can be expressed as a function of the fraction of time (77/) dedicated to producing with feed / in the furnace:
Fm. = (ts. + im. y' D. -ts^ -Tf.
(10)
Based on those variables, an alternative mathematical programming model can be stated as follows: maxz = ^ P m . - r / . (11) i
ST: b.-Xm.-ts^=crtsrb,+a.-are''""' Pm. -ts. + Pm. T„ = P, D. -Xnif -r^,. - Cm,. Fm, -W, + Fm,. T„ = D,. -ts, -Tf.
\/i V/ Vr
(12) (13) (14)
i
Flo. < Fm. < Fup. Under these circumstances, the decision variables are tsi and Frtii (the later in correspondence with Tf^. The optimum is also found and the corresponding results are given in Table 2 (F2). Thus, this new formulation gives a better MOF value for the same problem. The main reason is that there is no constraint related to the cyclic operation {Tcyde variable and associated constraints). It should be noted that in both cases (Fl and F2), the contribution to profit (Pm,'s) obtained processing raw materials A and C is substantially higher than B. Additionally, the lower bound constraints over feed B and C are active. For the specific problem where the lower bounds on Fm, are zero (which is likely the more common case), the results are shown also in Table 2 (F3). It can be seen that the improvement could reach about 35%, and that the upper bound in the most profitable feed is now the active constraint. 2.3 LP formulation (F4) Using the concepts introduced previously, a useful solution approach is proposed as a practical alternative to the last case. According to the information for every feed, the individual optimal values for tSi can be evaluated using the equation 2, and then the correspondent Fm/'s using the equation 1. This allows dealing a priori with the nonlinear terms of the problem. After that, the second formulation F2 is simplified to an LP formulation F4: max z = ^ Pm^Tf, (16) ST :
Xvy, = 1
(18)
791 Such an approximation reduces the number of decision variables to a half (only the Tfi or what is equivalent, Fm^. Additionally, as a consequence of the constraint imposed over the Tfi set (Eq. 18), is possible to eliminate another decision variable. The numerical solution for this example is shown in Table 2 (F4). The resulting z is only 1.8 % lower than the optimal (F3).
2. On-Line Approaches Suppose that the on-line information needed to compute lOF at a given time interval k is available. Suppose again that lOF(k) monotonally decreases. Under such circumstances, there will be an optimal period of time kopt for performing the corresponding maintenance task (Fig. 1). According to that, a simple solution is given by answering at every period k: ^'Should we stop for maintenance at this period or the following?". The answer can be obtained just by comparing of MOF(k) and MOF(k+I), at every interval k. Naturally, if MOF(k) < MOF(k-\-l) it is better to wait. Otherwise, when MOF(k ) > MOF(k+I) then we should stop at period k. It can be observed than this is equivalent to find when: MOF(k) - MOF(k-\-I) = 0, which is the discrete form of equation 2, and hence provides the optimal solution. This affirmation holds, even when Cm and Tm are functions of t rather than constant values. This kind of approach, termed Real Time Evolution (RTF, Sequeira et. Al, 2001b), relies on an evolutionary basis rather than formal optimization and has been successfully applied for optimizing continuous process in real time. The implementation of the mathematical programming result (F4) using a dynamic simulation environment (ASPEN CUSTOMER MODELLER) is shown in Figure 3. However, there are two main reasons for disagreement between the mathematical programming results and that obtained after its implementation over the plant. In the first place, a model mismatch. This will be the case for "bad" values of the model parameters a,, bi and c, for the considered example (or even structural mismatch). Secondly, as these parameters are determined by statistical techniques, usually least squares fitting, they are just averaged values. Therefore, the instant plant behavior will vary according to the degree of deviation observed during the adjustment stage. As a consequence, the results will vary according to the implementation methodology, because the previous factors will have more or less influence on the global behavior. A Montecarlo simulation was performed in order to reflect the model mismatch and the instant uncertainty over, at, bt and c, parameters. The formulation F4 was then applied in two different ways: First, using the policy given by the fixed tsi values resulting from formulation F4. Second, following the RTE procedure to on-line determinate these values. Figure 4 shows the difference between the objective function values (ZRTE and Zf4) obtained for different values of e (model mismatch) and a (uncertainty), where can be seen the expected benefits obtained when implementing an RTE system.
3. Conclusions This works presents both, a way for off-line calculation and a methodology for the online implementation of production scheduling problems relative to maintenance tasks. The advantages of the proposed approach are its simplicity and robustness, which make
792 it interesting for application to industrial cases where a DCS has been already installed. Certainly, this approach may not guaranty global optimum but, depending on the case, this will be largely compensated because its robustness achieved over the two key aspects always present: model uncertainty and model mismatch. In any case, future work includes investigating methodologies needed for reaching the global optimum online. However, the difference between the optimum and the proposed approach is likely to diminish with the problem complexity (in terms of number of feeds and furnaces) as well as when the parameter values of different feed-reactor pairs are similar.
\{Mwn4i\^ I'M
I 10
Figure 3: Simulation results when applying F4 over the plant
Figure4: Benefits obtained using RTE in the presence of model mismatch and uncertainty
4. Acknowledgements One of the authors (S. E. S.) wishes to acknowledge to the Spanish "Ministerio de Ciencia y Tecnologia" for the financial support (grant FPI). Financial support from CICYT (MCyT, Spain) is gratefully acknowledged (project REALISSTICO, QUI-99-1091).
5. References Buzzi, G., M. Morbidelli, P. Forzatti and S. Carra, 1984, Int. Chem. Eng. 24,441. Georgiadis, M. C , L, Papageorgiou, S. Machietto, 1999, Comp. Chem. Engng. SuppL, S203. Jain, V. and I. Grossmann, 1998, AlChEJ. 44, 1623. O'Donnell, B. R., B. A. Barna and C. D. Gosling, 2001, Chem. Eng. Prog. 97, 56. Sequeira S. E., M. Graells and L. Puigjaner, 2001a, Computer-Aided Chemical Engineering, Vol. 9: ESCAPE-U, Eds. R. Gani and S. B. J0rgensen, Elsevier, Amsterdam. Sequeira, S. E., M. Graells and L. Puigjaner, 2001b, Ind. Eng. Chem. Res. (submitted).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
793
Operation Decision Support System using Plant Design Information Yukiyasu Shimada\ Hossam A.Gabbar^ and Kazuhiko Suzuki^ Department of Systems Engineering, Okayama University ^Frontier Collaborative Research Center, Tokyo Institute of Technology
Abstract Operation support system (OSS) has been investigated to support the operator decision making in abnormal plant condition. The usefulness of OSS depends on the efficiency of the utilized operation decision method. This paper proposes an enhanced operation decision support system (ODSS), as an important function in OSS, using plant design information. At the plant design stage, the potential factors of dangerous situations are clarified and the causal relationships between causes and effects are investigated as a result of the risk assessment practices. Safety design and operating procedures of abnormal situations are examined based on such knowledge, which can also be useful to decide the appropriate operation at the plant operation stage. This means that plant design rationales can be reflected into the operation decision making. Accordingly, the system can show the reason of selecting the next operation. Prototype system of ODSS is developed based on the proposed operation decision method and applied to HDS process as a case study.
1. Introduction Plant operation is automated by introducing computer-controlled system; this including chemical plants. Such automation needs minimum intervention from the operator, especially in normal operation. While the reliability of the devices and equipments, which composes chemical plant, are improving and many safeguards system are developed, most of the accidents occuring in actual plants are due to operator mistakes. Therefore, the development of operation support system (OSS) to support the operator decision making in abnormal plant conditions is essential. In case of process malfunction, expert operator infers the cause of malfunction and its effect on plant condition and decides the approproate operation based on these plant condition (SCEJ, 1999). The usefulness of OSS depends on how the operation decision making methods are efficient. From the other side, most of the safety problems, related to plant operation, are discussed at the plant design stage, and are embodied through the risk assessment, instrumentation design, and safety facility design. However, it is pointed out that operators at the plant operation sites perform their daily work without having such safety management information, which includes design intent, objective and rationale. In order to overcome such limitations, this paper proposes the enhanced operation decision making method using the plant design information, which is implemented in a prototype system for operation decision support system (ODSS).
794 Plant design Plant information Risk assessment Safety design
Process plant
s—
(1) Data acquisition (2) FRAME
:\C=^
[B] Plant design
&
Operational KB
(4) Fault diagnosis
Historical information Heuristic knowledge
j'ast operation
(3) Monitoring
[A] Op-scheduler (DMSforOSS)
I
(6) User-interface
(5) Simulation
Operator
Fig. I System architecture of operation support system.
2. Outline of Operation Support System In case of process malfunction, expert operator infers the cause of malfunction and its effects on plant condition and decides the appropriate operation based on this plant condition. Vedam et al. (1999) proposed the framework of OSS highlighted by a dotted line in Fig.l. The roles of each module are summarized as follows; (J) Data Acquisition: This module acquires on-line data from the plant and provides it to other modules. (2) FRAME (Fault paRAmeter Magnitude Estimation): The magnitude and the rate of change of the root causes are estimated. (3) Monitoring: The process data for the presence of abnormalities are monitored. (4) Fault diagnosis: The root causes of detected process malfunction are identified. (5) Simulation: The consequences of a detected abnormal situation are estimated. (6) User-interface: The status of the process and the output results from each module are communicated to operator through this module (Shimada et al., 2000). And the plant condition and corresponding operation are indicated to operator. [A] Op-scheduler (Data Management System, DMS includes ODSS in this paper): This module manages the exchange of information among modules. The appropriate operation is decided based on the output results of each module. [B] Knowledgebase (KB): KB stores the plant design information, heuristic operation information and is searched to carry out each module. The above system architecture will efficiently suppor operator's judgement. Vedam et al. focused to assist the operator in quantitative diagnosis and assessment of current and future consequences of abnormal situation. The main target of this paper is the realization of ODSS, which is an important function within OSS. The proposed ODSS decides the next operation by referring to plant design information, including safety management information, based on the output results from each module. This idea has been realized in prototype system of ODSS.
795
3. Operation Decision Making At the plant emergency situation, operator has to judge whether the operation to be continued or stopped considering safety factors. In this proposed solution, operation decision making is performed by searching the information about the abnormal operation based on the plant condition. 3.1 Operation Category Plant abnormal operation can be classified as three major categories: (a) recovery operation, (b) partial shutdown operation (PSD), and (c) total shutdown operation (TSD). Recovery operation should be considered first that returns the plant back to normal condition by taking a measure against root cause of malfunction or by using the any suitable prevention means. PSD stops part of the plant temporary to protect the fault propagation within certain area under consideration of effective restart. TSD stops the whole plant safely to avoid the crucial problem. The proposed ODSS decides the appropriate operation in these operation categories and instructs operators with detailed operation procedures. 3.2 Knowledgebase (KB) of plant design information During the plant design stage, safety measures are considered and studied comprehensively to ensure the plant safety. These safety measures include information about plant abnormal operation and design rationale, which can be used to support operation decision making. In this proposed method, the results of risk assessment and the safety design information as well as the plant information are stored in the KB. (1) Plant information (plant structure, process behavior and operation): Plant information on plant structure, process behavior and operation is stored in the KB. Such information is needed to analyze how the process malfunction propagates within the plant. Abnormal operation procedures can be derived from normal operation procedure and used as the information on the operation support procedure. (2) Results of risk assessment (process malfunction, causes and effects): Information about the process malfunction's cause identification and its effect on plant condition can be acquired using risk assessment techniques such as HAZOP, FT A, etc.. Such information can be used to check the relationships between the malfunction and causes/effects. Also, the results of risk assessment, including information about the severity of process malfunction and fault propagation speed, can be used to analyze the severity of events/situation during the plant operation. (3) Safety design information (prevention and protection means): At the safety design, well-balanced safety facility design can be carried out based on information of fault propagation models and likelihood and severity of hazardous impact event. Recently this safety design is carried out based on IPL (Independent Protection Layer) concept (AIChE/CCPS, 1993). In this research, we have studied the design of the prevention means (Ex. stand-by pump) for high reliable plant operation, and protection means (Ex. Depressuring system) to protect against the fault propagation. The information collected and developed during the design of both the prevention and protection means can be used effectively to support operator deciding making.
796 Step (1) I
I>etect malfunction
I Detection means
Data acquisition, Monitoring
Step (2)
Step (2) Predict the effects
Identify the cause
FRAME, Simulation
Fault diagnosis
NO
Severe?
Step (3) YES
Remove r-d ., . .. ^ Failed the root cause Success Normal operation
Severity (Propagation speed* if Magnitude* j^)
Failed Success
Failed Success
TSD Success
2 restart operation for restart
Fig. 2 Basic procedure of operation decision making. 3.3 Operation Decision Making Procedure Previously, it has been pointed out that OSS makes no sense if it cannot identify the cause of the process malfunction. In this research work, the algorithm of operation decision making to protect against fault propagation is proposed, even if the cause cannot be identified as, shown in Fig.2 (Shimada et al., 2001). Steps (1) & (2): Detect malfunction and infer the cause and its effects When process malfunction is detected, the cause and effects are identified by SDGmodel-based and knowledge-based reasoning, or by using simulation techniques (Vedam et al., 1999). These methods are out of scope in this paper. Steps (3) & (4): For recovery operation If the effects are not severe and the root cause of the malfunction can be removed easily, the ODSS decides the removal operation to recover the plant back to normal condition. The information (KB) on recovery operation against the causes of the malfunction is searched. If the root cause of malfunction cannot be identified, or the measure for the root cause fails, the switch to prevention means is considered. If the prevention means such as stand-by pump can be available, then the ODSS can decide to switch to them as a recovery operation. Step (5): For shutdown operation If the recovery operation in steps (3) and (4) fails, or its effect on plant condition is severe, the shutdown operation is considered to protect against the fault propagation. First the ODSS tries to select the PSD to protect against the fault propagation.
797
OffGas
Feed
N^iitha Steam RefLixPump
Redundancy Pimp
PiDduct Diesel BTM Pump
Fig. 3 Stripping area of Hydrodesulfurization Plant. If the speed of fault propagation is high and there is a possibility that the process malfunction may expand in the whole plant and lead to the accident, the ODSS selects the TSD such as addition of shortstop. In this proposed method, OSS can suggest the operation according to the intents of plant safety design, because plant information and safety design information considerations at the plant design stage are included in the KB. The output results from ODSS is instructed to operator through the use-interface and operator can decide the appropriate operation fmaly. The different features from the conventional methods are to systematize the design information, including safety management information, and to use them for operation decision making positively. 3.4 Prototype for Operation Decision Support System A prototype ODSS is utilized as an experimental testbed for the proposed method. The environment is being implemented in Visual Basic (VB). The stripper area of HDS (Hydrodesulfurization) plant process is used as a case study, as shown in Fig.3. HDS plant is well-known process for removing sulfur from refinery distillates through a reaction with hydrogen. The stripper area removes H2S, which is produced at previous reaction area and produces the high purity diesel as the product. Overhead vapor from stripper is partially condensed in stripper overhead receiver and a part of it is taken out as by-product. Naphtha. In this process area, a redundancy pump is designed as a prevention mean for reflux pump. It is assumed that the temperature malfunction of stripper (Stripper-T(+)) in Fig.3 occurs. Fig.4 shows the candidate operations, which were displayed as output, within the developed ODSS. The left window shows the corresponding operation of: cause, temperature control of stripper as a recovery operation, and an emergency shutdown operation as a PSD. The right window shows the corresponding operation of: effect, the protection of cavitations as recovery operation, PSDs against stripping error and the prediction of pump cavitations. When operator selects one operation, the detailed operating procedures and the objectives (types) of such operation are displayed.
798
File(F)
Vtew
^^1
Opto
V3 i J i i C«uses / E«»cU
|sWgper-T_
For Cauaaa
ForCauMsJForEflvcU | Tamperatur* manual control ESO for Stripping Error
For Effact*
|
Pravant Cavitations ESO for Stripping Error {ESO for Pump Cavitations
Nama
Find 2 operation candidataa Find 5 fault propagation patha (Including 3 accidont accnario)
Zili
PREVENTION ESO ESO
E
h'anv Control (-) of Strippw
F^
0 0 0
^
Type
iPREVENTION
1 I
Operation Procadurs 3 causes / 5 effects
Close control valve C\M>3 Increaae Flow rate of Steam
Fig. 4 Output results from operation decision support system.
4. Conclusion This paper proposes an enhanced operation decision making method using plant design information, which is considered as a positive step towards the development of an effective operation support system (OSS). The presented approach can be used in plant abnormal operation to decide the appropriate next operation. The ideas in this paper have been successfully tested in an example of HDS continuous plant. The proposed operation decision support system (ODSS) can suggest the appropriate operation and explain the reason of selecting such operation. This will enable plant operator to understand the safety management information in clearer way, which will have a positive impact on reducing the human errors during plant operation. Also the offered solution can help process designers to debate the different design rationales during the plant design stage.
Acknowledgement This work was funded by Japan Society for the Promotion of Science, Japan.
References AIChE/CCPS, 1993, Guidelines for Engineering Design for Process Safety SCEJ, Society of Chemical Engineer Japan, 1999, Chemical Engineering Report, 38 (in Japanese). Shimada,Y., H.A.Gabbar and K.Suzuki, 2000, Study on Designing the Operation Support System, Proc. of PSAM5 (Osaka), 4, 2637. Shimada,Y., H.A.Gabbar, K.Suzuki and Y.Naka, 2001, Advanced Decision Method for Plant Operation Support System, Proc. of Loss Prevention 2001 (Stockholm), 619. Vedam,H., S.Dash and V.Venkatasubramanian, 1999, An Intelligent Operator Decision Support System for Abnormal Situation Management, Computers & Chemical Engineering, Suppl., S577.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
799
Supply Chain Optimization Involving Long-Term Decision-Making Jehoon Song\ Jin-Kwang Bok^, Hyungjin Park\ and Sunwon Park^ ^Department of Chemical Engineering, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Korea ^Samsung SDS, 707-19 Yeoksam-dong, Gangnam-gu, Seoul 135-918, Korea
Abstract This paper deals with the optimal design problem of multiproduct, multi-echelon supply chains for Supply Chain Management. The supply chains are composed of plants, warehouses, distribution centers, and customers. The locations of the plants and the customers are fixed. The locations and capacities of warehouses and distribution centers should be determined. The transportation and production of products should be determined, too. A mixed integer linear programming is used for the mathematical modeling. The objective is to minimize total cost of supply chains. We propose more detail modeling techniques in the cost calculations for the transportation of products and the installation of warehouses and distribution centers on the basis of the recent research on the supply network design by Tsiakis et al. (2001).
1. Introduction Recently, many companies have tried to seek various methods to implement SCM. However, the basic formation can be explained with the Supply Chain Operations Reference-model (SCOR) proposed by Supply Chain Council (SCC). SCOR divides the activities of supply chains into 4 categories shown in Figure 1. It illustrates that the plan category has influence on the supply chains more comprehensively than remaining three categories. Among many activities in the plan category, supply chain optimization that means determination of optimal location of facilities like plants, warehouses (WHs), and distribution centers (DCs), supply route, and transportation method is one of key issues of SCM (Simchi-Levi et al., 2000). Typical supply chains are shown in Figure 2. Actually, IBM has experienced cost reduction effect by 20-30% through supply chain optimization (Quinn, 1997). In this paper, we solve a supply chain optimization problem focusing on the points as follows on the basis of the work of Tsiakis et al. (2001) that reflects recent research trends. 1) The distance between echelons should be considered as well as the transportation volume of products when the transportation costs are calculated. The transportation cost should be expressed as a discontinuous piecewise linear function of the transportation volume, and a continuous piecewise linear function of the distance between echelons. 2) The installation costs of WHs and DCs should be linear functions of their capacities. The calculated initial installation costs are paid by installment considering the compound interest rate.
800 3) The single source assumption is applied only to the links between DCs and customers. Table 1. Distance from Plant to WH Plan (km) Suppliers Customers) WH ' Plant Source Make > Delivery 1 2 3 4 1 2 3
Figure 1. Supply Chain Operations Reference process.
299 453 390
72 418 240
206 153 313
155 95 359
Table 2. Distance from WH to DC (km) WH DC 2 3 4 1 1 96 201 153 281 376 193 303 280 2 471 437 97 145 3 99 244 4 30 248 Plants Warehouses
Distribution Centers,
' Customers Figure 2. Supply chains. Table 3. Distance from DC to customer (km) CI C2 C4 C3 C5 DCl 429 156 323 269 58 DC2 282 351 216 139 160 DC3 365 197 168 139 65 DC4 109 216 432 22 357
C6 208 406 338 132
C7 39 319 151 151
C8 188 181 333 210
C9 134 28 270 366
CIO 309 294 541 284
Cll 110 158 297 126
2. Problem Description We deal with the supply chains including three products, three plants, four candidate WHs, four candidate DCs, and eleven customers. The location of plants and customers are fixed. To satisfy customers' demands for each product at the least expense in supply chain, the location of WHs and DCs, production in plants, and the capacities of WHs and DCs should be determined. The transportation links and transportation volume among plants, WHs, DCs, and Customers should be also determined.
3. Mathematical Model 3.1 Objective Function The objective of optimization is to minimize total cost in supply chains. Total cost = {installation cost of WHs} + {installation cost of DCs} -i- {production cost in plants} + {operation cost in WHs} + {operation cost in DCs} + {transportation cost from plants to WHs} + {transportation cost from WHs to DCs} + {transportation cost from DCs to Customers}
801 3.2 Constraints The constraints used in this problem are on the basis of Tsiakis et al. (2001). To avoid simple duplication, the categories of constraints are just referred. But in detail, we explain the cost calculation techniques considered in this paper. Network composition Transportation Mass Balance Capacity Transportation cost In general, the transportation cost of products between two echelons can be expressed as a function of transportation volume and the distance between them.
D l DaD2
D3 DbD4
Q l QuQ2
Q3 QVQ4
Transportation volume(t)
Figure 3. Transportation cost as a linear function of the distance.
D l DaD2
D3 DbD4
Distance
Figure 4. Transportation cost as a linear function of the transportation volume.
Q1QUQ2
Q3 Q v Q 4
Q5
Transportation v o l u n e (t)
Figure 5. Transportation cost as a Figure 6. Transportation cost as a continuous piecewise linear function of the discontinuous piecewise linear function of distance. the transportation volume. CTR = f(Q,D) (1) where CTR: Transportatin cost, Q: Transportation volume, D: Transportation Distance. If we calculate roughly, the transportation cost is proportional to the transportation volume with constant transportation distances, and to the transportation distance with constant transportation volume. CTR = kDQ (2) where k: proportional constant. Then, since Q is a decision variable, and D is a parameter, eqn. 2 can be expressed as a divided form.
802 CTR = kqQ (3) kq = kdD (4) where kq: transportation cost per unit transportation volume, kd: transportation cost per unit transportation volume and distance. That is to say, if D is given as a parameter, constant kq can be obtained since kd is also constant. Therefore, if only decision variable Q is determined, the transportation cost is obtained. Figures 3-4 illustrate this relation graphically. In Figure 3, if the transportation distance is Da, kqa is fixed by eqn. 4. This kqa means the slope in Figure 4. If transportation volume Qa is determined, transportation cost Cu is obtained by eqn. 3. In the same way, if distance is Db that is different from Da, different slope kqb is obtained. If transportation volume is Qv at this time, the transportation cost is Cv. Realistically, however, the transportation cost is not linearly proportional to the transportation volume and the distance. Based on the economy of scale, the larger transportation volume, the lower transportation cost per unit volume at constant distance, and the farther transportation distance, the lower transportation cost per unit distance at constant transportation volume. Furthermore, the transportation cost is expressed as a continuous piecewise linear function of transportation distance, and as a discontinuous piecewise linear function of transportation volume. Figures 5-6 illustrate this relation graphically. In Figure 5, the number of possible ranges of transportation distance is S, and the distance of the point where the slope changes is Ds. The slope in the interval [D,./, DJ is kds. Then, based on the economy of scale, the slopes have the features as follows. kd^_,>kd^, V5 (5) Generally, kds forms a sequence that has a certain rule such as geometric sequence. If the sequence {kd^} is a geometric sequence, the general term of the sequence is like this.
H = kd.CR'-'
(6)
where CR: commom ratio. If the distance Da (D^.y < Da < D,) is given, the transportation cost per unit transportation volume is like this. s
s-\
kqaj obtained in this way is a basic slope of the transportation cost function of transportation volume in Figure 6. In Figure 6, the number of possible range of transportation volume is /?, and the volume of the point where the slope changes is Qr. The continuous slope in the interval [Qr-y* Qr] expressed as a dashed-line is kqr. Then based on the economy of scale, the slopes have the features as follows. kq^_,>kq^, \fr (8) However, the real slopes of the cost for the interval [Qr.h Qr] are discontinuous. They are smaller than continuous slope by some ratio. So they make discontinuity. Let the ratio h. Generally, kqr forms a sequence that has a certain rule such as geometric sequence. If the sequence [kqr] is a geometric sequence, the general term of the sequence is like this. The first term kqal is from eqn. 7.
803 kqa^ = kqa^CR"^ (9) If the distance Qu (Qr.i < Qu < Q^) is given, the transportation cost per transportation volume is like this. Cu=kqa^h(Qu-Q^) + Ca^ (10) However, since Qu is not a parameter but a decision variable, it is possible to exist in any interval of R intervals. Accordingly, a new set of binary variables Zr that denote if Qu is in a certain interval [Qr.h QA is introduced. Therefore, the transportation cost is determined by the equations as follows.
a-iZ.
vr
(11) (13) (14)
kqa^ = kqa^CR''^
(15)
R
(16)
Table 4. Variation range of transportation cost rate based on transportation volume and distance.
Range 1 Range2 Range3 Range4 Ranges
Plant-WH 0-20 20-40 40-60 60-80 80-100
Transportation Volume (ton/week) Unit WH-DC DC-C 10 0-20 20 20-40 30 40-60 40 60-80 50 80-100
Distance (km) cost ($/ton) 400 600 700 750 775
100 200 300 400 500
Table 5. Production cost at each plant and handling cost for each product at warehouses and distribution centers ($/ton). Product PDl PD2 PD3
Production cost P2 P3 PI 62.27 33.33 92.56
59.45 61.44 35.44 37.55 90.01 88.79
WHl 4.25 5.28 4.98
WH2 4.55 4.06 4.93
WH3 4.98 4.25 4.06
Operation cost WH4 DCl
DC2
DC3
DC4
4.25 5.28 4.98
4.55 4.06 4.93
4.98 4.25 4.06
4.93 4.55 5.28
4.93 4.55 5.28
Installation cost In general, the installation costs of WHs and DCs can be expressed as a function of the capacity. CI = f(G) (17) where CI: installation cost G: capacity. The installation cost is proportional to the capacity. CI=kgG (18) where kg: installation cost per unit capacity. However, this cost is initial investment cost. For calculation with operating costs, it is necessary to convert into the payment per unit time. Simply, the cost divided into usage time is used, but we use installment-paying method in this problem. If the initial installation cost of WHs or DCs is A, the period is n (520 weeks), and the rate of interest is r (weekly 0.001), then the installment paid every week is expressed as eqn 19.
804
r(l + r)" a = (l + r ) " - !
(19)
Therefore, the time-based installation cost of WHs and DCs is expressed as follows. (20)
(1 + r r - l ^
where CJ: installation cost per unit time. 3.3 Results The optimization model involves 1160 binary variables, 2622 continuous variables, and 4320 equations. This MILP model was solved with CPLEX in GAMS 2.50. 1416 seconds of CPU time was taken to solve the model. The optimal supply chain configuration is shown in Figure 7. Selected data are shown in Table 6.
4. Conclusions Supply chain otimization problem is an essential part in SCM. We addressed supply chain optimization problem including multiproduct, multi-echelon facilities. A MILP model based on the work by Tsiakis et al. (2000) was used, but we focused on the realistic cost calculation. We considered the distance and transportation volume with continuous/discontinuous piecewise linear functions. The compound interest rate was considered for the calculation of installation costs. Considerations of Multi-period and uncertainty are needed for further study.
Acknowledgement This work was partially surpported by the Brain Korea 21 Projects.
References Quinn, F.J., 1997, The Payoff!, The Supply Chain Series Part Six, Logistics Management & Distribution Report, Dec. 1. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E., 2000, Designing and Managing the Supply chain, McGrawHill, Singapore. Tsiakis, P., Shah, N., PanteUdes, C C , 2001, Design of Multi-echelon Supply Chain Networks under Demand Uncertainty, Ind. Eng. Chem. Res. 40, 3585.
Table 6. Transportation volume (ton/week)
Plants Warehouses
Distribution Centers Customers
Figure 7. Optimal chains configuration.
supply
PDl PD2 PI 40 40 P2 WH4 40 78 40 P3 40 PDl PD2 DCl C5 34 7 C7 7 C8 5 Cll 10 34 DC4 CI C4
43 12
PDl PD2 PD3 43 49 DCl 22 83 20 WH4 DC2 75 DC3 45 44 40 PDl PD2 PD3 21 11 DC2 C2 9 7 C3 28 50 27 C6 C9 15 CIO 3 20
PD3 55 0 20 PD3 5
16 33
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
805
Modelling Liquefied Petroleum Gas Storage and Distribution S.L. Stebel, F. Neves-Jr., L.V.R. Arruda, J.A. Fabro, L.C.A, Rodrigues E-mails: {stebel, neves, arruda, joao, lcar}@cpgei.cefetpr.br CEFET-PR, CPGEI Av. Sete de Setembro, 3165, 80230-901 Curitiba, PR, BRASIL Tel.: +55 41 310-4707 Fax.: +55 41 310-4683
Abstract This paper presents two models for the Liquefied Petroleum Gas (LPG) transfer and storage operations in a refinery. First, a simulation model is proposed, based on Petri Nets, which integrates the continuous and discrete parts of the process. The second model uses mixed integer linear programming techniques for the optimisation and scheduling of the system. The models allow the visualization and simulation of the problem, helping the system operator to easily test and correct scheduling plans despite the complexity of the operations. Results from the simulation model, when applied on the optimisation model, can then be used to reduce the severity of the problem.
1. Introduction The optimisation of operational processes is critical to the success of petrochemical organisations. The optimal allocation of productive features takes into account monetary, physical, and operational restrictions imposed by the structures and production processes. Optimal allocation is a priority for the organizations. In several activities of business, the classical problem of optimisation appears: how can critical features be used most efficiently? Oil refineries have used computational features directed at the activities of planning and scheduling since the 1950's. The techniques commonly used for these activities are based on linear programming (Williams, 1999). This methodology is still predominandy employed in the modelling of chemical plants, despite its modelling capacity limitations. This paper focuses on the development of two approaches for LPG distribution and storage operations. The main goal is to provide an auxiliary tool for the decision making process. The Petri Nets model enables a diagnosis of critical points of scheduling. The objective of the second approach proposed is minimising operational costs. The operational and physical constraints, as well as product demands, are considered in both approaches. Since storage plays a major role in the scheduling of continuous processes, the proposed approaches are based on the different possible states of the spheres. These are: ready to receive, receiving, ready to deliver, delivering, and product analysis. The developed models allow a better storage and distribution system understanding, providing a simulation tool. A graphic process simulation facilitates the scheduler's task,
806 allowing the scheduler to easily experiment and correct the scheduling, despite the complexity of the operations in a real life scenario. Moreover, the modelling process allows the detection of bottlenecks and provides mathematical problem analysis.
2. Problem Deflnition LPG is basically a mix of hydrocarbons with 3 (C3 - propane) and 4 (C4 - butane) carbon atoms, receiving its name from the fact that it can be liquefied by compression at room temperature. This product may be used as domestic fuel for cooking and heating. The fact that LPG can be liquefied at relatively low pressure facilitates the storage of large amounts using, in general, spherical tanks, known simply as spheres. In a typical refinery, the catalytic cracking process is the principle method of producing LPG with smaller amounts being produced by the crude distillation column, delayed coking, etc (Pinto and Moro, 2000). The deficit of Liquefied Petroleum Gas (LPG) is a persistent problem in some Brazilian refineries. The Araucaria refinery, located in southern Brazil, is an example. It only produces ^^ of its LPG demand. The final quarter is supplied by tankers, which unload in a harbour. The product is then conveyed to the inland refinery by a pipeline system. The refinery is responsible for the LPG distribution to small delivery bases as well as to a pipeline of finished products (Schechtman et al., 2000). In order to manage storage constraints some operational decisions must be taken by the scheduler. The decisions are currently taken based on personal experience, with the aid of manual calculations. Due to storage and distribution complexity, the operational decision making process becomes a difficult procedure (Stebel et al., 2001a). Some decisions to be taken by the scheduler are shown in figure 1. The overall scheme of the LPG storage and distribution area is shown in figure 2.
Figure J: Schedulermade decisions
807 LPG PRODUCTION
C4 STORAGE
HARBOUR
O
oo oo o oo oo
LOCAL MARKET
1
*~~
LPG and C3 STORAGE
OTHER MARKETS
Figure 2: LPG storage and distribution
3. Simulation Model The LPG Storage and distribution process is continuous. The spheres are lined up with the pipeline and then loaded (or unloaded) continuously, in accordance with the flow of each product involved in the operations. However, the decision operations can be considered discrete events in time, for example, at the beginning of the receiving operation. Therefore, the LPG storage and distribution process modelling can be done using Petri Nets. To achieve this, t-temporised nets (Reisig, 1985) were used. Utilising this net, the time for the accomplishment of a transition is placed at the transition, determining a minimum time until a transition is qualified and can be triggered. The duration of each process task is codified by the time definition (receiving, period of product analysis, and delivery). The Petri Nets allow dynamic simulations of problem instances. The model allows the visualization and simulation of a problem. It helps to test and correct scheduling plans despite the complexity of the operations, as well as making it possible to identify bottlenecks within the system. The model is separated in two main parts, one for the receiving of gas and another for its delivery. The receiving model has two sources, that are (Stebel et al., 2001a):
808 • the constant gas flow (C3 and C4) from process units; • the flow from external sources, represented by a harbour, which transfers LPG to refinery using a pipehne. After the spheres receive any product it is necessary to wait some time until the product can be delivered to customers. This permits water separation and draining as well as product analysis (Pinto and Moro, 2000). The delivery process is also carried through two pipelines that are in the LPG area. One of the pipes sends gas to the local market and the other to other markets (located in another state). Local companies can receive gas in three ways: C3, LPG, C3 + C4 (mixed in line). C3 and C4, when mixed in line, must be sent at the same time and the same quantity. The delivery to other markets must be of a minimum amount of gas. Due to the fact that the pipeline is used by a lot of products. The Petri Nets model was divided into blocks, LPG receiving process, storage and sending. The processes of receiving and sending were modelled taking into account their continuous character (Drath et al., 1998). Therefore, the intermediate states of the processes can be visualized by these models. This was possible through discretisation of the gas flow and using timed transitions. Each token, in this part of the model, represents an amount of 50 m^ of gas. This value represents the highest common divisor between the various flow rates of a pipeline system. The simulation software used to implement the Petri Nets model was the Visual Object Net+ +(Drath, 2001).
4. Optimisation Model The optimisation model is based on mixed integer linear programming with a uniform discretisation of time (Stebel, 2001b). The representation of time and the model structure itself are the two main reasons for building the model (Pinto and Moro, 2000). When the problem has a large scheduling time horizon, it is not practical to use a uniform discretisation of time, which makes problem solution a difficult and timeconsuming task. The problem in question has a scheduling horizon of between one and two days, allowing a simulation with discretisation units of one, two or four hours. The proposed approaches are based on the different possible states of the spheres causing an increase in the model size. The considered model has, as its objective function, the minimisation of the costs involved in this area of the plant. These costs are: changeover (between two spheres), storage, and electricity. These costs were normalised to represent the qualitative aspects of the system. The model will be more efficient if the costs are derived from another model. The specialist information will be represented in a rule based system. Through this, the different costs will exert a distinguishing influence on the model. The constraints are derived from operational procedures, material balance, flow rate, and
809 demand. The total number of constraints and variables depends on the input data of the problem. Shown in (1) is an example of model constraint (Stebel, 2001b). Each sphere, at all times, should be either: receiving {ERe,o,p,t)y analysing a product (EAgj), delivering (EEed,p,t)^ ready to receive {LRei), or ready to deliver (LEej). These variables are binaries and they represent the different possible states of the spheres:
11 ER,_„_p, + EA,, + ^
\fee
^
EE,^,^p, + LR,, + LE,, = 1
(1)
EPROPj
The LINGO (UNDO, 1999) solver was used to implement the MILP optimisation model. The simulations were done on a PC (Pentium III 850 MHz). In order to reduce the CPU time, the relative optimality criterion was set to a non-zero value and was tested to reduce the size of the search tree providing significant timesaving.
5. Model Integration And Results It is possible to make an analysis of the feasibility problem with the simulation model. It is also possible to determine the minimum resources to be carried by all the operations. However this analyses does not determine an optimal point of operation. This is only possible through the optimisation model. The structure of the optimisation model enables the insertion of the amount of resources to be simulated. For each simulated resource the number of the binary variable can be increased. Therefore, using minimum resources, optimal scheduling can be identified without the necessity of generating binary variables, which are not used. Consequently allowing for a reduction in the size of the optimisation model. An example is the number of spheres necessary to carry out all the operations. This number is used in the optimisation model to search for the best answer, however, only searching in a narrow search tree. Table 1 shows an example of this. From the simulation in Petri Nets model it was possible to perform all the operations with six spheres (4 propane and 2 butane), but this model cannot guarantee optimal solution. This is possible through the optimisation model. If all spheres are used (7 propane and 3 butane) the CPU time will be Imin 27s with simulation in contrast to 7min 12s without. This difference occurs because of the increase in binary variables. Table 1: Computational Results
number of 0-1 total spheres variab. variab. 6 342 1768 10 576 3656
nodes objective constraints iterations CPU time function lmin27s 30476 4340 22 106 7minl2s 71259 9054 106 26
6. Conclusions The LPG scheduling problem lies basically in the determination of the best policy for the utilisation of storage resources. Petri Nets checks the feasibility of the modelling of
810 a continuous process through discrete events. This model integrates the continuous part of the process (load and unload) to the discrete part of operational control. It allows the visualization and simulation of some instances of the problem. This problem can be modelled as a MILP model with uniform discretisation of time. The user's interface is carried out through a spreadsheet, where the data input is kept and the Gantt chart is presented. With the change of some restrictions, these models can be applied to other refineries. The results obtained are better than those obtained currently by the operator with the use of manual calculations. Not only can the developed models be used to test receiving and sending plans to find the most suitable scheduling, but also to detect and correct possible problems arising. Furthermore, the modelling process allows for the detection of problem bottlenecks. These models can result in costs reduction or an increase in the plant profit. Integrating models can facilitate obtaining results more efficiently, but at the moment this is done manually.
Acknowledgments The authors acknowledge the financial support from the Brazilian National Agency of Petroleum (PRH-ANP/MCT PRHIO CEFET PR).
References Drath, R., U. Engmann and S. Schwuchow, 1998, Hybrid Aspects of Modelling Manufacturing Systems by Using Modified Petri Nets, International Federation of Automatic Control - Preprints of the 5th IF AC Workshop on Intelligent Manufacturing Systems, Gramado - RS, Brazil. Drath, R. Short User's Guide for Visual Object Net + +. Obtained in September 2001 at http://www.systemtechnik.tu-ilmenau.de/~drath. UNDO, 1999, UNDO: The Modelling Language and Optimizer - User's Guide, UNDO Systems Inc. Chicago, Illinois. Pinto, J.M., Moro, L.F.L., 2000, A Mixed Integer Model for LPG Scheduling, In: European Symposium on Computer Aided Process Engineering-10, Comp. Aided Chemical Engineering (8) (S. Pierucci(Ed)), 1141-1146, Elsevier, Amsterdam. Reisig, W., 1985, Petri Nets - An Introduction, Springer-Verlag. Schechtman, R., Vieira, J.V.C, Moreira, J.G.S, Costa, L.S., Nascimento, D.L., 2000, LPG Demand Outlook: 1999-2004, Proceedings of the Rio Oil & Gas Expo and Conference (in Portuguese). Stebel, S.L., Fabro, J.A., Neves Jr, F., Arruda, L.V.R., Tazza, M., 2001a, Simulation of the LPG Transfer and Storage Process by Petri Nets, Proceedings of the V Brazilian Symposium of Artificial Intelligence (in Portuguese). Stebel, S.L., 2001b, Modelling of the Liquefied Petroleum Gas Storage and Distribution in an Oil Refinery, Master Thesis, CPGEI / CEFET-PR, 100 pages (in Portuguese). Williams H.P., 1999, Model Building in Mathematical Programming, 4* ed. Chichester (England): John Wiley & Sons Ltd.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
811
A Successive Mixed Integer Linear Programming approach to the grade change optimization problem for continuous chemical processes Rob Tousain and Okko Bosgra Mechanical Engineering Systems and Control Group Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands [email protected]
Abstract To enable flexible, market-oriented operation of continuous chemical processes the control of transitions between different operating conditions (hereafter called: grades) should be enhanced. Dynamic optimization is believed to be a major enabling technology in this respect. In this paper, we formulate the grade change problem as an economic optimization problem using a finite horizon evaluation of the added value of the processing plant. The resulting objective function is discontinuous due to the transitions between grade-regions which makes standard, gradient-based optimization methods unsuited for solving the problem. A new. Successive Mixed Integer Linear Programming approach is proposed and its potential is demonstrated on an example: the optimization of transitions in a binary distillation column.
1. Introduction The operation of many multi-product (or multi-grade) continuous chemical processes is subject to rapidly changing market conditions and strong competition. Ideally, the production should continuously track the most favorable market conditions to guarantee good margins. The moderate status of most current industrial process control systems is a main obstacle in doing so. The result is that many multi-product plants still are operated in a highly inflexible fashion, running through a predetermined and fixed sequence of product types, also called a slate, or product wheel, as described in e.g. (Sinclair, 1987). One way to improve the flexibility of the operation is by the use of model-based dynamic optimization technology. Trajectories resulting from such an optimization effort can be combined with Advanced Process Control to achieve automated and well predictable control of process. The actual implementation of the optimal trajectories will not be treated in this paper, instead we will focus on the definition and the solution of the grade change optimization problem. The use of model-based optimization in switchability analysis for multi-grade processes was discussed by White et al., 1996, where the feasibility problem was reformulated as a minimization problem with arbitrary quadratic penalties and solved using a gradientbased algorithm. A similar approach was used by McAuley and MacGregor, 1992, for the optimization of grade transitions in a polyethylene plant. In this paper, we consider the optimization of grade transitions with economic cost functions. The new approach that is described in this paper proceeds via a mixed integer
812 description of the grade regions and the corresponding production rates. The feasibihty of the proposed method is illustrated by means of an example, grade change optimization for a binary distillation column.
2. Grade change problem formulation 2.1 The plant We assume the continuous chemical manufacturing plant to be described by the following system of differential algebraic equations (DAE): A : = / ( A : , U , y),
0 = g ( x , u , y),
z = C^X'}-C^u
(1)
where x are the states, y the algebraic variables, u the inputs and z the so-called performance variables, i.e. all variables that are required for the performance evaluation (objective and constraints) of the plant. The operating constraints are expressed as follows
fiz)<0
(2)
Many continuous chemical plants can be operated in different production grades. A production grade can be identified by a set of specific characteristics of the end products or the mix of end products that is produced. Examples of product grades are grades of high-density poly-ethylene and certain purities of a distillation product. We define a production grade fif by a corresponding set of inequality constraints Rg{z)<0
(3)
The operation of a multi-grade plant can be characterized in terms of finite-time quasistationary tasks and transitions between those. 2.2 Quasi-stationary tasks: static optimization The operation of the plant during the quasi-stationary intervals is assumed to be realized according to a static optimization of the process economics, where the objective function to be minimized is given as follows
Uz) = -S.P"X(2) + S.P'-^C,(z)
(4)
Y^(z) is the yield of end product e. C^(z) is the consumption of raw material or utility r . The prices of the end products and the raw materials/utilities are given by respectively p^'^ and p^ "^. 2.3 Transition tasks: dynamic optimization We consider two production grades g and h and a transition between those. Initial and target conditions are given by the stationary optimal operating conditions for the two production grades, respectively (x^ ,u^) and ( x ^ , u ^ ) . In literature, no consistency exists as to the general definition of the grade change optimization problem. From a theoretical viewpoint it might be argued that the optimal
813 changeover strategy is given by the solution of a time-optimal control problem, see e.g. (Lewis and Syrmos, 1995). However, note that the classical time-optimal control problem forces all states to reach the desired end-point as soon as possible. This does not imply in general that the fastest transition between grade g and h is achieved because the grades are defined by their corresponding sub sets of the state space. Also, the time-optimal formulation does not honor the fact that valuable end products and often invaluable off-spec materials are produced during the transition. Finally, market situations may exist in which there is no incentive to implement the fastest transition possible. For example, during periods of low market demand it may be more advisable to minimize transition costs instead. An alternative formulation, which can be seen as a generalization of the economic grade change optimization problem for different market situations is proposed here. The corresponding finite-time economic objective function is the following:
J = Jl-J,^cc'YMt))+2j'CMt))dt
(5)
where T is a fixed end-time, which should of course be chosen larger than the minimum transition time, a^ and p' are weighting parameters. Emphasis on transition time can be introduced by selecting all )3 *" 's equal to zero and the a^ corresponding to the product that is being produced in the target grade equal to a large value. Emphasis on transition economics can be introduced by selecting a^ = p ^ ' ^ , and jS'^ = p ^ ' ^ . In the most general setting, a number of different end products is produced in each grade. The yield of end product e then is given as follows: y,[z) =^^Gnz)M'^P(z) (6) where P(z)
is a vector function representing all material flows from the plant and
M J', a row vector with zeros everywhere except for a * T at at most one location, assigns the material flows to the e^^ end product for grade gf. G ^ are so-called grade variables and are defined as follows
[0,
otherwise
The grade change optimization problem is defined as to minimize the objective (5), subject to the model equations (1), the path constraints (2), the initial and endconditions corresponding to the departure grade and the target grade and with the product flows given by (6) and (7). The choice of the economic objective function makes the optimization problem discontinuous. Next, we will describe how this optimization problem can be solved.
3. Solution approach 3.1 Exploration of possible approaches In this work we focus on the sequential approach to dynamic optimization (see e.g. (Vassiliadis, 1993)). The standard sequential approach uses an outer loop gradient-based Nonlinear Programming tool (e.g. Sequential Quadratic Programming or Generalized Reduced Gradient) to solve the finite-dimension optimization problem that results after
814 control parametrization. This approach assumes the objective as a function of the parameters to be twice continuously differentiable which does not hold for the economic grade change problem. To circumvent this, we proposed in an earlier publication to approximate (7) by smooth functions (e.g. 'arctan'-functions) and we presented a tailormade sequential optimization routine for solving the resulting smooth but strongly nonlinear problem (van der Schot et al., 1999). This approach works well on many examples, however it may get stuck in poor local minima since it uses gradient-based algorithms on a problem that is strongly nonlinear by nature. The approach we present here treats the discontinuities through the introduction of a set of binary decision variables in the optimization problem. 3.2 A Successive Mixed Integer Linear Programming approach The crucial step in this approach is the introduction of grade variables Gf E {0,1} at time instances iT^, i = 0...T / T ^ . The following constraints enforce that the grade variable for which the corresponding grade constraints (3) are satisfied, is set to one: R ^ ( z J - ( l - G f ) 9 ^ <0, ^^Gf = 1 (8) where z. = z[iT^). This is a standard technique for coupling real and binary variables, see e.g. (Bemporad and Morari, 1999). Note that Gf enters these equations linearly. For feasibility of (8) we require the fixed parameter Q^ to be larger than the maximum that is attained by R (z.) on the feasible set in which z^ lives and that all grade regions are adjacent. Next, we introduce new continuous decision variables Y^. which represent the flow of end product e during operation of the plant in grade g . We require (9)
Y'eA
where the fixed parameter Yf^ should be chosen larger than the maximum end product flow. Yf^ relates to the material flow P(z) as follows l^le^eYf,
=PU<)
(10)
where N ^ maps the flow of end product e to the material flow in grade g . Finally, the total end product yield at time iT is given by y . = V Y ^. - Using the integer description of the grade regions and after discretization of the objective (5) (using e.g. the Riemann sum) and the path constraints (2), a mixed-integer nonlinear progamming (MINLP) formulation of the transition problem can be derived. Several approaches towards solving such MINLP's exist; most popular methods are branch and bound and cutting plane methods, see e.g. (Floudas, 1995) for an overview, however their applicability is generally limited to small problems only. For most problems, h , R , and P will be linear functions, leaving the process model (J, g) the only remaining nonlinearity. Therefore, we propose a successive linearization approach for the problem at hand. The linearization of the MINLP can be obtained by substituting the nonlinear dynamics by the linearized-time-varying (LTV) dynamics that describe the behavior
815 along the solution that results from the previous iteration. The LTV dynamics can be obtained for example by sampling the solutions of the sensitivity equations (St0ren and Herzberg, 1994). The linearized (inner loop) problem then is a Mixed Integer Linear Program (MILP) which is solved in each iteration of the outer loop. Well-established Branch&Bound techniques can be used to solve the MILP inner loop problem to a desired accuracy. Only local minima can be guaranteed.
4. Application to a binary distillation column grade change problem As an example of the grade change optimization we consider a model of a 20-tray binary distillation column based on the CONSTILL example by Ingham et al. 1994. The inputs are the reflux ratio (Uj) and the reboiler duty (u2)- The performance variables are respectively the top purity, the bottom impurity, the distillate flow, the bottom outflow, the reflux ratio and the reboiler duty. We distinguish 3 different top-products, 2 different bottom-products and hence 6 product grades given in the following table.
9 1 2 3
Constraints
(Rg(z))
0.00< z ^ <0.98
0.00
0.00
0.05
0.98< Z ^ <0.99
0.00
g 4
Constraints ( i ? ^ ( z ) ) 0.98
0.05
5
0.99
0.00
6
0.99
0.05
We consider a changeover from grade 3 to grade 5 with the economic objective (5) and prices of the top product chosen equal to [1,2,4] for rising purity and of the bottom product [1.5,0.8] for rising impurity. Reboiler duty costs 0.25 per unit. The discretization interval used in the piecewise constant control parametrization and in the Riemann sum approximation of the objective is of length 0.3 hr. The optimization horizon has length 6 hr. Additional constraints on the rate of change of the inputs are imposed, respectively 0.25 and 200. The end point constraint is relaxed by 0.001 in order to ensure feasibility of the optimization. Implementation of the optimization algorithm is done using gPROMS (for function and gradient evaluations) and GAMS with CPLEX (for the MILP optimization). The optimization converges in 7 iterations. Progress in each iteration is measured using a specific merit function that is based on the smooth approximation of the grade regions proposed in (van der Schot, 1999). Details are omitted. The optimization results are plotted in Figure 1 (in between the vertical rulers). Due to the economic attractiveness of grade 5 there is a clear incentive to establish the grade change (in this case equivalent to a change of the top purity) as quickly as possible. The rate-of-change constraint on the reflux ratio is bounding the performance.
5. Conclusions A formulation of the grade change problem for continuous chemical processes as an economic optimization problem is presented. This formulation leads to a discontinuous optimization problem which cannot be solved using the standard sequential or
816
2800, I 2600 ^2400 •° 2200 2
3 4 5 time [hr.]
Figure 2: Optimal trajectories of the controls (left two images) and the performance variables for a transition from grade 3 to grade 5. simultaneous dynamic optimization method. Using control parametrization, a Mixed Integer Linear description of the grade regions, and a Riemann sum approximation of the economic objective the problem can be transformed into a Mixed Integer Nonlinear Program (MINLP), which we can solve using a successive linearization approach. Search directions are computed from the Mixed Integer Linear Program (MILP) that results after linearization of the dynamics. The feasibility of the approach is demonstrated on an example: the optimization of grade changes in a binary distillation column.
References Bemporad, A. and Morari, M., 1999, Control of systems integrating logics, dynamics and constraints. Automatica, vol. 35, p. 407-427. Floudas, C.A., ,1995, Nonlinear and mixed-integer optimization: fundamentals and applications. Topics in chemical engineering. Oxford University Press. Lewis, F.L. and V.L Syrmos, 1995, Optimal Control, Wiley, New York. Ingham, J. and I.J. Dunn, and E. Heinzle, and J.E. Prenosil, 1994, Chemical Engineering Dynamics, VCH, New York. McAuley, K.B. and J.F. MacGregor, 1992, Optimal Grade Transitions in a Gas Phase Polyethylene Reactor, Aiche Journal, vol. 38, p. 1564-1576. van der Schot, J.J., R.L. Tousain, A.C.P.M. Backx, and O.H. Bosgra, 1999, Computers and Chemical Engineering Supplement, p. S507 - S510. Sinclair, K.B., 1987, Grade change flexibility defined, determined, compared. Proceedings fifth International Conference on Polyolefms, Houston, Texas, USA. St0ren, S. and T. Hertzberg, 1994, The sequential linear quadratic programming algorithm for solving dynamic optimization problems - a review. Computers and Chemical Engineering, vol. 19, p. 495-500. Vassiliadis, V.S., 1993, Computational Solution of Dynamic Optimisation Problems with General Differential-Algebraic Constraints, University of London. White, v., J.D. Perkins, and D.M. Espie, 1996, Switchability analysis. Computers and Chemical Engineering, vol. 20, p. 469-474.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
817
An Integrated Framework for Multi-Objective Optimisation in Process Synthesis and Design H. Alhammadi, G.W. Barton, J.A. Romagnoli and B. Alexander Laboratory for Process Systems Engineering Chemical Engineering Department, University of Sydney NSW 2006 Australia
Abstract A multi-objective optimisation framework is used to identify trade-offs between various goals in flowsheet design/synthesis. Life Cycle Assessment (LCA) is used to handle the environmental considerations. The methodology is demonstrated using a vinyl chloride monomer (VCM) plant with varying degrees of heat integration as a case study.
Introduction Industry is required to operate any process so as to satisfy economic, environmental and social objectives, while at the same time being readily operable. In general, the major challenge (both at the design and the operational stage) lies in resolving the conflicts between these objectives (Miettinen, 1999). A major task for process system engineers is to develop tools that assist in the trade-off scenarios arising in such multi-objective situations. Our ultimate goal is an integrated approach allowing all relevant objectives to be accounted for during the detailed design of a processing plant. To this end, we present here a framework for such a methodology that incorporates both economic and environmental objectives for assessing various levels of heat integration for a given process. Life Cycle Assessment (LCA) is a methodology for estimating the environmental impacts associated with a given product, process or activity (Consoli et al., 1993). It is a comprehensive technique that covers both "upstream" and "downstream" effects of the activity or product under examination, thus often being referred to as "cradle-to-grave" analysis. Being an accepted (and widely used) tool in this area (Azapagic and Clift, 1999), it was employed in this study to map the environmental impact potential of any given alternative. Energy integration techniques are today an accepted means of both improving process economics and reducing environmental impacts (Rossiter, 1994; Linnhoff, 1994), with 'pinch analysis' being a simple means of determining the minimum heating and cooling requirements for a given process configuration. In this paper, thermal pinch analysis is included to demonstrate the incorporation of energy integration within a multi-objective optimisation framework. An obvious extension here is to include techniques for devising and assessing 'practical' energy integration networks within the methodology. However, it is regularly noted in the literature (Morari, 1983; Bildea and Dimian, 1999) that the inclusion of energy integration generally makes a process more difficult to control. Thus, another (on-going) extension to the methodology is the inclusion of process controllability as part of the multi-objective framework.
818 HYSYS
PROCESS MODEL
i
F=. HEAT INTEGRATION
4.
EXCEL
MASS & ENERGY DATA\
"NON CONVERGED" DESIGN VARIABLES
ENVIRONMENTAL OBJECTIVESs
V
ECONOMIC
y OBJECTIVES o
MULTI-OBJECTIVES OPTIMIZER "CONVERGED" DESIGN VARIABLES
Figure 1. Structure of the proposed framework In this paper, a vinyl chloride monomer (VCM) process is used to demonstrate our approach.
Framework and methodology The proposed framework extends the work of Alexander et al. (2000), so as to include energy integration analysis within the multi-objective optimisation environment. Figure 1 is a schematic of the proposed methodology's structure, showing the inter-linking of the software tools used and the flow of data between them. Process Model: The process is modelled within the Hysys® simulator (Hyprotech Ltd). This was selected as the simulation environment as it provides for both steady-state and dynamic modelling, while permitting the ready exchange of data with other software packages (including equation-oriented simulators, such as gPROMS). Mass and energy data from the Hysys® model are transferred to/from MS-Excel® using an object link and embed (OLE) communications protocol. Environmental Model: In Excel, LCA is used (via impact potentials) to analyse the environmental impacts of the process and to formulate the environmental terms to be used in the multi-objective optimisation problem. All the upstream/external activities, from the extraction of raw materials to the provision of site utilities, are considered in this study. It should be noted that the LCA study was completed as a "cradle-to-gate" analysis, whereby the final usage and disposal phases of the various products are not considered. The environmental burdens of the upstream activities are determined using the SimaPro® commercial LCA database. The LCA analysis first performs an inventory
I
819 analysis, followed by an environmental impact assessment. The impacts covered in this environmental model are global warming potential (GWP), ozone depletion potential (ODP), eutrophication potential (EP), acidification potential (AP), summer smog potential (SP), human toxicity potential (HTP) and process energy potential (PEP). Economic Model: Also within Excel®, calculations are performed (using the mass and energy data transferred from Hysys®) to formulate an economic objective based on the operating profit. Heat Integration: Similarly, all heating and cooling requirements for the process are obtained from the data transferred from Hysys®, and are tabulated in Excel®. Here, a pinch analysis is performed to determine the minimum utility requirements for any given operating point. These values are used in the economic and environmental models as 'best estimates' for an energy integrated version of the process. Multi-Objective Optimisation Algorithm: The £-constraint method is used to solve the multi-objective optimisation problem and to generate individual points on the Pareto curve. Note that a Pareto curve is a set of 'non-inferior' solutions defining a boundary beyond which none of the objectives can be improved without sacrificing at least one of the other objectives. The trade-off between objectives can, thus, be visualised through the Pareto curve over the set of design alternatives. The basic strategy for the 8constraint method is to select one of the objectives (eg operating profit) to be optimised and to convert all other objective functions {eg the environmental impacts) into constraints by setting an upper bound (81, ..., 8n) for each of them (Miettinen, 1999). The algorithm for the 8-constraint approach then solves a problem posed as follows, Max Objectivej Subject to:
1. Objective; < 8i ; i=l, ..., n : i ;«^j 2. Mass and energy balance constraints
where the Pareto curve is generated by parametrically varying the upper bound on the constrained objectives over each entire range, and solving the above for each case.
Description of Case Study Figure 2 is a simplified block diagram of a typical vinyl chloride monomer (VCM) plant (McPherson et a/., 1979; Cowfer and Gorensek, 1997). This integrated process produces VCM from ethylene, chlorine, oxygen and a portion of the by-product hydrogen chloride (HCl). The major sections of this plant are as follows: (1) A direct chlorination of the ethylene to produce ethylene dichloride (EDC). (2) An oxy-chlorinator to produce EDC by reacting ethylene with oxygen and HCl. (3) The two crude EDC streams are mixed and purified in a pair of distillation columns (essentially to remove water and unwanted reaction by-products). (4) The pure EDC undergoes (partial) thermal cracking in a pyrolysis furnace to yield VCM and HCl. (5) VCM is separated from the HCl and EDC in another pair of distillation columns. Note that a portion of the HCl is recycled to the oxy-chlorination reactor to make EDC, while any unconverted EDC is recycled (via the purification) to the furnace.
820 1 Chlorine CI.
^
w
Ethylene
^
W
C2H4
Oxygen
EDC Cracking
Direct Chlorination
1
p»
—'W
t
EDC Distillation
t
w
^ ^
Oxychlorination
EDC
Distillation
VCM Product
HCl
T ^
HCl Snlit
-•
HCl Product
Figure 2. A simplified block diagram of a typical VCM plant The process design variable selected here for (multi-objective) optimisation was the portion of HCl recycled to the oxy-chlorinator. It would be a straightforward extension to the framework to include multiple design variables - however, in this paper only a single variable was considered for ease of demonstration. Also, as the environmental potentials in this case all trend in the same direction, the impact potential most sensitive to this design variable {ie GWP) was chosen as an exemplar. The economic objective chosen was the operating profit, that is, the difference between the total value of the products and the total cost of the raw materials and utilities. Each objective function was normalised (over the specified range for the recycled HCl) and scaled so that 0 and 1 represents the best and worst value of the objective. The e-constraint method was used to solve the multi-objective optimisation problem and obtain the Pareto curve. Here, the economic objective was optimised while the environmental objective was converted into a constraint with a specified upper bound. This multi-objective optimisation problem was performed for three cases. In the first case, no heat integration was considered, while in the second the process was examined for optimal heat integration with the minimum heating and cooling requirements being determined using pinch analysis. These minimum utilities were then used in the environmental and economic models to formulate their respective objectives. In the final case, a single heat exchanger around the pyrolysis reactor was considered.
Results and Discussion The Pareto curves for all three cases are shown in Figure 3. The curves for cases one and two provide the lower and upper bounds for all possible levels of heat integration at all operating points. The 'optimal heat integration' curve shows the maximum possible reduction achievable for the two objectives. However, as noted previously, this level of heat integration may well be impractical, as the heat exchanger network (HEN) required to realise it could make the resultant process extremely difficult to operate/control. For example, a HEN was realised for a single point on the optimal heat integration Pareto curve but this required nine exchangers and the splitting of two streams - not really a practical option. Thus, moving from the 'no heat integration' curve towards the 'optimal heat integration' curve involves other trade-offs to be considered (in addition to economic and environmental) - those of plant controllability and/or operability.
821 - No Heat Integration — K — O p t i m a l Heat Integration
0
0.25
0.5
Partial Heat Integration
0.75
GWP Figure 3. Pareto curves of the three cases Table 1: Annual values of the objectives for various VCM plant designs
HCl
No Heat Integration GWP Profit
Recycled %
10^tonCO2/yr
M$/yr
20 40 60 80 100
2842 3010 3243 3548 3935
160 163 168 175 183
Partial Heat Integration GWP Profit M$/yr lOMonCOz/yr 2762 2920 3140 3428 3790
164 167 173 180 190
Optima Heat Integration Profit GWP 10^tonCO2/yr
M$/yr
2441 2515 2618 2750 2911
175 181 189 201 216
Also shown on Figure 3 is the Pareto curve for the case where a single heat exchanger is employed around the pyrolysis reactor (using the hot reactor effluent to heat the cold reactor feed). As expected, this example of partial heat integration resulted in a Pareto curve that lies between the two extremes. The extent to which the Pareto curve has shifted by the inclusion of this one exchanger is less than a quarter of what is achievable, leaving scope for considerable improvement (remembering that the 'best' point on Figure 3 is the origin) but at the price of a (likely) reduction in process controllability and/or operability. Table 1 summarises the numerical results given on the plots in Figure 3. Such a table quantifies the trade-offs possible between the economic, environmental and degree of heat integration objectives. In this table, the impact of employing different levels of heat integration is tracked as you progress along a row (ie for a given percentage of HCl recycled). Similarly, the trade-off between economic and environmental objectives as a function of the HCl recycled (for a given level of heat integration) is tracked as you proceed down a column.
822
Conclusions In this paper, a methodology has been proposed that incorporates both economic and environmental considerations within a multi-objective optimisation framework that permits (for a given process) the inclusion of various levels of heat integration. The methodology as it stands enables us to draw 'boundary' Pareto curves corresponding to the maximum and minimum levels of heat integration for all operating points achievable by the process. It is also possible to use the proposed approach to draw the Pareto curve for any HEN between the calculated limits, and thus to quantify the trade-off between economic and environmental objectives. It was noted that improved energy efficiency generally increases plant complexity and may well have significant impacts on plant operability and/or controllability. Thus, it will be necessary to explicitly incorporate the effects of increased energy efficiency as you move between the two 'boundary' Pareto curves. The inclusion of such operability and controllability analysis into the general framework is the next step in this work, so as to enable assessment of 'practical' HEN alternatives. Acknowledgements The authors wish to thank Professor J. Petrie for his LCA contribution, while Hasan Alhammadi acknowledges the financial support of The University of Bahrain. References Alexander, B., Barton, G.W., Petrie, J. and Romagnoli, J.A. (2000). Process Synthesis And Optimisation Tools For Environmental Design: Methodology And Structure, Computers & Chemical Engineering , 24, 1195-1200. Azapagic, A. and Clift, R., 1999, The Application of Life Cycle Assessment to Process Optimization, Computers & Chemical Engineering, 23, 1509-1526. Bildea, C. and Dimian, A., 1999, Interaction between Design and Control of a HeatIntegrated Distillation System with Prefractionator, Trans IChemE, 77, 597-608. Consoli, F., Allen, D., Boustead, I., Fava, J., Franklin, W., Jensen, A., de Oude, N., Parrish, R., Perriman, R., Posdethwaite, D., Quay, B., Seguin, J. and Vigon, B. (eds.), 1993, Guidelines for Life-Cycle Assessment: A *'Code of Practice". SET AC, USA. Cowfer, J. and Gorensek, M., 1997, Vinyl Chloride: Encyclopaedia of Chemical Technology, 24, 851-882. Linnhoff, B., 1994, Use Pinch Analysis to Knock Down Capital Costs and Emissions, Chemical Engineering Progress, August, 32-57. McPherson, R., Starks, C. and Fryar, G., 1979, Vinyl Chloride Monomer - What You Should Know, Hydrocarbon Processing, March, 75-88. Miettinen, K., 1999, Nonlinear Multi-objective Optimisation. Kluwer Int. Series. Morari, M., 1983, Flexibility and Resiliency of Process Systems, Computers & Chemical Engineering, 7, 423-437. Rossiter, A. P., 1994, Process Integration and Pollution Prevention. In: Pollution Prevention via Process and Product Modifications, AIChE, 90, 12-22.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Elsevier Science B.V. All rights reserved.
823
An approximate Optimal Moving Grid technique for the solution of Discretized Population Balances in Batch Menwer M. Attarakih\ Hans-Jorg Bart' and Nairn M. Faqir^ ^Kaiserslautern University, Faculty of Mechanical & Process Eng., Institute of Thermal Process Eng., POB 3049, D-67653 Kaiserslautern, Germany. "^University of Jordan, Faculty ty of Eng. & Technology, Technology Chemical Eng. Department, 11942, Amman, Jordan.
Abstract The numerical solution of droplet population balance equations by discretization is known to suffer from inherent finite domain errors (FDE). A new technique that minimizes the total FDE during the solution of discretized population balance equations (DPBE) using an approximate optimal moving grid for batch systems is established. This optimal technique is found very effective for tracking out steeply moving population density with a reasonable number of size intervals. The present technique exploits all the advantages of its fixed counterpart by preserving any two moments of the evolving populadon. The technique is found to improve the predictions of the number density, zero and first moments of the population.
1-Introduction The population balance equation (PBE) for a well-stirred batch vessel could be written as (Ramkrishna, 2000): —r at
+
r dv
= p{M(v,r),v,r}
(1)
where n(v,t) is the average number of droplets per unit volume of the vessel at time t. The first term on the left hand side denotes the rate of accumulation of droplets of size V, and the second term is the convective flux along the droplet volume coordinate. The term on the right hand side is the net rate of droplets generation by coalescence and breakage. Despite the importance of Eq. (1) it rarely has an analytical solufion. So, in general numerical solutions are sought where several methods are proposed in the literature. Kumar and Ramkrishna (1996a) critically reviewed the available methods where they concluded that the methods conserving both total number and volume of droplets are not only computationally efficient but are also accurate. These authors made great achievement in the discretization of the PBE by introducing a general framework of discretization. Their method preserves any two moments of the population, and converts the PBE into set of discrete partial differential equations. The developed methods are called the fixed and moving pivot techniques, where the latter is used in the present work due to its generality and accuracy.
824 Nevertheless, this discretization is by no means exact, and it is inherently associated with the so-called finite domain error (FDE) resulting from trying to use a finite droplet volume to approximate the infinite one. As so far, only Sovova and Prochazka (1981) tried to investigate rigorously the effect of the FDE on the accuracy of the solufion of the DPBEs. They studied droplet breakage and coalescence in batch vessels at steady state and tried to estimate the FDE by extrapolating both ends of the droplet distribution. The main drawback of this technique is the general uncertainties associated with extrapolation and its lack of general relations to predict the time dependent FDE. The objective of this work is to develop an approximate optimal moving grid technique for batch systems, based on the minimization of the total FDE. The proposed technique has the ability to conserve any two moments of the distribution. A general equation is also derived for the total FDE by approximate discretization of the general PBE.
2-The discretized PBE using the moving pivot technique In the moving pivot technique, the droplet volume is discretized according to the partition (grid) V^ ={^min'^^2' '^M'^maxl ^^^ ^he /th interval is denoted by// = [v,, V.^j) . Kumar and Ramkrishna (1996b) derived the DPBEs, which conserve the total number and droplet volume for droplet breakage in batch vessel: /V7
dt
: A, yv,(0+ X ^o,ar,yV,(0, k=ux
/ = l,2,
M
(2)
1 ^ ^777;T S ^^U'^ -^,^o,a)r.^.(0, / = i,2, M (3) dt N,(t),trli where jc, is the representative size of the population in the interval /, and it is called the pivot, A^, is the total number of droplets associated with this pivot, Vi=i^i,i,i-Xi^o,i,i)^i^ ^i =(^ojj~^)^i^ ^o.u^"^ ^ua^re given by Kumar and Ramkrishna (1996b) and F, is the breakage frequency. dx (t)
-Tr^"^^
3-The Finite Domain Error In discretizing an equation defined over an infinite domain an inherent error is incurred due to the failure of taking into account the portion of the funcfion lying outside the domain of discretization. This error is termed the total FDE and is represented by (Sovova and Prochazka, 1981): £o(t)= J n(v,r)^v+ j n{vj)dv 0
(4)
v^
Note that for a given number of intervals, M, and interval width, zlv, an optimal minimum droplet volume, v^,>„ exists and could be found by differentiating Eq. (4) with respect to v^i^ and set the result equal to zero:
«(vL,0-^n(vL,0 = 0
(5)
According to Eq. (5), the optimal minimum droplet volume must decrease as function of
825 time to account for the increasing number density at the lower size range. This suggests the use of optimal moving grid for droplet breakage, which moves from the upper to the lower size ranges as function of time. Consequently, Eqs. (2) , (3) and (5) must be solved simultaneously at each instant of time to find such an optimal moving grid. Unfortunately, the solution is iterative by solving Eq. (5) at each integration step, and might mask the benefits gained by using the optimal moving grid. To compensate for this drawback, an approximate optimal moving grid technique is derived in the following section.
4-An approximate optimal moving grid technique The total finite domain error, as defined above, will be close to the minimum value when both residuals are equal, which leads to an approximate optimal minimum droplet volume and hence optimal moving grid. This optimal moving grid should keep the number of intervals constant during grid movement, and hence redistribution of the population between the old and the newly formed grids is essential. This should be performed by conserving any two moments of the population in order to be consistent with Eqs. (2) and (3). Now consider a typical geometric grid ( Vi(t)=cf'^Vmin(t) ) at two instants of time: t and t-\-At where the optimal minimum droplet volume moves from v^^^ (t) to v^^^ (r + A/). Let y^"^ (t) be the fraction of droplets at the pivot jc, (r) to be assigned to the pivot Xi (t + A/) and y^l^ (t) be the fraction of droplets at the pivot X. (t) to be assigned to the pivot .X.^i {t + At) . These fractions are found such that both number and volume of these droplets are conserved after redistribution. Accordingly, the discrete lower and upper residuals at this instant of time are given by: (6)
FDEUt + At) = y:''N,{t) FDE^{t-\-M)=
2 r r ' ^ A ^ M (0 + 7 ^ ^ , ( 0
(7)
where only the (M+7)th term in the summation above has a significant value for sufficiently large M or geometric factor a. The optimality condition implied by Eq. (5) could be approximately satisfied by forcing both sides of Eqs. (6) and (7) to be equal, which after some algebraic manipulation yields: ^min V ^ ^^) ^
(7-1
a
a
cr-1
/g\
826
5-Estimation of the lower and upper residuals To estimate the lower residual, Nd(t), it could be assumed that the first interval will only receive broken droplets from higher ones or from droplets within the interval itself with no droplets are lost through breakage from this interval (Laso et al., 1987; Hill and Ng, 1995). Consequently, an unsteady state number balance on this interval yields: dA^ it) ^ — r ^ = ^(vL (0)ro N, (t) + 5^ ;ro,o,, r , N, (/)
(9)
Since the width of the interval [Vmin^ v^^^ (t) ] is very small due to the geometric grid used in discretization, the pivot X(/t) is fixed at the middle of this interval. Similar arguments for the IM+I interval leads to the following equation: ^
^
= A.,yV,,(r)
(10)
and the {M+l)th pivot could be derived from Eq. (3). So Eqs. (6) and (7) along with Eqs. (9) and (10) define completely the lower and upper residuals at any instant of time for specified grid parameters a and M.
6-Numerical results and discussion By using a geometric grid, it should be mentioned that when the last interval is completely passed due to the grid movement, the new and the old interval boundaries completely coincide except for the first boundary. This suggests that the number densities could be updated only when v^^^^ (t) is less than or equal to v^it) to exclude any numerical inaccuracies due to population redistribution. This strategy is adopted in the present solution algorithm using uniform daughter droplet distribution and linear breakage frequency over a relatively long period of time, t=]00 (arbitrary time units) to illustrate the steepness of the number density. The solution algorithm is implemented using an exponential initial condition and a number of intervals, M=75, and G=2.0. The analytical solution is given by by Ziff and McGrady (1985) for binary breakage. We start by comparing the exact and numerical FDE as well as the optimal minimum droplet volumes. Fig. 1-a shows these results, and it can be seen an excellent agreement between the numerical and exact FDE is obtained. As expected the optimal fixed FDE increases with time due to the failure to account for the increase in number density in the small size range as droplet breakage proceeds. This is actually equivalent to a loss of number of droplets from the system. To compensate for this, we let the grid move in an optimal manner as shown in Fig. 1-b, where the exact minimum droplet volume is depicted along with that predicted using the optimal moving grid algorithm. First the agreement between the optimal piecewise minimum droplet volume and the exact one is also excellent even when the grid moves so fast. Second the great influence of the optimal grid movement on the reduction of the total FDE is obvious when compared to the fixed grid (Fig. 1-a). Fig. 2-a shows the exact and numerical average number densities at the final time of simulation where, excellent agreement is perceptible. Also, one could see how the optimal moving grid leaves the approximately empty intervals (large sizes) to accommodate the increasing number densities in the small size range as
827 expected. Fig. 2-b shows the clear discrepancies between the discrete zero moment of the distribution using fixed and optimal moving grids respectively. As expected for a long time of droplet breakage, the number density becomes increasingly sharp. Failure to include the small size range of the population will induce appreciable errors in the total number density as a result of increasing the total FDE. The mean droplet volume is also over predicted, however to a small extent, when fixed grid is used because large number but small volume of droplets are lost at long times of breakage due to the increase in the total FDE. It should be mentioned that as the number of intervals decreases the sum of the residuals becomes the main source of the discretization error (relative to the integration error). Under these circumstances minimizations of these residuals (FDE) is the only way to reduce the discretization error if coarse grid is to be maintained. 0.0006
0.0002
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Figure 2: The effect of the optimal grid movement on: a- The average number density, b- The zero moment and c- The mean droplet volume using geometric grid with factor a=2.0andM= 15
7-Conclusions An optimal moving grid technique is developed for the solution of the DPBEs for droplet breakage in batch systems base on the minimization of the total FDE. The redistribution algorithm, on which this technique is based, is consistent with DPBEs by
828 preserving any two moments of the distribution. Moreover, ordinary differential equations are derived to estimate the total FDE of the droplet distribution, which shows excellent agreement with the analytical solution studied in this work.
Nomenclature FDE^, FDE^
lower and upper residuals based on zero moments of the distribution
M Ni(t) n(v,t)dv V, v' ^mm.» ^max.
total number of intervals used in droplet volume discretization total number of droplets in the /th interval, at time t number of droplets in size range v to v+dv, at time t per unit volume droplet volumes minimum and maximum droplet volumes
*
•
^min ' ^max.
Optimal minimum and maximum droplet volumes
V Xi(t) t
droplet growth rate characteristic volume of the droplet population in the /th interval time
Greek Symbols r(v) YT^
^ YT'^""
droplets breakage frequency fractions of droplet assigned to the /th pivot
At £o(t) rji
time increment total finite domain error based on zero moment of the distribution the /th eigenvalue of the pivot equations.
A-
the /th eigenvalue of the number density equations
G t^(v')
geometric grid factor number of droplets produced when droplet of volume, v^, is broken
References Hill, P. J. and Ng, K. M., 1995, New discretization procedure for the breakage equation. AIChE J., 4 1 , 1204-1216. Kumar, S. and Ramkrishna, D., 1996a, On the solution of population balance equations by discretization-I. A fixed pivot technique. Chem. Engng. Sci., 5 1 , 1311-1332. Kumar, S. and Ramkrishna, D., 1996b, On the solution of population balance equations by discretization-II. A moving pivot technique. Chem. Engng. Sci., 5 1 , 1333-1342. Laso, M., Steiner, L. and Hartland, S., 1987, Dynamic simulation of liquid-liquid agitated dispersions-I. Derivation of a simplified model. Chem. Engng. Sci., 42, 24292436. Ramkrishna, D., 2000, Population Balances. San Diego: Academic Press. Sovova, H. and Prochazka, J., 1981, Breakage and coalescence of drops in a batch stirred vessel-I Comparison of continuous and discrete models. Chem. Engng. Sci., 36, 163-171. Ziff, R. M. and McGrady, 1985, The kinetics of cluster fragmentation and depolymerisation. J. Phys. A: Math. Gen., 18, 3027-3037
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
829
Restructuring the Keywords Interface to Enhance CAPE Knowledge Acquisition via an Intelligent Agent F. A. Batzias ^ and E. C. Marcoulaki'a,b ' Department of Industrial Management, University of Piraeus, Greece ' Department of Physics, Faculty of Applied Sciences, NTUA, Greece
Abstract This work proposes an improved KeyWord Interface (KWI) to enhance the efficiency of information retrieval when using an advanced search engine, as an intelligent agent. This can be achieved by restructuring the KWI into a new hierarchical structure based on an {n domains} by {3 levels} arrangement of keywords (AZX3 KWI), forming a loose/adaptive semantic network. The hierarchical levels used in the suggested implementation are set of species, logical category, and holistic entity. As an illustration, the method is applied to an example of literature survey concerning a welldocumented process engineering field. The results of the proposed technology are compared with the outcome of general-purpose search-engines built in common academic publication databases. The comparison reveals the advantage of intelligent searching in creating a local base according to the orders/interests of the researcher.
1. Introduction Intelligent Agent (lA) technologies have been proposed recently, and partially been implemented, for the collective acquisition of information from large, unstructured, and heterogeneous information spaces like the Internet (O'Meara and Patel, 2001; Crestani and Lee, 2000; Zacharis and Panayiotopoulos, 1999; Etzioni and Weld, 1995; Maes, 1994). The quest for information is hereby assumed to support the initiation, progress and termination stages of a scientific research program, as demonstrated in Figure 1. Filtered/semiprocessed information
f
Researcher
Order to agent Communication line (acknowledgement)
Compact information storing
Intelligent Agent (lA)
Combined information Intelligent searching] Standard searching
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Figure 1: General scheme of information retrieval through an intelligent agent (IA)
830 At present, there is a vast amount of information on Computer - Aided Process Engineering (CAPE) available on Internet platform, in the form of relevant publications, data, applications, casestudies, etc. This information can be used to construct a CAPE knowledge base, to be interfaced with an available process engineering software tool. The base can be continually enriched and periodically restructured, either exclusively via the software designer, or inclusively via an lA and the designer. The former constitutes the usual present practice, while the inclusive scheme is mostly met on experimental stage in small-scale application. Regardless of the updating schemes applied, the KeyWord Interface (KWI) remains the main communication protocol and the most popular tool in the current database search engines. This work considers the inadequacies of conventional KWFs to cooperate efficiently with the lA's. Evidence of this inadequacy is provided by searching the Internet for prespecified targets. A solution is to define a more concrete communication protocol to improve the efficiency of the interaction between the researcher and the search engine. The proposed interface assumes a pyramid-like structure of keyword information levels, so the user can move uphill/downhill according to the abstraction of the search query.
2. Motivation and problem description A scientific program is a problem describable in linguistic terms providing appropriate keywords and key-phrases for information retrieval. The main difficulty is that these terms are not unique and each of them frequently belongs to a class or category represented by a wider concept, possibly used as a substitute for the keyword. Consider the quality control of the electrochemical anodization of aluminium, as a typical CAPE research program. There are over 40 low level keywords describing defects that might appear on the product surface (e.g. burning, cracking, spotting, blooming, crawling, blistering, traffic marks, wood grain). A survey is carried out in ScienceDirect ® (SD), a very reliable and extended database of relevant literature that uses the LEXIS®-NEXIS(5) search logic. Using the combination anodi! AND alumin! AND (burn! OR bloom! OR blister! OR crawl!) nothing is retrieved (until November 2001). The truncation is used to find the root word plus all the words made by adding letters to the end of it. Inefficiencies in returning results can be attributed to the use of (i) a synonym, Q.g. fretting corrosion instead of traffic marks or polishing rings instead of wood grain, and/or (ii) a roundabout description, e.g. ''catastrophic local dissolution of the anodic coating due to overheating'' instead oi burning. A commonsense way of extending the search domain is to conduct a more abstract search, i.e. by supplying the general keyword defect. However, when SD is called to process the combination alumin! AND anodi! AND defect only 6 articles are obtained, while alumin! AND anodi! AND pit! returns 53 and alumin! AND anodi! AND crack! returns 32 articles. Note that the intersection of the resulting lists for the two last queries contains 8 articles. As the search is carried out within the fields Title-AbstractKeywords, it is safe to conclude there is no hierarchical structure in the keywords field for the SD database entries to conform to.
831
A solution for improving the communication protocol might be a hierarchical structuring of the keywords assigned to each article, either by the author or/and by a reviewer of the Base, according to a controlled vocabulary. This standardization seems unattainable in a huge central Base, partially due to lack of resources and partially to the complexity of categorization in many cases. It could though be realized (i) as an additional service offered by the Base in special areas, like Materials Technology, where standardization/ categorization is already at a satisfactory level due to increased interest from industry for quality control purposes, and/or (ii) when an ad hoc decentralized local base is structured to serve as an information aid for a research program. This work develops an algorithmic search procedure to support the semiautomatic parallel creation of the KWI and the local Base. The KWIs of the documents stored in this local Base form a network of interconnected terms with hierarchical structure, i.e. an ontology. CAPE-related knowledge fields are used in the implementation and illustration of the software tool.
3. Algorithmic procedure The proposed hierarchical KWI structure is based on an {n topic domains}x{3 levels} arrangement of keywords (nx3 KWI), forming a loose/adaptable semantic network. The n domains considered here to be of interest to the process engineer are three, namely materials, processes and management. The levels of methodological approach assigned to each domain correspond to: set of species, logical category and holistic entity. In the aluminium anodization example discussed previously, the holistic entity in the materials domain is quality, which includes the desired and undesired features of the products. The logical categories can be defined by the general keywords defect and property. The set of species is a collection of low-level word-descriptors (e.g. cracking, blistering). In ihc process domain, the holistic entity may be the collection of all the unit operations met in aluminium treatment (e.g. anodization, dyeing, sealing), a logical category is a member of this collection (e.g. anodization) and the set of species is the collection of alternative process schemes (e.g. anodizing with sulfuric acid). Similarly, the management domain includes design decision methods applied on the various abstraction levels of the process domain. The developed algorithmic procedure for the creation of a {nx3 KWI}-based Thesaurus is illustrated in Figure 2 and includes the following steps: 1. Description of the subject under investigation in proper linguistic terms 2. Creation of keyword structure according to width/depth of concepts, including the nx3 novel KWI, and information storing (function of temporary/permanent memory) 3. Determination of the appropriate keyword combination within the set of species 4. Determination of the appropriate keyword combination in logical category terms 5. Searching in field, host, site, document 6. First evaluation of documents obtained as an output of searching 7. Second evaluation of documents - marking and sorting according to relevance 8. Presentation of dependence of marks assigned on the number of corresponding combinations found within the full text of each document (scatter diagram)
832 9. Choice of the appropriate function to correlate relevance with the number of keyword combinations retrieved in the full text - correlation parameters estimation 10. Cluster analysis of the new keywords extracted from the retrieved documents 11. Sorting of new keywords according to their relative frequency and elimination of the most frequent keyword in case of rejection via the feedback route 12. Identification of key phrases 13. Removal of stop and indifferent words (e.g. and, or, with, etc) 14. Determination and expansion of critical root words 15. Creation/enrichment of the nx3 KWI as a communication protocol 16. Local Base creation by storing the information and interconnecting the KWIs R. What is the type of interface to be followed? A. The number of relevant documents retrieved is: too big, too small, satisfactory? Pj. Is the most frequent keyword accepted for incorporation in the structure of stage 2? W. Is the desired change an elimination/substitution of keyword or an addition of field? S. Is the fitting satisfactory? Q. Is there another correlation function available? The creation of the local knowledge Base under the semantic network of the structured
I I activity node / \ decision node satisfactory
Figure 2: Flowchart of procedures adopted for restructuring the KWI and creating a local knowledge Base
833 keywords takes place in: (i) stage 2, via the route [...->! l^Pj^2—>...-^11 ->...], and (ii) stages 15, 16. The procedure terminates when a proposed incorporation is rejected a-times successively. Note that direct information retrieval via the route [2->3->4-^5->16] can be done by means of a simple search engine, possibly using a tailored internal indexing system. The algorithm is hereby implemented for bottom-up action, i.e. for extending the set of species assigned to a certain logical category, and the logical categories assigned to a certain holistic entity.
4. Results and discussion Example results of a bottom-up action are shown in the screenshot of Figure 3 based on the initial query alumin! AND anodi! AND crack!. The choice shown here concerns the inclusion of the new keyword pit! in the local Base of stage 16 as a member of the set of species. Initially, pit! is proposed for incorporation, as it is the next most frequent term encountered in stage 11. The decision for accepting it in node Pj is guided by information concerning the origin of this entry, i.e. the journal/database name and the frequency of appearance of this source (multiplicity). High multiplicity signifies increased importance of this source with respect to the proposed keyword. Lower multiplicity (as it is the case here) indicates a more widely diffused term, i.e. of higher interdisciplinary value. 6i ^Kteridlrtg the Set of species The Route 12 -> Pj -> 2 - Action 3
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Figure 3: Screenshot depicting a process of extending the keyword structure, by incorporating a new keyword member in the set of species
834 An example of a fervent field for the application of the proposed hierarchical KWI structure is that of materials design. The KWI could facilitate the management and assessment of information tabulated in property databases, and provided by computeraided molecular design synthesis tools (e.g. Marcoulaki and Kokossis, 2000; Marcoulaki et al., 2000). The function (e.g. solvent) of the material may present the holistic entity. The logical categories could classify the molecular configurations performing the described function into aromatic, straight-chain, branched-chain etc. Each logical category contains a set of molecular configuration species (tabulated or designed) and anticipates the introduction of novel entries to extend the bank.
5. Conclusions and recommendations The paper shows that several inadequacies appear when conventional KeyWord Interfaces (KWIs) are used in searching through a web-based database using an intelligent agent. The situation can be improved by using a structured communication protocol employing controlled vocabulary. A new hierarchical KWI configuration is proposed, assuming an [n domains} x {3 levels} arrangement of keywords, forming a loose/adaptive semantic network. The three hierarchical levels discussed here are set of species, logical category and holistic entity. An algorithmic procedure is developed for the domain of materials, which can be similarly applied to the domains of processes and management. The application of this procedure in an example search of the aluminium anodization literature, is found successful in enriching the hierarchical levels and creating a local knowledge Base in a continuous intelligent mode.
Acknowledgements The authors kindly acknowledge the contribution of an anonymous reviewer, and the financial support provided by the Research Center of the University of Piraeus.
References Crestani, F. and P. L. Lee, 2000, Searching the web by constrained spreading activation. Information Processing & Management 36 (4), 585 Etzioni, O. and D. Weld, 1995, Intelligent agents on the Internet: fact, fiction and forecast, IEEE Expert 10 (4), 44-49 Maes, P., 1994, Agents that reduce work and information overload, Commun. ACM 37 (7), 30 Marcoulaki, E. C. and A. C. Kokossis, 2000, On the development of novel chemicals using a systematic synthesis approach. I - Optimisation framework, Chem. Eng. Sci. 55(13),2529 Marcoulaki, E. C , A. C. Kokossis and F. A. Batzias, 2000, Novel chemicals for clean and efficient processes using stochastic optimisation. Computers Chem. Eng 24, 705 O'Meara, T. and A. Patel, 2001, A topic-specific web robot model based on restless bandits, IEEE Internet Computing, March-April 2001, 27 Zacharis, Z. N. and T. Panayiotopoulos, 1999, A learning personalized information agent for the WWW, Proc. ACAI-99 on Machine Learning and Intelligent Agents, Chania, Greece: EETN, 39
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
835
Mining Textual Project Documentation in Process Engineering A. Becks ^, J.-C. Toebermann^ ^ Fraunhofer Institute for Applied Information Technology FIT, D-53754 Sankt Augustin, Germany, [email protected] ^ RWTH Aachen - Computer Science V (Information Systems), D-52056 Aachen, Germany, [email protected]
Abstract The aim of Knowledge Management is to systematically create, maintain and distribute intellectual capital. To analyse numerical or structured data, techniques like "online analytical processing" and data mining methods are used. However, a lot of existing coq^orate information like manuals, guidelines, patents and project documentation is captured in textual, i.e. unstructured, form. Specific requirements for mining such textual data were analysed and reviewed and an according tool "DocMINER" was developed and evaluated. Using an example session with project documentation from the process engineering domain its features and usage are described and its potential to lower the necessary effort during typical work steps are demonstrated.
1. Introduction High efforts in time and money are often spent to re-capture or even re-invent already existing knowledge. Knowledge Management can be seen as a way to systematically create, maintain and distribute intellectual capital. An important technological aspect is supporting companies in their efforts to analyse and learn from their existing proprietary data and documentation. Techniques for analysing numerical or structured data include "online analytical processing" (OLAP) and a broad spectrum of data mining methods. However, a lot of existing corporate information is captured in textual, i.e. unstructured, form. Examples are manuals, guidelines, patents and project documentation. Consequently, text mining is an emerging field of research. Important aspects of text mining comprise document clustering and visualization. Graphically displaying complex information directly appeals to the powerful human visual perception, enabling users to easily identify patterns and trends in data collections. We developed the tool "DocMINER" (Document Maps for Information Elicitation and Retrieval) which uses a so-called document map for displaying the semantic similarity structure of document collections. Its motivation is to ease access to corporate text collections for analysis purposes. Its development was based on an empirical investigation of typical document analysis tasks in knowledge-intensive industries (Becks 2001).
836 In this paper we introduce document maps and then sketch specific requirements of text mining as well as the architecture and design of our tool. Using a detailed example session with project documentation from the process engineering domain, its features and usage are described and its potential to lower the necessary effort during typical work steps are demonstrated. By this example we show that the tool provides effective access to relevant parts of the documentation and enables the user to easily understand the relationships among the complex material.
2. Document Maps and the DocMiNER System Document Maps present the semantic structure of a document collection by using a suitable metaphor for intuitively visualizing 'document similarity'. The concept of document or group similarity is usually reflected by a notion of distance: the more similar two documents or document groups are, the closer they appear in the visualization. In literature different metaphors and granularities for visualization have been proposed, e.g. 2 or 3-dimensional scatter plots (Chalmers et al 1992, Wise et al. 1995) and 3D landscapes (Davidson et al. 1998) for presenting inter-document similarities, or category maps (Chen et al. 1996) for presenting the group structure. These visualizations enable the user to easily study the topical density or relatedness in non-trivial text collections. In general, the application of document maps is useful when an explorative access to text collections is required. This is important especially in the context of knowledge management (Becks et al. 2001): here, the user is often interested in gaining insight into a text collection as a whole and in analysing aspects, e.g. figuring out relationships of single documents. Mostly, the user is not able to specify his information needs adequately and precisely in such a context. In the DocMINER project (Becks 2001) we have designed and evaluated a document map system for visually aiding text collection analysis tasks in knowledge management. DocMINER supports an adaptable framework for generating a graphical overview, using a modular combination (fig. 1) of algorithms from classical information retrieval, spatial scaling, and self-organizing neural networks. The basic method allows a finegranular (dis-)similarity analysis of specialized document collections. It can be tailored to domain-specific needs since the module for assessing the similarity of documents is exchangeable. For example, statistical methods (Salton 1971) to assess document
user interface:
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Figure 1: Modular approach for generating document map
837 similarity are usually well suited, but for some types of document collection improvements are possible using terminological (or description) logics (Meghini et al. 1996). Additionally, a semantic refinement extension based on fuzzy rules enables the analyst to incorporate a personal 'bias' into the map generation process. DocMINER tightly integrates the graphical map display with explorative and goaldirected interaction methods. Documents are represented as points in a map that conveys a picture of the document collection structure, i.e. the grouping of similar documents and distribution of related document groups. Neighbouring points in bright shaded areas represent similar documents. Grey borders separate these areas: the darker the border, the stronger the separation, and thus the more dissimilar the single documents or document groups. DocMINER comprises several interaction methods, which help to explore a given collection of documents step by step and to successively gain information on different levels of granularity. Its interface design was guided by following rules: overview first, zoom and filter, then details-on-demand (cf Shneiderman 1996). Documents can be opened by point-and-click. System features include different zoom, scaling and sub-map functions, means to define and assign document symbols (e.g. to support the analysis of a priori given document classes), an annotation function, automatic map labelling and textual document group summaries, and a tight coupling with a query-driven retrieval interface.
3. Mining Project Documentation: An Example Session We describe a detailed example session in which our tool is used to get familiar with a complex collection of project documentation, i.e. to understand relationships and dependencies of subprojects. Such a scenario typically arises when an employee gets involved in an existing long-term project. The example documentation is taken from a collaborative research project that aims at improving development processes in process engineering. The collection is made up of the documentation of 12 projects, each with various subtasks, so this example incorporates 120 documents. Assuming that there is an accessible prepared pool of project documentation a new DocMINER project is defined. Then keywords are automatically extracted from the documents by an indexer; the inter-document similarity is assessed by keyword cooccurrence, and the document map is generated (for details cf Becks 2001). Assume further, that one document is a ^starting' document, which was identified as clearly relevant to the employee's task within the project. That document is selected from the document list within DocMINER and is accordingly highlighted in the map. Additionally, its text is displayed for a quick orientation. From this map, fig. 2, a first document grouping is easily identified. Examples are: • a group of relative ^similar' documents within the white to light grey 'plateau' at the centre of the map. The 'starting' document lies somewhat distinctly but still within this 'plateau'. • the documents at the 'eastern' part are grouped internally, but more significantly they are all clearly separated from the other documents by an approximately vertical dark shaped boundary.
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the few documents within the very dark area at the midst of the 'western' part are strongly separated from all other documents.
To get a first overall impression of the collection's topical structure, visually identified groups of documents are selected. The names of the documents and their most characteristic terms (in relation to the other documents) may be displayed directly in the map for an easier assessment. Additionally a term profile, e.g. the most significant terms of selected documents, may be displayed. It follows that the 'central plateau' with the 'starting' document is about supporting the flowsheet editing and evolution process. Figure 3 shows another step where it turns out, that the common theme of the documents in the 'north-eastern' corner is extrusion. Now it is possible to: • colour-code documents according to discovered or predefined categories, e.g. red-mark all documents in the 'central plateau' • make a more detailed investigation of document groups, e.g. zooming in the map at densely populated parts • investigate 'bridging' documents between otherwise distinct document groups • search for terms and highlight them in the map For example a search for "extrusion OR extruder" reveals the highlighted documents in fig. 3. One of the retrieved documents lies in the 'central plateau'. Clicking on this document and scanning the displayed text shows that it considers the integration of the extrusion simulation within the flowsheet editing support. Having gained a good understanding of the document collection a lot is already learned about the project and subproject themes as well as their relationships and inter-
839 goas
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4. Conclusions Data mining techniques are more and more frequently used on numerical or structured data to discover new knowledge and the benefit of such techniques is well proven. However, knowledge captured in textual documentation is also a very valuable information source for any organization, but methods and tools to explore and exploit such data are less mature. We identified specific requirements and developed an appropriate tool with a powerful visualization and user-interaction for text mining. The tool was successfully used in many case studies, e.g. analysing usage scenarios for chemical process simulators (Becks et al. 1999) and for condensing and maintaining internal and external technical documentation of foundry simulator manuals (Becks et al. 2000). In the given example session we showed that the tool provides effective access to relevant parts of documentation typical for research in process engineering. The tool enabled the user to easily understand the relationships among the complex material. The benefit in reducing efforts needed during settling into a new project seems obvious, and was additionally confirmed in a laboratory study (Becks 2001). First contacts are established for a broader industrial evaluation, among others in chemical engineering companies. The modular approach of the tool allows an easy adaptation to specific tasks. For example, modifications to enable an improved mining of "structured" textual data like simulation input files will be assessed. Furthermore, at
840 Fraunhofer FIT we are actually working on a tight integration of "DocMINER" with various other text access tools in conjunction with information brokering aspects. From this we expect a higher degree of flexibility and advanced adaptive features.
5. Acknowledgements This work was supported by the DFG in its focused doctoral programme on Informatics and Engineering at RWTH Aachen and within the framework of the collaborative research center "SFB 476 IMPROVE".
6. References Becks, A., J. Koeller, 1999, Automatically Structuring Textual Requirement Scenarios. Proc. 14^ EEEE Int. Conf. on Automated Software Engineering, Cocoa Beach, Florida, USA Becks, A., 2001, Visual Knowledge Management with Adaptable Document Maps, PhD-thesis, RWTH Aachen, Germany Becks, A., M. Host, 2000, Visuell gestutztes Wissensmanagement mit Dokumentenlandkarten, Wissensmanagement July 2000 (in German) Becks, A., C. Seeling, 2001, A Task-Model for Text Corpus Analysis in Knowledge Management. Proc. 8* Int. Conf. on User Modeling, Sonthofen, Germany Chalmers, M., P. Chitson, 1992, Bead: Explorations in Information Visualization. Proc. 15* Int. ACM SIGIR Conf. on Research and Development in Information Retrieval, Copenhagen, Denmark Chen, H., Ch. Schuffels, R. Orwig, 1996, Internet Categorization and Search: A SelfOrganizing Approach. Journal of Visual Communication and Image Representation, Vol. 7, No. 1 Davidson, G.S., B. Hendrickson, D.K. Johnson, Ch.E. Meyers, B.N. Wylie, 1998, Knowledge Mining With Vxinside: Discovery Through Interaction. Journal of Intelligent Information Systems, Vol. 11, No. 3 Meghini, C , U. Straccia, 1996, A Relevance Terminological Logic for Information Retrieval. Proc. 19* Int. ACM SIGIR Conf. on Research and Development in Information Retrieval, Zuerich, Switzerland Salton, G. (Ed.), 1971, The SMART Retrieval System - Experiments in Automatic Document Processing. Prentice Hall, New Jersey Shneiderman, B., 1996, The Eyes Have It: A Task by Data Type Taxonomy for Information Visualization. Technical Report 96-66, Institute for Systems Research, University of Maryland Wise, J.A., J.J. Thomas, K. Pennock, D. Lantrip, M. Pottier, A. Schur, V. Crow, 1995, Visualizing the non-visual: Spatial analysis and interaction with information from text documents. Proc. IEEE Information Visualization 95
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
°^^
Multilevel Dynamical Models for Poly disperse Systems: a Volume Averaging Approach Bela G. Lakatos Department of Process Engineering, University of Veszprem H-8200 Veszprem, Hungary
Abstract The paper presents the fundamental elements of multilevel dynamical models of polydisperse systems of chemical engineering. A three-level model is derived and generalised by means of the volume averaging method, modified appropriately for dispersed systems. The model errors and computational aspects are analysed and discussed.
1. Introduction Solid-fluid heterogeneous systems often are modelled by pseudo-homogeneous models taking into account the effects of processes of the disperse elements by means of effectiveness factors (Datar et a/.,1987, Ramkrishna and Arce,1989) When, however, the time scales of the disperse solid and continuous fluid phases are of similar order of magnitude then the pseudo-homogeneous models, especially in dynamic conditions, may lead to significant errors. A characteristic feature of the solid-fluid systems is the solid and fluid phases exhibit significantly different length scales, mostly differing from each other by some orders of magnitude. This property provides a basis for application of the point sink approximation treating the dispersed elements as point sinks immersed in the environment formed by the continuous phase. Adsorption on porous adsorbents (Ruckenstein,1971, Bathia, 1997, Liu et a/.,2000, Quintard and Whitaker,1993) and heterogeneous catalytic reactions in porous catalytic particles (Do and Rice, 1982, Burghardt et a/., 1988, Keil,1996, MoUer and O'Connor, 1996) have been modelled in this way, but only few works have been concerned with the special features of this modelling approach. Neogi and Ruckenstein,1980 examined the problem using ensemble averaging, Ramkrishna and Arce, 1989 studied the spectral properties of the operator representing the models, and recently, Lakatos (2001) analysed the problem in terms of multilevel dynamical models, formulated by means of the volume averaging technique. The aim of the present work is to further develop the concept of multilevel dynamical models, and to discuss the computational aspects and model errors, arising in applying the multiscale volume averaging.
2. Multilevel Models: General Development Consider the three-phase system shown schematically in Fig. 1 in which the interfacial surfaces between the phases are complex and may vary in time. Let the characteristic length scales of the phases, termed a-, /3- and y-phases, respectively, differ from each
842
Figure I. Three-phase system exhibiting three-level spatial hierarchy other significantly. Then, in relation of the a- and ^phases, the a-phase is considered continuous and the jS-phase is termed dispersed, but, in turn, in relation of the /J- and yphases, the ^phase is the continuous one and the y-phase is considered dispersed. Let I/A be a scalar quantity which in the phases is denoted by I/A^^, i/^p and i/Zy. The variation of Xjf inside the phases is described by the balance equations dt
• +^°(//)=^/'
(1)
i = cc.P.}
where y is the flow density and ;r, is the volumetric source density of y/. The transport across the a/J- and Py- interfaces is described by the boundary conditions ^ij°iji-Pi¥i^ijhn,jo(j.-pjy/jW^j)=:a,j,
ij=a,p
and
ij = P,y
(2)
where Wij is the velocity of the //-interface, a;^ denotes surface source density of quantity I//'on the //-interface, and w^ is normal unit vector to the //-interface. Let us assume that it is possible to define such spatial averaging volumes V^ = constant L^
and
Vp = constant L^ ,
(3)
for the a- and ^phases, associated with coordinates Xa and xp, that the conditions
"^a « ^a « ^a
^^^ ^p « Lp « A^
A^ oc A^ and A^ oc A^
(4) (5)
are satisfied. Then, following the procedure presented by Lakatos (2001) for a two-level model, the molecular (single) level mathematical model of the system can be converted into a three-level one by means of the modified volume averaging technique. In this case, the phase average (..)^ of the intensive quantity i/x'in the a-phase is defined in the usual way (Whitaker,1967, Slattery,1967, Gray, 1975)
843
{Va)a(Xa^t)=—
(6)
jVadV
where \4= V'aa+ V/3a, Vaa and l/y3a are the partial volumes of the a- and j3-phases in l^, respectively. The phase average (..)^ of quantity i/in the j3-phase takes the form {Vp)aiXa^^)=
(7)
j{¥p)pn^{Vp.x^j)VpdVp 0
where (.)p denotes the average of i/zover a j3-phase element (particle): (8)
{Vp)p=^l¥pdV.
In Eq.(7), the function n^: RQ xR^ XRQ -> RQ is called the population density function of the j3-particles which in the present case is determined in the following way: np is such a function that the equality ^^T /
1 ^
\
/
\
]8[Vph(Vp.t.xJdVp = - X n V )
(^)
^ ^=1
0
is satisfied for each continuous and bounded function g(.), where K is the number of pparticles. By means of this function V(^nplypj,X(^)dVp
expresses the number of par-
ticles having volume (v^, V^ 4- dV^ ) at the moment of time t in the averaging volume Va associated with coordinate jc^. The spatial averaging (..)^ in relation of the j5- and yphases is derived analogously. Applying now, in turn, the averaging operators (..)(^ and {..)p to Eqs (l)-(2), and taking into account that, because of the relations (3)-(5), (10)
{{X)p={-)a
as well as the appropriate volume averaging and general transport theorems, we obtain the following hierarchy of the model equations. The motion of i// in the a-phase, i.e. on the a-level is described by the equation ^ p lllOA
i
"
(11)
^ UlOA
AR[V,
where the left hand side terms of Eq.(l 1) describe the variation of quantity i/A in the aphase, while the right hand side terms describe the changes of y/ due to the variation of the volume of /3-particles, the transfer of y/ through the aj3-interface, and the production of i/Aby the Gap surface source density, respectively. Here, the population density function is determined by the population balance equation - + V o ( ( v ^ ) p n ^ j + — - —f-np dt '' '"^ ^' dW, dt
\ = {np)pnp
(12)
844 describing the beiiaviour of )3-particles, represented on the a-level as point sinks immersed and moving in the a-phase. Similarly, equations on the /J-level are
0
0
A^[V,)
A,(vJ
and '-"y
{n,)pn^.
(14)
Finally, equation on the y-level dxi/y
I
\
describes the variation of quantity y/ inside the y-particles. Here, qy denotes some nonconvective component of the flow density that may be of complex nature, depending on the structure of particles. Eqs (11)-(15) are completed with the appropriate boundary and initial conditions. The boundary conditions for Eqs (13)-(14) describe the connection of the system with the environment while the boundary conditions for Eq.(15) describe the connection between the internal world of a y-particle and its continuous phase environment.
3. Model Errors In developing multilevel dynamical models by using the volume averaging technique some approximations have been made, yielding different errors. Namely, model errors, characteristic for that procedure, are produced by • averaging nonlinear source functions, • closing the averaged equations using approximating closure models, • assuming that particles represented as point sinks are immersed in homogeneous environment given by the average value, • intersecting particles by the surface of the averaging volume. For the sake of illustration, Fig.2.a shows what is the difference between the approximated y/p and the real y/preai profiles in a )3-phase particle immersed an ideal and realistic environment. At the same time, Fig.2.b shows that from the particles intersected those, which have mass centre inside the averaging volume, are accounted for, but those that have mass centre outside do not. A detailed analysis and quantitative predictions of the model errors of this approach will be presented elsewhere.
4. Three-Level Model: Adsorption in Bidisperse Solids A three level model describes a fixed bed adsorber with bidisperse adsorbents, i.e. in which the adsorbent pellets consist of small porous (micro) particles. Here, the a-level is formed by the gas phase, where the pellets are treated as point sinks, and is described
845
O Particles not intersected Q Particles intersected liaving mass center outside Va 9 Particles intersected liaving mass center inside Va
a) ! b) Figure 2. Effects of) the point sink approximation and b) of intersection of some particles by the averaging volume by the axial dispersion model
dc^{x^,t) _ ^ dt
d^c^(x^j)
= D^
dc^(x^j)
dxl
3(1-£)^
dx^
dcp(rpj)
(16)
dro
RR£
The spherical adsorbent pellets form the /5-level, where the microparticles, often distributed in size, are considered point sinks dcp{rp,t) dt
Dp
d
3(1-g^:
dro
rp drp
Ry£p
D,
dCy(ry,rQj)
dr^
n(Ryj)dRy .=R.
(17) Finally, the internal world of the microparticles forms the y-level in this model. The complex structure of microparticles often yields intensive dynamic interactions between the adsorbate and the solid surface (Luikov,1980), so that adsorption proceeds under dynamic conditions, i.e. local relaxation phenomena should also be taken into consideration. Consequently, the transport and adsorption in the micropores is described by the equations involving also the local relaxation time scale T« of the adsorption process dcyiry,rp,t) dt
Dy d ^ dr^
2
dcy(ry,rpj)
da{t,Cy,dCy jdt) dt
'
a{t) = f{c
)^T^-
dCy
(18) Here in Eqs (16)-(18), we used the following notation: c - concentration, x - axial coordinate, r - radial coordinate, R - radius, D - dispersion or diffusion coefficient, v - linear velocity, £ - porosity, a - adsorbed species,/- static adsorption equilibrium.) The set of nonlinear partial differential equations (17)-(18), with the appropriate initial and boundary conditions, can be solved numerically by using multilevel computational schemes, determined in principle by the hierarchical structure of the model, shown schematically in Fig.3. A three-level orthogonal collocation scheme, has proved suitable for
846
Figure 3. Three-level computational structure corresponding to the hierarchy of the system shown in Fig.l. solving Eqs (16)-(18), similarly to that developed for catalytic reactors with biporous catalysts (Lakatos, 2001). Also, this scheme illustrates that systems exhibiting such hierarchy usually are manipulated at the a-level. Then, accounting for the possible motion of the lower level elements and the detailed description of their internal world simultaneously produces contradictory requirements. Particles, due to their independent motion in the environment represented by the higher level, with significantly different internal histories may be present in the same averaging volume which provides a number of difficulties in computations. In such cases, often hybrid, i.e. combination of deterministic and stochastic, computations should be applied. Acknowledgement The author would like to thank the Hungarian Research Foundation for the financial supporting this work under Grant T034406.
5. References Bathia, S.K., 1997, Che, Engng Sci., 52, 1377. Burghardt, A., J. Rogut, and J. Gotkowska, 1988, Chem. Engng Sci. 43, 2463. Carbonell R.G. and S. Whitaker, 1983, Chem. Engng Sci. 38, 1795. Datar, A., B.D. Kulkarni and L.K. Doraiswamy, 1987, Chem. Engng Sci. 42, 1233. Do, D.D. and R.G. Rice, 1982, Chem. Engng Sci. 37, 1471. Gray, W.G., 1975, Chem. Engng Sci. 30, 229. Hu, X. and D.D Do, 1995, AIChE Journal, 41, 1581. Keil, F.J., 1996, Chem. Engng Sci. 51, 1543. Moller, K.P. and C.T. O'Connor, C.T., 1996, Chem. Engng Sci. 51, 3403. Lakatos, B.G., 2001, Chem. Engng Sci., 56, 659. Liu, F., S.K. Bathia and I.I. Abarzhi, 2000, Computers chem. Engng, 24, 1981. Luikov, A.V., 1980, Heat and Mass Transfer. Mir, Moscow. Neogi, P. and E. Ruckenstein, 1980, AIChE Journal, 26, 787. Quintard, M. S. and Whitaker, 1993, Chem. Engng Sci. 48, 2537. Ramkrishna, D and P. Arce, 1989, Chem. Engng Sci. 44, 1949. Ruckenstein, E., A.S. Vaidyanathan and G.R. Youngquist, 1971,Chem. Engng Sci.,1305. Slattery, J.C, 1967, AIChE Journal, 13, 1066. Whitaker, S., 1967, AIChE Journal, 13, 420.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
847
Open Software Architecture For Process Simulation: The Current Status of CAPE-OPEN Standard Jean-Pierre Belaud Laboratoire de Genie Chimique (LGC, UMR CNRS 5503), Equipe Analyse Fonctionnelle des Precedes, INPT-ENSIACET 118 route de Narbonne, 31077 Toulouse Cedex 4, France [email protected] Michel Pons ATOFINA Centre Technique de Lyon BP 32 F-69492 Pierre-Benite Cedex, France [email protected]
Abstract Traditionally simulation environments have been closed monolithic systems and the resulting bottlenecks in interoperability, reuse and innovation have led to the search for a more open and interoperable solution. The CAPE-OPEN (CO) effort, launched in January, 1997, is a standardisation process for achieving true plug and play of process industry simulation software components. The resulting CO standard is now being widely disseminated to the chemical engineering community. It relies on a technology that integrates up to date concepts from the software field such as a component-based approach. A number of software components based on this technology have been developed and are already available. Thanks to this new generation of CAPE tools, it is expected to reach cheaper, better and faster design, operation and control of processes. The CAPE-OPEN Laboratories Network (CO-LaN) consortium is in charge of managing the lifecycle of the CO standard.
1. Introduction and Objectives The integration of external know-how in traditional simulation environments has to deal with proprietary code, and requires painful and expensive activities, while open software architectures are the way forward for the next generation of CAPE tools as demonstrated by (Braunschweig et al., 1999). The CO standard is the result of a worldwide collaboration between chemical and petroleum refining industries, academics, and CAPE software suppliers, all with a common goal of defining a standard for a component-based approach to process simulation. Thus any compliant component can be integrated instantly in any compliant environment. The standard is open, multiplatform, uniform and available free of charge. Version 1.0 of the CO standard was released in February 2002. This paper explains how it will be subsequently managed by the CO-LaN. The associated technology and the
848 scope of this standard are then described. Finally we present some CO-compliant commercial and non-commercial implementation of this technology.
2. CAPE-OPEN Laboratories Network The (CO-LaN, 2001), a non-for-profit organisation, maintains the CO standard, disseminates information on the standard, releases software tools to test CO compliance and publishes interoperability test results. With more than 20 members in year 2001 (operating companies, vendors and academics), it provides a service to the CAPE community in all aspects of the CO standard. The bylaws of the CO-LaN ensure that it is open to the entire CAPE community for access and membership. Membership is definitely not required to use CO compliant software components or to develop such components. However the CO-LaN organises the CAPE community with respect to the CAPE-OPEN standard. The CO-LaN operates through a web portal where visitors find or will soon find the CAPE-OPEN documentation, as well as additional resources for implementing and using CO compliant components (FAQ's, discussion board, how-to's, software migration support, etc.). Members of the CO-LaN find the services that help them to develop new standard interfaces or improve existing ones, as well as dedicated help for implementing CO compliant software components. Special Interest Groups will be created and maintained by the CO-LaN for refining or extending the CO standard, following a careful analysis of the value creation brought to users and to vendors by each development and improvement. The SIGs will be managed in such a way as to bring these developments quickly to the market, updafing as necessary the documentation and the tools delivered to the CAPE community.
3. CAPE-OPEN Technology Key elements of CO technology are openness, interoperability, standardisation process and service. This technology integrates today's concepts in software domain, development tools and web enabled skills. It embodies the CO formal documentation set, the CO architecture, and the CO system model. We focus below on the current potential for process simulation development engineers, so the CO system model is detailed while the CO formal documentation set and the CO architecture are only introduced. 3.1 CO formal documentation set The CO standard follows a specific versioning system using a unique and global version number and is composed of a set of documents. These documents are organised according to a CO formal documentation set. This documentation set includes six blocks: General Vision, Technical Architecture, Business Interfaces, COSE Interfaces, Common Interfaces and Implementation Specifications. • General vision contains documents that should be read first to get the standard general information, such as general requirements and needs.
849 • •
•
•
•
Technical architecture integrates the horizontal technical materials and defines an infrastructure for a process simulation based on the CO standard. Business interfaces contain all vertical interface specification documents. These interfaces are domain-specific interfaces for the CAPE application domain. They define CO components involved in a CO process simulation application. COSE (CAPE-OPEN Simulator Executive) Interfaces refer to horizontal interface specifications. They are interfaces for simulation environments such as simulator executives. Within this category, services of general use are defined such as diagnostics and material template system in order to be called by any CO components through a call back usage. Common interfaces enclose horizontal interface specification documents for handling concepts that may be required by any Business and COSE interfaces. This is a collection of interfaces that support basic functions and are always independent of Business and COSE Interfaces. Implementation Specifications contain the implementation of the Business, COSE and Common Interfaces specifications for a given distributed computing platform. All documents from Business, COSE and Common Interfaces are abstract specifications which create and document a conceptual model in an implementation neutral manner. Thus the design of CO is independent from any computing platform. It has the ability to be extended to any platform. The Implementation Specifications are available for (D)COM and CORBA through the Interface Definition Language libraries. In order to produce CO compliant software components, any software developer has to use these official libraries.
3.2 CO architecture The CO architecture elements describe technical objectives and terminology and provide the infrastructure upon which supporting Business, COSE and Common Interfaces are based. This identifies the technologies associated with the CO standard, includes the object model which defines common semantics, and shows the reference model which embodies the CO interfaces categories, CO (compliant) software components and communication mode. That is based on the distributed component (heterogeneous) system and the object-oriented paradigm. The involved technologies are the UML notation (Rumbaugh et al., 1997), the OMG CORBA (OMG, 2001) and Microsoft (D)COM (Microsoft, 2001) middleware, as well as the Unified Processes and object-oriented programming languages (Meyer, 2001). The wide-scale industry adoption of this CO architecture provides application developers and end-users with the means to build web-enabled interoperable simulation software systems distributed across all major hardware, operating system and programming language environments. 3.3 CO system model The CO system model represents the UML design model of standard. It defines the scope. The physical view of this model allows extraction of the CO software components and shows their dependency relationships. The logical view of this model organises the services, identifies the CO packages and CO interfaces, and designs the related structural organisation. The standard distinguishes two kinds of software components: Process Modelling Components (PMCs) and Process Modelling
850 Environments {PMEs), the latter making use of the services provided by the PMCs. Typically the PMEs are environments that support the construction of a process model and that allow the end-user to perform a variety of different tasks, such as process simulation or optimisation (Pantelides et al., 1995). Among the standardised PMCs there are: Thermodynamic and Physical Properties, Physical Properties DataBases, Unit Operations, Numerical Solvers and Sequential Modular Tools. • Thermodynamic and Physical Properties component: In the area of physical properties, CO focuses on uniform fluids that are mixtures of pure components or pseudo-components, and whose quality can be described in terms of molar composition. The physical properties operations that have been provided with standardised interfaces are those required for the calculation of vapour-liquid or liquid-solid equilibria or subsets thereof, as well as other commonly used thermodynamic and transport properties. A key concept is that of a Material Object. Typically, each distinct material appearing in a process (in streams flowing between unit operations, as well as within individual unit operations) is characterised by one such object. Each unit operation module may interact with one or more Material Objects. To support the implementation of the above framework, the CO standard defines interfaces for Material Objects as well as for thermodynamic property packages, calculation routines and equilibrium servers. • Unit Operation component: CO defines a comprehensive set of standard interfaces for unit operation modules being used within modular and steady-state PMEs. A unit operation module may have several ports that allow it to be connected to other modules and to exchange material, energy or information with them. In the material case (which is also the most common), the port is associated with a Material Object. Ports are given direcfions (input, output, or input-output). Unit operation modules also have sets of parameters. These represent information that is not associated with the ports but that the modules wish to expose to their clients. Typical examples include equipment design parameters (e.g. the geometry of a reactor) and important quantities computed by the module (e.g. the capital and operating cost of a reactor). • Numerical Solvers component: As explained by (Belaud et al., 2001a) the CO standard focuses on the solufion algorithms that are necessary for carrying out steady-state and dynamic simulation of lumped systems. In particular, this includes algorithms for the solufion of large, sparse systems of non-linear algebraic equafions (NLAEs) and mixed (ordinary) differenfial and algebraic equafions (DAEs). Algorithms for the solution of the large sparse systems of linear algebraic equafions (LAEs) that often arise as sub-problems in the solution of NLAEs and DAEs are also considered. The CO standard introduces new concepts, such as models and the equation set object (ESO), which is a software abstracfion of a set of non-linear algebraic or mixed (ordinary) differenfial and algebraic equations. The standard ESO interface enables access to the structure of the system, as well as to information on the variables involved. The equafions in any model may involve disconfinuities. Disconfinuous equafions in a models are represented as state-transition networks (Avraam et al., 1998). • Sequential Modular Specific Tools component: A key part of the operation of sequenfial modular simulation systems is the analysis of the process flowsheet in
851
•
order to determine a suitable sequence of calculation of the unit operation modules (Westerberg et. al., 1979). Thus, typically the set of units in the flowsheet is partitioned into one or more disjoint subsets (maximal cyclic networks, MCNs) which may then be solved in sequence rather than simultaneously ("ordering"). The units within each MCN are linked by one or more recycle loops which are converged iteratively via the identification of appropriate "tear streams". The above tasks are typically carried out using a set of tools that operate on the directed graph representation of the flowsheet. The CO standard defines standard interfaces for the construction of these directed graphs, and for carrying out partitioning, ordering, tearing and sequencing operations on them. Physical Properties Databases component: CO defines how a database of recorded physical property values and model parameters can be connected to flowsheefing and other engineering programs. This interface deals with physical property data at discrete values of the variables of state (temperature, pressure, composition), as far as measured, correlated or estimated values are concerned.
4. Delivering Components Software component developers are working either to bring new CO compliant products to the market place or to make existing software components CO compliant. In either case, these software components can be for commercial sale, for proprietary use within an organisation, or for proprietary delivery to a specific client. Commercial software such as detailed by (Belaud et al., 2001b) and non-commercial software based on the CO standard are already available. Only a few are listed in order to illustrate results and potentials of the CO standardisafion effort. 4.2 Process modelling environment development The CO technology is now delivered in commercial process simulation software: Hyprotech has developed the HYSYS Unit and Thermodynamic CO sockets, which make HYSYS.Process/Plant version 2.2 (and subsequent versions) a CO compliant PME. AspenTech has implemented a socket for CO Thermodynamic and Physical Properties components in Aspen Plus 10.2 and Aspen Properties 10.2 (and subsequent versions i.e. 11.1). Aspen Plus 10.2 also implements a socket for CO Unit Operation components. Process Systems Enterprise has released a new version of their gPROMS tool with a CO Thermodynamic socket. BELSIM SA has done the same for their VALI III data reconciliation tool. 4.3 Process modelling components development Several PMCs are already implemented and many more are being developed. Some are only prototypes or for internal use only (for example ProSim SA delivered, exclusively to TotalFinaElf, a CO compliant thermodynamic server) while others are releases to the CAPE marketplace. In its CAPE-OPEN kit, for demonstration purposes, Hyprotech is distributing one Property Package and two Unit Operations which are CO compliant. AspenPlus can be used to create new CO physical property packages, which can be integrated in any CO compliant PME. Infochem has made its MultiFlash tool CO
852 compliant and has successfully tested it with Hysys.Process, AspenPlus and gProms. From the academic side, (Belaud et al., 2001a) from LGC-INP Toulouse institute have developed a CO-compliant Numerical Solvers component called Numerical Services Provider which supplies LAE, NLAE and DAE objects and acts as a real framework that makes up a reusable design for disseminating any solver algorithm through the CO standard. Furthermore, INPT has demonstrated a Sequential Modular Specific Tools PMC.
5. Conclusion The CO standard gives process engineers more flexible process modelling tools by allowing simulation with software components from multiple sources, assembled easily in a simulation environment. Any CO compliant software component can be integrated in any CO compliant simulation environment by "plug and play". The CO standard benefits software component developers by increasing the usage of CAPE tools and reducing the development time thanks to the CO technology. This technology is based on the distributed component heterogeneous system and modern software development techniques. A non-for-profit organisation, CO-LaN, promotes and maintains the standard.
References Avraam M., N. Shah and C.C. Pantelides, 1988, Modelling and. Optimisation of General Hybrid Systems in the Continuous Time Domain, Comput. Chem. Engng., 22S, S221-S228, 1998 Belaud J.P., K. Alloula, J.M. Le Lann and X. Joulia, Open software architecture for numerical solvers, 2001, Proceedings of European Symposium on Computer Aided Process Engineering-11, Elsevier, pp 967-972, 2001. Belaud J.P., B. L. Braunschweig, M. Halloran, K. Irons and D. Pifiol, 2001, New generation simulation environment: Technical vision of the CAPE-OPEN standard. Presentation at the AIChE Annual Meeting, Reno, NV, Nov 4-9, Modelling and Computations for Process Design, 2001. Braunschweig B. L., C. C. Pantelides, H. I. BriU and S. Sama, Open software architectures for process modelling: current status and futures perspectives 1999, FOCAPD'99 Conference, Breckenridge, Colorado, 1999. CO-LaN, 2001, CAPE-OPEN Laboratory Network web site: www.colan.org Meyer B., 1997, Object-Oriented Software Construction, 2nd edition, Prentice Hall, 1997 Microsoft, 2001, Microsoft web site about COM: www.microsoft.com/com OMG, 2001, Object Management Group web site about CORBA: www.corba.org Pantelides C.C. and H. I. Britt, 1995, Multipurpose Process Modeling Environments. In L.T. Biegler and M.F. Doherty (Eds.), Proc. Conf on Foundations of Computer-Aided Process Design '94. CACHE Publications, Austin, Texas, pp. 128-141, 1995. Rumbaugh J., I. Jacobsen and G. Booch, 1997, Unified Modeling Language Reference Manual, Addison Wesley, 1997. Westerberg A.W, H.P. Hutchinson, R.L. Motard and P. Winter, 1979, Process Flowsheeting. Cambridge University Press, Cambridge, U.K, 1979.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
853
An Open Software Architecture for Steady-State Data Reconciliation and Parameter Estimation Chouaib Benqlilou\ Moises Graells^, Antonio Espufia' and Luis Puigjaner^* Universitat Politecnica de Catalunya, Chemical Engineering Department, ^ETSEIB, Av. Diagonal 647, E-08028 - Barcelona (Spain) ^EUETIB, Comte d'Urgell, 187, E-08036 - Barcelona (Spain) E-mails: [chouaib, graells, aec, lpc]@eq.upc.es
Abstract In this paper a flexible and open architecture design for Parameter Estimation and Data Reconciliation (PEDR) software application is proposed by de-coupling it according to the functionalities involved. In the proposed approach the different components that are involved in this application and their interactions are specified and tested. The proposed architecture aims at an improved efficiency and upgrading of the PEDR application by allowing the exchangeability and connectivity of the present components in an easy and consistent way.
1. Introduction Accurate process measurement is increasingly vital for modelling, monitoring and optimisation of chemical and related processes. Once gross errors are filtered, accurate process data may be obtained using Data Reconciliation (DR), a technique that "corrects" measured data by minimising the variance with respect to a process balance model. The same procedure may lead to model adjustment by determining the best model parameter values matching a set of reliable data. Thus, Parameter Estimation (PE) and DR are based on optimisation techniques that require the following types of elements: mathematical models of the process or processes, sets of process variable measurements and/or reconciled data and mathematical solvers. Being an aggregation of different technologies, a PEDR tool should not be conceived as a monolithical application since it will not be possible to exchange one component without affecting the performance of others. This could be a serious drawback for taking full advantage of a growing number of tools, such as equation-based simulators, optimisation algorithms, databases, Distributed Control System (DCS), etc. Therefore, a flexible design requires all the PEDR components to be as de-coupled as possible. This work introduces an open modular architecture for a PEDR software executive allowing the management of different sets of process data, process models and available mathematical solvers through standardised interfaces. This leads to the exchangeability of the corresponding software components. Furthermore, the communication between
854 the corresponding objects has been established using Common Object Request Broker Architecture (CORB A) as middleware in order to handle data across the network. Case studies are provided showing the great flexibility of the system for handling different plant structures and situations by matching various predefined or newly introduced process models with the corresponding available data sets. Moreover, the system could be also upgraded by plugging new optimisation solvers to it, when available.
2. The system architecture 2.1 Multi-module vs. monolithical applications The performance of numerical solvers and process modeling tools has a direct impact on the results of Data Reconciliation and Parameter Estimation. Thus, the continuous improvement of such tools requires a flexible and modular structure for a PEDR system so that upgrading of this system may be easily achieved by replacing old modules. Since, these modules could be supplied from different providers, they may present a possible software and hardware incompatibility. Therefore, their integration and incorporation in a monolithical structure leads to high effort both, in cost and in implementation. These drawbacks may be overcome by using separated components that inter-operate through a well-specified interface. The use of standardised interfaces for open communication between software components has emerged as a promising challenge for the software application incompatibility (CAPE-OPEN, 2000). Currently, different vendors and industrial companies are already incorporating the CAPE-OPEN interfaces. By using the results of CAPE-OPEN, it is relatively easy to include and integrate modules from various suppliers and in a heterogeneous environment. 2.2 Architecture design 2.2.7 Modules The monolithical application should be de-coupled as far as possible for an open and flexible architecture in such a way that each one of the present components could be replaced without affecting the application performance. The PEDR system can be functionally de-coupled into five main components: the PEDR Manager, process model, sets of process or reconciled measurements, mathematical solvers, and the PEDR client component as is shown in Fig. 1. 1.
2.
The main purpose of the PEDR Manager module is to gain access to the measured data and to the process model description in order to build up the optimization problem by interacting among the required modules (e.g. database and solvers). Furthermore, it is the responsible for variable classification, estimation of unmeasured observable variables and Gross Error detection (GED) tasks. The main purpose of the process model component is to generate a model description of the process under consideration. It is important to notice that only steady-state processes are considered. The fundamental building block employed for this purpose is an equations-based model. However, any other available model (black box, simulation software package) may be used.
855 The process data module is responsible for acquiring measured process variable values for on-line as well as for off-line applications. These process variable data are often stored intermediately in a relational database or obtained directly from a DCS. The mathematical solver module is responsible for the resolution of the DR or PE problems. The solver may be accessed by direct methods or via file (e.g. MPS, MathML...). The PEDR Client component prepares the output generated by the PEDR Manager for easy use by the customer.
PEDR Client
Customer MutUh
FVPFT
PEDR Manager
Process Model
Process Data
Solver
MySql
CONOPT
Figure 1: Architecture design for PEDR. 2.2.2 Modules interaction In the first step, the database, the solver and the PEDR Manager are registered in order to make them accessible to their clients (there can be more than one client placed remotely). Once the servers are registered, the client component contacts and initiates the communication with the PEDR manager component through a specified interface. The client component initiates the PEDR Manager by asking it to reconcile/estimate a given set of process variables supplying the references of the following information: process measurements, the corresponding process models and the appropriate solvers. Then the PEDR manager accesses the process measurement and the process model, prepares the optimisation problem, and interacts with the solver to obtain the optimisation results that are finally offered to the client (Fig. 2).
3. Implementation Component software and object-oriented approaches, which view each component in the above architecture as a separate object, were adopted. All the communications between objects are handled by CORBA and implemented in Java (Orfali et aU 1998). The sequence diagram (Fig. 2) represented in Unified Modeling Language (UML) shows the temporal sequence of steps to be followed in order to perform the DR or the PE. Test prototype software has been developed to demonstrate the use and benefits of the proposed component architecture and the specification of the open interface proposed.
856
Figure 2: The sequence diagram.
4. Interfaces 4.1 The Model and Solver interfaces For the sake of standardization, the interfaces to communicate the PEDR Manager with the solver and model modules should follow the CAPE-OPEN specifications (CAPEOPEN, 1999). The process model interface specification assumes that the process model is described by a set of continuous equations, while the solver interface specification assumes a mathematical programming problem. 4.2 The database interface This is not a general-purpose interface for process measurement data but only a specific one according to the requirement of a PEDR module. The proposed interface permits
857 both to introduce and to retrieve data. This data could be measured, reconciled or estimated, and are grouped into sets of experiments (corresponding to experiments on the plant in different operating conditions). 4.3 The PEDR Manager interface Finally, despite its internal modularity, PEDR manager had to expose a common interface to be invoked by any external client. Fig. 4 shows the interface in UML that is being proposed within the GLOBAL-CAPE-OPEN project. Furthermore, the PEDR Manager provides a graphical and user-friendly interface (Fig. 3) designed according to the methods that the Manager exposes. First of all, a PEDR Client can choose to perform either a DR or a PE task. Then, it selects the measured data to be reconciled or used for parameter estimation, the required mathematical model to be used and the appropriate solver for solving the resulting optimization problem. Finally, the Client could ask the system to solve the problem. PEDR Manager 1 M e a s u r e ._1 Measure 3 M e a s u r e ,_4
1 Balance. 1 1 Balance._2 Balance. 4 Balance. 5 Balance. 6
<•
Reconciliation
C
Estimation Solve
Jj
Resufts... Add / Edit. Reconcile
data set
Add/Edit... M e a s u r eJ 2
j
Exit
Add / Edit .
according to model
Balance_3
Sending data... Solving...
Figure 3: The Graphical User Interface for the PEDR Manager.
using solver
NLPJ2
858 «lnterface» IPEDRManager %3etModelRed() ^GetDataRefO ^GetDegreeaRedundancyO %3etVariableType() %3etVariableVarianoe() %3ietVariableVaIue() ^GetParameterValue() ^GetObjectiveFunctionValueO %3etSoverRef() %teconcile() ^Estimate{)
Figure 4: The PEDR Manager interface.
5. Conclusions The main objective of the presented work has been the design and implementation of a software architecture for distributed Data Reconciliation and Parameter Estimation applications. The task of maintaining and supporting new process modelers, databases and/or solvers within the PEDR packages could be justified in cost and implementation using the proposed open software architecture. In this work the specification of the Data reconciliation and Parameter Estimation interfaces are conceived for steady state. However, the general approach adopted makes them likely to be of application for the dynamic case. Thus, the extension to the dynamic case can be implemented with relatively low effort without making any essential change to the other components.
Acknowledgment Financial support received from European community (project Global-Cape-Open IMS 26691) and the Spanish ''Ministerio de Educacion, Cultura y Deporte" (project REALISSTICO QUI99-1091, CICYT) are gratefully acknowledged.
6. References CAPE-OPEN 2000, Conceptual Design Document (CDD2) for Global-CAPE-OPEN Project. Available on the Word Wide Web at the URL http://www.global-capeopen.org/. CAPE-OPEN 1999, Open interface specification (Numeric Solvers) for-CAPE-OPEN Project. Available on the Word Wide at the URL http.7/www.global-capeopen.org/CAPE-OPEN standard.html. Orfali,. R. and D. harkly. Client/Server programming with Java and CORBA. John.Wiley & Sons, Inc., 1998.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
859
TRIZ and the evolution of CAPE tools From FLOWTRAN® to CAPE-OPEN-^® and beyond Bertrand Braunschweig, Institut Fran^ais du Petrole, 1 & 4 avenue de Bois Preau, 92852 Rueil Malmaison Cedex, France, [email protected] Kerry Irons, The Dow Chemical Company, Engineering Sciences, 1400 Bldg, Midland MI 48667. USA, [email protected]
Abstract The paper looks at trends in evolution of software tools with a particular emphasis on CAPE applications. Using results from TRIZ (Theory of Inventive Problem Solving), we show why and how CAPE software follows a major trend towards distributed adaptive heterogeneous components. Consequently, and thanks to the CAPE-OPEN communication infrastructure, we show that CAPE models can be made of distributed autonomous agents that we nickname "cogents".
!• Introduction In the middle of the 20th century, Genrich Altshuller, a Russian engineer, analysed hundreds of thousands of patents and scientific publications. From this analysis, he developed TRIZ, the theory of inventive problem solving, together with a series of practical tools for helping engineers solve technical problems. Among these tools and theories, the substance-field theory gives a structured way of representing problems, the patterns of evolution show the lifecycle of technical systems, the contradiction matrix tells you how to resolve technical contradictions, using the forty principles that describe common ways of improving technical systems. For example, if you want to increase the strength of a device, without adding too much extra weight to it, the contradiction matrix tells you that you can use principle 1, segmentation, or principle 8, counterweight, or principle 15, dynamicity, or principle 40, composite materials. TRIZ is now used by a wealth of Fortune 500 companies in support of their innovation processes. When Atshuller developed TRIZ, he could not think of software components. These objects just did not exist. But software components are technical objects too. They are in fact some of the most complex technical objects produced by man. Many of the TRIZ tools and theory elements only relate to concrete objects (e.g. principle 11, prior counteraction, or principle 32, change the colour). But some of the principles can be applied to software. Among these, a famous one is principle 26, copying : a simplified and inexpensive copy should be used in place of a fragile original or an object that is inconvenient to operate. Another one, principle 7, nesting or « matrioshka », introduces the modular approaches to software development. We really like two principles, 1: Segmentation, and 15: Dynamicity, which are also considered to be patterns of evolution because of their importance. Segmentation shows
860 how systems evolve from an initial monolithic form into a set of independent parts, then eventually increasing the number of parts until each part becomes small enough that it cannot be identified anymore, such as in a powder. Further evolution based on this principle leads to similar functions obtained with liquids, gases or fields. Think of a bearing with balls suspension, replaced by microballs, then by gas suspension and finally by magnetic field. Dynamicity introduces flexibility and adaptation by allowing the characteristics of an object, of an environment, or of a process, to be altered in order to find an optimal performance at each stage of an operation. Think of a traffic light that adapts its period depending on the traffic flow. If you look around you, you will find examples of segmentation and dynamicity in technical objects. Here are a few in process engineering and in computer science: • Segmentation : process operating companies are now experimenting with microscale processes that fit on a chip or on a PC board; • Dynamicity : feedback control loops are everywhere in process plants; there are multilevel controllers too; • Dynamicity and segmentation : what is now called ''mass customisation' with end products being adapted towards the needs of their individual customers; • Dynamicity : simulated counter-current processes operating in non-equilibrium mode; • Segmentation: mainframe computers were replaced by networks of smaller computers, and then by clusters of PCs; • Segmentation and dynamicity : in numerical modelling, local adaptive grid methods allow precise and efficient modelling and simulation of unit operations.
2. The evolution of CAPE tools CAPE software, as other technical objects, follows the TRIZ trends of evolution by becoming segmented and dynamised. We present the first three stages of evolution. - Stage 1: Monolithic to FORTRAN subroutines and modular architectures The first CAPE software developed in the sixties and seventies were large monolithic systems. Developed in FORTRAN, they were designed as large multipurpose programs sharing data through COMMON declarations and using internal or external subroutines. Modular programming helped in facilitating maintenance and debugging, so it was quickly adopted. In-house code such as IFP's PGGC\ BP's Genesis, or commercial software such as FLOWTRAN are examples of stage 1 of evolution. They remained as such until recently when developers started to cut those systems into smaller pieces that would fit together. - Stage 2: from object-oriented tools to component architectures Modularity, object-oriented programming, component software, and n-tier architectures are the current paradigm for CAPE software development, and can be considered as the ^ Programme General de Genie Chimique
861 second stage of evolution. Although there are other component-based architectures, The CAPE-OPEN interoperability architecture, based on object orientation and middleware, is the best representative of this stage, as it appears to be the dominant one being adopted by a majority of players. CAPE-OPEN is now accepted as a standard for communication between simulation software components in process engineering. This leads to the availability of software components offered by leading vendors, research institutes, and specialized suppliers which enable the process industries to reach new quality and productivity levels in designing and operating their plants. See (Belaud et al., 2002) for more information. It is also important to mention the CAPE-OPEN Laboratories Network (CO-LaN) and its public catalog of components. CO-LaN, a non-profit society, was created by the Global CAPE-OPEN (GCO) project to become the internationally recognised testing and process management organisation for the CAPE-OPEN standard, and more generally to encourage the use of CAPE software tools in industry, administration and academia. More specifically, the main activities of the CO-LaN are to disseminate, to maintain and manage evolution of the specifications, to facilitate component and interoperability testing, and eventually to keep a public catalog of compliant components. Information on each CO software component is stored in a database which can be searched by visitors of the CO-LaN Web site (www.colan.org). The database can be searched, for example, by type of calculation performed by a piece of software, thus enabling a process engineer to find the right tool to perform any piece of process engineering work, while being sure that the found piece of software will plug easily in his/her CAPE-OPEN compliant simulation environment. Through the component directory, the CO-LaN will provide a mean for buyers and sellers to meet as well as an appropriate classification of software tools. - Stage 3: from dynamic components to software agents The third stage will be the one of dynamicity, as the needs for self-adaptation become increasingly important, in order to match the increasing diversity in usage. Self-adaptation can be obtained using current software technologies, such as Enterprise Java Beans and web services, which allow software components to discover their environment at runtime and to seamlessly integrate within these environments. Current architectures, even though they allow distributed computing on heterogeneous hardware platforms, share the same paradigm for control and co-ordination: a central piece of software controls and co-ordinates execution of all software modules and components that together constitute the model and the solving mechanism of a system. One example is the central piece of software that is usually called the "simulation executive", or "COSE" in CAPE-OPEN architectures. Its tasks are numerous: it communicates with the user; it stores and retrieves data from files and databases; it manages simulation cases; it helps building the flowsheet and checks model topology; it attaches physical properties and thermodynamic systems to parts of the flowsheet; it manages the solving and optimisation algorithms; it launches and runs simulations; etc. All other modules (e.g. unit operations, thermodynamic calculation routines, physical
862 property data sets, solvers and optimisers, data reconciliation algorithms, chemical kinetics, unit conversion systems etc.) are under control of the simulation environment and communicate with it in a hierarchical manner, as disciplined soldiers will execute their assignments and report to their superiors.
Figure J: a COSE-centric architecture
3. Beyond CAPE-OPEN: COGENTS co.gent (-J3nt): n. pi. [CSG.
863 A user will describe the simulation problem using the CAPE-ML language and interface, communicating with an enhanced COSE which understands CAPE-ML, knows the COLaN component catalogue, and is able to discuss with the individual cogents. The COSE will look for possible components in the library, then will start discussing with all potentially useful components - they will probably communicate with each other too - and establish one or several networks of cogents in order to solve the modelling problem. The COSE will then eventually run these networks, confront them with real process data, and possibly choose among the networks of cogents, following a quality criterion.
Figure 2: A distributed architecture Such an agent-based distributed system will enable interoperability, inter-working, openness and integration of CAPE applications and services across platforms. This will enable businesses and organisations to deploy agile and integrated systems in support of the development of new value chains. Of course, the cogents concept demands significant evolution in general computer hardware and software capabilities. The requirements for added computing power to effectvely implement the cogents concepts are perhaps obvious. A (r)evolution in operating systems and/or software interface standards is necessary to allow the use of smart software as proposed for cogents.
4. Conclusion The scenario that we draw will become real only in a few years time. But we can start working on it. Companies and research organisation who take this new step will gain a competitive advantage and be ready to offer Application Service Provider (ASP) CAPE services taking advantage of the current software technologies and of the many combinations offered by the internet.
Acknowledgements Many thanks to Rafael Batres, Jean-Pierre Briot, Alexis Drogoul, Eric Fraga, Zahia Guessoum, Wolfgang Marquardt, Didier Paen, Pascal Roux, Sergi Sama, Philippe Vacher, Lars von Wedel, Aidong Yang, for their contributions to COGENTS ideas. Some
864 of the ideas in this paper are developed in the new EC-funded 1ST project, "Cogents: Agent-Based Architecture For Numerical Simulation", contract IST-2001-34431. cogents interface
cogents framework
> OS, Middleware, COM, CORBA, TCP/IP Figure 3: a possible cogents architecture; note that cogents can be either CAPE-OPEN components wrapped with agent facilities, or autonomous agents acting on behalf on CO components.
5. References Altshuller G. (1954) 40 Principles : Triz Keys to Technical Innovation, Uri Fedoseev, Steven Rodman (Translator), Lev Shulyak (Translator) Batres, R. et al (2001), A life-cycle approach for model reuse and exchange, ESCAPE-11 conference, Kolding, Denmark Belaud J.-P., Braunschweig B., Pons M., (2002) Open Software Architecture For Process Simulation : The Current Status of CAPE-OPEN Standard, in this ESCAPE-12 conference. Braunschweig B., (2000) Architectures ouvertes pour I'ingenierie de procedes: le standard CAPEOPEN, rapport Arago 28, Observatoire Fran^ais des Technologies Avancees, Oct. 2000 Braunschweig B., Gani R. (editors), (2(X)2) Software Architectures and Tools for Computer Aided Process Engineering, Elsevier (in print). Briot J.-P., Demazeau Y., (2001) « Principes et Architecture des systemes multi-agents», Hermes Science PubUcations, collection IC2, in print. CO-LaN, CAPE-OPEN Laboratories Network web portal, www.colan.org Guessoum Z., Briot J.P., (2002) From Active Objects to Autonomous Agents, Submitted to IEEE Concurrency - Special Series on Actors & Agents Salamatov Y. (1999) TRIZ : The Right Solution at the Right Time. A Guide to Innovative Problem Solving., Insytec B.V. Sycara K., Klusch M., Widoff S., Lu J., (1999) Dynamic Service Matchmaking Among Agents in Open Information Environments, SIGMOD Record (ACM Special Interests Group on Management of Data), Vol. 28, No. 1, March, 1999, pp. 47-53. Von Wedel L., (2000) An Object Model for Chemical Process Models, WP 5: Advancing Open Process Engineering Modeling Concepts and Exchange Language, internal Global CAPEOPEN document, December 2000
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
865
Automatic Integration of High-Index Dynamic Systems E.F.Costa Jr^, A.R.Secchi^ and E.C.Biscaia Jr.^* ^Programa de Engenharia Quimica - COPPE/UFRJ - Rio de Janeiro - RJ - Brazil ^Departamento de Engenharia Quimica - UFRGS - Porto Alegre - RS - Brazil * Author to whom correspondence should be addressed, [email protected]
Abstract A computational code for automatic characterisation, consistent initialisation and integration of generic DAE systems with index up to 3 is presented in this contribution. Two dynamic models are presented to illustrate the use and to characterise the performance of the code. The developed code has been tested in several dynamic simulations, the successful results and the minimal participation of the user during the simulation are encouraging characteristics to incorporate this code to processes dynamic simulators.
1. Introduction When solving DAEs one must concern about the index of the system and about the consistency of the initial conditions. The index of a DAE system is the number of times that all or part of the system must be differentiated with respect to time (independent variable) in order to convert it into an explicit set of first order ODEs. Index 1 systems may be solved with modified ODE methods, while higher index systems (systems with index 2 or greater) require specific methods. Generally, higher index systems must be reducted to an index 1 problem and then solved using standard integration codes. During the index reduction, some extra algebraic equations are obtained which generally correspond to derivatives of the original algebraic equations. Those hidden algebraic equations along with the original DAEs compose the extended system. Consistent initial values must satisfy not only the DAE system but also the resulting extended system. A computational code for automatic characterisation of DAE system has been developed by Costa Jr. et al. (2001). In the present contribution, that code has been improved making possible the integration of DAE system with index up to 3 using the DASSLC computer code (Secchi, 1992). The Jacobian matrix has been calculated using automatic differentiation tools, resulting in a higher integration speed. It should be pointed out, that on this code the equations of the system could be presented on their original form avoiding any further manipulations.
866
2. Code Description In order to use the automatic differentiation code ADOLC (Griewank et al., 1996), the code was written in C++. This computational language is also compatible with the code DASSLC, written in C, used to integrate the extended system. To use the present code, the user should write a program compatible with the syntax of DASSLC code as described in Secchi (1992). The code accomplishes the following stages for the characterisation and resolution of the DAE systems: (1) determination of the DAE system graph through numerical perturbation; (2) execution of the algorithm proposed by Pantelides (1988) seeking the equation sets that should be differentiated; (3) construction of the extended system by automatic differentiation; (4) determination of the extended system graph; (5) selection of the variables subset specified by the user; (6) checking up consistency by structural analysis of the variables subset (If this subset is structurally inconsistent returns to stage 5.); (7) checking up consistency by numerical analysis of the variables subset and determination of the initial conditions of the problem (If this subset is numerically inconsistent returns to stage 5.); (8) construction of the Jacobian matrix and determination of the index 1 system.
3. Numerical Examples 3.1 The Start-up of a Batch Distillation Column The model equations of this system, as presented by Costa Jr. et al. (2001), are: nij=Vi„ y i „ , j + L i _ , ^ - L i ^ - V , y,^,
(la)
Ni=I-ni,j
(lb)
Nihi=Vi+,H,+,+Li_,hi_i-L,hi-ViHi+Qi
(Ic)
yi,j=^K(Ti)
(Id)
Nihi=I°f,ni,jhj(Ti)
(le)
Hi=IJf,yyHj(Ti)
(If)
Vi+i=Li
(Ih)
where: ng (=12) and nc (=5) are the numbers of trays and components of the column, respectively; i is the stage index (i is equal to 0 for the condenser and np+1 for the reboiler); j is the component index; y represents molar fraction in vapour phase; h and H are enthalpies in liquid and vapour phases; Vj and Lj are the vapour and liquid fluxes
867 leaving the i^ stage; n is the amount of a component in a stage; N is the stage holdup; Q is the heat removed from a stage and T represents temperature. The expressions to calculate the values of h, H e K can be found in Holland and Liapis (1983). This system has 223 equations and, as exposed in Costa Jr. et. al. (2001), the Equations (lb), (Id), (le) and (Ig) must be differentiated once (index 2). The resulting extended system has 335 equations and 405 variables. Thus, 70 variables should be assigned in the initial condition. There are405C7o (approximately 10^^) possible subsets to be assigned, and only some of them are feasible. The feasible initial subset has been specified as the amount of component j in every stage i (nij), according to Table 1. Figures la and lb show some dynamic simulation results.
Table 1 - Specified initial conditions subset Component
C3H8
i-C4Hio
n-C4Hio
i-C5Hi2
n-C5Hi2
no,j (j = 0,...nc-l)
0.2
0.6
1.0
0.8
1.4
nij (i=l,...np and j=0,...nc-l)
0.05
0.15
0.25
0.2
0.35
1.7
5.1
8.5
6.8
11.9
ni3j 0=0,... nc-1)
2
1.5 o E
^
no,o
"^^"^
"0,1
1 0.5
H A\no.3
"0,2
[ Ho.^V^^
0.0
1.5 time (h)
3.0
1.5 time (h)
Figure 1 - Condenser dynamic simulation: (a) Number of moles of different components(left); (b) Heat duty(right). Figure la reproduces the results of simulation reported in Holland and Liapis (1983). After 3 hours, the column is near to its steady state and the heat removed from the condenser is the equal to the heat added in the reboiler, as presented in Figure lb. 3.2 Product Purity Control of a Batch Distillation The simplified mathematical model of this system, as presented by Logsdon and Biegler (1993), is presented below: M o = - V / ( R + l)
(2a)
868 MoXij = v(xoj - yoj + R(xij - xoj)/(R +1))
(2b)
MiXij = V(yi_ij - y i j +R(Xi^lj -Xij)/(R + 1)), i=l,...,np
(2c)
Mn+lXn+l,j = V(yiij -Xn+ij)
(2d)
X n y i J ^ l ,i=0,...,np+l
(2e)
Xn+u =0.998
(2f)
where, V is the molar vapour flow rate; Mj is the molar holdup in the i^ stage; R is the reflux ratio; np is the number of trays in the column; i is the stage index (i is equal to 0 to the reboiler and np+1 to the condenser); j is the component index; x and y represents molar fractions on liquid and vapor phases. The molar fraction of the last component, Xi,nc, has been eliminated by using the Equation (3), and yij has been eliminated through the thermodynamic equilibrium relation, as presented in Equation (4). Xi,nc = l - X f L V x i j , i=0,...,np+l
(3)
yij=K(Ti,Xij)xij
(4)
,i=0,...,np+l
The model has nc(npH-2)H-2 equations. There are (nc-l)(np+2)+l differential equations and the remaining ones are algebraic. The (nc-l)(np+2)+l differential variables are Xij (j=l,...,nc-l; i=0,...,np+l) and MQ, and the (np+3) algebraic variables are Tj (i=0,..., np+1) and R. The model is a high-index DAE system because the algebraic variable R does not appear in any algebraic equation. The operating conditions, proposed by Logsdon and Biegler (1993) and used in this work, are presented in Table 2.
Operating Conditions (Logsdon and Biegler, 1993) V = 120 mol/h Mo = 100 mol np = 10 trays Mi = 1 mol, i = l,...,n-i-l
XQ.I = 0.55, cyclohexane Xo,2 = 0.45, toluene
The characterisation of this system by the developed code has been the same reported by Logsdon and Biegler (1993). The equations that should be differentiated once are (2d) and (2e) for i=np (last tray). The Equation (2f) should be differentiated twice (index 3). Thus, in the extended system there are four new equations and two new variables: Xj i j and TJQ • The number of degrees of freedom is 11 because there are 41 unknowns (variables and their time derivatives) and 30 equations. The user must specify a subset of variables among the variables of the original system. Then, the user must choose 11 from the 39 original variables. There are more then 1.6 billions of possible subsets. In this stage, the user physical knowledge is very valuable because only few subsets are consistent. For example, it is not possible to choose, at the same time, the composition and the temperature of a stage. Besides, the specification of the purity in the Equation (2f) determines all variables of the two last stages of the
869 column (last tray and the condenser). The values of the variables of these two stages are presented in Table 3. If a chosen subset does not follow these conditions, then it will be infeasible.
Table 3 - Variables values of the last two column stages Xio,i-
0.9949
^10,1 =
oh-'
Tio = 80.83 °C
tio
= 0 T/h
T i i ==80.77 T
^11,1 -:
^11,1
0.9980
= oh-^
t i l = O^C/h
For any supplied initial condition, the values of the variables in the last two column stages will always be those supplied in Table 3. The consistent initial condition subset used in the present work is presented in Table 4.
Table 4 - Inital subset Ho = 100 mol Xo,i= 0.55 R=l
Ti = 89.80 °C T2 = 87.50 °C T3 = 85.40''C T4 = 83.80 ^C
T5 = 82.60 "C T6 = 81.70 °C T7 = 81.10 °C T8 = 81.00 ^C
This subset is structurally and numerically feasible. However, the variables of Table 4 cannot assume arbitrary values because the solution of the extended system can be out of the real domain. Another feasible subset can be obtained changing the variable R by the variable T9. In this situation, meaningless results have been obtained, e.g. negative values of the reflux ratio. Other feasible subsets can be also specified. The numerical solution of the extended system, using the initial subset of Table 4, is presented in Table 5.
Table 5 - Obtained initial conditions X4,i = 0.8549 xo,i =-0.2040 h-^ X41 =-1.0015 h-^ xi,i = 0.6070 x i i = 0.2927 h"* X2,i = 0.6970 X2,i =-1.8847 h'^ X3,i = 0.7845 X31 =-1.8111 h-^
X5,i = 0.9100 X51 =-0.5111 h"^
X71 =-1.1423 h-^ xgi = 0.9866 xgi = 0.0284 h'^ X9,i =0.9911 X91 = 0.0210 h'^
X6.1 = 0.9526 X61 =-0.4696 h-^
To = 91.34 °C
X7,i = 0.9817
T9 = 8 0 . 9 r C
After the determination of the consistent initial conditions, the code automatically builds the index 1 extend system (without recompilation due to the use of automatic
870 differentiation) to be integrated by DASSLC. The simulation results of the column production phase are presented in Figure 2.
time (h)
time (h)
Figure 2 - Simulation results: (a) molar fraction of ciclohexane(left); (b) R (right). As expected, the reflux ratio increases with the time. The small decrease between 0.08 and 0.18 is resulted of the initial conditions considered. The reflux ratio increases indefinitely with the time in order to assure the product purity. Its value is approximately 40,000 after 5 hours of production, resulting in a very low distillate production rate.
4 Conclusions In this present contribution the code developed by Costa Jr. et al. (2001) has been improved making possible the direct numerical integration of DAE system with index up to 3 through the DASSLC computer code. Two dynamic models have been presented to illustrate the use and to characterise the performance of the code. The new version of the code requires minimal intervention of the user during the simulation. This new feature shows the potentiality of incorporating the code to processes dynamic simulators.
5 References Costa Jr., E.F., R.C. Vieira, A.R. Secchi and E.C. Biscaia, 2001, Automatic Structural Characterization of DAE Systems, Proc. of IT European Symposium on Computer Aided Process Engineering - ESCAPE 11, 123. Griewank, A., D. Juedes, H. Mitev, J. Utke, O. Vogel and A. Walther, 1996, ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++, ACM TOMS 22 n^2, 131-167, Algor. 755. Holland C, A.I. Liapis, 1983, Computer Methods for Solving Dynamic Separation Problems, McGraw Hill Publishers. Logsdon J.S. and L.T. Biegler, 1993, Accurate Determination of Optimal Reflux Policies for the Maximum Distillate Problem in Batch Distillation, Ind. Eng. Chem. Res. 32, 692-700. Pantelides, C.C, 1988, The Consistent Initialisation of Differential-Algebraic Systems, SIAM J. Sci. Stat. Comp. 9, 213-231. Secchi, A.R. (1992). "DASSLC: User's Manual, a Differential-Algebraic System Solver", Technical Report, http://www.enq.ufrgs.br/enqlib/numeric.DASSLC, UFRGS, Porto Alegre, RS/Brazil
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
871
Modeling work processes in chemical engineering from recording to supporting M. Eggersmann, R. Schneider, W. Marquardt* Lehrstuhl fur Prozesstechnik, RWTH Aachen, D-52056 Aachen, Germany
Abstract Design processes in chemical engineering are complex and highly creative. They show a large potential for improvement. In order to improve these work processes different steps including recording, formalization, analysis, and implementation are necessary. They will be described, together with their interactions, in this contribution.
1. Introduction In a chemical engineering design project many software tools are used to support engineering activities. The focus of these tools is either on specific design tasks like simulation and optimization or on the management of information for example in a database. Not much attention has been paid to the work processes or workflow during process design (Subrahmanian et al., 1997). Our long term research objective is to improve the work processes by reengineering (similar to business process reengineering) and to support the individual designer during the design process by computer-based aids. For both objectives a thorough understanding of work processes in chemical engineering is necessary. In this paper, we present a conceptual approach towards workflow support. In order to understand design processes it is essential to identify their characterizing features which are necessary to describe them in sufficient detail. In our opinion at least five elements are needed. Core elements are the activity to describe what has been done and the order of activities, which defines the control flow. Besides, the results of an activity and the data resources needed to perform it, have to be known. These are the input and output information of an activity resulting in an information flow. Another important element is the person performing the activity, or more precisely the role a person is performing in a design process. In the next section we will summarize a procedure which can be followed to reach an improvement of work processes in chemical engineering. The consecutive sections describe the most important parts of such a procedure including recording of design processes, as well as their analysis and formalization.
2. An Approach towards Workflow Support Figure 1 shows a simplified overview of the steps of the workflow support approach, under consideration of the elements introduced in Section 1. The first step towards * Correspondence should be addressed to W. Marquardt, [email protected]
872
Workflow Modeler
Designer
Workflow Engineer
Formaliza -tion
Figure. 1: An approach to workflow support workflow support is to acquire knowledge about industrial work processes in process design. Whereas predefined design procedures may exist on a coarse level internally in companies or in the open literature (e.g. Douglas, 1988), not much knowledge exists about work processes on a finer level of granularity. This is mainly due to the creative character of design processes. Therefore recording of actual industrial work processes is necessary in the first place for understanding it (Bucciarelli, 1994). In principle, there are two different ways of recording: (i) The designer who has the best (tacit rather than explicit) knowledge about design processes protocols what he is doing during the design (self observation), (ii) The designer is interviewed after completing (part of) the design process by a person who has some background on interview techniques and the domain of interest, namely chemical engineering. We will call this person a workflow engineer. Besides the designer and workflow engineer, a workflow modeler is necessary who transforms the (usually informal) results of the recording or the interviews in a, what we call, semiformal workflow model. This model has to be easily understandable. Therefore a graphical representation is used, which facilitates the understanding. To eliminate misunderstandings this model is validated by the designer and the workflow modeler together.
873 On the basis of the agreed workflow model the workflow engineer analyzes the work process, to identify problems and to find possible ways of reengineering the work process or of formulating requirements for computer support. From the analysis of a certain number of design processes it should be possible to define standard workflows which have proven to be effective and can serve as some kind of template for further projects. These standard procedures could be directly implemented without the use of information technology: A project manager for example uses the standard workflows to organize his project or a designer can employ a certain design procedure to design a process unit. Obviously work processes can be even better supported by the application of computer tools. A software tool which captures information about various best practice workflows could guide a designer by suggesting appropriate activities and advising him how to perform those. A prerequisite for such a support system is a proper formalization of the work processes in a format which can be processed by a computer. This task has to be fulfilled by the workflow modeler. In contrast to a human-readable representation such a format must be unique and unambiguous and must not require or allow interpretations. After the application of new - hopefully better - design procedures their success has to be evaluated, again starting with the recording of the work process. In Figure 1 this is indicated by the feedback information flow from performing of new design processes to recording.
3. Recording and Representing Work Processes In order to record and model work processes a method is needed which is easy to use and to understand as in most cases the designer is not an expert in work process modeling. When a project manager wants to record and represent the workflow in a particular design project he is responsible for, not too much extra time should be necessary to accomplish this. Also, when a workflow engineer is supposed to conduct interviews in order to elicit information about the work processes, the interviewee (the designer) needs to be able to understand the workflow representation so that he can give direct feedback for discussion and verification purposes. These requirements are addressed by the modeling formalism C3 suggested by Killich et al. (1999). The name C3 refers to cooperation, coordination, and communication as the formalism is capable of representing these three features, which are typical for design processes. Although C3 is based on UML (Unified Modeling Language, Booch et al., 1998), a detailed knowledge of UML is not necessary to understand the models, as the reader may have noticed by intuitively understanding the content of Figure 1, which is modeled in C3. The elements of C3 are roles, activities, input and output information, control flows, information flows, synchronous communication, and tools. The last two are not used in the example of Figure 1. All acfivities are assigned to a role within a so-called swim lane. The solid arrows represent control flows which indicate the order of the activities. Within each swim lane, the temporal order is indicated by the arrangement of the single activities and their connecting control flows. Information flow from an activity where information is produced to other activities which use this information is represented by a dashed line. Such an information flow is completed by the specification of the informafion transfered within an associated rectangular box. An information flow between two or more
874 activities indicates which activities can only be performed after the completion of a previous one. The main elements for modelling the order of activities in C3 are, however, the control flows, whereas information flows provide additional information about the dependency between those activities. Tools used within an activity are indicated by an associated block arrow (not shown in Fig.l). The elements are predefined, but the user can fill them with arbitrary information. Besides the ease of understanding its representation, C3 has the ability to be used directly as an interactive modeling technique during interviews (Scheele and Groeben, 1984). It allows a structured representation of the work process the interviewee has in his mind: the interviewer uses certain cards to represent the interviewee's answer; these cards represent the central elements of C3 like activities, information, and tools. The interviewee himself writes additional information on the cards. The cards are then pinned together with other informal, written comments on a large roll of wallpaper, which is then analyzed and (semi-)formalized by the workflow modeler. Finally the model is verified by the workflow modeler and the interviewee together.
4. Analysis and Implementation of Work Processes Typically, knowledge about work processes in the domain of process design is only implicit. Writing down and modeling these processes has the benefit that the designers become more conscious about the way they solve problems as part of a design team. The C3 model of a particular design process can be employed to analyze the work processes, similar to the application of SADT models (Structured Analysis and Design Technique, Ross and Schoman, 1977). The C3 models may be used for example to identify potential ways for shortening the design process in the sense of concurrent engineering. By identifying the information flow it is possible to judge if one activity can be performed concurrently to others. Activities can be identified which lack good computer support. This way, necessary requirements on tool functionalities can be defined. The work process model clearly shows the interfaces between different roles in the design team or even between different departments and the information transfer between them. Hence, it can also be used to discover outsourcing opportunities. Outsourcing of specific design activities can be planned by assisting in defining interfaces between external and internal roles. The benefits of such an analysis can be demonstrated by means of a case study. We modeled the workflow during conceptual design of a nylon6 process by means of the C3 formalism (Bayer et al., 2001a). The resulting work process model consists of more than 100 activifies and involves seven roles. In a conventional design process the polymer processing subprocess is being developed after the polymer reaction subprocess has been specified. The C3 model of the case study helped to discover that the polymer processing part can be designed much earlier. The implementation of this insight significantly decreases the duration of the design process and hence the time to market. Additionally, concurrent engineering allows an earlier investigation of the effect the polymer processing unit has on the rest of the process. Problems and potentials for improvement can be identified earlier. In our case the knowledge about the polymer processing permitted a more economic design of the separation section.
875
5. Formalization In the approach presented here the C3 notation is used for modeling past processes; its main feature is an easily understandable graphical representation. However, it is not unique nor unambiguous and therefore has to be interpreted by a human user. In order to use workflow models in a computer, they have to be unambiguous, and a more strict formalization is required. Standard work processes have to be modeled which are not related to a specific execution but rather can be used as templates for new work processes. These standard workflows are an abstraction from previous workflows. According to the object-oriented paradigm (Rumbaugh et al., 1991) the previous workflows can be seen as instances and the standard workflows as classes. We adapted this view within CLiP (Conceptual Lifecycle Process Model) (Bayer et al., 2001b, Eggersmann et al., 2000), an object-oriented data model covering product data of the design and the work processes leading to the design. The work process model consists of three layers (instances, classes, and metaclasses). The instance layer contains those activities which have already been performed and are therefore well known. Standard workflows and standard activities, which can form templates for future activities are represented on the class layer. The metaclass layer includes the modeling concepts for the other two layers. However, it is not yet clear how certain object-oriented concepts like e.g. inheritance between classes can be applied to work processes. The differences between C3 and CLiP can be explained by the example in figure 2. On the instance level the activity ''Design CSTR for PA6 production" is modeled using the C3 formalism. Here it is represented what one specific person (the reaction expert) did and that he used the PA6 polymerization kinetics as input information and Pro II as a tool. By abstracting this activity, an activity "Design reactor" can be defined which is modeled within CLiP. We call this an activity class because the specific activity instance (and possibly further activities) do carry the same attributes and methods as this class. On the class level in CLiP no specific actor is modeled but the skill he is supposed to possess. This is necessary if activities should be assigned according to the skills of the designer. In the C3 model the skills are only implicitly represented by the role. Whereas the control flow is explicitly modeled in C3 it is only implicitly contained in the class level of CLiP by the definition of the information. The possibility to use alternative tools - Polymers Plus and Pro II - can also be modeled within CLiP. instance level (C3) reaction expert John Smith PA6 polymerization kinetics
< Design CSTR 1
for PA6 production
<J[ Prod
metaclass level (CLiP)
«nnetaclass» input information
"metaclass" activity
1
1 «metaclass» skill
Class level (CLiP) [input information:Teaction kinetics^
I tool::pro II | Reactor size = 10 nrr^
1
«metaclass» tool
«metaclass» output information
«nnetaclass» goal
[tool::polymers plusl
|goal::determine reactor size^
|activity::Design reactor] |
Figure. 2: C3 and CLiP model of an example activity
actor
|
[output information::reactor size| |skill::polymerization knowledge]
876 The abstraction and formalization must be done by a human and cannot be done automatically because it involves modeling and domain knowledge. Whereas C3 is easy and fast to use, modeling with CLiP requires both more time and more modeling knowledge. An approach to facilitate the transition from C3 models to CLiP is to extend the C3 graphics by a storage of links between e.g. activities and information.
6. Conclusions and Future Work We have presented the steps necessary to support work processes in chemical engineering design and their interactions. A system is being implemented which currendy supports the graphical representation and in the future shall be extended to facilitate the formalization. Open issues are still the transition of a textual description of the work process by a designer to a C3 model, and the abstraction of C3 models to classes of work processes in CLiP. Additionally, the analysis requires further systematization and the possibilities of computer support have to be evaluated.
Acknowledgements This work is supported by the DFG, Deutsche Forschungsgemeinschaft, in the CRC 476 TMPROVE'. The authors thank B. Bayer, C. Foltz, and M. Wolf for many fruitful discussions.
References Bayer, B., M. Eggersmann, R. Gani, R. Schneider, 2001a, Case Studies for Process Design, In: Braunschweig, B., R. Gani, Software Architectures and Tools for Computer Aided Process Engineering, Elsevier Publishers, to be published. Bayer, B., C. Krobb, W. Marquardt, 2001b, Technical report, Lehrstuhl fur Prozesstechnik, RWTH Aachen, LPT-2001-15. Booch, G., J. Rumbaugh, I. Jacobson, 1998, The Unified Modeling Language User Guide, Addison Wesley, Reading. Bucciarelli, L.L., 1994, Designing Engineers, MIT Press, Cambridge, Massachusetts. Douglas, J.M., 1988, Conceptual Design of Chemical Processes, McGraw-Hill, New York. Eggersmann M., C. Krobb, W. Marquardt, 2000, A Modeling Language for Design Processes in Chemical Engineering. In: Laender, A.H.F., S.W. Liddle, V.S. Storey (Eds.): Lecture Notes in Computer Science 1920, Springer, Berlin, 369-382. Killich, S., H. Luczak, C. Schlick, M. Weissenbach, S. Wiedenmaier, J. Ziegler, 1999, Behavior & Information Technology, 18, 325-338. Ross, D.T., K.E. Schoman, 1977, IEEE T. Software Eng., SE-3, 1, 6-15. Rumbaugh, J., M. Blaha, W. Premerlani, F. Eddy, W. Lorensen, 1991, Object-Oriented Modeling and Design, Prentice-Hall Inc., Englewood Cliffs, New Jersey. Scheele, B., N. Groeben, 1984, Die Heidelberger Struktur-Legetechnik (SLT), Beltz, Weinheim. Subrahmanian, E. S.L. Konda, Y. Reich,. A.W. Westerberg, the N-dim group, 1997, Comp. Chem. Engng., 21, Suppl., S1-S9.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
877
A Multi-Cellular Distributed Model for Nitric Oxide Transport in the Blood Nael H. El-Farra, Panagiotis D. Christofides and James C. Liao Department of Chemical Engineering University of California, Los Angeles, CA 90095-1592
Abstract In this work, a multi-cellular, spatially distributed model, that describes the production, transport and consumption of Nitric Oxide (NO) in blood vessels and the surrounding tissue, is developed. In contrast to previous modeling efforts, such as continuum and single-cell models, the current model accounts explicitly for the presence of, and interactions among, a population of red blood cells (RBCs) inside the blood vessel. Using mainly experimentally derived parameters, the model equations, subject to the appropriate boundary conditions, are solved for the NO concentration profile using an efficient finite-element algorithm.
1. Introduction
Despite the well-documented significance of NO, as a versatile reactive free radical that participates in a diverse array of vital biological functions in humans and animals, the transport of NO from its site of synthesis to its target remains a complex process that is not yet completely understood. For example, in order to exercise its regulatory effect in the circulatory system, NO must diffuse from the blood vessel wall to the external surrounding tissue. Due to its reactivity, however, NO may be consumed and degraded by numerous reactions before it reaches its target. In particular, NO diffuses into the lumen of the blood vessels where it is believed to react at very high rates with hemoglobin in red blood cells (Cassoly and Gibson, 1974) . The extremely fast kinetics of this reaction suggest, however, that most of the NO produced in the blood vessel wall would be consumed in the blood, thus severely limiting NO diffusion into surrounding tissue and compromising NO regulatory effects. This paradox has triggered a significant search, with the aid of coupled mathematical modeling and experiments, for the specific mechanisms that must exist to reduce NO consumption in the blood (see, e.g., (Lancaster, 1994; Vaughn et al., 1998; Vaughn et al., 2001)). Despite these significant contributions to the modeling and analysis of NO transport, the particulate nature of the blood, namely the presence of, and interactions among, a population of RBCs inside the lumen of the blood vessel (as opposed to a homogeneous medium or just a single RBC where NO reacts with Hb), has not been investigated in any of the existing models so far. This factor contributes to further reduction of NO uptake in the blood by diminishing the actual exposure of NO to Hb further, compared to that predicted from the continuum model, and introducing additional diffusional barriers in the extracellular spaces between cells in the population. Motivated by these considerations, together with the recent advances in numerical methods and the development of efficient tools for the numerical simulation of complex systems, we develop in this study a detailed, multi-cellular, distributed model for NO transport, based on diffusion-reaction principles, that explicitly takes into account the presence of, and interactions among, a population of red blood cells inside the blood vessel. The model
878 equations consist of several sets of PDEs that describe the production, diffusion, and consumption of NO in the tissue surrounding the blood vessel, the vessel wall, a cell-free zone near the vessel wall, the extracellular space between the population of RBCs, the membrane surrounding each cell, and, finally, the intracellular space inside the RBCs. The parameters for the model are obtained from experimental data reported in the literature (see, e.g., (Vaughn et al., 1998; Vaughn et al., 2001) ) and the entire set of PDEs is solved for the NO concentration profile using efficient finite-element algorithms. The detailed model offers a more realistic understanding of the key mechanisms for NO transport and consumption, which, in turn, has crucial clinical implications for the development of practical technologies (e.g., artificial blood substitutes) that can be used to treat diseases attributed to imbalances in NO transport in humans.
2. Mathematical Modeling of NO transport
2.1 Governing equations To provide a detailed, yet computationally tractable, model that captures the main features of NO transport in blood vessels and the surrounding tissue, we divide the system into three major compartments: the abluminal region, the endothelium, and the lumen. The lumen is further divided into a cell-free zone, extracellular spaces between the cell population in the vessel, the membrane surrounding each cell, and, finally, the intracellular spaces (see Figure 1). The system is modeled using polar coordinates. Abluminal region (1) space (4) Endothelium (2)
Blood vessel wall Cell-free zone (3)
Figure I: Geometry of the blood vessel showing the various compartments of the model and a distribution of RBCs.
The concentration of a diffusing reacting substance, such as NO, is described by the species mass balance. For NO, this balance, in its general form, can be written as D^n^'C^n-^C 'NO
v + R,
(1)
where V is the vector gradient operator, V^ is the Laplacian operator, C^o is the concentration of NO, D/vo is the diffusion coefficient, and R/^o is the rate at which NO is produced or consumed by reaction. Two processes are involved in the transport of NO: the first term on the right hand side represents the diffusion of NO; the second represents the transport of NO by a molar averaged velocity. By focusing on the steady-state behavior and neglecting convective transport, we have that VCm -v = 0 and dCj^ol^^ 0, respectively. The system can therefore be treated as a two-dimensional problem, with NO concentration varying only in the radial (r) and azimuthal {&) directions. The azimuthal variations in NO concentration arise from possible non-uniform consumption of NO inside the vessel due to a non-uniform (random) distribution of cells. The balance
879 between NO diffusion and reaction in all compartments can be written for cylindrical coordinates as
Eq.2 is used for all compartments, although the value of Dj^o rn^Y differ depending on the intrinsic transport resistance in each region. Also, the expression for Rj^o rnay differ in each region depending on whether NO is being produced or consumed and depending on the rate laws of the chemical reactions taking place in each region. Following (Vaughn et al., 1998), NO is assumed to be consumed by a second order reaction in the abluminal region, so Ri^o takes the form R^Q = -k^i^CJjQ , where kab is the reaction rate constant. In the endothelium, NO is also consumed by the above second order reaction but, in addition, it is produced by an enzyme that is partially bound to the membrane of endothelial cells. The expression for R/^Q in this compartment therefore takes the form R^Q = -k^i^C^Q + Q^Q , where Qi^o is the total NO production rate per unit area of the endothelium. In the extracellular vascular lumen and inside the RBC membrane, NO is transported by diffusion (with a different diffusivity in each region) and R^^o is taken to be zero. Finally, inside each RBC, NO consumption by hemoglobin can be expressed by the rate equation given in (Vaughn et al., 1998)/^y^o ^-kjju^NO^Hb ^~^\,IU^NO^ where k2ju and kjji, are the rate constants and Cnb is the hemoglobin concentration in the red blood cell, which remains essentially constant so the reaction can be considered pseudo-first order in NO with the reaction rate constant kju, = k2ju CmIn order to solve the resulting set of partial differential equations for NO concentration, we need to specify the boundary conditions. In the radial direction, one boundary condition is implied by the no-flux condition at the center while far from the vessel wall we have that the NO concentration changes slowly; therefore dCNO 9C.NO =0 (3) dr .=0 ^'' The rest of the boundary conditions in the radial direction are obtained by invoking continuity of the NO concentration profile and matching the fluxes at the interfaces between the various regions. For the azimuthal direction, periodic boundary conditions that express continuity of both the NO concentration and NO concentration gradients in the azimuthal direction are used. 2.2 Model parameters and numerical solution The main parameters in the model include: 1) the diffusion coefficient of NO in the abluminal region, the endothelium, and the extracellular vascular lumen (all assumed to be the same), Dext. 2) the diffusion coefficient of NO in the RBC membrane, D^^^, 3) the diffusion coefficient of NO inside the RBC Dint, 4) the NO production rate Qf^o^ 5) the rate constants for NO degradation in each region kab, k2,iu and /:;/«. The values of these parameters have been derived from experimental data reported in the literature (see (Vaughn et al., 1998; Vaughn et al., 2001) for details and references) and are given in Table 1. Other parameters used in the numerical simulations include the radius of the blood vessel, /?, the thickness of the cell-depleted zone, 5, the thickness of the
880 endothelium region producing NO, e, the effective radius of the RBC, a (modeled as a sphere), the effective thickness of the RBC membrane, 5RBC» and the concentration of hemoglobin inside the RBCs, Cnb- The values of these parameters are given in Table 1. Using the software FEMLAB, a fmite-element algorithm was developed to discretize the spatial domain of the problem and numerically solve the model equations subject to the above boundary conditions and model parameters. Because the concentration of NO in the endothelium and close vicinity is of particular interest, an adaptive meshing (variable grid spacing) technique was used to allow a larger number of elements near the endothelial surfaces, where the concentration gradients are expected to be steep. In all simulations, the mesh was continuously refined to insure that the solution is gridindependent. Also, although the second boundary condition in Eq.6 applies to an infinite domain, it was implemented on the outermost elements of our finite mesh, typically 2,000-4,000 jLim from the vessel axis. Table 1. Model parameters \ De,r Dint
P
' mem ^ab
^Llu QNO
R 5 £
a 5RBC
Cub
3300 ^im^s'^ 880 fxmV^ 450 pims'^ 0.05 \JM^ S^ 2.3x10's"' lO.exlO'^"^ |imol.|LimV^ 50 ^im 2.5 ^m 2.5 ^im 3.39 |Lim 0.0078 |im 23|LiM
3. Simulation results To gain some insight, from the developed model, into the factors governing NO mass transport, we solve the model under the following four different scenarios: 3.1 Homogeneous case In this case, the NO-hemoglobin interaction is assumed to take place uniformly everywhere in the vascular lumen. To simplify the presentation of our results, we will focus only on the mean NO concentration (averaged over ^ as a function of distance from the vessel axis. The resulting profile in this case is depicted by the solid line in Figure 2. It is clear from this profile that NO transported across the endothelium-lumen interface diffuses only very little into the lumen before it is scavenged by hemoglobin and completely depleted (note the zero concentration of NO inside the blood vessel). As expected, the NO concentration profile exhibits a maximum in the endothelium where NO is produced. However, by comparing the NO concentration gradients at this point, on the lumen side and the smooth muscle side, it is clear that the blood acts as a sink for NO, where the majority of NO produced in the endothelium flows into the lumen (note the steep gradient inside the lumen).
881 0.08 0.07
0.06 ^ 0.05 0.04 0.03 0.02 0.01
•1 1
'1 •
Or 0
50
100
150 200 250 300 Distance from vessel axis (jam)
350
400
Figure 2: Mean NO concentration profiles as a function of distance from the vessel axis for continuum case (solid), with population of RBCs (dashed), with RBCs+cell-free zone (dasheddotted), and with RBCs+cell-free zone+RBC membrane (dotted).
3.2 Particulate model Since hemoglobin is packaged inside red blood cells, and is not free floating in the blood vessel, NO consumption by hemoglobin takes place only inside red blood cells, and not everywhere in the blood vessel. Furthermore, the diffusion of NO in the extracellular space between RBCs offers an additional barrier that could slow down NO uptake by the RBCs. To analyze these effects, we explicitly incorporate a population of RBCs in the vascular lumen compartment of our model. The RBCs are modeled as spheres, with an effective radius, a, that are distributed randomly throughout the lumen (see Table 1). The number of cells included depends on the hematocrit of the blood, which is the volume fraction of RBCs in the blood. Under normal physiological conditions, the hematocrit is around 50%. For our two-dimensional model, this number is equivalent to the fractional coverage, by cells, of the lumen compartment. For each cell, a mass balance of the form of Eq.2 describing the diffusion and consumption by hemoglobin of NO inside the cell is included in the model.. The resulting mean NO concentration profile for this case is shown by the dashed profile in Figure 2. As expected, the concentration profile exhibits a similar shape to the continuum case; however, the endothelial and abluminal NO concentrations increase, as more NO (not consumed by RBCs) is now available to diffuse into the smooth muscle tissue. 3.3 Cell-free zone In the above calculations, the RBCs were assumed to be randomly distributed throughout the vascular lumen and, therefore, allowed to get arbitrarily close to the vessel wall. However, due to the hydrodynamic effects of blood flow, the RBCs tend to migrate towards the center of the vessel, thus creating a cell-depleted zone or layer near the vascular wall where NO has to travel across before it reaches Hb inside the RBCs. To account for this effect, we include a thin layer near the vascular wall of our model where no RBCs are present. The thickness of this layer depends on fluid mechanical considerations and is expected to be around 2.5|im for a 100|Lim -diameter vessel (Vaughn et al., 1998). Solving the model equations for this case, the resulting mean NO concentration profile is shown by the dashed-dotted profile in Figure 2. It is clear from
882 this figure that under these conditions both the endothehal and abluminal NO concentrations increase almost twofold over that predicted form the continuum model. 3.4 RBC membrane permeability So far in our model, we have neglected the potential role of the RBC membrane in reducing NO uptake in the blood by assuming that the RBC membrane is highly permeable to NO. Recently in (Vaughn et al., 2001), however, combination of an experimental technique, that overcomes experimental diffusional limitations, together with model analysis have been used to show that the RBC possesses an intrinsic mechanism that can slow down NO uptake. The results of this work point to the RBC membrane as a potential source for this mechanism. To investigate this effect in our model, we explicidy model the RBC membrane by enclosing each cell in the population with a thin layer whose thickness corresponds to the effective thickness (based on spherical geometry) of the RBC (see Table 1). Inside these layers, NO is transported by diffusion and is not consumed by hemoglobin. The value of the NO diffusion coefficient inside the membrane is computed directly from the permeability estimate obtained in (Vaughn et al., 2001) using the film theory approximation
n ^mem
^£l^^Rin£n.(r -^ ^^
c. \^NO,ext ^RBC
~r
]-p
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-r NO,ext
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)
r4^ v'+;
where C^o.ext is the NO concentration at the interface between the membrane and extracellular space, C/^o.cyt is the NO concentration at the interface between the membrane and the intracellular region (cytosol), and P^em is the NO permeability of the membrane. By adding the membrane, the model now includes all six compartments depicted in Figure 1, and accounts for the host of transport barriers, analyzed above, combined. The mean NO concentration profile for the full model is shown by the dotted profile in Figure 2. As expected, when the full host of transport barriers are considered together, the increase in the endothelial and abluminal NO concentrations (over those obtained from the continuum model) is larger than the increase observed when fewer barriers are accounted for. Finally, we note that the results of this detailed model have been used to assess, quantitatively, the contribution of each mass transport barrier considered here to the overall resistance to NO uptake in the blood (see (El-Farra et al., 2002) for details).
4. References Cassoly, R. and Q. H. Gibson, ^'Conformation, co-operativity and ligand binding in human hemoglobin," / Mol. Biol.. 91, 3301-3313 (1974). El-Farra, N. H., P. D. Christofides, and J. C. Liao, ''Analysis of nitric oxide transport barriers in blood vessels," Annals. Biomed. Eng., submitted (2002). Lancaster, J., '^Simulation of the diffusion and reaction of endogenously produced nitric oxide," Proc. Natl Acad. Sci. USA, 91, 8137-8141 (1994). Vaughn, M. W., K.T. Huang, L. Kuo, and J. C. Liao, ''Erythrocytes consumption of nitric oxide: competition experiment and model analysis," Nitric Oxide: Biology and Chemistry, 5, 18-31 (2001). Vaughn, M. W., L. Kuo, and J. C. Liao, "Effective diffusion distance of nitric oxide in the micocirculation," Am. J. Physiol. Heart Circ. Physiol., 274, H1705-H1714 (1998).
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
883
Systems Engineering Approaches To Gene Expression Profiling Sanjeev Garg and Luke E.K. Achenie [email protected], [email protected] Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269, USA
Abstract
Several data analysis algorithms exist for the analysis of gene expression data resulting from cDNA microarray experiments. From a biological perspective, all of these have a number of disadvantages, some of which are addressed in this study. Dimensionality reduction, a priori specification of the number of classes and the need for a training set are a few of these disadvantages. To address these issues, we propose two novel approaches based on systems engineering principles. The proposed algorithms do not (1) require a training set, (2) require the a priori specification of the number of classes and (3) perform any dimensionality reduction. The first algorithm has been used on three gene expression data sets (yeast cell cycle data, human fibroblast response to serum data and the cutaneous melanoma data) from the open literature, while the second has been used on the fibroblast data set. The results found in general are at least in excellent agreement with studies in the open literature or they reveal further knowledge, which was not available previously. The present study, thus, establishes the viability and strength of the proposed algorithms for gene expression data analysis.
1. Introduction Many different techniques are reported in the open literature for gene expression data analysis resulting from cDNA microarray experiments, namely data reduction techniques such as principal component analysis (Raychaudhuri et al., 2000), multidimensional scaling plots (Bittner et al., 2000), clustering techniques such as hierarchical clustering (Eisen et al., 1998), self-organizing maps (Tamayo et al., 12), knowledge-based support vector machines (Brown et al., 2000), "gene-shaving" (Hastie et al., 2000) and many more. These techniques have many advantages, but have some inherent disadvantages from a biological perspective. Any form of data reduction technique is generally biased and thus can result in the loss of meaningful biological information. Hierarchical clustering is best suited for data, which follow a hierarchical pattern; however it is not well suited for gene expression data analysis in general (Hastie et al., 2000). Self-organizing maps require the a priori specification of the number of cluster centers, which might not be known in many instances. Support vector machines need a training data set to learn the class information. The difficulty lies in the fact that the training data might not be a true representation of the whole data set. In this study, the proposed algorithms address most of the above issues associated with existing techniques in gene expression data analysis for class discovery and class prediction as applied to gene expression data. The proposed algorithms can be used efficiently in a
884 unified strategy to perform genome wide classification and functional annotation of genes with previously unknown functionality.
2. Formulation Gene expression matrix is an NxM matrix, where N is the number of genes and M is the number of attributes or time points. Typical numbers for N and M are in thousands (or hundreds) and hundreds (or tens), respectively. The goal of different data analysis algorithms is to group genes with similar gene expression patterns in different clusters. This is desired as genes with similar expression patterns may have similar functions based on homology. The first of the proposed data analysis techniques is based on the Growing Neural Gas algorithm developed by Fritzke (1994a, 1994b and 1995). The growing neural gas technique employs a model that uses a growth mechanism of the growing cell structures (Hastie, 1994b) and the topology generation of competitive Hebbian learning (Martinetz, 1991). Starting with a small number of units (step 1), new units are inserted (step 8) successively in the network (subsequently referred to as the net) using the error measurements. These error measurements are gathered during the adaptation process (steps 4-7). In the proposed algorithm, the maximum number of cells in the net is fixed to be the total number of genes in the data set. In addition, the maximum number of connections for each cell is fixed to be one less than the number of cells in the net. A new performance criterion is defined and is used as a stopping criterion for the growth of the net (step 10). This is in contrast with strategies that employ the maximum number of cells. A step (step 11) has also been added to calculate the 'class-center' positions in the input space, based on the given gene expression data and the resulting net topology. An iteration counter is defined and is updated to keep track of the number of input data read by the algorithm. We define a normalized mean error as the ratio of mean error value to the number of experimental attributes in the gene expression data as a measure of 'how good the clustering is'. The steps of this algorithm can be more formally given as: Step 0:
Step 1: Step 2:
Initialize the set A to contain two units a and b (with zero initial errors) at random positions w^ and w^, in /f^ (M-dimensional feature space), that is A = \a,b]. Initialize the connection set C , C C A x A to the empty set as C = 0 . Get an input signal x from the given data. Find the nearest unit Si and the second nearest unit S2 \S^ ,S2^ A) by:
1 - arg XEA mm pc - w^ s. = arg mm \\x - w^ Step 3:
Step 4:
;ce>\\{5i}"
Create a connection between 5; and si if it does not exist already: C = C U K^j, ^2 )}• ^^^ ^'^ ^§^ ^^ ^'^ connection between si and si to zero. Add the squared distance between the input signal and the nearest unit in input space to the error variable:
885
Step 5:
Step 6: Step 7: Step 8:
^1 -^1 II -^i II Adapt the reference vector of sj and its direct topological neighbors by fractions Ei, and En, respectively, of the total distance to the input signal: w^ <—w^. + £ . ( x - w ) and w
Step 9:
•
Insert connections connecting the new unit r with units q and/, and remove the original connection between q and/ Decrease the error variables of q and/by a factor a:
•
Initialize the error variable of r from q and/:
E^
Decrease the error variables of all units by j3 :
£,^£^(l-i8),0
If the performance criterion (£max ^ ^set) is not yet fulfilled go to Step 1 else continue. Calculate the 'centers' by averaging the reference vectors of each unit c and its direct topological neighbors. n
^center ~
{(w,+ Y^,)/in+\)}.
Similar concepts are used in a second algorithm, the ''adaptive centroid algorithrn'\kCk), to group genes with similar expression patterns. An analogy is made and used from the center of mass calculation for a heterogeneously distributed mass. An overall centroid is calculated as the average expression pattern of all the unclassified genes at a particular stage. The gene nearest to this centroid, with distance equal to minimum distance, is located and a local search for other similar genes is performed. The centroid of all genes belonging to this local region is calculated and constantly updated. Local search is continued till the distance between this centroid and the initial gene located is less than the minimum distance (centroid condition) or the local distance for search is less than the minimum distance
886 (distance condition). When any of these conditions is violated a new cluster location is found by adapting the centroid of unclassified genes to a new location based on the average of unclassified genes at this stage and the gene nearest to it. The process is repeated and the centroid location is iteratively adapted till all the genes are classified. This algorithm is currently being reformulated as a mixed integer-programming model.
3. Results and Discussion The data sets chosen for the first algorithm are: (a) a subset of yeast cell cycle data by Spellman et al., (1998); (b) a subset of human fibroblast response to serum data by Iyer et al., (1999); (c) data set for the study of cutaneous malignant melanoma by Bittner et al., (2000). The yeast cell cycle data corresponding to the genes for which all the experimental attributes are available (other than the alpha factor arrest and release) is chosen. The subset of the data for human fibroblast response to serum (available in the public domain) is selected for analysis in this paper. The first of the proposed algorithms is run with the following parameter values on these three data sets: f^, = 0.1, £;, = 0.01, a - 0.001, j3 =0.001, A =1000 and a^ax = 100. It is observed that the algorithm performs equally well in the three data sets chosen and is able to extract the biologically meaningful class information correctly. For the yeast data set, nine class-centers are observed. Based on the average gene expression, a few of these centers can be grouped together to give seven centers. The normalized mean error values range from 0.0508 to 0.0806. In addition, the low error values signify a close grouping of different class members around the class center. The algorithm results in fifteen class-centers when applied to the human fibroblast data. The higher number of class centers in the present study can be attributed to the further resolution or division of classes, into subclasses, reported in the previous study. The normalized mean error for fibroblast data ranges from 0.08 to 0.25. The error rates are again observed to be low and this proves that the class members are closely grouped together. Based on the average gene expression patterns these centers can be grouped together into nine different centers. The gene expression patterns in the present study are in excellent agreement with the clusters "A-D, G-/" as reported in a previous study (Iyer et al., 1999). In the latter study, the superposition of the two clusters "E and F ' results in a similar expression patterns found in the present study. We can therefore say that the two studies are in excellent agreement except that our approach led to a better resolution of the classes. The cutaneous melanoma data was divided into two sets to avoid computer memory limitations during analysis. The even-numbered genes from the parent set were put into one group, while the odd-numbered genes were put into another group. Both groups were analyzed independently. The even-numbered gene group resulted in five class-centers. Based on these expression pattern we observe that we can differentiate the 19 tissue samples as in the published study. We can therefore classify the different cutaneous melanoma samples based on their gene expression patterns. This is in agreement with the findings of Bittner et al. (2000). The normalized mean error for these class centers lies in the range of 0.1395 - 0.7623. Three class-centers are observed in the odd-numbered gene group. The two sets have similar overall expression patterns. It should be noted that the two groups belong to one parent data set and should therefore, ideally, follow the same pattern. The difference in the resulting number of centers in the two groups analysis can be attributed to the fact that there is only a small fraction of genes that have different critical gene expression and are able to classify the samples in two classes. This fact is also in accordance
887 with the findings in (Bittner et al, 2000). The normalized mean error for these class centers lies in the range of 0.1761 - 0.5716. The high error values seen in both of these groups are not a concern as the actual gene expression values for some samples greatly exceeded their fixed value of 50.00 in the analysis. Thus, the percentage error was quite low in these cases too. The second algorithm ("ACA") was run with the fibroblast data sets. It is observed that 17 clusters are obtained by this method, as shown in Figure 1, but most of the genes are classified within first seven clusters and the remaining ten clusters have very low cardinalities. This can be seen as oudiers being classified as individual clusters. Thus, this algorithm outperforms the previous algorithm in classifying the oudiers. Class prediction (the assigning of functionality to genes with previously unknown functions) can be done based on the assumption that genes with similar gene expression patterns may have similar functions (similarity by homology). Thus, genes with known function can define the function of other genes in the same class, whose function is currently unknown. Fibroblast/Serum Data "ACA"
Time Points, Last Unsync
Figure 1. Average Gene Expression Patterns for Individual Classes
4. Conclusions The proposed algorithms learn the true number of functional classes, which are also biologically significant, without requiring a priori information about the number of centers, as in self-organizing maps (Tamayo et al., 1999) or in support vector machines (Brown et al., 2000). The algorithms are unbiased as they use only the Euclidian distances and local error measures for the learning process and no assumption is made about the correlation of different genes. There is no dimensionality reduction and hence no knowledge loss, as in principal component analysis or multidimensional scaling. No assumption about the data distribution is made and, thus, the algorithms are expected to work for both hierarchical and non-hierarchical data sets. Therefore, the present study establishes the feasibility of the proposed algorithms for class discovery and shows that these algorithm excel over other algorithms by not having any of the disadvantages compared to reported algorithms for data analysis of gene expression data. Class prediction is done based on class membership.
assignment by homology, genes with previously unknown functionality are assigned new functions similar to other genes in that class. This is envisaged to be a good starting basis for further analysis and performing experiments by molecular biologists.
5. References Bittner, M., Meltzer, P., Chen, Y., Jiang, Y., Seftor, E., Hendrix, M., Radmacher M., Simon R., Yakhini, Z., Ben-Dor, A., Sampas, N., Dougherty, E., Wang E., Marincola, P., Gooden, C, Lueders, J., Glatfelter, A., Pollock, P., Carpten, J., Gillanders, E., Leja, D., Dietrich, K., Beaudry, C, Berens, M., Alberts, D., Sondak, V., Hayward, N. and Trent, J., 2000, Molecular classification of cutaneous malignant melanoma by gene expression profiling. Nature 406, 536-540. Brown, M.P., Grundy, W.N., Lin, D., Cristianini, N., Sugnet, C.W., Furey, T.S., Ares, M. Jr. and Haussler, D., 2000, Knowledge based analysis of microarray gene expression data by using support vector machines. Proc. Natl Acad. Sci. USA 97,262-267. Eisen, M.B., Spellman, P.T., Brown, P.O. and Botstein, D., 1998, Cluster analysis and display of genome-wide expression patterns. Proc. Natl. Acad. Sci. USA 95, 1486314868. Fritzke, B., 1994a, Fast learning with incremental RBF networks. Neural Processing Letters 1(1), 5. Fritzke, B., 1994b, Growing cell structures - a self-organizing network for unsupervised and supervised learning. Neural Networks 7(9), 1460. Fritzke, B., 1995, A growing neural gas network learns topologies. Advances in Neural Information Processing Systems 7, 625-632. Hastie, T., Tibshirani, R., Eisen, M.B., Alizadeh, A., Levy, R., Staudt, L., Chan, W.C, Botstein, D. and Brown, P., 20(X), 'Gene shaving' as a method for identifying distinct set of genes with similar expression patterns. Genome Biology, 1, 1-31. Iyer, V.R., Eisen, M.B., Ross, D.T., Schuler, G., Moore, T., Lee, J.C.F., Trent, J.M., Staudt, L.M., Hudson, J. Jr., Boguski, M.S., Lashkari, D., Shalon, D., Botstein, D. and Brown, P.O., 1999, The transcriptional program in the response of human fibroblasts to serum. Science 283, 83-87. Martinetz, T.M. and Schulten, K.J., 1991, A "neural gas" network learns topologies. In Kohonen, J., Makisara, K., Simula, O., and Kangas J., Editors, Artificial Neural Networks, 397-402, North-Holland, Amsterdam. Raychaudhuri, S., Stuart, J.M. and Altman, R.B., 2000, Principal components analysis to summarize microarray experiments: application to sporulation time series. Pac. Symp. Biocomput. 5,455-466. Spellman, P.T., Sherlock, G., Zang, M.Q., Iyer, V.R., Anders, K., Eisen, M.B., Brown, P.O., Botstein, D. and Futcher, B., 1998, Comprehensive identification of cell- cycleregulated genes of the Saccharomyces cerevisae by microarray hybridization. Mol. Biol. Cell 9,3213-3291. Tamayo, P., Slonim, D., Mesirov, J., Zhu, Q., Kitareewan, S., Dmitrovsky, E., Lander, E.S. and Golub, T.R., 1999, Interpreting patterns of gene expression with self organizing maps: methods and application to hematopoietic differentiation. Proc. Natl. Acad Sci. USA 96, 2907-2912.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
889
A Concurrent Engineering Approach for an Effective Process Design Support System Solomon S. Gelete, Rene Banares-Alcantara and Laureano Jimenez University of Rovira i Virgili., Av. Paisos Catalans 26;. 43007 Tarragona, SPAIN. Tel.: •4-34-977-559673; Fax: x9667/21. e-mail: rbanares, sgelete, [email protected]
Abstract The Concurrent Engineering (CE) methodology applied to chemical process design has the potential to improve the performance of chemical process design. This work presents a software prototype, CEPD/MODEL, that uses commercial software (e.g. HYSYS and AXSYS) and some tools developed in-house (e.g. CEPD-ART and CEPDDOC) for the application of Concurrent Engineering to process design. Among the several requirements for a successful application of Concurrent Engineering in design we examine in some detail tool integration, organisational requirements related to the workflow and document management requirements. The paper presents the initial stages of the development of CEPD/MODEL, and how it can be used as a design support system.
1. Introduction and Objectives The adaptation of a CE methodology to chemical process design has the potential to reduce design time and cost, and improve the quality of the designed chemical process, impact assessment and decision integration. These ideas have been applied successfully for more than 15 years in the aerospace, automotive, communications, and hardware/software industries with savings of up to 50% in development time and 30% of development costs (Rossenblat, 1991). Recent analyses of the impact of the application of new CAPE tools and techniques (which include the application of the CE approach) on the European process industries have estimated conservatively a reduction of 10% in investment costs and 5% in operation costs, i.e. around 9.8 and 10.95 bn euros/yr respectively (Perris & Bolton, 2001).
2. Requirements for the application of CE in process design There are many definitions for the term "Concurrent Engineering" (CE). The definition adopted in our work is taken from the Concurrent Engineering Network of Excellence (CE-NET, 1998): ''Concurrent Engineering is a systematic approach for the integration and parallelisation of the design of a product and all its related processes (e.g. building, production, decommissioning) by concurrently taking into account upstream and downstream requirements'".
890 The term integration relates to the incorporation of requirements that come from interand intra-organisational context, for example the alignment of the goals of the people and organisations involved in the development of a product. In turn, the term parallelisation refers to the concurrent execution of processes. The CE approach is similar to the "tiger team" approach in small organisations, where a small group of people works closely for a common endeavour. However, for larger organisations the tiger team concept needs to be modified and restructured because team members can be at geographically different, networked locations, requiring far-reaching changes in the work culture and ethical values of the organisation (Tad, 1992). An example of these changes is to ensure that the hierarchical position of an individual is not a barrier to information exchange. At this point it is also important to define the term Front End Engineering Design (FEED). FEED is a stage in the process life cycle encompassing the whole of its conceptual design and parts of its detailed design (including the specification of parts of the control, safety and environmental systems of the process). At FEED about 50% of process design task is completed and 80% of the cost of the design is committed (Battikha, 1996). FEED places a strong emphasis on an increased involvement of the client and on the revision of alternatives. Its output is a set of documents known as a bid package, typically consisting of a PFD, a P&ID, simulation cases and cost estimations, equipment data sheets, requests for quotation responses, equipment and piping layouts and 3D models. The application of CE ideas and techniques involves a complex interaction of social and technological factors, see Figure 1. The current state of the art suggests that a generic information management system satisfying all the requirements neither exists nor will it be developed in the near future. However, the technology exists to implement substantial parts of CE and improve significantly current practice. Cultural, organisational and human factors needed for CE require new technological support. At the same time, new technologies will not be successful without taking into account their implications in the social factors (Kimura et al., 1992). Therefore, to improve business practices the application of new technology is needed, but improvements in technology will have a limited impact unless the underlying business processes also change to exploit these new capabilities (Bafiares-Alcantara, 2000). cultural organisational (e.g. business processes) human (e.g. psychology)
Figure 1. Factors influencing the implementation of Concurrent Engineering.
891 Among the several requirements for a successful application of Concurrent Engineering in design we examine in some detail the: • Organisational requirements: Related to the workflow, i.e. an ordered sequence of activities that conform a project. • Document handling: An analysis of the types of documents used and generated during the process design workflow, their possible handling mechanisms and the important features to consider when developing a document-handling tool.
3. CEPD/MODEL as a Prototype to Test the Application of CE 3.1. CEPD/MODEL CEPD/MODEL is a software prototype designed to test the applicability of the Concurrent Engineering methodology during process design. As a precondition to fulfil the previous objective CEPD/MODEL must meet the requirements of a CE environment, in particular with regard to the integration of tools, transfer of information and document management. The workflow in CEPD/MODEL consists of the following steps: 1. Process simulation for conceptual design and as a tool in the final steps of process synthesis. 2. Generation of PFDs and P&IDs. 3. Integration with equipment detailed design applications. 4. Costing. 5. Generation of datasheets and reports. 6. Revision and work approval. 7. Document handling. 3.2. Requirements of CEPD/MODEL One of the important targets of CE implementation is the availability of a document management system, i.e. a system to maintain access and manipulate documents quickly, securely and cost effectively. It has been reported that mismanaged files may account for up to half of the "unexpected" project delays (Linton & Hall, 1992). Manual procedures to manage paper documents are simply inadequate for electronic documents mainly because of poor security. Furthermore, just storing the files on a PC hard drive does not mean that they are being efficiently managed because electronic revisions, copies, backups and releases multiply each day. For this reason it is essential to have a document management system with features helpful to facilitate the automation of the CE process, these might include (Gelete, 2001): • Document organisation and access (finding a document and allowing the examination of its contents). • Flexibility in organisation (regrouping documents without physically moving them). • Easy to access profile information. • Document search capability and document security. • Allowing a quick comparison for revision.
892 • Automation in the workflow. The other basic aspect is the availabihty of the process design tools and their features related to integration and automation. The growth in technology and information management points to the availability of new process design tools and means of communication and interaction in the market. However, their application in the area of concurrent engineering for process design needs a substantial improvement in terms of interfacing the applications, document handling, management and automation. CEPD/MODEL encompasses a number of activities and tools (commercial and developed in-house) and their integration, see Table 1. Table L Activities and tools in CEPD/MODEL Activity Process simulation
Tools used by CEPD/MODEL
CAD
HYSYS® AUT0CAD-R14®
PFD and P&ID generation
AXSYS®
Detailed design of equipment Costing
Various, e.g. HTFS (planned)
Datasheet and report generation Work revision and approval Document management
ICARUS 2000® (planned) AXSYS®, Excer^, Access^^ CEPD/ART CEPD/DOC
Three of these packages warrant some further explanation: • AXSYS® is commercial database software that provides storage and use of process stream, unit operation, equipment sizing and cost estimates data. It can also generate automatically PFD and P&ID diagrams from a HYSYS® output based on customisable rules. • CEPD/ART is a software tool being developed for the revision of data extracted from AXSYS®, e.g. lists the changes done to the approved data for design revision purposes. It is being implemented in Microsoft Access^^, Visual Basic^^ and Except. • CEPD/DOC is a network-based document-handling tool for file storage and retrieval. It is being implemented in Microsoft Access^^ and Visual Basic^^. It is currently being expanded into a document management system compliant with the requirements listed in Section 3. The components of CEPD/MODEL are organised around a database system as shown in Figure 2.
4. Case study A full test of the workflow and tool integration capabilities of CEPD/MODEL is under development. In the meantime, and based on data from four sections of the refinery operated by REPSOL-YPF in Tarragona, a test of the integration and capabilities of the tools and of the potential of CE in the area has been completed as a case study. The test is done using data taken from a section of the Olefin (U 661) refinery plant of Repsol, in which the production of monomers such as ethylene and propylene is completed. It also
893
Pn)cess Simulation HYSYS
Equipment Design [TBA]
Figure 2. Structure ofCEPD/MODEL produces sub-products such as ethane, propane, hydrogen, methane, C4s, aromatic components and fuel oil by pyrolysis. This part of the plant takes naphtha as a feed and also propane, butane or residues of isomers generated in other parts of the site or imported from other sites. The CEPD/MODEL testing simulation is done considering a section of the processes listed below. 1. Precooler , process gas drying and de-ethaniser. 2. Hydrogenation of acetylene. 3. Separation and purification of light gases. 4. Purification of ethylene. The workflow of the test case in CEPD/MODEL includes: 1. Simulation of the whole section using HYSYS Plant. 2. Simulation data loading from HYSYS to AXSYS, and development of PFD and P&ID diagrams. 3. Detailed design work in AXSYS, including PFD and P&ID, completion and approval at the logout as a checking and approval of the original work. 4. Loading of the design data for the sake of later revision, and also its approval status information into CEPD-ART. 5. Design document handling using CEPD-DOC by manual data storage. This includes typical information about equipment from the company and registration and retrieval checking. 6. Data and information exchange with AUTOCAD R-14 for design specification format, using AXSYS customisation. 7. Data sheet and reports generation with the help of AXSYS Significant improvement is expected by the application of CE for the improvement of process design workflow and by organising individual activities within the team work such as simulation, PFD and P&ID drawing, revision of design work and documentation and data sheet specification preparation. The considerations in the case study could also indicate that CE application can be automated if these activities are integrated and appropriately organised. Design revision time and cycle is the other concern that requires attention. CEPD- ART as a tool able to record and display the change in name, code, type (among others) in the approved file and the revision work, is thought to be helpful in reducing the revision time and work.
894
5. Conclusions There are several advantages to be obtained from the development and application of a software model such as CEPD/MODEL, the main one being the increased understanding of alternative workflows for the FEED stage of process design. We also expect to obtain the same sort of advantages observed in other areas where CE has been applied, i.e. time reduction, improved workflow management, and an increased automation of parts of the design process. Valid conclusions could be drawn from the case study which show the future potential of CE application in process design: • A Concurrent Engineering approach automation is a short-term possibility. Process design tool integration shows a potential in the area of process design workflow management and also in the improvement of organisation and work performance during process design. • During the FEED stage of process design, considerations of other important process steps in the overall workflow can apparently be possible. This points to automation of the design process as an ideal target. • The CEPD/MODEL approach is compliant with the CE approach as it permits task assignment and accomplishment through distant communication. An important extension in the development is assisting the process design revision through transfer of relevant data and information from the records of the detailed design stage to the simulation/conceptual design stage. It is envisaged that this reverse flow of design information will increase the level of automation of the revision process.
6. References Banares-Alcantara, R., 2000, "Concurrent Process Engineering. State of the Art and Outstanding Research Issues", http://capenet.chemeng.ucl.ac.uk/ CAPE.Net website. Battikha, N.E., 1996, "Applying Front-End Engineering to Control Projects", ICI Explosives, PE Magazine. CE-NET, 1998, CE Network of Excellence, http://esoce.pl.ecp.fr/ce-nt/ Gelete, S.S., 2001, 'A Concurrent Engineering Approach for an Effective Process Design Support System", Tesina, Dept. of Chem Eng, Univ. Rovira i Virgili. Kimura, F., T. Kjellberg, F. Krause and W. Wozny, 1992, Conclusions of the First CIRP Int. Workshop on CE for Product Realisation, Tokyo, Japan. Linton, L. and D. Hall, 1992, "First Principles of Concurrent Engineering - A Competitive Strategy for Product Development", CALS Technical Report 005. Perris, T. and L. Bolton, 2001, "CAPE-21. Feasibility Study", http://CAPE-21.ucl.org.uk. RossenblaU, A., 1991, "Concurrent Engineering", IEEE Spectrum, July. Tad, A.D., 1992, "Advancing CE using STEP", Concurrent Engineering Research, DARPA Project report, West Virginia.
Acknowledgements We acknowledge the support of Hyprotech for the use of HYSYS® and AXSYS® under a university agreement, and to REPSOL-YPF for access to some of its process data.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
895
Agent-based Refinery Supply Chain Management N. Julka, I. Karimi, R. Srinivasan Department of Chemical and Environmental Engineering National University of Singapore
Abstract The refinery business involves tasks spanning across several departments and requiring close coordination and collaboration among them. Present-day business decision support systems are tuned to individual departments and their work processes and are incapable of exploiting the intricate interdependencies among them. There is a need for a decision support system that provides a structured way to make decisions at the enterprise level. In this paper, we address this critical need by developing an agentbased decision support system for refinery supply chain management. Software agents are used to emulate the internal departments of the refinery. These agents gather relevant information from across the enterprise, make decisions based on the embedded business policies and mimic the different business processes of the modeled enterprise. Uncertainties in decision-making are captured by stochastic elements embedded in the agents. Through this, the dynamics in the supply chain is emulated and the performance of the supply chain measured through department-specific key performance indicators.
1. Introduction Decision-making is distributed across various departments in a refinery. Each department solves its own locally focused problems. But local improvements do not necessarily ensure that the overall process attains an optimum. Many times, the departmental objectives may conflict, and all decisions may not contribute positively to the overall performance of the refinery. Furthermore, the decision support systems available for these problems are disjoint and thus inadequate. They are unable to (1) integrate all the decision-making processes of a refinery (2) interface with the present systems in place (3) incorporate dynamic data from various sources and (4) assist various departments (e.g. operations, procurement, storage, logistics, etc.) at the same fime. The need for integrated decision-making is obvious and is the new approach to manage business. The challenges associated with this approach include the difficulty of modeling the entire system and integration with present decision support systems for individual departments or tasks. In addition, the opportunity that the Internet offers in exchanging data seamlessly needs to be exploited in the B2B decision-making processes. In case of a refinery supply chain, information on the web includes the prices of crude, logisfics, products, etc. The crude procurement process involves several departments and is critical to the refinery's bottom line. This paper describes a decision support system for refinery supply chain management. Refinery supply chain management solutions have predominantly focused on various sub-problems of the complete refining business. Existing research addresses the opfimization of refining operations, pooling and blending, planning and scheduling, and
896 a limited integration of some aspects. Rigby et al. (1995), Amos et al. (1997) and Adhya et al. (1999) have addressed the pooling and blending problem for crudes and stored products in a refinery. Pinto et al. (2000) review the research on planning and scheduling. Zhang et al. (2001) propose a model to integrate the oil liquid flow system, the hydrogen system and the steam and power system for simultaneous optimization. Sullivan (1990) emphasizes the integration of the operations and process functions, with blend control and optimization strategies in a refinery, to achieve global optimization. Apart from manufacturing, business processes such as oil trading, logistics and product delivery are also important. Refineries need an integrated supply chain vision to achieve the desired competitive advantage in the present day dynamic business environment. An approach using the optimization of a model of all refinery supply chain entities, their relationships and associated processes is a challenging and computationally intensive problem. On the other hand, an approach using simulation allows only scenario modeling, and analysis is left to the user. A new approach is to blend optimization and simulation to create decision support systems (Padmos et al., 1999).
2. PRISMS In an earlier paper (Julka et al., 2000), we proposed an agent-based framework for the modeling and analysis of a general supply chain. We classified the elements of supply chains as endties and flows. Entities include the operators in a supply chain, e.g. manufacturers and their internal departments, oil suppliers, internet exchanges, etc. These are modeled as emulation agents. Material and information are modeled as commodity and message objects. These objects are exchanged between agents to simulate the material and information flow in the supply chain network. The business processes of each entity in the supply chain are embedded into its agent in the form of grafcets. Each task that the entity performs is modeled as a thread in the grafcets. Threads are triggered by events such as the arrival of a message from another agent or the occurrence of a landmark event. Task threads have steps and transitions representing the various activities performed by the agent such as processing information, acquiring data, recording history and delegating tasks to other agents. Adding task threads to the grafcets chart increase the functionality of an agent. This agent-based supply chain modeling framework provides an environment where all business processes can be emulated in an integrated manner. Various supply chain scenarios can be configured; simulated and analyzed using this model based on user defined metrics or key performance indices (KPIs). The Petroleum Refinery Integrated Supply Chain Modeler and Simulator (PRISMS) utilizes this framework and is developed using Gensym's G2.. It models a refinery with procurement, sales, logistics, storage and operations departments. The refinery supply chain consists of oil suppliers, 3rd party logistics providers (3PL) and electronic exchanges for oil trading. The next section describes the modeled refinery and its crude procurement process. 2.1 Refinery Model Our objective was to develop a system that allows the modeling and simulation of a refinery supply chain. The system should assist in evaluating the impact of different policy and planning parameters on the overall working of a refinery based on userdefined metrics. The entities in the refinery supply chain are:
897 1. 2. 3.
Oil Exchange: Oil suppliers submit postings about crudes available and their prices. Oil Suppliers: Sell crude oil to the refineries. 3PLs: Transport crude from the oil supplier terminals to the refinery.
A brief overview of the internal departments of the refinery is as follows. 1. Procurement department coordinates the crude procurement process. It retrieves availability postings from the exchange and decides on crudes to consider for purchase by taking into account the crude properties, refinery targets, product data and logistics information. 2. Sales: The sales department provides the departments with the present and forecasted product prices and demands. 3. Storage: The storage department schedules the tanks and jetty. It also issues the requested amounts of crudes to the operations department. 4. Operations: This department decides the crudes to be run through the refinery every day. It also decides the various operating parameters for the refining process. 5. Logistics: The logistics department arranges 3PLs for the transport of crude from the oil supplier terminal to the refinery. It performs this through a bidding process. For modeling purposes, the refinery business process is divided into three sub-processes crude selection and purchase, crude delivery and storage, and crude refining. 2.1.1 Crude Selection and Purchase Crude procurement is a key business process in the refinery management. It is also very critical to refinery operations, as shutdown costs are huge and to be avoided under all circumstances. Most refineries purchase many crudes and process them in various mixes. Products with different quality can be obtained from the same crude mix by blending products and varying cut points. Volatility in crude/product prices, crude availabilities and product demands all impact the optimum crude mixes, purchased and refined. Jetty and storage tank scheduling, choice of logistics, uncertainty in ship arrivals, etc. impact the crude procurement and scheduling process. The crude procurement thus is a multi-dimensional process with a single objective of maximizing profit. Figure 1 maps the crude selection and purchase sub-process. The process is initiated based on the present stock of crudes and the estimated ship arrival schedule. Based on a forecast of product prices and demands, the procurement department evaluates the crudes available on the exchange and computes their netback values. The procurement team selects a set of crudes (crude basket) based on these netback values and sends the crude basket to the operations department. The operations department refines the basket based on any operational constraints or previous experience and returns it back to the procurement department. The procurement department compiles the locations and times of all crude pickups and forwards it to the logistics department. 2.7.2 Crude Delivery and Storage The delivery and storage sub-process has three steps (Fig 2a) - dispatch of a ship for crude pickup, arrival of the ship at the pickup terminal and loading of the oil in the ship, and actual arrival of the ship at the refinery jetty and the unloading of oil from the ship. 2.1.3 Crude Refining The crude refining process represents the actual processing in the refinery (Figure 2b).
898 EXCHANGE
LEGEND
OIL SUPPLIER
Figure 1. Modeled Refinery's Crude Purchasing Process
In PRISMS, the user creates a scenario by defining the refinery configuration, selecting the refinery supply chain entities, setting the planning parameters and specifying the simulation details. The prices of crudes and products and the costs of logistics are modeled stochastically and are also input to the system. Once a scenario is configured, the refinery operation is simulated for a specified number of days. The results of simulation are stored as attributes of the respective agents in PRISMS. The messages exchanged among the agents capture the communication between various entities of the refinery supply chain. PRISMS allows the user to add key performance indices (KPIs) such as production profiles, inventory profiles, crude quality profiles in storage tanks, supply demand curves, etc. by defining procedures to compute them. Other assumptions of the modeled refinery are: (1) The refinery has five products: Gas to C4, Gasoline, Kerosene, Gas oil and Residue (2) Only one type of crude is procured in each cycle (3) There is only one jetty for the refinery (4) Crude is unloaded directly into storage tanks (5) The refining of crude is modeled as a single day batch process, i.e. crude is released to the operations in the morning and the products are produced on the same day (6) A tanker takes one day to unload all crude into storage tanks, and. (7) Crude mixtures have linear relationships with respect to the products.
3. Case Studies PRISMS has been implemented to simulate the refinery supply chain described above. Here, we present three case studies to illustrate PRISMS' ability to support the following types of decisions: 1. Policy changes: Evaluate and compare refinery business policies 2. External changes: Understand refinery's response to business environment changes 3. Plant configuration: Evaluate impact of changes in plant configuration
899 Study 1: Impact of Procurement Policy In each procurement cycle, crudes are purchased in discrete packets comprising one crude each. The amount of crude to be purchased in each packet is an important decision for the refinery. Two different practices are possible - fixed packet size and packet size based on the present inventory, schedule of crude arrivals and demands for products. In this study, we evaluated the impact of the fixed vs. variable packet size policies. For the former, a packet size of 125 kbbl was used. Six runs were performed for each scenario using a four-month horizon. The average throughput was calculated using the closing stocks on each day of production. Simulations revealed that the use of variable packet size reduced the average throughput by about 6% (106.9 kbbl/d to 100.3 kbbl/d) and decreased the average inventory level by about 6% (1.18 Mbbl to 1.12 Mbbl). Standard deviations for the throughput were 21.1 kbbl/d and 23.7 kbbl/d for the fixed and variable packet sizes respectively, while those for the inventory level were 0.17 Mbbl and 0.18 Mbbl. This reveals that the variable packet size policy lowers inventory levels while resulting in the same cycle-to-cycle variability as indicated by the similar values of the standard deviation. Study 2: Effect of Demand Fluctuation The ability to handle demand fluctuations is a key to a refinery's economic performance. Production targets are decided based on product demands estimated by the sales department. In this study, we evaluated the robustness of the refinery operation to volatile demand patterns and compared it against a relatively static demand. The average demand in both cases (volatile and static) was kept same, but the standard deviadon changed. The refinery's ability to match the target producUon was quantified by the Root Mean Square of Percentage Shortfall (RMSPS) in daily producUon. It was observed from six runs of each scenario that the refinery business process could not effectively handle a volafile demand as the RMSPS was almost double (13.47 kbbl) that in the static demand scenario (6.56 kbbl). Study 3: BeneHt of Extra Storage Capacity To enhance the refinery's inability to handle demand volatility, the option of increased capacity (additional 15% by adding another tank) in the tank farm was studied. It was observed that in this case, the RMSPS reduced to 9.49 kbbl and the additional inventory helped the refinery match the spikes in the demand.
4. Conclusions Integrated supply chain management is crucial in today's business environment. The lack of decision support systems has led to business processes and policies being evaluated in isolation without consideration of their impact on the overall business performance. In this paper, we have proposed a system for emulating supply chains and illustrated it using the crude procurement process in a refinery. Individual departments of the refinery are modeled as intelligent agents that emulate the different business processes of the department. The system can be used to study the effects of internal policies of a refinery upon KPIs. A key application of the system is for comparing different business policies under a variety of business scenarios in order to idenufy the ones suitable for actual implementation in the enterprise. Future work will include embedding detailed solufion techniques used by individual departments and studying
900 the effectiveness of these with reference to the whole refinery. Additional case studies dealing with oil trading decisions and long-term contracts are also planned.
References Adhya, N., M. Tawarmalani and N.V. Sahinidis. A Lagrangian Approach to the Pooling Problem, Industrial Engineering and Chemistry Research, 38 pp. 1956-1972. 1999. Amos, F., M. Ronnqvist and G. Gill. Modelling the pooling problem at the New Zealand Refining Company, Journal of the Operations Research Society, 48 pp. 767-778. 1997. Julka, N., R. Srinivasan, I. Karimi, N. Viswanadham and A. Behl. Enabling Framework for Decision Support Systems in the e-Commerce Era. In AIChE, 12 -17 Nov. L.A., U.S.A.2000 Padmos, J., B. Hubbard, T. Duczmal and S. Saidi. How i2 integrates simulation in supply chain optimization. In Proceedings of the 1999 Winter Simulation Conference, Dec 5-8. Phoenix, Arizona, USA, pp. 1350-1355. Pinto, J.M., M. Joly and L.F.L. Moro. Planning and scheduling models for refinery operations. Computers and Chemical Engineering, 24 pp. 2259-2276. 2000. Rigby, B., L.S. Lasdon and A.D. Warren. The evolution of Texaco's blending systems: from OMEGA to StarBlend, Interfaces, 25 (5), pp. 64-83. 1995. Sullivan, T.L. Refinery-wide blending control and optimization. Hydrocarbon Processing, May pp. 93-96. 1990. Zhang, J., X.X. Zhu and G.P. Towler. A Simultaneous Optimization Strategy for Overall Integration in Refinery Planning, Industrial and Engineering Chemistry Research, 40 (12), pp. 2640-2653. 2001. LEGEND
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Figure 2. Refinery (a) Crude Delivery and Storage Process and (b) Crude Refining Process
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
901
Using Continuous Time Stochastic Modelling and Nonparametric Statistics to Improve the Quality of First Principles Models Niels Rode Kristensen^, Henrik Madsen^ and Sten Bay J0rgensen^ ^^ Computer Aided Process Engineering Center, Department of Chemical Engineering ^^ Mathematical Statistics Section, Informatics and Mathematical Modelling Technical University of Denmark, DTU, DK-2800 Lyngby, Denmark
Abstract A methodology is presented that combines modelling based on first principles and data based modelling into a modelling cycle that facilitates fast decision-making based on statistical methods. A strong feature of this methodology is that given a first principles model along with process data, the corresponding modelling cycle can be used to easily, rapidly and in a statistically sound way produce a more reliable model of the given system for a given purpose. A computer-aided tool, which integrates the elements of the modelling cycle, is also presented, and an example is given of modelling a fed-batch bioreactor.
1. Introduction The increasing use of computer simulations in analysis and design of process systems and recent advances in model based process control and process optimisation have made the development of rigorous dynamic process models increasingly important over the past couple of decades. Particularly in view of the increasing focus on batch and fedbatch operation in many areas of the process industry, the ability of such process models to describe nonlinear and time-varying behaviour has also become more important. Altogether, these developments have necessitated faster development of new and improvement of existing first principles models, i.e. models based on physical insights and conservation balances. The purpose of this paper is to show how continuous time stochastic modelling and time series analysis tools based on nonparametric statistics can be used to facilitate this. Continuous time stochastic modelling is a grey-box approach to process modelling that combines deterministic and stochastic modelling through the use of stochastic differential equations (SDE*s) and has previously been described in Kristensen et al. (2001a). Other previous contributions in the area of greybox modelling include the work of Madsen and Melgaard (1991) and Bohlin and Graebe (1995) and references therein. The outline of the paper is as follows: In Section 2 the overall methodology is described in terms of a modelling cycle, some details of the individual elements of this cycle are given, and a computer aided tool that facilitates the use of the overall methodology is briefly described. In Section 3 a case study is presented that shows how the
902 methodology can be used to improve the quality of a first principles model of a simple fed-batch bioreactor. Conclusions are given in Section 4.
2. Methodology The overall methodology can be described in terms of Figure 1, which shows the proposed continuous time stochastic modelling cycle described in the following. 2.1 Model construction The first step in the modelling cycle deals with construction of the basic model, which is a continuous-discrete stochastic state space model consisting of a set of SDE's describing the dynamics of the system in continuous time and a set of algebraic equations describing measurements at discrete time instants, i.e. (1) dx^ =f(x^,u^,/,e)/r +
nn I
expl - ^ £ ^
i=\
^'k\k-
^^^K\k-i^
^(^oh)
exp(-^£e5:e^£e)
(3)
^detXe V2^^
VV
by further conditioning on the initial conditions yo' in the individual experiments. z\ and R\\k.i are the mean and covariance of the innovations from an extended Kalman filter at the k'ih sample in the /'th experiment, and £e and Le are the deviation from, and the covariance of a prior estimate of the parameters. A more detailed account of this formulation is given in Kristensen et al. (2001a), and details about the algorithms behind the corresponding estimation methods can be found in Kristensen et al. (2001b).
Figure 1. The continuous time stochastic modelling cycle.
903 2.3 Statistical tests and residual analysis The third step in the modelling cycle deals with assessing the quality of the model once the unknown parameters have been estimated. The estimators described above are all approximately Gaussian, meaning that t-tests can be performed to test the hypothesis that a parameter is marginally insignificant. The test quantity is the value of the estimate of the parameter divided by the standard deviation of the estimate and is approximately t-distributed with a number of degrees of freedom that equals the number of data points minus the number of estimated parameters. To test the hypothesis that some parameters are simultaneously insignificant, several tests can be applied, e.g. a likelihood ratio test, a Lagrange multiplier test or a test based on Wald's W-statistic. These test quantities all have the same asymptotic x^-distribution with a number of degrees of freedom that equals the number of parameters to be tested for insignificance, but in the context of the proposed modelling cycle Wald's test has the advantage that no re-estimation is required. Details about the derivation of this statistic are given in Hoist et al. (1992). Another important aspect in assessing the quality of the model is to investigate its predictive capabilities by performing cross-validation and examining the corresponding residuals. Depending on the intended application of the model this can be done in a onestep-ahead prediction setting or in a pure simulation setting, and one of the most powerful methods is to compute and inspect the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) of the residuals to detect if there are any significant lag dependencies, as this indicates that the model is incorrect. Nielsen and Madsen (2001) recently presented extensions of these linear tools to nonlinear systems, the lag-dependence function (LDF) and the partial lag-dependence function (PLDF), which are based on the close relation between correlation coefficients and values of the coefficients of determination for regression models and which extend to nonlinear systems by incorporating nonparametric regression in the form of additive models. In the context of the proposed modelling cycle the ability of the LDF and the PLDF to detect nonlinear lag-dependencies is particularly important. 2.4 Model validation The last step in the modelling cycle deals with model validation or invalidation, or, more specifically, with whether, based on the information gathered in the previous step, the model is invalidated with respect to its intended application or not. If the model is invalidated, the modelling cycle is repeated by first changing the structure of the model in accordance with the information gathered in all steps of the previous cycle. 2.5 A computer aided tool for continuous time stochastic modelling To facilitate the use of the proposed modelling cycle, a GUI-based computer-aided tool, called CTSM, has been developed, cf Kristensen et al. (2001b). Within CTSM models of the kind (l)-(2) can be set up, unknown parameters can be estimated using a variant of (3), and statistical tests and residual analysis can be performed. CTSM is very flexible with respect to the data sets that can be used for estimation, as features for dealing with occasional outliers, irregular sample intervals and missing observations have been implemented. CTSM runs on Win32, Solaris and Linux platforms, and on Solaris platforms the program supports shared memory parallellization using OpenMP for improved performance.
904
3. Case study: Modelling a fed-batch bioreactor To illustrate how the proposed modelling cycle can be used to improve the quality of a first principles model, a simple example is given. The process considered is a fed-batch bioreactor described by a simple unstructured model of biomass growth, i.e.
As)x (sp-^y
s
dt +
V
^x^ s
0
0
722
0
0
^33
(4)
^to,
'e ^ (5)
yv(o,522) yv(o,533)
where X is the biomass concentration, S is the substrate concentration, V is the volume of the fermenter, F is the feed flow rate, Sr (=10) is the feed concentration of substrate, Y (=0.5) is the yield coefficient of biomass and iMS) is the growth rate. On, (J22, 033, Sn, S22 and S33 are stochastic parameters. Two different cases are considered for /ifij, corresponding to Monod kinetics with and without substrate inhibition, i.e.
^{s)=^.
^{S)=li[
(6)
K^S^^S^K^
(7)
S-^K,
In the following the model consisting of equations (4), (5) and (6), with A^2=0.5, is regarded as the true process to be modelled, and using the true parameter values in Table 2 two sets of data are generated by stochastic simulation. One data set is used for estimation and the other is used for validation. The model consisting of equations (4), (5) and (7) is regarded as an original first principles model, which in the context of the modelling cycle is the basic model. Using the estimation data set, the unknown parameters of the model are estimated with CTSM, giving the results in Table 1. Table 1. Estimation results using the incorrect model structure. Parameter True value Estimate Std. Dev. t-score Significant
Xo 1 1.042 0.014 72.93 Yes
So 0.245 0.250 0.010 24.94 Yes
Vo 1 0.993 0.001 689.3 Yes
^ max
Ki
-
-
0.737 0.008 96.02 Yes
0.003 0.001 2.396 Yes
Oil
022
0-.?.?
Sii
S22
S33
0 0.104 0.018 5.867 Yes
0 0.182 0.010 18.26 Yes
0 0.000 0.000 1.632 No
0.01 0.008 0.001 6.453 Yes
0.001 0.000 0.000 3.467 Yes
0.01 0.011 0.003 3.801 Yes
Results of marginal t-tests show that the only insignificant stochastic parameter is G33, whereas a^ and G22 are significant. This in turn indicates that the deterministic parts of the equations for X and 5 in (4) are incorrect in terms of describing the variations in the estimation data set. To investigate this further, residual analysis is performed. One-stepahead prediction results on the validation data set are shown in Figure 2 and Figure 3
905 shows the SACF, SPACF, LDF and PLDF for the corresponding residuals. There are no significant lag dependencies in the residuals for yi and ys, whereas in the residuals for y2 there is a significant lag dependence at lag 1. This is an additional indication that the equation for 5 in (4) is incorrect. A final piece of evidence that something is wrong is gathered from the pure simulation results in Figure 2. The information now available clearly invalidates the model, particularly if its intended purpose is simulation, and the modelling cycle is repeated by modifying the structure of the model.
Figure 2. Cross-validation results. From left to right: One-step-ahead prediction and pure simulation using the incorrect model structure and one-step-ahead prediction and pure simulation using the correct model structure. (Solid: Predicted values, dashed: true yi, dotted: true yi, dash-dotted: true y^).
Figure 3. One-step-ahead prediction cross-validation residuals and corresponding SACF, SPACF, LDF and PLDF using the incorrect model structure. (Top: yi, middle: yi, bottom: y^). Table 2. Estimation results using the correct model structure. Parameter True value Estimate Std. Dev. t-score Significant
Xo 1 1.004 0.010 101.0 Yes
So 0.245 0.262 0.008 32.75 Yes
Vo 1 1.003 0.007 143.3 Yes
f^max
Kj
(711
^22
033
Sii
S22
S33
1 0.999 0.009 109.4 Yes
0.03 0.030 0.007 4.240 Yes
0 0.000 0.000 0.003 No
0 0.000 0.000 0.005 No
0 0.000 0.000 0.003 No
0.01 0.009 0.001 7.142 Yes
0.001 0.001 0.000 7.391 Yes
0.01 0.011 0.001 7.193 Yes
The information available suggests that the deterministic parts of the equations for X and S in (4) are incorrect, i.e. those parts of the model that depend on IJ(S). Replacing (7) with the correct structure in (6) and re-estimating the unknown parameters with
906 CTSM, the results shown in Table 2 are obtained. Marginal t-tests indicate that all three stochastic parameters, Oj], 022 and o^j, are now insignificant, and the hypothesis of simultaneous insignificance cannot be rejected when performing a test based on Wald's W-statistic. Additional evidence that the modified model is correct is gathered by performing residual analysis. One-step-ahead prediction results on the validation data set are shown in Figure 2, and the SACF, SPACE, LDP and PLDF (not shown) for the corresponding residuals show no significant lag dependencies. A final piece of evidence of the validity of the modified model is gathered from the pure simulation results in Figure 2. In summary, if the intended purpose of the original model was simulation or infinite-horizon prediction, e.g. for use in an MFC controller, it has been now been invalidated and a more reliable model has been developed. However, if the intended purpose of the original model was one-step-ahead prediction, it might still be suitable.
4. Conclusion A methodology has been presented that combines modelling based on first principles and data based modelling through the use of stochastic differential equations and statistical methods for parameter estimation and model validation. The methodology features a modelling cycle that can be used to easily, rapidly and in a statistically sound way develop a reliable model of a given system. A computer-aided tool, called CTSM, which integrates the elements of the modelling cycle, has also been presented.
5. References Bohlin, Torsten and Stefan F. Graebe (1995). Issues in Nonlinear Stochastic Grey-Box Identification. International Journal of Adaptive Control and Signal Processing 9, pp. 465-490. Hoist, Jan, Ulla Hoist, Henrik Madsen and Henrik Melgaard (1992). Validation of Grey-Box Models. In: Preprints of the IF AC Symposium on Adaptive Systems in Control and Signal Processing, Grenoble, France, pp. 407-414. Kristensen, Niels Rode, Henrik Madsen and Sten Bay J0rgensen (2001a). Computer Aided Continuous Time Stochastic Process Modelling. In: European Symposium on Computer Aided Process Engineering -11 (Rafiqul Gani and Sten Bay J0rgensen, Eds.), pp. 189-194. Kristensen, Niels Rode, Henrik Melgaard and Henrik Madsen (2001b). CTSM 2.0 User's Guide. DTU, Lyngby, Denmark. Madsen, Henrik and Henrik Melgaard (1991). The Mathematical and Numerical Methods used in CTLSM. Technical Report 7/1991. IMM, DTU, Lyngby, Denmark. Nielsen, Henrik Aalborg and Henrik Madsen (2001). A Generalization of Some Classical Time Series Tools. Computational Statistics and Data Analysis 37(1), pp. 13-31.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
907
Moving mesh generation with a sequential approach for solving PDEs Y. I. Lim\ J. M. Le Lann^ and X. Joulia^ ^CAPEC, DTU, 2800 Kgs. Lyngby, Denmark ^'^Laboratoire de Genie Chimique (LGC, UMR-CNRS 5503), INPT-ENSIACET 118 route de Narbonne, F-31400 Toulouse, France
Abstract In moving mesh methods, physical PDEs and a mesh equation derived from equidistribution of an error metrics (so-called the monitor function) are simultaneously solved and meshes are dynamically concentrated on steep regions (Lim et al., 2001). However, the simultaneous solution procedure of physical and mesh equations suffers typically from long computation time due to highly nonlinear coupling between the two equations. Moreover, the extended system (physical and mesh equations) may be sensitive to the tuning parameters such as a temporal relaxation factor. It is therefore useful to design a simple and robust moving mesh algorithm in one or multidimension. In this study, we propose a sequential solution procedure including two separate parts: prediction step to obtain an approximate solution to a next time level (integration of physical PDEs) and regriding step at the next time level (mesh generation and solution interpolation). Convection terms, which appear in physical PDEs and a mesh equation, are discretized by a WENO (Weighted Essentially Non-Oscillatory) scheme under the conservative form. This sequential approach is to keep the advantages of robustness and simplicity for the static adaptive grid method (local refinement by adding/deleting the meshes at a discrete time level) as well as of efficiency for the dynamic adaptive grid method (or moving mesh method) where the number of meshes is not changed. For illustration, a phase change problem is solved with the decomposition algorithm.
1. Introduction For the numerical solution of dynamic systems involving steep moving fronts, a uniform fixed-grid structure is computationally inefficient, since a large number of meshes are required for an accurate solution. In such cases, methods which automatically adjust the size of the spatial step, namely moving mesh methods, are likely to be more successful in efficiently resolving critical regions of high spatial activity. Moving grid methods have important applications for a variety of physical and engineering problems (e.g., solid/fluid dynamics, combustion, heat transfer and phase changes) that require extremely fine meshes in a small portion of physical domain. Successful implementation of the adaptive strategy can increase the accuracy of the
^ To whom correspondence should be addressed, e-mail: [email protected].. Fax. +45 4593 2906
908 numerical approximation and also decrease the computational cost (Furzeland et al., 1990). The moving grid methods (Miller & Miller, 1981; Dorfi & Drury, 1987; Huang & Russell, 1997), where the number of meshes is kept constant, could be very powerful due to the continuous grid adaptation with the solution evolution. However, intrinsic coupling between physical PDEs and mesh equations leads to a large system of nonlinear equations and much calculation time. The simultaneous solution techniques have been well presented and tested in the literatures (Dorfi & Drury, 1987; Furzeland et al., 1990; Blom & Zegeling, 1994; Li & Petzold, 1997; Huang & Russell, 1997; Li et al., 1998), using a central or ENO discretization method for convection terms and well-known monitor functions as an error metric. In contrast, there are few articles on the sequential solution approaches where physical and mesh equations are solved separately or sequentially. Mackenzie & Robertson (2000) proposed a two-step iterative method for fully discretized PDEs, using a Runge-Kutta ODE integrator. In the solution update step, the solutions are linearly interpolated. Tang & Tang (2000) used conservative-interpolation in the solution update step. The solutions of underlying PDEs and a mesh equation are obtained independently using a Runge-Kutta or BDF ODE integrator. For the two sequential approaches, physical PDEs considered do not contain a mesh convection term (dx/dt or x). Therefore, the two static sequential methods may not be adequate for stiff systems. Huang & Russell (1999) proposed a sequential approach based on the Lagrangian description involving mesh movement for two-dimensional mesh equations. However, using their algorithm, it is not easy to determine the next time stepsize for integration of physical and mesh equations. In next section, we present a new decomposition algorithm that allows us to easily implement on stiff systems in using a conventional BDF (Backward Differentiation Formula) ODE integrator.
2. Decomposition algorithm The decomposition algorithm is to decouple the calculation of the mesh position from the solution. There are two advantages of the decomposition algorithm (i.e., sequential solution procedure). First the size of the discretized system (ODEs or DAEs) that arises at each time step is smaller. This is of great importance for the extension to multidimensional problems and for a reduction of the calculation time. The second advantage is that decomposition allows flexibility in the choice of iterative methods used to calculate the grid and the solution. We propose a sequential solution procedure including two separate parts: prediction step to obtain an approximate solution to a next time level (integration of physical PDEs) and regriding step at the next time level (mesh generation and solution interpolation), as shown in Fig. 1. The theoretical background of this approach relies on the Sequential Regularization Method (SRM) proposed by Ascher & Lin (1996, 1997). 2.1 Prediction step Given the physical solution (u"), the mesh (x") and the time stepsize (At") at time t=t". Normally, initial meshes (x^) are uniformly given. To obtain new initial mesh positions
909 equi-distributed, it is recommended to update the mesh and the solution with a very small time stepsize (At) at the first time level (t=0).
Initial Condition u«, x°,t=t»,At Initialization of Mesh t=t"+At"and x=0 x" <-x"^\ u"<-u"^^
x^ from x^ and u^, x =0
I'^steorODEs/DAEs Prediction step (physical PDE integration) Find 5"^' from x", x and u" on x= x°+ X (t-t") Re-initialization x=(x"^^-x")/At" r^ ^tPn • AF.st/OnF<:
Regriding step (Equidistribution & interpolation) . n+l
Find x""^ with u"^'=f(x", S""', x""')
Convergence test
no
Il5"^^-u"^1l < e yes Fig. 1. Dynamic moving mesh algorithm in the two-step solution procedure. A new solution (ii"^^) is computed by integrating time-dependent ODEs of Lagrangian description from t=t" to t=t"-hAt", using x" as the initial mesh. li-u^x = f(u,x,x)
(1)
In the above equation, the flux correction term caused by mesh movement, u^x, may become a considerable value, when the mesh is updated after a relatively large time step (At"). In this case, the convection term affects stability and accuracy of the whole equation (1). It is nature that the convection term (u^) is discretized by a high resolution
910 upwinding scheme to enhance accuracy as well as stability (Li & Petzold, 1997). In our study, the fifth-order WENO scheme (Jiang & Shu, 1996) is employed in the upwinding sense. At the first iteration, mesh points are fixed with time, i.e., x =0. Hence, the mesh position x" is not equidistributed any more with regard to the new solution ii"^^. The meshes will be redistributed in the regriding step. 2.2 Regriding step In this step, the solution u"^' is interpolated on new meshes x"^^ during mesh calculation, where a mesh equation derived from an error-metrics (i.e., monitor function) is required. To efficiently generate moving meshes, the monitor function must be appropriately selected according to problems considered. For the phase change problems, the integrable monitor function improves accuracy and temporal performance over the arclength or curvature monitor functions. To our knowledge, the integrable monitor function is useful for systems in which a value indicating discontinuous regions (e.g., phase change temperature or enthalpy) can be prescribed. The mesh equation based on an integrable monitor function (Mackenzie & Robertson, 2000) is presented here. For the first and last mesh points (XQ and XN) fixed, each mesh Xi (i=l...N-l) is defined by: (Xi-Xo)+-y^sinh-'(^2(Xi-x*))--^(l-i/N)sinh'V2(Xo-x*))^^2
/ V
> ^^2
sinh V 2 ( X N - - X * )
(2) =0
)
where N is the total number of meshes and x* is a point estimated (or given) as a phase change boundary. The parameters (jii and ILI2) are positive constants that govern the smoothness or clustering of the grid around the point x*. The regularization parameters are not sensitive to problems, to our knowledge. In our study, these values are set to jii=^i2=200. Equation (2) can be solved easily by a Newton's iteration for nonlinear algebraic equations. The solution u""^^ is updated by using a conservative piecewise cubic Hermite interpolant (EOIBEF/EOIBBF subroutines in the NAG numerical library). At the first iteration, the solution u"^^ on x"^^ is obtained from ij"^^ not considering mesh movement (x =0). Iteration must be performed between the two steps, after the convergence test, ^(u""^^ - u"""^)^ < e . 2.3 Iteration between the prediction step and the regriding step From the second iteration, mesh movement x can be calculated on the basis of x""^^ obtained in the previous iteration: ^„^x
x_ At"
911 In this iteration, the physical equation (1) is integrated with x" that keeps constant during integration. The starting values of iteration, u" and x" remain constant. There is rarely any need to use a very strict tolerance for the convergence of the grid points. To our knowledge, a satisfactory solution can be obtained after 2-4 iterations. A numerical test follows in the next section.
3. Numerical study We consider a heat conduction-diffusion PDE with phase change, where two phase change interfaces develop with time along a spatial direction (x) and enthalpy profiles (H) are discontinuous at the interfaces: ~
at " dx
(4)
+ cp(T)
with cp(T) =
f0.336 + 3.457T, T<0.6
(5)
[1.708+ 1.220T,T> 0.6 0.780T, T < 1
H(T)=. 0.780
(6)
0.780T + 0.331, T > 1 Refer to Mackenzie & Robertson (2000) for boundary, initial conditions and a differentiable enthalpy function. Mesh evolution of 40 points with time and the enthalpy profiles are shown in Fig. 2. The phase change boundaries are well captured and near them meshes are concentrated, as shown in Fig. 2 (a). The enthalpy profiles agree well with the reference solutions (solid lines in Fig. 2 (b)) obtained by the 600-fixed grid system. (a) m e s h evolution
0.25
0.5 mesh, X
(b) enthalpy profiles
0.75
Fig. 2 Numerical results for the sequential moving mesh method based on the WENO scheme and the integrable monitor function.
912
4. Conclusion With the same number of meshes, the moving mesh methods improve solution accuracy over the fixed mesh methods at the cost of the computational time. The fixed grid system often requires a large number of meshes and a long computational time in order to obtain numerical results with accuracy. Through the sequential moving mesh method with an appropriate monitor function, an accurate solution is obtained at a small number of meshes for a short computational time. For the mesh convection term, the WENO (Weighted Essentially-Non Oscillatory) discretization scheme is used to enhance the moving mesh method for which the central discretization has been commonly used. The sequential approach, where the physical equation and the mesh equation are solved separately, can reduce the computational time, since strong nonlinear coupling between the two equations is avoided.
References Ascher, U. and P. Lin, 1996, Sequential regularization methods for higher index DAEs with constraint singularities: the linear index-2 case, SIAM J. Num. Anal., 33, 1921. Ascher, U. and P. Lin, 1997, Sequential regularization methods for nonlinear higher index DAEs, SIAM J. Sci. Comput., 18, 160. Blom, J. G. and P. A. Zegeling, 1994, Algorithm 731; a moving-grid interface for systems of onedimensional time-dependent partial differential equations, ACM Trans, on Math. Software, 20, 194. Dorfi, E. A. and L. O'C. Drury, 1987, Simple adaptive grids for 1-D initial value problems, J. Comput. Phys., 69, 175. Furzeland, R. M., J. G. Verwer and P. A. Zegeling, 1990, A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines, J. Comput. Phys., 89, 349. Huang, W. and R. D. Russell, 1997, Analysis of moving mesh partial differential equations with spatial smoothing, SIAM J. Num. Anal., 34, 1106. Huang, W. and R. D. Russell, 1999, Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM Sci. Comput., 20, 998. Jiang, G. and C. W. Shu, 1996, Efficient implementation of weighted ENO schemes, J. Comp. Phy., 126, 202. Li, S. and L. Petzold, 1997, Moving mesh methods with upwinding schemes for time-dependent PDEs, J. Comput. Phys., 131, 368. Li, S., L. Petzold and Y. Ren, 1998, Stability of moving mesh systems of partial differential equations, SIAM J. Sci. Comput., 20, 719. Lim, Y. I., J. M. Le Lann and X. Joulia, 2001, Moving mesh generation for tracking a shock or steep moving front, Comp. Chem. Eng., 25(4-6), 653. Mackenzie, J. A. and M. L. Robertson, 2000, The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method, J. Compt. Phys., 161(2), 537557. Miller, K. and R. N. Miller, 1981, Moving fmite elements I, SIAM J. Numer. Anal., 18, 1019. Tang, H. and T. Tang, 2000, Moving mesh methods for one and two-dimensional hyperbolic conservation laws, preprint. Department of Mathematics, The Hong Kong Baptist university, available in the website: http://www.math.hkbu.edu.hk/~ttang/.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
913
An Optimized Strategy for Equation-Oriented Global Optimization Ricardo M. Lima and Romualdo L. Salcedo Departamento de Engenharia Quimica, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal
Abstract An optimized strategy for simulation and optimization of steady-state processes, under an equation-oriented environment, is presented. Equation-oriented environments apply a solution procedure to solve the entire system of non-linear algebraic equations arising from the mathematical model describing these processes. The difficulty in solving these systems may change drastically by specifying different independent variables for the degrees of freedom, as it has long being recognized. An algorithm that chooses the decision variables by minimizing the number and size of the subsystems of equations that need to be simultaneously solved for, while allowing for the inclusion of functional constraints, is used (Salcedo and Lima, 1999). With this algorithm, optimum sets of decision variables and the corresponding solution strategies are obtained. This paper describes the implementation of this approach linked with a simulated annealing-based global optimizer. The proposed strategy was applied to the optimization of a reactor-extractor system and to a more difficult absorber-stripping system with heat integration. With these examples we pretend to compare different optimization procedures for each test case, respectively solving the entire system of equations, solving some smaller subsystems (a local optimum for the simulation step) or solving for the global optimum of the simulation step (which may correspond to a sequential solution). By optimizing the simulation step much more accurate results as well as significantly reduced CPU times are obtained, in comparison with simultaneous solution strategies, suggesting that this may be a powerful tool for global optimization.
1. Introduction The simulation of steady-state chemical processes can be performed by three different approaches: sequential modular, simultaneous modular and equation-oriented. The ideas behind them, as well as the advantages and disadvantages of each approach are well known and described by several authors (Biegler et al.; 1997, Tolsma and Barton, 2000). The main idea behind the equation-oriented approach is the application of a solution strategy to all equations that describe the process. This set of equations is usually a non-linear system, very often with a sparse structure. The simultaneous solution of these systems is the key task of equation-oriented environments, which however require a proper initialisation (Zitney and Stadtherr, 1988). For the optimization of steady-state processes, the equation-oriented approach can be interfaced Author to whom correspondence should be addressed. E-mail: [email protected]
914 with deterministic global search algorithms (Grossmann, 1994), or with stochastic algorithms (Salcedo, 1992; Cardoso et al., 1997, 2000). With stochastic algorithms, the solution of these optimization problems proceeds at two different levels: a) the simulation level that performs the solution of the system of equations and the calculation of a performance index; b) and the optimization level, which receives the performance index and is responsible to update the decision variables. This paper describes the implementation of an equation-oriented simulation approach integrated with an optimization framework suitable for the global optimization of non-linear programming (NLP) problems.
2. An equation-oriented environment for global optimization The main idea behind the proposed approach is the optimization of the solution of the simulation step within a global optimization procedure. This is performed using the algorithm described by Salcedo and Lima (1999). Here the simulation step of an optimization is posed as an optimization problem itself, whereby a cost function, i.e. a performance index related to the difficulty of solving non-linear systems of equations, is associated with the choice of decision variables. The algorithm couples a combinatorial optimization algorithm with a tearing/partitioning algorithm (Ledet and Himmelblau, 1970). The global optimum for the simulation step may eventually correspond to a sequential solution, but it may also correspond to the solution of smaller subsystems (Salcedo and Lima, 1999). In any case, it is expected that choosing different decision variables will provide for a more efficient simulation procedure than the solution of the entire system of equations, no matter how efficient this might be, viz. even employing algorithms that exploit sparcity. Simulated annealing-based optimizers are suitable algorithms for the global optimization of NLP problems due to their capability to escape local optima and to find solutions close to the global optimum. However, these optimizers need a high number of function evaluations (Grossmann and Biegler, 1995), which usually results in a great computational burden. Although the development of faster convergence is possible, when the simulation step involves the iterative solution of models the computational requirements can still be quite large. The present paper shows the implementation of an equation-oriented simulation strategy for the global optimization of NLP problems, which is able to: i) solve efficiently systems of non-linear equations; ii) minimize convergence problems; iii) minimize CPU times when couple with stochastic global optimizers. The proposed approach for the global optimization of NLP problems is shown in Fig.l, and is based in four steps: pre-processing of the data; optimization of the simulation step; symbolic manipulations; and numerical optimization. The next sections highlight the details of the implementation of this approach. 2.1 Data pre-processing The equation-oriented environment proposed needs the user to specify the mathematical model that describes the chemical process in simple text files. One essential task of this step is the automatic build-up of the occurrence matrix, where the information about functional constraints is also stored.
915 2.2 Optimization of the simulation step The choice of the decision variables is automatically performed by the combinatorial global optimizer and the corresponding output set is obtained. The combinatorial algorithm is responsible for the choice of the decision variables and the tearing/partitioning algorithm is responsible for obtaining the cost function of the simulation step. This cost function is based on the minimization of the number of subsystems to be simultaneously solved for, while allowing for the inclusion of functional constraints (Salcedo and Lima, 1999). The output set gives information about the sequence of solution and the occurrence of irreducible subsystems. In the proposed approach, whenever the global optimum for the simulation step corresponds to a sequential solution, there is no need of a numerical method to solve the system of equations (apart from possibly solving nonlinear functions in one variable). However, it may be impossible to fmd a sequential solution, and in these cases a numerical method is needed to solve the subsystems of equations. These may be solved either by a direct iterative process using tearing variables, or by a simultaneous approach using Newtontype algorithms. The latter approach was retained in the present work, since direct Combinatorial optimizer I I I I
Equations Objective function Constraints (numeric) Parameters
Tearing/partitioning algorithm
N
Mathematical model Text files
I I I I
Structural information Functional information Subset of decision variables
j
Y
Numerical methods
MatLab'*' Set up of the occurrence matrix Optimum simulation procedure Fortran
MSIMPSA
Global Optimum
Optimization of the simulation f
^
ZT "^
^
I Decision variables | I J
.
Output set I I Solution sequence I I Presence of subsystems I
MatLab® Manipulation of equations
Figure 1. Flow diagram of information to perform an optimization using the simulation/optimization framework. iteration often fails to converge (Lin and Mah, 1978). At the present we have implemented two numerical methods: a globally convergent Newton method from Press et al (1992) and a modified Newton algorithm combined with a trust region method from MINPACK, available from netlib. These were compiled into libraries interfaced with the MatLab® environment, where the model equations and constraints are initially written by the user. 2.3 Symbolic manipulations This level receives the information contained in the output set. At this point, the Symbolic Toolbox from MatLab® (using the kernel from Maple®) is used to explicit variables whenever possible. If it turns out to be impossible to explicit one variable, a Newton-based method is applied to solve a non-linear function in a single variable. The
916 automatic generation of several Fortran subroutines which comprise the entire mathematical model are also generated at this level. 2.4 Optimization The optimization of the NLP problem is performed using MSIMPS A. This optimizer is a robust simulated annealing-based algorithm suitable for the global optimization of nonconvex NLP and MINLP constrained functions (Cardoso et al. 1997, 2001). Largescale problems (large non-linear systems) may need the application of sparse-type solvers to improve the numerical solution of irreducible subsystems. The Fortran routine created in the previous level is built to be compliant with MSIMPSA, and all the compilations and links between the routines are automatically done and completely transparent to the user.
3. Case studies The first case study represents a reactor-extractor-recycle system taken from Lee (1969). The second is an absorber-stripping system with heat integration, described by Umeda (1969) and more recently used as a test case by Ferreira and Salcedo (2001). For both cases the equations and a detailed description about these processes can be found in the cited literature. With these examples we pretend to compare different optimization procedures for each test case: i) solving the entire system of equations by a Newton-type method; ii) solving explicitly some equations followed by some smaller subsystems using Newton's method (a local optimum for the optimized simulation); iii) solving for the global optimum of the simulation procedure. A proper initialization of the variables to solve the system of equations was achieved by adopting the role of pseudo-decision variables for the initial estimates, leaving this function to the optimizer (Salcedo, 1989). Whenever the Newton-based methods were used, the error criterion was set to 10'^ both for the convergence of the function values and for the variables. The performance evaluation of the different solution procedures was carried out through the study of the results of 10 optimization runs for the simultaneous solution of all equations and 100 runs for the global optimum of the simulation step (each different run corresponds to a different starting point, which is randomly generated by MSIMPSA). 3.1 Case I This case represents a reactor-extractor-recycle system composed by five isothermal continuous stirred tanks reactors followed by three crosscurrent extractors with recycle. The process is described by a system with 45 algebraic equations with 53 continuous variables, resulting in 8 degrees of freedom. The optimization of the solution procedure did not produce a sequential solution, but rather the solution of two subsystems, one with 2 equations and one with 12 equations. Figs, la-b show that the simultaneous solution of all equations presents much higher computational times and lower accuracies in arriving at the global optimum (24.77 kg/h). 3.2 Case II This example represents an absorber-stripping system with heat integration. The process is described by 40 algebraic equations and 45 continuous variables, resulting in a NLP
917
H E
l.E-05
l.E-04
l.E-03
l.E-02
mjfMl^^
- S i m u l s d . of 2 subsystems
-Sinnul. sol. of 2 subsystems
-Simul.sol. of d l system
-Sirrxjl. s d . o f d l system
l.E-01
l.E+00
l.E+01
0
1.E+02
10
20
30
40
50
60
70
80
90
100
Runs
% Accuracy
(a) (b) Figure 2. (a) Results for case I. (b) Results for case /, elapsed time for each run 100 90 80
-Seqjentidsdution - S imul. S d . d 2 s ubs ys terns - S imul. s d . of d l equations
-Seqjentidsdution - SimJ. Sd. of 2 s ubB ys tem - S i m i . sd. of dieqjdkns
70 60
S
50 -I
^
40
l.E-04
E
l.E-03
l.E-02
l.E-01
% Accuracy
l.E+00
l.E+01
l.E+02
f
0
10
20
30
40
50
60
70
80
90
100
Runs
(a) (b) Figure 3. (a) Results for case II. (b) Results for case II, elapsed time for each run. problem with 5 degrees of freedom. In this case, the global optimum for the solution procedure is a sequential solution (Salcedo and Lima, 1999). One local optimum for the solution procedure was obtained by choosing a different set of decision variables, resulting in an output set with one subsystem in seven equations, plus one subsystem with four equations. Figs. 2a-b show that the results are even more dramatic, where the optimized simulation corresponds to a sequential solution. The optimum value for this case corresponds to a profit of -$173,392.
4. Conclusions The present paper shows that it is possible to increase the chance of convergence to the global optimum, while drastically decreasing the CPU times, by optimizing the simulation step of the optimization problem. This is performed by associating a cost function to the simulation step, and optimizing this step by choosing different sets of decision variables (a combinatorial problem). This combinatorial problem was solved by the same global optimizer that was used for the numerical optimization of the original NLP problem, viz. the MSIMPSA optimizer. A completely automated procedure has been build around this paradigm, using the MatLab® symbolic toolbox
918 and Maple" kernel, with Fortran 77 coding to interface with the Fortran based MSIMPSA stochastic global optimizer. The results of the proposed approach, for a reactor-extractor-recycle system and an absorber-stripping system with heat integration, show that much more accurate and faster results may be obtained rather than by using a simuUaneous solution approach. This should be especially important when using stochastic algorithms for global optimization, due to their inherent large computational burden. Problems with a large number of decision variables could also benefit from the combinatorial optimization by simulated annealing, and this is currently under study. Acknowledgement- This work was partially supported by FCT-MCT (Funda9ao para a Ciencia e a Tecnologia - Ministerio da Ciencia e Tecnologia), under contracts PRAXIS XXI/BD/21481/99 and PRAXIS XX1/3/3.1/CEG/2641/95 employing the computational facilities of Institute for Systems and Robotics (ISR)-Porto.
5. References Biegler, L.T., Grossman, I.E., Westerberg, A.W., 1997, Systematic Methods of Chemical Process Design, Prentice-Hall, NJ. Cardoso, M.F., Salcedo R.L., Azevedo S.F. and D. Barbosa, 1997, A Simulated Annealing Approach to the Solution of MINLP Problems, Comput. Chem. Eng., 21, 12, 1349. Cardoso, M.F., Salcedo R.L., Azevedo S.F. and D. Barbosa, 2000, Optimization of Reactive Distillation Process with Simulated Annealing, Chem. Eng. Sci., 55, 5059. Ferreira, E.C. and R. Salcedo, 2001, A Tool to Optimize VOC Removal During Absorption and Stripping, Chem. Eng, 108, 1, 94. Grossmann, I.E. and Biegler, L. T, 1995, Optimizing Chemical Processes, CHEMTECH, p. 27. Grossmann, I.E., Daichent, M.M., 1994, New Trends in Optimization-Based Approaches to Process Synthesis, 5th Int. Symp. on PSE, En Sup Yoon, Ed., Kyongju, Korea, p.95. Ledet, W.D. and D.M. Himmelblau, 1970, Decomposition Procedures for Solving Large Scale Systems, Adv. Chem. Eng., 8, 186. Lee, E. S., Optimization of Complex Chemical Plants by a Gradient Technique, 1969, AIChE J., 15,3,393. Lin, T.D. and R.S.H. Mah, 1978, A Sparse Computation System for Process Design and Simulation, Part I. Data Structures and Processing Techniques, AIChE J., 24, 830. Motard, R.L., Shacham, M. and E.M. Rose, 1975, Steady State Chemical Process Simulation, AIChE Journal, 21, 3, 417. Press, W.H., Flannery, B. P., Teukolsky, S. A., Veterling, W. T., 1992, Numerical Recipes in Fortran 77, The Art of Scientific Computing, Cambridge University Press, New York, vol.1. Salcedo, R., 1989, Application of Random Search to the Optimization of Complex Systems, ChQmpofS9,24c]-24clO. Salcedo, R.L. and R. Lima, On the Optimum Choice of Decision Variables for Equation-Oriented Global Optimization, 1999, Ind. Eng. Chem. Res., 38, 4742. Salcedo, R.L., Solving Nonconvex Non-linear Programming and Mixed-Integer Non-linear Programming Problems with Adaptive Random Search, 1992, Ind. Eng. Chem. Res., 31,1, 262. Tolsma, J.E. and P.I. Barton, 2000, DAEPACK: An Open Modeling Environment for Legacy Models, Ind. Eng. Chem. Res., 39, 1826. Umeda, T., 1969, Optimal Design of an Absorber-Stripper System, Ind. Eng. Chem. Proc. D.D., 8, 3, 308. Zitney, S.E. and M.A. Stadtherr, 1988, Computational Experiments in Equation-Based Chemical Process Flowsheeting, Comput. Chem. Eng., 12, 12.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
919
Nonlinear Analysis of gPROMS Models Using DIVA via a CAPE ESO Interface M. Mangold*, K.D. Mohl", S. Gruner", A. Kienle*, E.D. Gilles*** Max-Planck-Institut fiir Dynamik komplexer technischer Systeme, SandtorstraBe 1, D-39106 Magdeburg, Germany Email: {mangold,kienle,gilles} @mpi-magdeburg.mpg.de Universitat Stuttgart, Institut fiir Systemdynamik und Regelungstechnik, Pfaffenwaldring 9, 70569 Stuttgart, Germany Email: {mohl,gruener}@isr.uni-stuttgart.de
Abstract The CAPE ESO interface of the process simulator gPROMS is used to pass model information to numerical methods contained in the simulation environment DIVA. By the interface, algorithms for the continuation of stable and unstable steady state and periodic solutions can be applied directly to gPROMS models. The use of the interface is illustrated by a detailed nonlinear model of an industrial reactive distillation column.
1. Introduction In academia, numerical bifurcation analysis has been widely accepted as a useful tool for the investigation of chemical processes. However, although nonlinear effects are frequently encountered especially in reactive chemical processes, methods for bifurcation analysis are still rarely used in industry. One reason may be that in the past most packages for bifurcation analysis were tailored to small systems of ordinary differential equations and hardly applicable to realistic models of industrial processes. To overcome this difficulty, in the last years a set of methods for nonlinear analysis was implemented in the simulation environment DIVA (Kienle, 1995; Mangold, 2000). The methods can cope efficiently with high order differential algebraic models of chemical plants. The methods comprise algorithms for the one parameter continuation and stability analysis of steady states and periodic solutions, and for the continuation of codimension-zero singularities in two parameters. Usually, models in DIVA are formulated in a proprietary modelling language. In order to make the architecture more open, now an interface to CAPE equation set objects (ESOs) (CAPE-OPEN, 1999) has been implemented, as they are provided by other simulation tools like gPROMS (PSE, 2000). It is possible to apply the bifurcation analysis and the other numerical methods available in DIVA to gPROMS models via this interface. The primary motivation to realize the interface comes from a joint research project between several research institutes and an industrial partner on the subject of nonlinear dynamics in reactive systems. The interface is intended to facilitate the exchange of modern numerical methods on one hand and applications on the other hand. Experiences with the interface will be reported in this contribution, which is structured as follows. In the next section, a brief introduction to numerical bifurcation analysis will be given. The methods available in DIVA will be summarized. Then the communication between gPROMS and DIVA via the ESO interface will be discussed. Finally, the use of the interface will
920 be illustrated by the example of an industrial reactive distillation column. First, the behaviour in the vicinity of the nominal operation point is investigated. The size of the operation window with respect to heat duty is analysed by one-parameter continuation. In the second step, a different set of operation conditions is used to demonstrate the nonlinear effects which may be encountered in a reactive distillation column.
2. Numerical Bifurcation Analysis in DIVA The simulation environment DIVA is a software tool for dynamic flowsheet simulation of chemical processes which has been developed at the University of Stuttgart (Mohl et al., 1997). The plant model is formulated as a linearly implicit system of differential algebraic equations of the type B(x,p)x
= f(x,ph
(1)
where x represents the state vector, p represents the parameter vector, B is a (possibly singular) left-hand-side matrix, and f is a right-hand-side vector. In addition to methods for steady state and dynamic simulation, optimization, and parameter estimation, DIVA contains a package for numerical bifurcation analysis. The package comprises algorithms for the one-parameter continuation of steady states and periodic solutions as well as for the two-parameter continuation of saddle-node and Hopf bifurcations. The numerics have been tailored to systems of high dynamical order. For a detailed description of the numerical methods the interested reader is referred to (Kienle et al., 1995; Mangold et al., 2000). In the following, only a brief idea of the methods will be given. The central element of the bifurcation package is a continuation algorithm used to trace the solution curve of an under-determined system of algebraic equations g(y)
= 0,
^ER'",JGR"^^
(2)
in an (m+7) dimensional space. A predictor-corrector algorithm with local parameterisation and step-size control is used. A simple application of the continuation algorithm is the computation of stable and unstable steady state solutions as a function of some distinguished model parameter p. In this case, g is the right-hand side vector/ of the model equations (1), andj consists of the state vector x and the parameter p. An eigenvalue monitor is used to determine the stability of the computed steady states and to detect singular points where one or several eigenvalues cross the imaginary axis. The singularities most frequently encountered in physical systems are saddle-node bifurcations (coincidence of two solutions) and Hopf bifurcations (stability change of steady state solutions and generation of periodic solutions). From bifurcation theory, necessary conditions for the singular points can be derived. Together with the steady state equations of the model they form an augmented equation system for the direct computation of the state vector x and the parameter;? at a singular point. The augmented equation systems are generated automatically by DIVA. In the framework of the continuation algorithm, they are used to trace the curves of singular points in two parameters. The resulting curves form the boundary of regions of qualitatively different behaviour in the parameter space. A further application of the continuation algorithm is the continuation of stable and unstable periodic solutions in one parameter. For that
921 purpose, the continuation algorithm is combined with a shooting method adapted to the special demands of high-order systems.
3. The CAPE ESO interface between gPROMS and DIVA 3.1 Architecture The CAPE-OPEN project (CO-LaN, 2001) proposes a standardized interface between a flow sheet simulator and an external (open) solver. The flowsheet simulator is to provide model information in the form of a so-called Equation Set Object (ESO) whose specifications were defined in (CAPE-OPEN, 1999). gPROMS 1.8.0 (PSE, 2000) is able to create such an ESO and is used as the simulator in this study. DIVA serves as the external solver. It sets the variables of the gPROMS model and receives information on residuals and Jacobians by the ESO interface. DIVA and gPROMS run as separate processes. The inter-process communication is handled by the object request broker OmniORB 3 by AT&T (AT&T, 2000). 3.2 Comparison of Computation Times The test example for the interface is the model of an industrial reactive distillation column which will be discussed in the next section. In the gPROMS formulation, the model consists of 250 differential and 5207 algebraic equations. The model size can be considered as typical for a realistic process model. In order to test the efficiency of the ESO interface, the model was also implemented in DIVA. Test runs were done on a 296 MHz Sun Enterprise under SunOS 5.6, and on a 1 GHz AMD Athlon PC under SuSE Linux 7.0. A computation for the bifurcation diagram in Fig. 2 takes 7.6 s of CPU time on the PC, if the model is implemented directly in DIVA, i.e. if the ESO interface is not used. The same computation takes 20.3 CPU s on the Sun. For the test of the ESO interface, gPROMS runs on the Sun and provides the residuals and Jacobians of the model under consideration. DIVA runs on the PC and carries out the nonlinear analysis. In this case, the same computation takes 10.5 CPU s on the PC plus 45 CPU s on the Sun. Obviously, the communication of the interface creates an overhead of computation time. However, in terms of time required to obtain computation results, the overhead seems still tolerable and is certainly preferable to the time consuming and error prone re-implementation of a complex model in a new software environment.
4. An Illustrative Example The use of the interface between DIVA and gPROMS will be illustrated by the example of an industrial reactive distillation column schematically shown in Fig. 1 (Fernholz et al., 2001; Grliner et al., 2001). The column consists of 63 trays including the reboiler and the condenser. The liquid feed contains the components B and C from which the product E is produced. In the liquid phase, two reactions take place whose simplified reaction scheme is given in Fig. 1. The lower boiling component A is removed as the top product. Its purity is specified in terms of a threshold for the top key component, the impurity C. The main product E is removed from the column as the bottom product. Its purity is determined by the bottom key component D. The model of the column consists of component material balances, quasi-static total material balances, and quasi-static energy balances for the trays. A constant volume hold-up and vapour-liquid equilibrium are assumed. Further details on the model can be found in (Gruner et al., 2001). As a first application example of continuation methods, the steady state behaviour of the column
922 top product: A
Vi
top key c o m p o n e n t : C
2 3 B,CL^ liquid feed
C h e m i c a l Reactions:
25
A+E r^ B+D A+D r^ B+C
k-1 k k+1
NS-1
^ X T s ^M ^XS re boiler
bottom product: E bottom key c o m p o n e n t : D
Q
Fig. 1: Scheme of the reactive column is studied in the vicinity of the nominal operation conditions. The bifurcation diagrams in Fig. 2 show steady state solutions as a function of the heat duty, which is one of the principal operating parameters. The left-hand-side diagram contains the steady state concentration of the key component C at the top of the column. The right-hand-side diagram shows concentration of the key component D at the bottom of the column. The horizontal dashed lines denote thresholds below which the specifications of the column are met. The vertical dashed lines indicate the nominal operation value of the heat duty. Obviously, in-spec operation of the plant is only possible in a very narrow window indicated by a grey band in Fig. 2. High performance controllers are required to keep the plant within this small operation window as shown in (Griiner et al., 2001; Fernholz etal.,2001). - -
top key comp. 11 specification f
— - -
top key comp specification
0.25
0.2 o E o 0.1
0.05
10
^,
20 30 heat duty
40
50
10
20 30 heat duty
40
Fig. 2: Steady state solutions as a function of the heat duty. Left diagram: concentration of C at column top; right diagram: concentration of D at column bottom.; grey band: operation window. The steady states in Fig. 2 are always unique and stable. However, instabilities and multiplicities of steady states are well-known patterns of behaviour in reactive distillation columns (e.g. (Mohl et al., 1999; Mohl et al., 2001). A second mode of operation is chosen to illustrate that the reactive distillation column studied here can
923 also show more complicated nonlinear behaviour. In contrast to the nominal operation conditions, the components A and E are now fed to the column. The distillate mass flow is chosen as the bifurcation parameter. The resulting bifurcation diagram is shown in Fig. 3. Multiple steady states exist in a rather large parameter region. Multiplicities of three and five steady states are found. Since the column is operated close to reaction equilibrium, the nonlinearities may be attributed to the interactions between phase equilibrium and reaction equilibrium. In literature, also the nonlinear relationship between mass and molar flow rates has been reported as a potential source of multiplicities (GUttinger and Morari, 1999). However, further investigations show, that they are not relevant for the nonlinearities in Fig. 3. The same qualitative behaviour can also be reproduced, if a constant molar volume for the liquid mixture is assumed. In addition to the multiple steady state solutions, a branch of periodic solutions is found. The periodic branch emanates from a subcritical Hopf bifurcation point, i.e. it is initially unstable. This means, that crossing the Hopf bifurcation by an increase of the distillate mass flow results in a sudden burst of periodic oscillations with large amplitudes. The branch of periodic solutions ends in a homoclinic orbit, which connects the stable and the unstable manifold of the saddle point marked "SP" in Fig. 3. Once the bifurcation diagram has been created, the existence of multiplicities can be easily verified by a dynamic simulation in gPROMS. For that purpose, the distillate mass flow is increased in a step-wise manner (see Fig. 4). Subsequently, it is reduced to the original value in the same steps. The different values of the mass flow applied in the dynamic simulation are indicated as vertical dotted lines in Fig. 3. The numbered labels in Fig. 3 and Fig. 4 denote corresponding solutions found by continuation (Fig. 3) and by dynamic simulation (Fig. 4). The dynamic simulation confirms for example the coexistence of the stable steady states "1" and "7" as well as "3" and "5", and the coexistence of the stable steady state "2" and the stable periodic solution "6".
O0.2I E
•
4000
4100
4200
4300
^
- stable steady state - - unstable steacty state • saddle-node bifurcation • HopI bifurcation • stable periodic orbit o unstable periodic orbit
4400 4500 4600 distillate mass flow
4700
4800
4900
Fig. 3: Bifurcation diagram of the industri- Fig. 4: Verification of the stable solutions al reactive distillation column created in in Fig. 3 by dynamic simulation in DIVA gPROMS
5. Conclusions The nonlinear nature of many chemical engineering processes requires adequate numerical methods for its analysis. The simulation tool DIVA offers a package of continuation methods for that purpose. In order to make the methods available to other simulators, an interface to CAPE-ESOs has been implemented in DIVA. The interface
924 has been used to apply bifurcation analysis to the model of an industrial reactive distillation column formulated in gPROMS. The example shows that the interface is efficient enough to analyse realistic process models of high complexity within reasonable time, even if the DIVA and the gPROMS process run on different computers. The additional computational burden caused by the inter process communication is acceptable and certainly negligible compared to the effort necessary to implement a detailed chemical process model on a different simulation platform. Furthermore, the interface proves to be an efficient way to transfer new numerical tools from academia to applications in industry. In a common project, academic as well as industrial partners can draw benefits from such an interface. The industrial partner can test new methods with comparatively low efforts. The frequently met obstacle of having to re-implement a model and get familiar with a new software tool is minimised. The industrial partner can bring his application examples into the co-operation without having to reveal all the details about his model like physical property correlations. For the academic partner, on the other hand, it is easier to get interesting application examples for his tools and he gets some feedback on the usability of the developed methods.
6. Acknowledgements The authors thank Bayer AG, Process Technology, Process Control, for providing the model of the reactive distillation column, and the group of Prof Marquardt, RTWH Aachen, for their support during the implementation of the ESO interface in DIVA. The financial support by the German Bundesministerium fur Bildung und Forschung (contract no. 03C0268B) is gratefully acknowledged. CAPE-OPEN was funded by the European Community under the Industrial and Materials Technologies Programme (Brite EuRam III), under contracts BRPR CT96-0293.
7. References AT&T , 2000, OmniORB - Free High Performance CORBA 2 ORB. http://www.uk.research.aU.com/omniORB CAPE-OPEN, 1999, Open interface specification numerical solvers. Ref Nr. CONUMR-EL-03, http://www.global-cape-open.org CO-LaN, 2001, CAPE-OPEN Laboratory Network web site, http://www.colan.org Femholz, G., M. Friedrich, S. Gruner, K.D. Mohl, A. Kienle, E.D. Gilles, 2001, Linear MIMO controller design for an industrial reactive distillation column. Presented at DYCOPS'2001, June 3-6, 2001, Chejudo, Korea. GrUner, S., K.D. Mohl, A. Kienle, E.D. Gilles, G. Femholz, M. Friedrich, 2001, Nonlinear control of an industrial reactive distillation column. Accepted for publication in Control Engineering Practice Guttinger, T.E., M. Morari, 1999, Predicting multiple steady states in distillation: singularity analysis and reactive systems, Ind. Eng. Chem. Res., 38, 1633 Kienle, A., G. Lauschke, V. Gehrke, E.D. Gilles, 1995, On the dynamics of the circulation loop reactor - numerical methods and analysis. Chem. Eng. Sci. (50), 2361 Mangold, M., K.D. Mohl, A. Kienle, E.D. Gilles, 2000, Nonlinear computation DIVA -methods and applications. Chem. Eng. Sci. (55), 441 Mohl, K.D., A. Spieker, R. Kdhler, E.D. Gilles, M. Zeitz, 1997, DIVA - a simulation environment for chemical engineering applications, in : Informatics, Cybernetics, and Computer Science (ICCS-97), Donetsk, Ukraine Mohl, K.D., A. Kienle, E.D. Gilles, P. Rapmund, K. Sundmacher, U. Hoffmann, 1999, Steady state multiplicities in reactive distillation columns. Chem.Eng. Sci. (54), 1029 Mohl, K.D., A. Kienle, K. Sundmacher, E.D. Gilles, 2001, A theoretical study of kinetic instabilities in catalytic distillation. Chem. Engng. Sci. (56), 5239 PSE, 2000, gPROMS User Guide, Process Systems Enterprise Ltd., London
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
925
Model Transformations in Multi Scale Modelling S. McGahey and I. Cameron CAPE Centre, Department of Chemical Engineering The University of Queensland, Australia 4072
Abstract Computer Aided Process Engineering (CAPE) requires computer based process models for most of its applications. Many of these applications require different attributes (accuracy, speed of solution, etc) of their models. For example, models that run in realtime sacrifice accuracy in exchange for speed, while offline applications do not need to be so concerned about the time spent in solving their models. Therefore it comes as no surprise that a number of models may exist to describe a process, v^ith each model having certain attributes that qualify it for a certain CAPE application. Such a collection of models is termed a Model Family A formalised method for the semi-automated creation of new members of a model family from existing members has been created, and is outlined in this document as six operations that manipulate the boundary-volume / connection structure of a model. Two of these operations, Merging and Demerging are believed to be new additions to the set of previously non-formalised meta-modelling operations; Aggregation, Disaggregation, Addition and Neglection. The major contribution of these operations is that they generate certain modelling information automatically when multiple boundary volumes are merged into a new boundary volume and the original boundary volumes lose their identity, reducing the amount of information required from the modeller.
1. Introduction Model development is an iterative process, in which many models are derived, tested and built upon until a model fitting the desired criteria is built. Subsequent modelling work may need to begin the search at the same place as the original model building began, rather than where it finished. This may be for a number of reasons, a common one being that the model currently in service is poorly suited for reuse because it is poorly understood (Foss et.al 1998). Previous work has tried to make models more understandable in a number of ways. For example, MODKIT (Bogusch et al. 1996) creates a record of the decisions that were made during the modelling process and MODEL.LA (Stephanopoulos et al. 1990) utilises a Process Modelling Language (PML) (Marquardt 1996) that is rich in process engineering, rather than mathematical, terminology. This work makes extensive use of a set of Process Modelling Objects (PMOs) from the PML "SCHEMA" (System for CHemical Engineering Model Adaptation) developed by Williams et al. (2001), to create an ordered object oriented representation, called a model description of a process model.
926 This contribution illustrates the use of the SCHEMA PMOs to develop two Meta Modelling concepts which give Process Engineering (PE) modellers added power in model building - especially for novel systems which will require a lot of model based work. The first concept is that a group of models describing the same process system are related, and form a group called a model family, which is described in Section 2. The second concept is that the relationships between members in such a model family can be described by and exploited with a set of model transformations, covered in Section 3.
2. Model Families Section 1 introduced the idea that a number of PE models may exist to describe the same system. Such a collection of models is termed a Model Family. Work by Aris (1994) and Rice and Do (1995) notes the existence of such related model sets, however it is not until Foss et al. (1998) and Bogusch et al. (1996) that explicit references to the information contained in these groups are made. In extension to their ideas of retaining model version information, the model families described here relate all of the models describing a particular system, including those developed in different modelling projects. Ongoing work by Williams et al. (2001) is exploring the viability of these model families for recording the various modes contained in models of hybrid systems. The families are constructed as follows. An initial model of a system, preferably one with a low level of detail, is presented as a starting point. New members of this model family are then derived from this model by performing appropriate model transformations (Section 3), as illustrated by the arrows between Models a, b, c and d in Figure 1. Each member of the model family needs to be a completely valid model, else the model family will quickly become crowded and meaningless due to the many models which cannot be used or related to by the user. In this figure, the model members are arranged in terms of the number of Boundary Volumes that each system contains, to show their relative structural complexity and provide some method of categorising the different family members. Models a, b and d have two Boundary Volumes while Model c has three Boundary Volumes.
2 BVs
/
Demerging
3 BVs Figure 1: Diagram of a model family arranged in terms of structural complexity
927 The resulting arrangement of models and transformations provides an indication of how the various models are related to each other, supplementing any documentation present in the individual models. This extra information can assist a modeller to choose the most appropriate model for further development by detailing what models are already in existence, and how they differ.
3. Model Transformations In most PE Computer Aided Modelling (CAM) systems, a simple PE modelling task such as assuming a Boundary Volume is negligible, or merging together a pair of Boundary Volumes, would require a long list of simpler tasks to be carried out. This is because the tools which have been designed for the various modelling tasks are built around existing CAM concepts, rather than being reinvented for the PE domain. Fischer and Lemke (1987) prove that by making a modelling package relevant to the domain in which it is used vastly improves the efficiency of the tasks performed with it. Therefore the introduction of the core modelling manipulations used in PE modelling as basic tools in a modelling package should drastically improve the efficiency of modellers using it. Recent work by the authors has investigated the typical modifications that are made to PE models during model development, and described them as a set of Meta Modelling operations, termed Model Transformations. This work aims to give these tools an accepted identity and form, and then implement them in the SCHEMA system mentioned in Section 1. Two transformations that may not have been noticed previously, Merging and Demerging (Sections 3.5 and 3.6) are brought to light in addition to the more obvious manipulations of Aggregation, Disaggregation, Addition and Neglection. The SCHEMA PMOs categorise model information in a very useful manner for these transformations, as will be seen in the following subsections. For the purposes of this discussion, they can be further divided up into two sets - the Structural Modelling Objects (SMOs, comprised of Models, Boundary Volumes and Connections) which describe the connectivity of the various elements in a described system, and the Internal Modelling Objects (IMOs, comprised of all that is not a SMO), which describe the internal elements of the SMOs. It is the introduction of these IMOs which sets this modelling system apart from other object oriented modelling languages which utilise concepts similar to the devices and connections of Marquardt (1996). Six set theory algorithms describe the SMO and IMO manipulations of these transformations, however space limitations mean that they can only be described briefly here, along with a few guidelines. The first two transformations require no explanation in terms of IMOs, and are nearly completely described by Figure 2. Addition and Neglection differ from these only in that they can also affect Assumption objects. Merging and Demerging involve intense IMO manipulations, and have a number of interesting issues which need to be considered when they are performed. 3.1 Aggregation Aggregation groups a set of Boundary Volumes and/or Models together, and encapsulates them inside a new Model. This is a very simple operation which involves
928 manipulation of the SMOs of the selected group of objects, and the creation of a new Model object. Figure 2a illustrates a set of Boundary Volume (labelled Vessel and Jacket) and Connection (unlabelled) objects before aggregation, and Figure 2b shows the same objects after they have been aggregated into a Model object labelled *'CSTR". All objects undergoing aggregation transformations retain their individual identities.
CSTR Vessel Vessel <—> Wall 4 — •
,
T
1
1 _.._.
i
.._..j J (c)
CSTR
Vessel <—>{
Jacket
±
Merged Heat Transfer Boundary Volume ^
^
(d) Figure 2 : Structural Modelling Objects as (a) individual entities, as (b) an aggregated group of entities, as (c) a group with an additional member, and (d) a group with a merged member 3.2 Disaggregation Disaggregation is the reverse of aggregation. This involves removal of the encapsulating Model object and replacing it with its contents. This can be visualised by taking the set of objects illustrated in Figure 2b, and then removing them from the Model object "CSTR", which is then disposed of and then reintroducing them as individual objects into the object which originally contained the "CSTR" Model. The resultant set of model objects are represented by Figure 2a. Once again, all modelling objects retain their individual identities when they are removed from their Model group. 3.3 Addition Addition is the operation of adding either a SMO or an assumption to a suitable object. An example of this operation on SMOs is the alteration of the object set shown in Figure 2b to that shown in Figure 2c. The modeller in this case has decided to enrich the model by adding information about a wall which separates the vessel and the jacket in the CSTR model. A second example of an additive operation is when a new assumption such as "vessel-is-isothermal" is applied to the vessel object of a CSTR. SCHEMA would then check that these additions to the model do not cause
929 inconsistency problems, such as conflicting assumptions or violations of model limitations. 3.4 Neglection Similar to Addition transformations, Neglection operations can involve SMOs or assumptions. A SMO neglection operation can be seen in the movement from Figure 2c to Figure 2b. In this example, a wall Boundary Volume is removed from the model, resulting in a model simplification. Neglection of assumptions is also possible. This illustrates the difference between transformation operations on Models, versus operations on Model Descriptions. If the operations were on a Model, then Neglection should always result in a simplified model, however neglecting a simplifying assumption, such as 'Vessel-is-isothermal" can result in a more complex model. The SCHEMA system deals with model descriptions. 3.5 Merging In this operation. Boundary Volume and/or Model objects lose their individual identities, and are merged into a new Boundary Volume object which attempts to describe the original objects. This results in a reduction in information richness. As an example, consider the set of SMOs illustrated in Figure 2c. If the modeller is only interested in the events in the vessel, but wants to consider the effects of the heat transfer to the jacket through the vessel wall, then they might ignore all non-energy events outside the vessel Boundary Volume. Merging of the Wall and Jacket Boundary Volumes into one large pseudo-balance, and only considering the heat flows in this pseudo-balance's interactions with its environment will result in the object set shown in Figure 2d. This has a similar effect conceptually to assuming thermal equilibrium between the wall and the jacket, except that index problems are avoided. In this transformation operation, the objects involved in the merger lose their individual identities to create a new single Boundary Volume object. This operation is not always reversible due to information loss. The automated construction of the IMOs and SMOs of this object from the IMOs and SMOs of the original merging Boundary Volumes and/or Models is the subject of continuing work. Some typical, non algorithmic rules follow: • Merging of Boundary Volumes which contain different phases will require the creation of a Pseudo-phase. All remaining connections will assume the new pseudo phase to be that phase they were connected to before merging for the purposes of determining appropriate transfer mechanisms. • Connections which have one end external to the Merging process must be incorporated into the new Balance Volume, with appropriate provisions made for them in the IMOs of the new Boundary Volume. • Connections which are anchored at both ends to Boundary Volumes or Models which are selected for Merging cease to exist after the Merging process. 3.6 Demerging This operation changes a Boundary Volume into a Model Object which contains other newly-defined Boundary Volume objects, resulting in a general increase in information
930 richness. This can be observed in changing the objects in Figure 2d into those in Figure 2c. Such a transformation can be compared to the addition transformation, with the consideration of a few algorithmic and conceptual guidelines, some of which follow: • The SMOs of the new Boundary Volume set must account for the SMOs of the original Boundary Volume - all connections leaving the transformation area must be accounted for in the new set of Boundary Volumes. • The overall effect of the new Boundary Volume set should have some similarity to the effects created by the IMOs of the original Boundary Volume.
4. Conclusions and Recommendations This work has highlighted the concept of model transformations and illustrated their utility as basic modelling tools. Two novel transformations, merging and demerging have been defined to further empower these model transformations. When developing models of differing levels (scales) of detail or dimension, these transformations serve to save a lot of time and effort by carrying over and/or modifying relevant information from one model to another. The use of a highly conceptual system of modelling objects (in SCHEMA) leads to models that are information rich and useful in advanced modelling operations. The use of model families aids in the understanding and development of models and makes use of already existing information which is normally wasted. This provides a record of the model development as well as a collection of models which can be drawn upon should future development be required.
Bibliography Aris, R., 1994, Mathematical Modelling Techniques, Dover Publications, Inc., New York Bogusch, R., Lohmann, B. and Marquardt, W., 1996, Computer-Aided Process Modeling with ModKit, Technical Report #8, RWTH Aachen University of Technology Fischer, G. and Lemke, A.C., 1987, Construction Kits and Design Environments: Steps Toward Human Problem-Domain Communication, Human-Computer Interaction, 3, 179-222 Foss, B.A., Lohmann, B. and Marquardt, W., 1998, A Field Study of the Industrial Modeling Process, Journal of Process Control, 8, 5-6, p325-338 Marquardt W., 1996, Trends in Computer Aided Process Modeling, Computers and Chemical Engineering, 20, 6/7, pp591-609 Rice, R.G. and Do, D.D., 1995, Applied Mathematics and Modeling for Chemical Engineers, John Wiley and Sons, Inc., New York. Stephanopoulos, G., Henning, G. and Leone, H., 1990, Model.LA A Modeling Language for Process Engineering - 1 . The Formal Framework, Computers and Chemical Engineering, 14, 813-846 Williams, R., Hangos, K.M., McGahey, S. and Cameron, I.T., 2001, Assumption Based Modelling and Model Documentation, 6th World Congress of Chemical Engineering, Melbourne
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
^^ 1
Application of Hybrid Models in Chemical Industry G. Mogk, Th. Mrziglod, A. Schuppert Bayer AG, Modelling & Methods, Leverkusen, Germany {georg.mogk.gm, thomas.mrziglod.tm, andreas.schuppert.as} @bayer-ag.de
Abstract The power of hybrid models as a combination of rigorous models and artificial neural networks (ANNs) was shown in several applications in different domains. This new technique is utilised in the area of chemical product development, process design and marketing applications for different demands. During a project in the last three years at Bayer hybrid modelling was advanced to a standard technique. This new modelling technique is completely integrated in the existing modelling software infrastructure for data based and rigorous models. In this paper an overview of the theory of hybrid modelling and the software implementation is given. The capabilities of hybrid models will be demonstrated on industrial application examples.
1. Introduction Model-based process control, simulation and optimisation are widely accepted to be key technologies for process improvement. In the wide area of fme chemicals as well as in biotechnology, however, the desired broad application of model-based process improvement, has not been performed yet. Although highly desirable closing the gaps in quantitative process knowledge for all subprocesses, and thereby allowing classical modelling techniques to be applied, is often not affordable. For more than 10 years Artificial Neural Networks (ANN) have been successfully applied in various application areas at Bayer. Serious problems in the practical use of ANN are for instance the lack of extrapolation capability and the problem to integrate scientific information. A significant improvement is achieved by the additional use of a-priori-known structural information about the process (see [2]). The central idea is that structural information is used to reduce the solution manifold of the system. This approach shall herein be called Structured Hybrid Modelling. A structured hybrid model (SHM) therefore consists of three components: • rigorous submodels describing the i/o (input/output)-relation of those subprocesses which are well understood • black-box submodels for those subprocesses where no rigorous model is available • a model flowsheet describing the i/o-structure of all the submodels, joining them together by mapping the i/o-structure of the real process. The new component with respect to the earlier, „simple'' hybrid modelling methods (e.g [1]) is the explicit use of the structure of the process in the generation of the model flowsheet. In theoretical and practical investigations since 1995 (cited in [2]) it has been
932 shown that by the use of hybrid models the extrapolation behaviour can be improved dramatically. During a project in the last three years at Bayer hybrid modelling was advanced to a standard technique. This new modelling technique is completely integrated in the existing modelling software infrastructure for data based and rigorous models. Furthermore new numerical strategies for fitting hybrid models are developed. As the mathematical foundations have been described elsewhere [2], in this paper we will focus our attention on a sketch of our software platform as well as on the demonstration of some real life application examples.
2. Theory of Hybrid Models The complexity of black box models increases in principle exponentially with the number of input variables of the model (curse of dimensionality). The incorporation of the model flowsheet, however, allows a splitting of one black box model for the entire process into separate, small submodels for parts of the process reducing the number of input variables to each black-box submodel significantly. We will demonstrate this superior quality of SHMs on correlated input data with the following artificial example. Let z : R^ —> R be a function with the known structure z(x, y) = u{x) + v(>;) -f- a -uix) • v(y)
(1)
where the functions u(x),v(y) and the parameter a are unknown. For a concrete situation we generated data with the settings u(x) = x^,
v(3;) = sin(-->;), a = 0.614
and xe[-l\l\
y = x-^y'X,
where Xe [-1;1J is a uniform distributed random number. For the case 7= 0.1 and 100 data points (figure la) we identified the function z in two different ways. Using the hybrid structure we identified the function w, v as onedimensional ANNs simultaneously with the parameter a.. It is important to stress that we hereby used only the information on x, y and z. The model was identified with the Bayer internally developed hybrid modelling software described in the next section. Second we ignored the knowledge of the hybrid structure and identified z as ANN from X and y with a commercial neural network software. The resulting functions are shown together with the exact solution in figure Ib-d. Inside of the area with the training data (which is only the small diagonal area inside of the square [-1,1]^ illustrated in figure la) all functions are identical. Outside of this region only the hybrid model is able to reproduce the original function. As can be seen, the values of the ANNs outside of the data region are arbitrary without any relation to reality. The error of the ANNs outside of the data region is more than a hundred times larger than the error of the hybrid model. This simple example illustrates that hybrid models in contrast to ANN are able to extrapolate. In our example the extrapolation region is ten times larger than the data region! It is important to remark that it is only possible to identify u and v up to an arbitrary constant c.
933 1
0,5 >
I I I ><1
0
-0,5 -1 -1
-0,5
0
0,5
1
X
Figure la: Distribution of the input components of training data
Figure lb: Exact solution
Figure Ic: Hybrid model
Figure Id: Neural Network
3. The Bayer HybridTool During an internal project since 1999 we have developed a specific software environment called HybridTool for the standardised generation and the identification of hybrid models. From a technical point of view a hybrid model is a network of databased submodels like ANNs and known rigorous submodels which have to be coded in MatLab. Several algorithms for the efficient identification of the databased submodels and of unknown parameters in the rigorous models are available. This allows a problem oriented choice of the best identification strategy as a combination of partial iterative and simultaneous model fitting. Thereby an automatic structure optimisation of all internal ANNs can be performed. We attached great importance to the efficient handling of incomplete datasets. Thereby the network topology is used automatically to improve the number of utilisable datasets. HybridTool has a modular design as shown in Figure 2. The model generation step can be performed with the help of a graphical user interface. The user interface is connected
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^
export as shared library
3.^
Standard Ap^icatikm Tools ^jHredicdtm
»visualisatioii
Figure 2: Modular organisation of the Bayer HybridTool to our internal data preparation tool and a component for the model and version management. With the help of the data preparation tool arbitrary data sources are available. Actually, the model identification takes place within MatLab, which is triggered automatically from the user interface. The identified MatLab model can be exported in two different ways. For our internal standard application tool which can dynamically handle arbitrary ANNs and hybrid models, the model is converted into a shared library (DLL). Otherwise the MatLab model can be directly integrated into an application specific software environment, e.g. to use the model for quality (online) prediction or for optimisation purposes.
4. Applications The potential of hybrid models and the features of HybridTool was demonstrated in several application at the chemical industry. Three of them are presented in the following. 4.1 Continuous polymerisation As an example for an industrial application a continuous polymerisation plant will be described. The polymerisation of monomers and comonomers had been performed in an tube reactor with low residence time. A quantitative modelling of the reactor on a rigorous basis was not possible. Using the chemical knowledge about the reactive part of the process a SHM for the reactor could be established. It can be described shortly as follows: The melt index (M/), which had to be modelled, depends on the average chain length
935
*'Ml„
Figure 3: Structure of the hybrid model for the continuous polymerisation process
MI^{c)
{yiddf
W
Using additional chemical information about the structure of the reaction network we arrived at a further reduction of the complexity of the SHM for the reactor subprocess as depicted in figure 3. This model was used to fit the melt index of the polymerisation plant with respect to the input variables and was compared with a pure ANN. For the underlying real process, the ANN alone required about 2000 data sets to predict the melt index properly, whereas the SHM could be used for melt index-prediction after training on only 200 data sets. Moreover, the prediction quality of the SHM was about five percent higher than that of the neural network. Due to the extrapolation behaviour, the SHM allowed the analysis of the impact of monomer impurities on the melt index also in parameter region where no process data are available. 4.2 Quality management for polymer compounding Among other things the quality of polymers depends on the formulation, the colorants and the quality of the raw materials which often shows strong variations. The aim of this application is to compensate the influence of quality variations with the help of formulation variations. Thereby the actual colour formulation has to be taken into account. Since the influence mechanism is partially unknown the situation is ideally suited to hybrid modelling strategies. Furthermore the number of input parameters is greater than 100. Hence the limited data information available makes it impossible to identify a pure ANN. Using the known network structure allows to reduce the number of inputs into the most complex ANN submodel to less than 20 and it is possible to generate accurate models for different quality parameters which are able to extrapolate. 4.3 Metal hydride process development The most important quality parameters for a specific metal hydride product are the crystallite size and the median of the particle size distribution. Thereby each particle consists of many crystallites. Relevant influence parameters are the reactor temperature and the concentration of the ingredients. For the hybrid structure we use the fact that e.g. the crystallite size depends only on the total amount of solved metal and some metal
936
6
8
13
12
14
Component 1 fg/l]
16
18
20
4
6
8
13
12
14
16
18
20
Component 1 fg/11
Figure 4: Comparison of the valid extrapolation region of the hybrid model and an ANN complexes. Therefore the hybrid model allows the extrapolation into a wide temperature range and permits to locate economical process regions (see figure 4).
5, Conclusion In the present paper SHMs with a given order for the interaction of black- and whitebox submodels are presented. The model flowsheet describing the connections between the submodels maps the real structure of the process explicitly. With respect to blackbox or unstructured hybrid models containing only one big black-box submodel lumping all unknown subprocesses, the functional structure of the SHM exhibits significant advantages over black-box and rigorous models. The extrapolation capability of SHM allows its use in process analysis, on-line investigating for example the effect of impurities or other disturbances on the product quality. In case of any process disturbance, the operators will try to keep the product quality constant through changing appropriate control parameters. Therefore the values of the process disturbances and control parameters are closely correlated and a retrospective analysis of the process dependence on the disturbances is impossible using only blackbox models. Appropriate SHMs, however, allow the extrapolation from correlated data, thereby rendering the analysis of the relation between product quality and process disturbances possible.
References [1] [2]
H.A.B. Te Braake, H.J.L. van Can, H.B. Verbruggen, Semi-mechanistic modelling of chemical processes with neural networks, Eng. Appl. Art. Intell. 11, (1998), pp. 507-515. A. Schuppert, Extrapolability of structured hybrid models: a key to optimization of complex processes, in: Proceedings of EquaDiff '99, B.Fiedler, K.Groger, J.Sprekels Eds., (2000), pp. 1135-1151
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
937
Simulation of Two-Dimensional Dispersed Phase Systems Stefan Motz, Natalie Bender and Ernst Dieter Gilles Institute for System Dynamics and Control Technology Pfaffenwaldring 9, University of Stuttgart, 70550 Stuttgart, Germany
Abstract In this contribution, the recently published Method of Space-Time Conservation Element and Solution Element for the numerical solution of conservation laws is compared to state of the art Method of Lines based schemes. The influence of the numerical methods on the dynamic behaviour of one- and two-dimensional dispersed phase systems is investigated. Guidelines for a proper selection of numerical methods for the treatment of population balance models are given.
1. Introduction In chemical engineering dispersed phase systems occur in a large variety. They play an important role in many industrial processes, such like e.g. crystallization, granulation, polymerisation or liquid-liquid extraction. A common characteristic of those processes is, that one or more dispersed phases of e.g. crystals, bubbles or drops, are embedded in a continuous medium. A suitable and commonly accepted concept for the modelling of dispersed phase systems is the population balance approach (Ramkrishna, 2000), which considers the dispersed phase as a population of individual particles that are distinguished from each other using some characteristic particle properties. The application of this approach leads in general to partial integro-differential equations. In this contribution, numerical methods will be discussed for the simulation of multidimensional dispersed phase systems containing both continuous changes of the particle properties (e.g. particle growth or aging) and sources and sinks due to the breakage or aggregation of particles. In the following, the recently published Space-Time Conservation Element and Solution Element (CE/SE) Method (Chang et al., 1999), which was originally designed for CFD problems, will be extended for the treatment of partial integro-differential equations and compared with state of the art Method of Lines based schemes using standard fmite volumes (upwind method) or flux limiters (Koren, 1993; Schiesser 1991). The numerical methods will be applied to one- and twodimensional crystallization models considering one or two particle properties, respectively (Gerstlauer et al., 2001).
2. Application of the CE/SE Method for One-Dimensional Problems For a brief introduction into the CE/SE method, a rather simple population balance accounting only for particle growth will be considered. Using a characteristic crystal length L as the property coordinate, this population balance can be formulated for a number density function F using the crystal growth rate G as
938 dF{Lj) dt
^
d{G(Lj)F{Lj))^^ dL
(2.1)
The application of the CE/SE method requires the discrete treatment of both the particle property coordinate L and the time /. As depicted in Fig. 1, hAL—»H * 4^ the considered domain will be subdivided into Conservation Elements (CE), which are shifted against each other. Around those CEs, Solution CEJ Elements (SE) are arranged. They have to be considered as L infinitesimal small elements or Lo h^h as connected lines. Figure I: Subdivision of the (L,t)-domain. Besides the differential form (2.1) of this conservation law, also the integral form
1fdFjLj)
^ dL
dCE)
= 0
(2.2)
CE"
which has to be valid for each CE, is taken into consideration. On the SEs, the number density function F, and the flux term GF will be approximated by a first order Taylor expansion around the center points {j,n) of each SE, which are marked by dots in Fig. 1. By an integration over the boundary of each CE in a row, and applying Gauss' divergence theorem, an explicit time marching scheme on a fixed staggered mesh can be calculated (Chang et al., 1999). This results in a second order method, because there are two marching variables at each mesh point {j,n), one for the solution of F, another for the partial derivative of F with respect to L. The stability of this explicit method only depends on the local Courant number
i^i^
2.1 The treatment of integral terms using the CE/SE method In order to include integral sink or source terms, which may account for breakage or aggregation phenomena, to the population balance (2.1), the CE/SE framework can be extended (Chang et al., 1999). The required integrations over the particle population can be carried out very accurately using the above introduced Taylor expansion.
3. Comparison of the CE/SE Method with Method of Lines Schemes In this section, the CE/SE method will be compared to state of the art Method of Lines based schemes using finite volumes with simple upwind discretization or with a flux limiter (Koren, 1993; Schiesser 1991). The three methods will therefore first be applied
939 to solve the simple population balance (2.1) as a test problem. Afterwards, the influence of the different numerical methods on the dynamic behaviour of a continuous crystallizer will be investigated. 3.1 A one-dimensional test problem accounting for particle growth Initial profile i '»' i -'" „»r For a reasonable comparison of the fluxllmlted Ish finite volume different numerical methods, a growing d. T u^ I >. Bparticle population with a constant « 1 S 5h "O ' growth rate of G = lfim/s is consider1E ' , ed. Therefore, the numerical solution of ^^ g^ w : standard ^ \ the population balance (2.1) will be 2 Itnftevdurhe / , 1 obtained by discretizing the particle i i - . ..^-'^ ^ • ^^,. length coordinate L in the range from 1 crystal longth L [mym] to iOOjJm into 100 finite volumes, Figure 2: Simulation of Eq. (2.1) respectively solution elements. The avvhiriQ different numerical methods. growth behaviour of an initially equally distributed number density F during 60s is then computed applying the different methods using MatLab^ and can be seen in Fig. 2. Since it is possible to calculate an analytical solution of Eq. (2.1), which would be the movement of the initial profile without changing its shape, the quality and accuracy of the numerically calculated number densities can easily be estimated. Here, the application of the standard finite volume scheme using simple upwind discretization results in an extremely decreased and enlarged profile after 60s of constant growth. The reason for this behaviour is the numerical diffusion introduced by this method. With the flux limited finite volume scheme this high amount of numerical diffusion can be reduced considerably. This results in a less broadened, sharper profile. In contrast to the two Method of Lines based schemes, the profile computed using the CE/SE method does not differ from the exact solution. This follows from the fact, that this method doesn't introduce any numerical diffusion for the problem in Eq. (2.1) (Chang et al., 1999). x10'
1
111
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,
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1
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-
•
^
'
•
"
^
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'
'
'
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'
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;
3.2 A one-dimensional population balance model for continuous crystallizers In order to investigate the influence of the discussed numerical methods on the dynamic behaviour of a complex chemical process, a continuous crystallizer will be considered. The mathematical model describing this crystallization process includes very detailed microscopic models for crystal growth and attrition of crystals due to crystal-sfirrer collisions (Gersflauer et al., 2001). The simulation will be performed with an initially Gaussian distributed number density F, with an initial Lso of SlOjum. In order to compare the computed simulation results, the progression of the mass median crystal size L50 will be considered as a measure of the dynamic behaviour of the crystal population. As can be seen in Fig. 3, the computed dynamic behaviour of the crystallizer differs significantly depending on the chosen numerical method. The simulafion using the CE/SE method results in a stable periodic behaviour that is MatLab 5.3, The Math Works Inc., 24 Prime Park Way, Natick, MA 01760-1500, USA
940
time [h]
Figure 3: Simulation of a continuous crystallizer applying different numerical methods.
determined by growth and attrition of crystals, i.e. the production of fragments, whereas the application of both of the Method of Lines based schemes does lead to a steady state. These considerable differences in the qualitative process behaviour result from the rather high numerical diffusion introduced by the finite volume schemes, which ultimately cause the damping of the oscillations. Thus, the CE/SE method has been proven to be an applicable numerical method, compared to the other here discussed schemes, in order to simulate particulate processes in an accurate way.
4. Application of the CE/SE Method for Two-Dimensional Problems In this section, the CE/SE method will be applied to simulate a two-dimensional problem. The considered population model accounts again for a crystallization process and uses two independent particle properties, the characteristic crystal length L and a molar energy Wp accounting for the lattice strain or the plastic deformation of the crystals (Gerstlauer et al., 2001). This two-dimensional crystallization model enables a more detailed simulation of the crystal growth, because the growth rate G{L,Wp) depending on both particle properties allows a physical description of the experimentally often observed growth rate diffusion. The derivation of the numerical scheme in case of this two-dimensional problem is straightforward, using the above described CE/SE methodology. The CEs now simply consist of three-dimensional volume elements in the (L,Wp,r)-domain and the SEs are two-dimensional plains that demarcate adjacent CEs. Again, an explicit time marching scheme can be derived by an integration over the surface areas of the CEs using Gauss' divergence theorem. The resulting marching variables at each mesh point are then the solution of F, and values for the partial derivatives of F with respect to L and Wp . 4.1 A two-dimensional test problem accounting for crystal growth The efficiency of the two-dimensional CE/SE method is tested by solving a simple population balance with the two particle properties L and Wp dFjUwpj) dt
3(GF(L,wp,r)) dL
d[v^^F{Uwpj))
(4.1)
941 with constant internal velocities G = 5 10'^IJm/s population after 10mln
and
dwp
~dr
(growth rate) = -0.0\
J/{mols)
(change of molar lattice strain). Starting with a narrow Gaussian distribution around 40jMn and l.SJ/mol, Fig. 4 shows the resulting particle population after lOmin. For this calculation, the property domain {L,Wp) was discretized into 50x50 solution Figure 4: Simulation of Eq. (4.1) with the CE/SE method. points. As can be seen from Fig. 4, this simulation can be performed very accurately using the CE/SE method, without any noticeable numerical diffusion. The application of this method is very efficient. The required CPU time for the calculation shown in Fig. 4 was only 405- on a state of the art Pentium 3 computer. 4.2 Simulation of the growth behaviour of attrition fragments For the simulation of the growth behaviour of attrition fragments, a growth rate G{L,WP) depending on both particle properties will be applied. The relaxation of the stressed fragments that result from crystalstirrer collisions will be described by an empirical approach. Therefore it will be assumed that the absolute stress energy stored in the crystal lattice is proportional to the crystal surface area {-1}), for more details about this two-dimensional molar lattice strain w . (J/mol} crystalliza-tion model the reader is referred to (Gerstlauer et al., Figure 5: The simulated growth behaviour oj 2001). The investigated growth attrition fragments. behaviour starts with an initial fragment formation according to Rittinger's law. This hypothesis states the relationship 310 ^PJrag
J ml mol
(4.2)
^frag
between the length Lj.^g of an attrition fragment and its molar lattice strain Wpj.^g . The changes of this fragment formation during 15minof crystal growth can be seen in
942 Fig. 5. This simulation demonstrates the applicability of the CE/SE method to complex, high dimensional population models using more than one particle property. As in case of the above discussed two-dimensional test problem, the performance of this numerical method is also very efficient, since it provides very accurate numerical solutions, in terms of low numerical diffusion, with acceptable CPU times (-- 505 on a Pentium 3 computer for the simulation in Fig. 5).
5. Conclusions and Outlook The main focus of the work presented here is the identification of proper numerical methods for the simulation of higher-dimensional dispersed phase systems that are mathematically described by population balances using more than one particle property. The presented comparison of an extended CE/SE method with state of the art Method of Lines based schemes for a one-dimensional test problem shows the superiority of the CE/SE method in terms of more accurate solutions with less numerical diffusion and higher computational efficiency. In case of a two-dimensional population model, the CE/SE method can also be applied successfully. The presented simulation of a continuous crystallizer shows a strong dependence of the computed dynamic behaviour on the selected numerical methods. Due to the rather high amount on numerical diffusion introduced by the applied finite volume schemes, the simulations with the Method of Lines based schemes result in a steady state, instead of the periodic solution that is computed with the more accurate CE/SE method. The better computational efficiency and higher accuracy of the CE/SE method also allow the simulation of more complex crystal growth models accounting for the often observed growth rate diffusion. Another important point in this context is the model based identification of, especially, two-dimensional growth kinetics from measured data. In order to obtain kinetics that are not polluted by the applied numerical methods, the therefore required accurate simulations can be performed applying the CE/SE method.
6. References Chang S.-C, X.-Y. Wang and C.-Y. Chow, 1999, The Space-Time Conservation Element and Solution Element Method: A New High-Resolution and Genuinely Multidimensional Paradigm for Solving Conservation Laws, Journal of Computational Physics, Vol. 156. Gerstlauer A., A. Mitrovic, S. Motz and E.-D. Gilles, 2001, A population model for crystallization processes using two independent particle properties. Chemical Engineering Science, Vol. 56. Koren B., 1993, A robust upwind discretization method for advection, diffusion and source terms, In Numerical Methods for Advection-Diffusion Problems, Eds. C.B. Vreugdenhil and B. Koren. Ramkrishna D., 2000, Population Balances - Theory and Applications to Particulate Systems in Engineering, Academic Press. Schiesser W.E., 1991, The Numerical Method of Lines, Academic Press.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
943
Experimental Study and Advances in 3-D Simulation of Gas Flow in a Cyclone Using CFD A.P. Peres\ H.F. Meier^ W.K. Huziwara\ M. Mori^ ^ School of Chemical Engineering, UNICAMP, P.O. Box 6066, 13081-970, Campinas-SP, Brazil, E-mail: [email protected] ^ Department of Chemical Engineering, FURB, P.O. Box 1507, 89010-971, Blumenau-SC, Brazil. E-mail: [email protected] ^ PETROBRAS, CENPES, Rio de Janeiro-RJ, Brazil.
Abstract Experimental results and a 3-D simulation of gas flow in a cyclone are presented in this work. Inlet gas velocities of 11.0 m/s and 12.5 m/s and measurements of local pressures were used to determine radial distributions of the tangential velocity component at five axial positions throughout the equipment. The aim of this work was to analyze an anisotropic turbulence model, the Differential Stress Model (DSM). First and higher order interpolation schemes and a numerical strategy were used to assure stability and convergence of the numerical solutions carried out using the computational fluiddynamics code CFX 4.4. The models showed a satisfactory capability to predict fluid dynamics behavior since the calculated distribution of velocity components match the experimental results very well.
1. Introduction Cyclones, such as those in FCC units, have been used as solid particle separators in large-scale chemical processes, due to their low building and maintenance costs and the fact that they can be used under severe temperature and pressure conditions. The design of new cyclones and the analysis of the actual equipment can be achieved using computational fluid dynamics (CFD) techniques in order to obtain higher collection efficiency and a lower pressure drop. In our recent studies, Meier and Mori (1999) and Meier et al. (2000), the models analyzed were the standard k-e, RNG k-e and the Differential Stress Model (DSM) and it was observed that turbulence models based on the assumption of isotropy, such as standard k-e and RNG k-e, were inapplicable to the complex swirling flow in cyclones. On the other hand, whenever a turbulence model that considers the effect of the anisotropy of the Reynolds stress is used, such as DSM, an adequate interpolation scheme must also be considered for the prediction of flow in cyclones. In this work, a 3-D simulation of the turbulent gas flow in the cyclone was carried out using the computational fluid-dynamics code CFX 4.4 by AEA Technology. The numerical solutions were obtained with a fmite-volume method and body fitted grid generation aiming at the analysis of an anisotropic turbulence model (DSM) using first and higher order interpolation schemes. The numerical strategy adopted assured stability and convergence of the numerical solutions.
944
2. Mathematical Modeling The time-averaged mathematical models along with the Reynolds decomposition governing mass and momentum transfers can be written as follows: V. (pv) = 0
at
a(pv)
-
^
—
(1)
_
(2)
+ V.(pvv) = pg + V. ( a - p v V )
The last term of Equation (2), pv'v', is a time-averaged dyadic product of velocity fluctuations and is called Reynolds stress or turbulent stress. Some difficulties are faced in relating the dyadic product of velocity fluctuations with the time-averaged velocities. In the literature this kind of problem is known as "turbulence closure," and it is still considered an open problem in physics. 2.1 Turbulence Model In engineering applications, there are two types of turbulence models. One is known as the eddy viscosity model, which assumes the Boussinesq hypothesis. Reynolds stress is related to time-averaged properties as strain tensor is related to laminar Newtonian flow. This model neglects all second-order correlations between fluctuating properties that appear during the application of the Reynolds decomposition procedure. The other is known as second-order closure, whereby Reynolds stress is assumed to have anisotropic behavior and also needs to predict the second-order correlation. The model used in this work, the Differential Stress Model (DSM), is known as secondorder closure and has one differential equation, or transport equation, for each component of Reynolds stress. These can generally be expressed by the following differential equation:
a(pvv) at
[
^
1
-T-
V.(pvVv) = V. p - ^ - v ' v ' ( V v V )
+ P-(t)—pel
(3)
in which P is a shear stress production tensor and is modeled as
p = -p r r v'v'(vv)%(vv)v'v'] 1
^^^
and (^ is the pressure-strain correlation given for incompressible flow defined as 0 = (^,+(^2
(5)
945
2 ^ P—PI ' 3
(6)
(7)
P in this case is the trace of P tensor and Cs(0.22), ODSCIO), CIS(1.8) and C2s(0.6) are the model's constants (Lauder et al., 1975). It is also necessary to include an additional equation for the dissipation rate of turbulent kinetic energy that appears in Equation (3), and this is written:
a(pe) + V.(pve) = V. p - ^ - ~ ( v ' v ' ) V e + Q - P - C 2 P — a' e ^ ^ at
(8)
in which Ci(1.44), C2(1.92) and k are obtained direcdy from its definition (k = l/2 v ' ' ) . 2.2 Numerical Methods In a general form, the numerical methods used to solve the models were the finite volume methods with a structured multiblock grid, generated by the body fitted on a generalized coordinate and collocated system. The pressure velocity couplings were the SIMPLEC (Simple Consistent) and PISO algorithms with interpolation schemes of first order, upwind and higher order, QUICK, Van Leer, CCCT and higher upwind. The Rhie Chow algorithm with the AMG solver procedure was also used to improve the solution and to avoid numerical errors like check-boarding and zigzag due to the use of collocated grids and numerical errors caused by no generation of orthogonal cells during the construction of structured grids, for more details on methodologies see Maliska(1995). The boundary conditions were uniform profiles at the inlet for all variables; no slip conditions at the walls; continuity conditions for all variables at the outlet, except for pressure where an open circuit condition with atmospheric pressure conditions were assumed. A laminar shear layer condition was also assumed for the wall with default models from the CFX 4.4 code.
3. Results The experimental study was conducted in an acrylic cyclone settled in a pilot unit belonging to Six/Petrobras in Sao Mateus do Sul, Brazil. The inlet velocities of clean air were 11.0 m/s and 12.5 m/s and the measurements of local pressures obtained with a Pitot tube were used to determine the radial distributions of the tangential velocity component at five axial positions throughout the equipment (two in the cylindrical section, 0.90D and 1.35D from the cyclone roof, and three in the conical section, 2.39D, 3.36D and 4.32D from the cyclone roof). The peak of the tangential velocity like a
946 Rankine curve typical of flows in cyclones was obtained. The grid used in the numerical simulations had about 72,500 cells. The experimental cyclone configuration and a typical 3-D grid are shown in Figure 1. Initially, the numerical solutions were obtained applying the upwind scheme for all variables to guarantee stability of the solution (one of the criteria adopted was a value of less than 10' for the euclidean norm of the source mass in the pressure-velocity coupling), but the results exhibited high numerical diffusion, as previously reported in the literature (Meier et al., 2000, Witt et al., 1999). The first solutions with upwind scheme were used as initial conditions. A higher order interpolation scheme was then introduced for the velocity components using a transient procedure. This has been found useful to overcome the difficulties of convergence presented by the DSM with higher order interpolation schemes. Nevertheless, it was observed that the steady state was achieved after about 1 second of real time. Numerical solutions were obtained for the inlet gas velocities (11.0 m/s and 12.5m/s) at five different heights and the behavior of the higher order schemes were all similar (higher upwind, QUICK, CCCT and Van Leer). More details of the interpolation schemes can be seen in Guidelines of CFX 4.4 (2001). Figure 2 shows the numerical solutions obtained for the inlet gas velocities of 11.0 m/s and 12.5 m/s and using the higher upwind scheme to illustrate the experimental data and the numerical results. Data obtained on the capability of the turbulence model to represent the radial distributions of tangential velocities throughout the cyclone was compared with the experimental data, and it was possible to verify a good agreement, especially for the velocity in the cylindrical section of the cyclone. In the conical section, the numerical results of tangential velocity were overpredicted, probably because of the imperfections in the acrylic surface of the conical section of the cyclone, which became evident during the experiments.
QM)
A afl)
Figure 1 - 3-D grid and geometrical dimensions of the cyclone.
947
30-
,.
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•
. -
\
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.
.
.
,
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/
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-
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.
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;
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]
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-
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. •
Figure 2 - Distributions of tangential velocity in the cyclone.
948
4. Conclusions The Differential Stress Model (DSM) with higher order interpolation schemes (higher upwind, QUICK, CCCT and Van Leer) showed great capability to represent the swirling flow in the cyclone, and no significant difference was observed between the higher order schemes used. Higher order schemes avoid numerical diffusion but introduce instability and convergence difficulties that can be minimized by using appropriate solution procedures. Transient procedure used in this work had been found useful to overcome these difficulties.
5. References Guidelines of CFX 4.4 User Guide, 2001, AEA Technology. Lauder, D.E., Reece, GJ., Rodi, W., 1975, Progress in the Development of a ReynoldsStress Turbulence Closure, J. Fluid Mech., 68, 537-566. Maliska, C , 1995, Transferencia de Calor e Mecanica dos Fluidos Computacional. LTC Editora, Rio de Janeiro, Brasil, 424p. Meier H.F., Mori M., 1999, Anisotropic Behavior of the Reynolds Stress in Gas and Gas-Solid Flows in Cyclones, Powder Technology, 101, 108-119. Meier H.F., Kasper, F.S., Peres, A.P., Huziwara, W.K., Mori, M., 2000, Comparison Between Turbulence Models for 3-D Turbulent Flows in Cyclones, Proceedings of XXI CILAMCE, 18p., Rio de Janeiro, Brasil. Witt, P. J., Mittoni, L.J., Wu, J. and Shepherd, I.C. 1999, Validation of a CFD Model for Predicting Gas Flow in a Cyclone, Proceedings of CHEMECA99, Australia. Acknowledgments The authors are grateful to PETROBRAS for the financial support that makes this work possible and to Eng. Alexandre Trentin from Trentin Engenharia for his help in the experimental study in the cyclone at SIX/PETROBRAS. Nomenclature
D diameter of the cyclone g gravity acceleration I identity tensor k kinetic turbulent energy p pressure P shear stress production tensor P trace of the tensor P t time V velocity vector w tangential velocity component e dissipation rate of turbulent kinetic energy ({) pressure strain correlation ^ viscosity p density a stress tensor Superscripts mean time-averaged property fluctuation property T transpose tensor or matrix
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
949
A New Algorithm for developing Dynamic Radial Basis Function Neural Network Models based on Genetic Algorithms Haralambos Sarimveis, Alex Alexandridis, Stefanos Mazarakis and George Bafas National Technical University of Athens Department of Chemical Engineering Greece
Abstract A new algorithm for extracting valuable information from industrial data is presented in this paper. The proposed methodology produces dynamic Radial Basis Function (RBF) neural network models and uses Genetic Algorithms (GAs) to auto-configure the structure of the networks. The effectiveness of the method is illustrated through the development of a dynamical model for a chemical reactor, used in pulp and paper industry.
1. Introduction Due to the always decreasing prices and increasing capacities of electronic data storage devices, today most industrial plants are collecting large volumes of process data in an every-day basis. However, in most cases the data remain unexploited, since the plant personnel rarely have the time and scientific background to work with the data and extract the important information. The desperate demand for new and more efficient algorithms to extract knowledge out of the plethora of available data was the motivation of this work. The proposed method is based on the powerful Radial Basis Functions (RBF) neural network architecture and employs Genetic Algorithms (GAs) to build dynamic models, using process input-output data. RBF networks are continuously increasing their popularity due to a number of advantages compared to other types of Artificial Neural Networks (ANNs), which include better approximation capabilities, simple network structures and faster learning algorithms. The most important part in the development of an RBF network model is the selection of the structure of the model and the hidden node centers. Most of the popular training techniques determine only the network parameters, whereas the network structure is obtained by trial and error (Moody and Darken, 1989; Leonard and Kramer, 1991). Another family of algorithms uses various methods to determine the structure of the network as a first step that is separated from the actual objective, which is the minimization of the prediction error (Chen et al., 1990; Musavi et al., 1992). However, none of the above algorithms guarantees the selection of the optimum number of hidden nodes. The inclusion of the structure selection in the formulation of the optimization problem is desirable, but results in a much more difficult problem, which cannot be easily solved
950 by standard optimization methods. Genetic algorithms are stochastic methods, based on the principles of natural selection and evolution, and offer an interesting alternative optimization technique for such complicated problems (Michalewicz, 1996). Boozarjomehry and Svrcek (2001) proposed a method based on genetic algorithms, for the automatic design of feedforward neural network structures. Billings and Zheng (1995) used genetic algorithms for the training of RBF networks. However their method restricts the potential node centers only among the set of training data. The proposed methodology uses GAs to determine the optimum number of centers and the network parameters simultaneously. The method gives more freedom in the selection of the hidden node centers, since it defines a multidimensional grid in the input space and every knot in this grid represents a potential node center. The proposed training algorithm is illustrated through the development of a dynamic RBF network model for a Kamyr digester, based on real process data.
2. RBF Networks and Genetic Algorithms - An overview RBF networks form a special class of neural networks, which consist of three layers. The input layer is used only to connect the network to its environment. The hidden layer contains a number of nodes, which apply a nonlinear transformation to the input variables, using a radial basis function, such as the Gaussian function, the thin plate spline function etc. The output layer is linear and serves as a summation unit. The typical structure of an RBF neural network can be seen in figure 1. The standard training procedure of an RBF network selects the structure of the network by trial and error. Determination of the network parameters involves two phases: In the first one, the centers of the hidden layer nodes are obtained, based on the /:-means clustering algorithm. In the second phase, the connection weights are calculated using simple linear regression. On the other hand, GAs which will constitute the basis for developing the new method are iterative stochastic methodologies, that start with a random population of possible solutions. The individuals with the best characteristics are selected for reproduction and their "chromosomes" are transferred to the next generation. In order to emulate the way nature works, some genetic "operators" are added to the algorithms, such as mutation, where a chromosome of a single individual is altered randomly, and crossover, where new individuals are born from a random combination of the old ones.
Input layer
Hidden layer
Output layer
Figure 1. Typical structure of an RBF network
951
3, Configuration of RBF networks using Genetic Algorithms The proposed methodology utilizes genetic algorithms to determine the optimum number of centers and the network parameters simultaneously, by minimizing an error function subject to the structure of the network, the hidden node centers and the parameters between the hidden and the output layer. The algorithm starts with an initial population of chromosomes, which represent possible network structures and contain the associated center locations. The centers are selected from a multi-dimensional grid that is defined on the input space. For each chromosome the weights between the hidden and the output layer are calculated using linear regression and the objective function is computed. New generations are produced by the standard genetic operators: crossover, mutation, deletion and addition. The algorithm stops after the specified number of generations has been completed. The chromosome which has produced the minimum value of the objective function is selected as the optimum neural network model. Each chromosome (solution) is represented by a matrix, where the non zero rows correspond to the hidden node centers. The detailed description of the algorithm that follows assumes that M input-output examples are available, which are split into two sets, X and Y. The number of input variables is represented by A^, while only one output variable is used. The algorithm can be easily generalized for more than one output variables, since a different neural network can be developed for each output. X is an M • N matrix, where each row corresponds to an input vector, and Y is an M • 1 vector, where each element corresponds to an output value. X and Y are scaled so that the values of all variables are positive. Additionally they are divided in three different subsets (Xi,Yi), (X2,Y2) and (X3,Y3) of size Mi, Mi and M3 correspondingly, namely the training, testing and validation set. Using the first set of the input data, Xi, the minimum and maximum values Xn^rnin Xn,max of every input variable Xn {n=J, 2,..., AO are found and the space between the pair (Xn,min. -^n.mox) IS dividcd iuto m„ subspaccs of range • Xn={xn.max-Xn,min)^ ^n- The number of subspaces m„ is usually the same for each input variable. Then the following parameters of the genetic algorithm are selected: the maximum number of hidden nodes K, the size of the population L, the number of generations G and the probabilities of crossover (p^), mutation (p^), addition (pa) and deletion (pj). The genetic algorithm can be described as follows: Step 1. Initialization: L matrices (chromosomes), Ci, C2,..., CL, of size K • A^ are created with all zero elements. For every matrix C/, (/=1, 2,..., L) a random integer number ki from 1 to A^ is selected. The ki first rows of the matrix C/ are replaced by n row vectors of size 1 • A^ that are the centers of the corresponding network. The rows below ki are left zero and do not correspond to a center. The elements of every row vector are given by the following equation: where n=7, 2,..., A^, k=l, 2,...,ki and r„ a randomly selected integer number between 1 and m„. At the end of this step, the matrices have the following form:
952
0
0
0
0
0
Step 2. Weight calculation: For every chromosome, C,, the output weights of the network are calculated by the equation w,=(A;.A,r.(A/'.Y.)
(2)
where A/ is the MixL matrix containing the responses of the hidden layer for the Xi subset of examples. It should be noted that the radial basis function used in the hidden nodes is the thin plate spline. Step 3. Error calculation: In step 2, L networks have been developed, defined by the pairs (C/, W/), 1=1,2,..., L The second subset of data (X2,Y2) is now used as a testing set, in the following manner: The predictions of the L RBF networks ^2A^^2,2^'"^^2,L
^^^ calculated given the input X2, and the corresponding error
values, El, are obtained:
^/= Kz-YJ
(3)
The chromosome with the minimum error is kept in a separate matrix B. The objective is to give more chances of surviving to the networks with the smaller error values. Therefore, the probability of selection pi of every chromosome, C/, is calculated by the following equation: f l^
Pi = V
' J
^
/F
(4)
and the cumulative probability by the equation: ^i
1P^
(5)
Step 4. New generation: In order to produce the new generation of L solutions, the following procedure is executed L times: A random number r in the range 0...1 is generated and the /th chromosome such that qi.j < r
953 even. If only one chromosome is selected then no crossover is executed. The chromosomes selected for crossover are grouped in pairs and for every pair all the data below a selected position are exchanged. The crossover position is selected randomly with the restriction that at least one of the two chromosomes in each pair has at least one non-zero row below it. The procedure is demonstrated with the example presented below, where the crossover operation is applied to chromosomes C/, C/+i and yields chromosomes Cj, C\_^^ in the new generation:
\c'u
<>
c[. C'z.l
0 0
0 0
c'u
cU
<.
0
0
4 0
<^'i.i
c',:i c',:^
0
0
<:
0
0
^4.2
^4,3
0J
Step 6. Mutation: For every non-zero element C^ ^, /:=1,2,...,A',
'VJ
c^l c'^1
0
<.
"^'u
c'::
0
0
<. - » <> <> c'.a c'.. 0
0
0
M=1, 2,...,A^
(M)
of each
chromosome C/, a random number r within the space 0...1 is selected. If Pm>r, the element is replaced using Eq. 1. Step 7. Addition and deletion: A binary value of 0 or 1 and a number r between 0 and 1 are generated randomly for each one of the chromosomes C/. If the binary number is 0, and pd>r all the data below a randomly selected position are deleted. If 1 is selected and Pa> r, 3. random number of non-zero vectors created as described in step 1, replace an equal number of zero rows, so that the total number of rows will not exceed K. Step 8. Replacement: The chromosome with the maximum error is replaced by chromosome B and the algorithm turns to step 2, unless the maximum number of iterations is reached. In the latter case, the algorithm stops and the network corresponding to the smallest prediction error is selected. Finally the model is validated using the third subset (X3,Y3), which has not been utilized throughout the entire training procedure.
4. Case study: Kappa number identification in a continuous digester Kamyr digesters are complex tubular reactors, where the delignification of wood chips takes place through combined chemical treatment and thermal effects. The kappa number, which represents the amount of residual lignin in the pulp is the most important quality variable in the process. The proposed method was applied in order to build a dynamic model, that can predict the next value of the kappa number, using as inputs hourly past values of the three manipulated variables, which are three temperatures along the reactor. For the particular application a set of noisy data from an industrial Kamyr digester was available. Because of very large retention times in the digester, past values of up to 12 hours were used, summing to a total of 36 input variables. The data set consisted of 400 data points, from which 185 were used for training, 100 for testing the model and 115
954 45 -
63 ^ 61 w 59 2 57g 55 |53 c 51 2 49
40 «
35 -
1 30-
]jl; IpuAj^^
ixi^
x>>r:^^n^
Y
5 2520 -
47 45
Real Values
15 101
201 301 Oenaration number
(a)
401
16
31
46
61 Time(h)
76
91
106
(b)
Figure 2. (a) Minimum SSE per generation (b) Kappa number prediction as a validation set. The algorithm was applied using ^^=40, L=20, pc=0.25, /7^=0.01, Pa=0.005 and p^O.OOS. The result after 450 generations, was an RBF network with 18 nodes. The training time in a Pentium IV 1400 Mhz processor was 18 mins. Figure 2a shows the reduction of the Sum of Squares Error (SSE) for the best network structure, as the algorithm proceeds. The neural network predcitions along with the real values, are shown in figure 2b for the validation set. Several runs of the algorithm with different parameter values resulted in similar results, showing that the method is insensitive to these parameters. Taking into account that the data are rather noisy, it can be seen that the model performs very well, thus rendering a set of collected data into an adequate model for the prediction of a crucial process parameter. For comparison purposes we used the same data set to train an RBF network with the standard MacQueen /:-means algorithm (Moody and Darken, 1989). The results showed that even by using the optimal structure with 18 nodes in the hidden layer, the SSE achieved by the standard method after 500 iterations, is 5% larger than the one obtained by the proposed algorithm.
5. Conclusions The paper presents a new methodology for data mining, which produces RBF network models based on input - output data. The proposed algorithm uses genetic algorithms to determine the optimum structure of the network, in contrast to the standard RBF training methods, where the structure is selected by trial and error. In order to test the methodology, dynamic process data from an industrial reactor were used to build an RBF network model, which predicts successfully a crucial product quality parameter.
6. References Billings, S.A. and G.L. Zheng, 1995, Neural Networks. 8, 877. Boozarjomehry R.B. and W.Y. Zvrcek, 2001, Computers Chem. Engng. 25, 1075 Chen, S., S.A. Billings, C.F.N. Cowan and P.W. Grant, 1990, Int. J. Control. 52, 1327. Leonard, J.A. and M.A. Kramer, 1991, IEEE Control Systems. 31. Michalewicz, Z., 1996, Genetic algorithms + Data structures = Evolution Programs. SpringerVerlag, Berlin Heidelberg Moody, J. and C. Darken, 1989, Neural Computation. 1, 281. Musavi, M. T., W. Ahmed, K.H. Chan, K.B. Paris, and D.M. Hummels, 1992, Neural Networks. 5, 595.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
955
Synthesis of large-scale models: Theory and implementation in an industrial case J. Pieter Schmal, Johan Grievink, Peter J.T. Verheijen Technical University Delft, Department of Chemical Engineering Julianalaan 136, 2628 BL Delft, The Netherlands
Abstract Four different model synthesis approaches from different authors are discussed and a variant was developed based on experiences gained with the modelling of a petrochemical plant. A qualitative improvement in model synthesis has been achieved by balancing formalism with practical manageability. Some practical implementation issues are given.
1. Introduction Over the years many people have contributed to development of the model building process (e.g. Aris 1994, Murthy et al. 1990, Marquardt 1995, Lohmann and Marquardt 1996, Hangos and Cameron 2001a). The development of models is similar to the design procedure of technical artefacts in general. Hence, the modelling procedure reflects the five steps of the generic design cycle: specifying the functional requirements, assessment of existing domain knowledge, synthesis of model structure, computation and analysis of behaviour, evaluation of model performance based on validation with plant data. The synthesis phase has received little attention in literature, as became apparent when the authors developed a model for a process that was part of a comprehensive cooperation project between industry and academia. This project, INCOOP (INtegration process unit COntrol and plant-wide Optimisation), researches the next generation of model based control and dynamic real time optimisation, techniques that rely on rigorous models. In this article we will discuss our experiences in model synthesis based on an industrial case study. First we will describe the case study, before we discuss the model synthesis theory and compare it with existing approaches. After this we will give guidelines for large-scale model building and discuss implementation issues before we end with conclusions.
2. Case study Within INCOOP a model of a petro-chemical plant has been built (figure 1). The plant contains heat integration, multiple recycles and multiple high purity separations, making it a complex non-linear plant. One single dynamic model was to be the base for the analysis of the plant behaviour for control, and optimisation. Furthermore, the model should function as plant replacement in the test phase of the new techniques. The model had to have high level of detail, sufficient accuracy, flexibility and robustness. The most
956 complex version of the DAE model of the plant contains 2.000 differential and 23.000 algebraic equations.
Reaction 1 Section [
Reaction Section
1 Recovery Section
P^ purification
Final purification
Figure 1: Block diagram of a petro-chemical plant
3. Model synthesis theory We will assume that we have a properly defined modelling goal and we do allow for some iteration in both goal and model structure. Although Aris (1994) states that the formulation is nothing more than rational accounting, especially in the case of largescale models some formalism is needed to minimize model errors. This formalism starts with defining the model building blocks used to set-up the model. Two formal descriptions of model building blocks can be found in literature. Willems (2000) distinguishes between nodes that correspond with the different blocks; edges with connections between these blocks, terminals with connections between nodes or devices and devices contain laws for change. One level consists of edges, terminals and leafs (a terminal that is not connected). Another level consists of terminals and devices only. A terminal can only be associated with two real variables that reflect a force and the accompanying flux. Marquardt's (1995) approach is also based on the idea of fluxes and forces. The fluxes are calculated in the connections and the forces in the devices. Marquardt splits the material entities in devices and connections, which in turn consist of composite or elementary elements. In Marquardt's approach a connection may well be a device on a lower level and an explicit distinction is made between (predominantly electrical) signals and phase flows. Other non-formal building blocks can be found in the method of Lefkopoulos and Stadtherr (1993) that only contains equations and Cameron and Hangos (2001a) that contains different levels of procedures, i.e. procedures and sub-procedures. The practical implementation led us to a more applicable approach. This approach consists of connections and devices with leaves to indicate one-sided connections. A connection can send multiple variables across the connection, not restricted to force flux combinations. A device may consist of devices coupled by connections, whereas a connection cannot be decomposed. The model process is to a large extent driven by specific variables that are deduced from the model goal. In our case study the model has wide range of applications from plant replacement to optimisation and thus a wide range of variables needs to be modelled. 3.1 Abstraction The abstraction is concerned with the translation of reality to a model and is thus one of the first steps in the synthesis. Every abstraction calls for a clear definition, place and character, of the system boundary. This character can be closed or open with respect to
957 a phenomenon or variable. Reductions on the other hand are performed within the model. Abstractions and reductions are a direct result of the model goal and the capabilities of both modeller and (numerical) solvers. The abstractions and reductions can be of the following types: Space (e.g. minimum length scale considered 1 cm for instance) Time (e.g. fastest process considered dynamically 10 s) Phenomena (e.g. no liquid entrainment in distillation) Forced (e.g. lack of knowledge) Solver based (e.g. ignoring very small numbers) Aris (1994) pays ample attention to abstractions and reductions, but in the sense of examples rather than an approach. Hangos and Cameron (2001a) point out that it is important to filter the set of all 'model controlling' mechanisms with respect to the modelling goal. In our case study we abstract our plant to figure 1 and define the system boundaries. For simplicity we start with boundaries at the walls of the equipment closed with respect to the surroundings except for explicit in- and outputs. The time scale we are interested in is in the range of seconds to days (time reduction). In a first step we considered two phases on our equilibrium tray and encountered an index problem later on that could not be handled by gPROMS, our modelling environment, forcing us to something different like describing the phases combined (solver based reduction). 3.2 Decomposition Decomposition of the problem reduces the complexity and ensures the problem becomes manageable. The decomposition can be done on two levels as indicated by Marquardt (1994): structural and equational. Structural decomposition The structural decomposition concerns breaking down the original problem in smaller blocks, which contain equations that have something in common depending on how the system was decomposed. For every structural decomposition we need to define an abstraction. The way we decompose the problem can be (for both levels) the following classes: • Locational: location (time, space or in process stream) is similar • Tree: natural connection (a direct physical, causal or mathematical connection) • Behavioural: similar behaviour • Temporal: similar time scales • Spatial: similar size • Functional: similar functions (both operational and mathematical) Orthogonal to these decompositions are the levels of detail and the levels of hierarchy. Equational decomposition The equational decomposition concerns the break down of equations in a block to sets of equations with a certain similarity. Hangos and Cameron (2001a) give a functional equational decomposition by splitting up the equations in one of the following classes: balance, transfer rate, property relation, balance volume relation, and equipment and control constraint.
958
V. p
Plant
Final purification
Level: 1
Figure 2: Locational and functional structural decomposition In our case study we break down the plant into sections as in figure 1 and next into devices. A distillation column is further decomposed into trays, a tray into a physical property part and a device specific part. This is a five-level (figure 2) locational structural decomposition with a functional structural decomposition on the last level. In our case we had two different types of physical property relations that could easily be switched due to the functional structural decomposition. A tree equational decomposition was used to decompose the equations (figure 3).
dM dt M
Ah ^—
F
F
in
out
Ku,
Ku, ^
f{T,x)
Figure 3: Tree equational aggregation 3.3 Aggregation In contrast with the decomposition we start with equational aggregation since we can only do structural aggregation once the blocks are composed. Equational aggregation The equations in a block must be set-up in such a way that the block can function according to its specifications, i.e. it must fulfil the sub-task it received during the structural decomposition. The equations must be written down in accordance with the way they were decomposed, because it determines the logic needed to minimize errors. The composed blocks are the devices of our model. Structural aggregation The structural aggregation combines the composed blocks, the devices, with the help of connections and creates connections to the surroundings via leafs. The connections describe the information that is shared between devices. Connections and leafs arise from the character of the system boundary of the device that is open. We choose to use a tree equational decomposition, therefore we start with the component balance (since we were among others interested in product purity). We write down the balance equation first and make sure all variables that occur in the balance are either specified or given by an additional equation (figure 3 for an example of a liquid vessel). Additional assumptions such as equilibrium on the tray could be incorporated at this stage. Next we construct the column by incorporating the other trays, condenser and
959 reboiler. To construct the purification section (figure 2) we couple all devices in this section before we hook it to the rest of the plant. 3.4 Level of detail As stated before orthogonal to the decomposition and aggregation above we have the level of detail. Changing the level of detail can be caused by a need for more accuracy or detail, but also by a need for faster simulations or less complexity. Since the systems functions as a whole, the lowest level of detail usually determines the level of detail for the whole system and the highest level of detail usually determines the speed of the simulation, but certainly not always! In our case study, we increased the level of detail by incorporating the heat-integration and decreased the level of accuracy in the physical properties to increase simulation speed, making the levels of detail more balanced over the system.
4 Implementation issues The way the equations are set-up can have a great influence on the (re-)initialisation and numerical stability. Physically correct alterations due to discontinuities or unwanted limit behaviour are preferred, but may not always be available. In our case study we helped the initialisation by reverting to two directional flows, despite the discontinuities we had to introduce, because the operating region of the model was greatly enlarged.
5 Discussion As stated by Foss et al (1998) the modelling process is poorly understood. This is caused by the fact that we are unable to predict the effect of some choices on the outcome of the simulation. It is the experiences of a modeller with respect to these sensitivities that play an important role during the model synthesis. Some experiences in the sense of pitfalls are given in Murthy et al. (1990). Table 1: Main differences Complementary Formality Number of components Strategy levels
Willems Yes High High 2
Marquardt No High Medium 2
Hangos Yes Low Low >2
Lefkopoulos No Low High 1
Presented Yes Medium Low 2
The main differences between the leading synthesis methods and our approach are given in table 1. Both Willems and Marquardt have defined a set of rules that should be obeyed for their devices and terminals or connections. In the case of Willems the terminals can only be associated with two real variables that are related in some sense, making the system unnecessary large (number of components). Marquardt's approach, in which a connection may become a device on a lower level, actually embodies multiple models and is therefore not complementary or has some overlap. Hangos* approach on the other hand is too unstructured (low formality) in the sense that there is no systematic way of setting up a consistent set of equations. Our approach is basically
960 using the advantages of both Willems, Marquardt and Hangos approach. It balances the need to avoid errors by formality and the manageability to limit model building time. The modelling tool heavily influences the modelling process (Foss et al. 1998), and thus usually determines the way we decompose our system. Since we used gPROMS in our case study, the model in the examples is well suited for equation-oriented solvers. Especially in the case of large-scale systems causality is usually difficult to determine and parallel equation-oriented solvers are then preferred. During the aggregation of equations the choice for certain equations is made. It is well known that conservation laws increase numerical stability (Gershenfeld 1999). Whether making the model more linear really helps is questionable, the numerical solver has less problems, but larger inaccuracies in variables occur. For structural aggregation it is important that the streams between the blocks are clearly defined and connecting streams contain the same set of variables. A useful sub-class of the forced reductions are the connection breakers. A connection breaker simply fixes some variables in the connection, e.g. a temperature. A proper decomposition in combination with modularity can help to change the level of detail from simple (at the start) to complex. The needed level of detail can best be obtained by investigating the levels below and above the current level of detail to see whether a step in either direction is useful. Finally, model misuse poses a serious threat in large-scale models. Assumptions should therefore be translated into constraints where possible (Hangos and Cameron 2001b).
6. Conclusions Using a model synthesis approach makes the synthesis qualitatively more efficient and robust to errors. The different model syntheses are comparable and the main differences are given in table 1. The key point of our approach compared to the others is that it balances the formality with practical manageability. Future work consists of invesfigation of the advantages of different decomposition strategies.
References Aris, R., 1994, Mathematical modelling techniques. General Publishing Company Ltd. Foss, B.A., B. Lohmann, and W. Marquardt, 1998, J.Proc.Cont. 8 (5-6), 325. Gershenfeld, N., 1999, The nature of mathematical modelling, Cambridge University Press. Hangos, K., and I. Cameron, 2001a, Process modelling and model analysis. Academic Press. Hangos, K., and I. Cameron, 2001b, Comp. Chem. Engng. 25, 237 Lefkopoulos, A., and M.A. Stadtherr, 1993, Comp. Chem. Engng. 17, 399. Lohmann, B., and W. Marquardt, 1996, Comp. Chem. Engng. 20, S213. Marquardt, W., 1995, Methods of Model-Based Control, NATO-ASI Ser. E, Applied Sciences 293, 3, Kluwer Academic Publ. Murthy, D.N.P., N.W. Page, and E.Y. Rodin, 1990, Mathematical modelling, Pergamon Press. Willems, J.C, 2000, Math. Computers in Sim. 53, 227.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) © 2002 Published by Elsevier Science B.V.
961
Prediction of the Joint Molecular Weight-Long Chain Branching Distribution in Free-Radical Branched Polymerizations p. Seferlis and C. Kiparissides Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 472, 540 06 Thessaloniki, Greece
Abstract A novel approach based on orthogonal collocation on finite elements (OCFE) techniques is proposed for the prediction of the joint molecular weight - long chain branching distribution in free-radical polymerization. The OCFE formulation reduces the size of the model, preserves the nature of the balance equations and ensures the closure of the overall material balance regardless of the selected kinetic mechanism. The proposed approach is successfully applied in the free-radical branched polymerization of vinyl-acetate. The OCFE formulation shows high accuracy, when compared to the method of the moments, with improved predictive ability in systems characterized by strong diffusion controlled phenomena and long-chain branching.
1. Introduction Most commonly employed polymer property indicators (e.g., mechanical strength, tear strength, rheological properties and so forth) are directly or indirectly linked with the molecular structural properties of the polymer chains (e.g., molecular weight distribution, MWD, long chain branching, LCB, and so forth), which are usually very difficult to measure on-line. Hence, the ability to accurately predict the molecular structural properties from mechanistic models becomes of significant importance. The mathematical models dealing with the prediction of the molecular weight distribution in free-radical polymerization are based on kinetic lumping methods (Crowley and Choi, 1997), continuous variable approximations, Z-transforms, polynomial expansion methods (Tobita and Ito, 1993), variations of the method of the moments for branched polymer systems (Pladis and Kiparissides, 1998), Monte-Carlo simulations (Tobita, 1993), discrete weighted Galerkin formulation (Wulkow, 1992; ledema et a/., 2000), and orthogonal collocation methods (Canu and Ray, 1991; Nele et ai, 1999). The proposed method employs orthogonal collocation on finite elements modeling techniques that preserve the nature of the balance eqautions and reduce significantly the size of the model, while providing accurate prediction of the joint weight chain length (WCL)-long chain branching (LCB) distribution. The effective computation of the WCL-LCB distribution becomes essential in the control of molecular weight properties via dynamic optimization of polymerization reactors.
962
2. Free-Radical Polymerization Kinetic Mechanism A general kinetic mechanism that describes the free-radical polymerization of branched polymers includes the following elementary reactions: Initiation: I—*^^*—>2R* Chain initiation: R * + M —^^—^ R j Propagation: R ^ + M —^^^^^ R ^^j Chain transfer to monomer: R ^ + M —^^^"^-^ Px "*" ^ i Chain transfer to solvent: R ^ + S —^^^^-^ Px + ^ i Chain transfer to polymer: R ^ + Py —^^^-> R y + Px Reaction with terminal double bond: R ^ + Py~ —^^^^—> R ^^y Termination by disproportionation: R ^ + R y —^^^—^ P^ + Py Termination by combination: R ^ + R y —^^^—^ P^^y The subscripts x and y denote the number of monomer units for the "live", Rx, and "dead", P^, polymer chains. Symbols I, M, and S denote the initiator, monomer and solvent, respectively. The kinetic mechanism includes propagation and termination reactions, molecular weight control reactions by transfer to monomer and solvent, and long-chain branching formation by transfer to polymer and terminal double bond reactions. The present kinetic mechanism assumes that Px=Px"^ without significant loss of accuracy (Pladis and Kiparissides, 1998).
3. Molecular Weight Distribution in Free-Radical Polymerization Orthogonal Collocation on Finite Elements Formulation A key characteristic of the OCFE formulation is the treatment of the discrete polymer chain length domain, Sf, as a continuous one. Hence, the concentrations of the "live" and "dead" polymer chains, become continuous variables. The OCFE chain length domain was divided into a number, NE, of finite elements, with boundaries at Co=l. Cb C2. •••» CwE-b CNE=Sf. In every element a number of n interior collocation points, [si, S2,..., Sn], was specified from the roots of the Hahn family of discrete orthogonal polynomials. The concentrations of the "live", R(S), and "dead", P(s), polymer chains within each element were approximated by continuous low-order Lagrange interpolation polynomials. R(S)=EW,:^^(S)R(S,J i=0
P(s)=XWiMs)P(s.J C H ^ S < C ,
j = l,...,NE
(1)
i=l
The tilde denotes approximation variables. The functions, Wij(s) and Wij(s) are Lagrange interpolation polynomials of order n+1 and n, respectively. The left boundary point of each element was also considered an interpolation point for the "live" polymer chains so that the boundary condition at chain length x=l was transferred throughout the
963 domain. Considering a batch polymerization reactor, the main requirement of the OCFE formulation forces the dynamic residual balances for the "live" and "dead" polymer chains to vanish at the selected collocation points, Sij. Residual balance equation for "live" polymer chains at the collocation points: 9^„ = R, = - ^ ^
{k J R 1 [ M ] + ( k J M ] + k J S ] ) | : R ( S ) | 5 ( S , , - l)
+ kp^(sj,i-l)-R(sj)}[M]-{k,jM]+kJs]}R(s,J+kJs,,jP(s,J}|;R(s) s=l
(2)
-k,pR(si,j)|^{sP(s)}-k^R(s,j|^R(s)-k,R(s,j)XR(s) s=2
-ka.R(si.j)|^P(s)+k,, ^ { R ( S , J - S ) P ( S ) } s=2
s=2
for i=0,...,nandj=l,...,NE Residual balance equation for ''dead" polymer chains at the collocation points:
^ p , „ = -
%-^+{k.[M]+kJs]}R(sJ+k,R(sj|:R(s)-k,p(sj|:R(s) s=l
^^
s=l
(3)
kn,{s„p(s,,,)}|:R(s)+k,R(sj|:{sP(s)}+lk./f^(s)R(s.,,-s)} 2 ., for i=l,...,n andj=l,...,NE. The individual elements of the discrete summation terms that appear in eqs 2-3 were approximated using the Lagrange interpolation polynomials for the given element partition. Hence, the residual balances were solely expressed in terms of the "live" and "dead" polymer chain concentrations at the collocation points. The WCLD was then reconstructed from the following relationship applied at the collocation points: WCLD(siJ=s,jP(sij)/|^{sP(s)} for i = l,...,n and j = l,...,NE /
(4)
s=2
The accuracy of the predicted WCLD mainly depends on the selected size of the chain length domain, the total number of finite elements, the partition of the domain in finite elements, and the total number of collocation points in the domain. OCFE formulation for branch classes The OCFE formulation was extended to the balance equations for the "live" and "dead" polymer chains of branch classes. Each branch class is defined as the fraction of the
964 entire polymer chain population with the same content of long chain branching (e.g., linear polymer chains, polymer chains with one LCB, two LCB etc.). The chain length domain for each branch class was partitioned in a number of finite elements, NEc, with n interior collocation points specified from the roots of the Hahn polynomials. In a similar fashion as in the overall balances, the residual balance equations for each branch class were forced to vanish at the collocation points (Seferlis and Kiparissides, 2002). The branch class residual balances were solved in conjunction with the overall residual balance equations (eqs 2-3). Such a solution approach eliminates the error in the prediction of the WCLD for each branch class arising from a poor selection of the assumed degree of long-chain branching. This is however true, only for kinetic mechanisms that do not contain reactions that reduce the branch class.
4. Methyl-Methacrylate Bulk Polymerization The outlined approach for the prediction of the WCLD was applied to the bulk polymerization of methyl-methacrylate (MMA) in a batch reactor. The polymerization is highly exothermic and exhibits a strong acceleration in polymerization rate due to gel-effect (e.g., the termination rate constant decreases with conversion). A total degree of polymerization equal to 656,840 monomer units was considered for the OCFE formulation. The chain length domain was partitioned into 42 finite elements with two collocation points in each finite element. Two operating scenarios were considered: i) isothermal operation and ii) one step change in temperature during the batch duration. Figure 1 shows the WCLD achieved at the end of the batch for the two operating scenarios. The isothermal reactor produces a unimodal distribution, while the second scenario produces a bimodal distribution. The WCLD calculated using the OCFE formulation was compared to the one obtained from the summation of the instantaneous WCLD, when reconstructed from the corresponding leading moments, with excellent matching results. Usually high monomer conversions or strong gel-effect phenomena cause considerable broadening of the associated WCLD. An empirical rule that alternatively uses the Schulz-Flory (polydispersity PD<4) or the
7
OCFE Schultz-Flory distr. Wesslaudistr. Combined SF-Wes di
•
"
•
Convers on. 85%
„
5
p.'
Ij
-2- 4 3
6
\
^5 $4
P,
3
4f
A*
2
1 -H»- OCFE 1 1 » Schultz-Flory distr Wesslaudistr 1 - Combined SF-Wes distr |
/\
6
0
•
.1 - /
1/
V
'^
V
2
\ !!!:.
.-.«.'.x::^
Conversion: 94 3%
1
Chain length, :
(a)
(b)
Figure 1. WCLD for MMA bulk polymerization (a. isothermal operation at 60 °C, b. temperature step change of-\-10°C after 125 minutes of operation).
965 Wesslau (PD>4) distributions for the instantaneous WCLD was applied.
5. Vinyl-Acetate Solution Polymerization The free-radical polymerization of vinyl-acetate was selected as a representative example of the production of branched polymers. Highly branched polymer chains are produced through the transfer to polymer and the terminal double bond polymerization reactions. Diffusion phenomena strongly affect the termination and propagation kinetic rate constants as monomer conversion increases. An OCFE scheme consisted of 72 finite elements and two collocation points per fmite element was employed. A total degree of polymerization equal to 668,580 monomer units was selected, large enough to allow an accurate approximation of the WCLD for high monomer conversion rates. The residual balance system (overall and 41 branch classes) was consisted of 2878 differential equations. Figure 2 compares the WCLD obtained from the OCFE formulation and the method of the moments at different monomer conversions. The overall WCLD with the method of the moments was calculated from the summation of the branch class WCLD reconstructed from their leading moments. A good agreement between the overall WCLD calculated from both methods is generally observed. Some small discrepancies are however present for short polymer chains. A closer analysis of the branch class distributions, suggests that the discrepancies are mainly attributed to the differences in the WCLD of the linear polymer chains. The reason is that the reconstruction of the WCLD for the linear polymer chains from its moments is prone to error due to the high polydispersity of the distribution. Such a behavior is more vividly observed at higher monomer conversion values (e.g., 90%). High conversion rates cause a dramatic reduction in the kinetic rate constants due to the gel-effect that subsequently results in the creation of a significant amount of linear oligomers (short polymer chains). The agreement between the two methods improves as the long-chain branching content increases. Figure 3 shows the joint WCL - LCB distribution for 41 branch classes.
10
Chain length
Chain length
(a)
(b)
10
Figure 2. Overall and branch class WCLD for OCFE (lines with symbols) and method of moments (plain solid and broken lines).
966 X 10
Branches
Chain length
Figure 3. Joint WCL-LCB distribution.
6. Conclusions A novel approach for the prediction of the weight chain length distribution based on the model reduction properties of OCFE techniques is presented. Comparison of the calculated WCLD using the OCFE modeling technique with the method of the moments verifies the accuracy of the proposed method. The power of the OCFE formulation as a prediction tool of the joint WCL-LCB distribution of complex polymer systems is evident from application to polymerization systems that are characterized by strong diffusion controlled reactions and high degree of long-chain branching.
7. References Canu, P., and W. H. Ray, 1991, Comput, Chem. Eng. 15, 549. Crowley, T. J., and K. Y. Choi, 1997, Ind. Eng. Chem. Res., 36, 1419. ledema, P. D., Wulkow M., and C. J. Hoefsloot, 2000, Macromol., 33, 7173. Nele, M., Sayer, C , and J. C. Pinto, 1999, Macromol. Theory Simul., 8, 199. Pladis, P., and C. Kiparissides, 1998, Chem. Eng. Sci., 53, 3315. Seferlis, P., and C. Kiparissides, 2002, submitted Chem. Eng. Sci. Tobita, H., 1993, J. Polym. Sci., 31, 1363. Tobita, H., and K. Ito, 1993, Polym. Reac. Eng., 1, 407. Wulkow, M., 1992, Impact of Computing in Sci. and Eng., 4, 153.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
967
Multiobjective Dynamic Optimization of Semi-Continuous Processes C. M. Silva, E. C. Biscaia Jr. PEQ/COPPE/UFRJ - Federal University of Rio de Janeiro - Brazil [email protected]
Abstract Multiobjective dynamic optimization of Chemical Engineering systems is carried out using an improved genetic algorithm. A fed-batch bioreactor for foreign protein production in an inducible bacterial system has been optimized. The target of this process is to maximize the production of the foreign protein and minimize the inducer consumption. An improved genetic algorithm is proposed to generate the optimal operating policies of this system. A new concept of fitness function, based on the ranking procedure, was adopted. A fuzzy penalty function method is used to incorporate the constraints into the fitness function. A new class of operators is introduced to enhance the algorithm performance, and the standard ranking procedure is extended to solve multidimensional problems. The simulated results demonstrated the efficiency of the algorithm to find the Pareto optimal set, offering a viable strategy to solve complex dynamic optimization problems.
1. Introduction Many chemical and biochemical processes are operated in semi-continuous mode. The dynamic behavior of such systems is usually highly complex and non-linear. In biotechnology processes, several phases can be distinguished during the operations, characterized by different substrate consumptions and metabolic production rates. Changes in the external environment often affect the internal composition of the cells as well as the cell morphology (Roubos et al., 1999). In order to specify the optimal control strategies for these processes, multiobjective optimization methods that take dynamic behaviors into account are required. Such methods can deal with conflicting targets that although influence the reactor performance in opposing ways should be achieved simultaneously. Several studies have been reported on multiobjective dynamic optimization of batch and semi-batch processes (Butala et al., 1992, Wajge and Gupta, 1994, Sareen and Gupta, 1995, Bhaskar et al, 2001). Different methods have been proposed to determine optimal control policies for bioreactors (Lee and Ramirez, 1996, Tholudur et al., 2000, Canto et al., 2001). Roubos et al.(1999) pointed out the limits of these methods on number of control variables and model complexity and suggested the use of evolutionary techniques. In this contribution, an improved genetic algorithm (GA) is proposed to conduct multiobjective dynamic optimization. A challenging case study presenting different feeding strategies is considered to evaluate the performance of the algorithm.
968
2. Multiobjective Genetic Algorithms The GA technique simulates a natural evolution process: the fittest species survive and propagate while the less successful tend to disappear. The multiobjective optimization procedure consists of a search for nondominated solutions. The concept of nondominance refers to the solutions for which no objective can be improved without worsening at least one of the other objectives. The nondominated solutions are superior to the others with respect to all targets, but comparatively good among themselves. Any of these solutions is an acceptable solution, as all are considered equivalent in dominance. The progress strategy is guided by the fitness evaluation, and consists of performing the population with genetic operators to generate the next population. Different adaptations of the original GA are presented in the literature (Cheng and Li, 1998, Toshinsky et al., 1999, Wang et al., 1998). A detailed background on the GA theory is reported in Goldberg (1989) and Busacca et al. (2001). 2.1. The proposed algorithm The multiobjective optimization algorithm developed is an improved version of the GA proposed by Cheng and Li (1998). The standard ranking procedure is extended to treat multidimensional problems. A new class of operators - niche, Pareto-set filter and elitism - is introduced to reduce the necessary number of generations. A fitness function and a fuzzy penalty method are also adopted. The algorithm operates in a continuous variable space, which is computational fast and stable in converging to global optima. The proposed GA procedure works through the following steps: a) creation of a random initial population; b) evaluation of the individuals and application of the penalty function method; c) ranking of the individuals, calculation of the fitness, registration of the best individuals; d) registration of all nondominated individuals in the Pareto set filter operator; e) selection of pairs of individuals as parents; f) crossover of the parents to generate the children; g) replacement of the individuals using the niche operator; h) genetic mutation; i) replacement of the individuals using the elitism operator. Penalty function method A fuzzy penalty function method has been adopted to treat constrained multiobjective optimization problems. This method incorporates the constraints into the objective functions by using a transferred function, which carries information on the point's position and feasibility (Cheng and Li, 1998). The penalization procedure consists of associating a finite value, established on the fuzzy logic theory, with the extent each constraint is violated. The largest amount violated of each point is used to determine the transferred function value of that point. As a result, all points in the feasible region present values between 0 and 1, while the infeasible ones are greater than the unity. Ranking procedure The ranking procedure consists of the classification of the individuals into categories according to the concept of dominance. First, all nondominated individuals of the population are identified and assigned rank 1. These individuals are virtually removed from the population and a new evaluation is conducted on the remaining individuals. The next set of nondominated points are identified and assigned rank 2. This procedure
969 continues until all the individuals are classified. Mathematically, the ranking method is conducted in four basic steps: a) the points are sorted according to the evaluation of an objective function, randomly chosen as a reference function; b) the sequence of points that simultaneously produces an increase on the reference function and a decrease on the other functions, or vice versa, are selected as potential candidates to the rank; c) repeat steps (a) and (b) until all the objective functions have been chosen as the reference function; d) all points selected at least (n-1) times in the step (b) will be assigned to the rank. These steps are repeated for each rank, until all the points are classified. Fitness function The fitness value represents a measure of each individual performance. Every individual belonging to the same rank class is considered equivalent and has the same probability of being selected for reproduction. The fitness function, F^t, is determined to each individual of the same rank k as follows (Cheng and Li, 1998): F, = ^^'''' SS
(1)
SS=-^h^r-k+^)Psk
(2)
where Nr is the highest rank of the population, P, is the population size and P^k is the population size of rank k. According to this fitness definition, the larger the population size at a rank is, the smaller the fitness of a point. Hence, the reproduction ratio of individuals at each rank depends on both the rank level and population size. Niche operator The niche operator determines which individuals will go to the next generation. The fitness of each child is calculated in the domain of its paternal population. The replacement of the parents only occurs if the child fitness exceeds the parents' inferior fitness. Otherwise, the parents will go to the next generation. This operator helps to avoid the genetic drift, which makes a population become clustered at certain regions, maintaining the appropriate diversity of the population. Elitism Operator The elitism procedure consists of the propagation of the best solutions to the next generation. For this purpose, it maintains an elitism file where the best solution of each individual objective function is registered. At each generation, this file is updated: if a better solution was generated it replaces the one stored. The individuals selected to the next population by the crossover and mutation procedures are submitted to the elitism operator. The points registered in the elitism file will randomly replace some of the candidates to the next generation. As a consequence, it increases the convergence of the optimization process as well as the robustness of the algorithm. Pareto-set filter operator All points assigned rank 1 are registered in the Pareto set filter at each generation. This file is dynamically updated by using a filter operator, in which the nondominated solutions of the current population are compared with those already stored in the file.
970 from the previous generations. A new evaluation is conducted in the filter, according to the following rules: a) all points in the filter identified as nondominated, are recorded in the Pareto set file. The dominated ones are discarded; b) if the number of points in the file is inferior to the population size, the new nondominated points are stored. Otherwise, if the file is full, the most similar points in the Pareto set file are replaced. At the end of the optimization process, the file itself comprises the Pareto optimal set and constitutes the result of the optimization. The computational load associated to the proposed algorithm relies on the number of objective functions of the problem. At each generation, the objective functions are evaluated just once for each point. Therefore, the total number of the objective evaluations is a function of the population size, the number of generations and the number of objective optimized. The algorithm is fully presented in a previous work (Silva and Biscaia, 2001).
3, Formulation of the Multiobjective Optimization Problem The case study deals with the production of induced foreign protein by recombinant bacteria. The mathematical formulation reported by Tholudur et al. (2000) has been modified to consider the reactor volume as a function of time: —
^RG-RI
(3)
(4)
^ ~ - ^ V dt
^
= ,,G,-ifK
(5)
4^-^,1,-r,XV
(6)
dt d(VP) -rpXV dt
(7)
where X, G, / and P are the cell mass, glucose, inducer and protein concentrations respectively, and V, the reactor volume. G/ and // are the glucose and inducer feed concentrations, and qc and qj the feeding rates {0 < qc ^ ^ and 0 < qj < i). ^ is the specific cell growth rate, r/, the specific inducer inactivation rate, and rp, the specific protein production rate. Y is the biomass yield on the substrate. The parameter function definitions and values are described in Tholudur et al. (2000). Aiming to increase the sensitivity of the model to the controls, a modified parameter function ju is presented:
1+A/ Krn +G
(8)
where PJ = 8.33 and P2 = 23.33. The process is conducted in 15 hours. Glucose and inducer are fed simultaneously at different rates, during the whole process. The optimization problem involves the maximization of the foreign protein production and the minimization of the total inducer consumption. The decision variables are the inducer initial concentration, /Q, the inducer feed concentration, /f, and the glucose and
971 inducer feeding rates, qdi) and qi(i) (/ = 1, 10). A total of 22 variables are manipulated. Two constraints are identified: the amount of inducer available, Ijotah is 3 g, and the fmal reagent volume, V(/y), is required to be greater than 10 dm^. This optimization problem is formulated as: maximize /i = P(tf) Vitf) (9) minimize
f2= I fj'^ qjiOdt-\-VQIQ
(10)
subject to
8i-lTo:ai(tf)<3g;
(11)
g2: Vitf) > 10 dm'
4. Results and Discussion The proposed algorithm has been applied to determine the optimal operating policies for the formulated problem. The optimization has been carried out using the GA parameters shown in Table 1, established in a previous sensitivity study (Silva and Biscaia, 2001). Table 7. Genetic algorithm parameters Population size 20 individuals Crossover probability 75% 5% Mutation probability
Number of children /crossover Number of generations
1 230
Figure 1 (a) shows the Pareto optimal set obtained. It can be observed that the objective functions chosen are affected in opposing ways by changes in the decision variables. A desirable increase of the total protein production corresponds to an undesirable increase of the total inducer consumption. Thus, this case constitutes a multiobjective problem and is suitable to test the proposed algorithm. Figure 1 (b) depicts the optimal feeding strategy for the maximum protein production. The other optimal strategies were omitted for the sake of brevity. No specific tendency was identified in any of these curves. The complexity of the feeding strategies reinforces the use of an evolutionary technique. Figure 2 presents the profiles of the protein production and inducer consumption, simulating the process under the optimal conditions for maximum protein production (max /i), minimum inducer consumption (min fi) and an intermediate result. The behavior of the profiles corroborates the optimization results: the optimal policy to maximize /i produces more protein and consumes more inducer; when minimizing /2, less inducer is used and less protein is produced; the intermediate policy presents intermediate responses. - Glucose •
0.5 _
0.4 ^
0.8
I 0.6
T3
=
•Inducer
B 2 0.4 -I
0.2-1 ,••'
CO
.--•••
0.2
0
G 3
4 Total ProtBin (g)
5
5
10
15
T i m e (h)
Figure 1. (a) Pareto optimal set; (b) Optimal feeding strategy for maximum protein production.
972
Time (h)
Time (h)
Figure 2. Simulation of the process using optimal values of the decision variables. The computational time taken to generate the Pareto optimal set has been considerable low. Although this parameter depends on the desired accuracy of the optimum, the CPU time required for this problem was inferior to 2 min, on a 1.3 Gb Pentium 4 computer.
4. Conclusions Our focus in this contribution has been to evaluate the performance of a proposed genetic algorithm in conducting multiobjective dynamic optimization. The algorithm has been applied to a fed-batch bioreactor. Different feeding strategies have been investigated in order to establish the optimal operating policies. The results of the challenging case study confirm the efficiency of the proposed optimization approach to solve dynamic control problems. The ability of the algorithm to seek tradeoff surface regions has also been demonstrated. It should be pointed out that the computational cost related to the proposed GA approach depends on the dimension of the problem and its level of complexity, regardless of the number of decision variables manipulated. It has been demonstrated that the proposed algorithm can successfully solve problems involving a large number of decision variables.
5. References Bhaskar, V., S.K. Gupta and A.K. Ray, 2001, Comp. Chem. Engng. 25, 391. Busacca, P.G., M. Marseguerra and E. Zio, 2001, Reliab. Engng Syst. Saf 72, 59. Butala, D., W. R. Liang and K. Y. Choi, 1992, J. Appl. Pol. Sci 44, 1759. Canto, E.B, J. Banga, A. Alonso and V. Vassiliadis, 2001, Com. Chem. Engng. 25, 539. Cheng, F.Y. and D. Li, 1998, AIAA J. 36, 1105. Goldberg, D.E,. Genetic Algorithms in Search, Optimization, and Machine Learning, AddisonWesley, Reading MA, 1989. Lee, J. and W.F. Ramirez, 1996, Chem. Engng. Sci., 521, 51. Roubos, J.A., G van Straten and A.J.B. van Boxtel, 1999, J. Biotech. 67, 173. Sareen, R. and S. Gupta, 1995, J. Appl. Pol. Sci. 58, 2357 Secchi, A.R., E.L. Lima and J.C. Pinto, 1990, Pol. Engng. Sci. 30, 1209. Silva, CM. and E.C. Biscaia, 2001, presented at 2nd Pan American Workshop on Process Systems Engn. (CEPAC/2001) and submitted to Comp. Chem. Engng. Toshinsky, V.G, H. Sekimoto and GI. Toshinsky 2000, Proc. Nucl. En. 27, 397. Tholudur, A., W.F. Ramirez and J.D. McMillan, 2000, Comp. Chem. Engng. 24, 2545. Wajge, R. and S. Gupta, 1994, Pol. Engng. Sci. 34, 1161. Wang, K., Y. Qian, Y. Yuan and P. Yao, 1998, Comp. Chem. Engng. 23, 125.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
973
A MILP approach to the optimization of the operation poUcy of an emulsiHcation process M. Stork, R. L. Tousain and O. H. Bosgra Mechanical Engineering Systems and Control Group Delft University of Technology The Netherlands
Abstract This work addresses the model-based optimization of the operation policy of an emulsification process. In this paper the choice of the stirrer speed as function of the time, for reaching a certain predefined drop size distribution (DSD) in minimal time, is studied. It is argued that "standard" gradient based optimization techniques will fail to solve this optimization problem. An approach is suggested that approximates the original minimum time optimization problem as a Mixed Integer Linear Program (MILP). The MILP can be solved using well-proven, standard optimization codes and its solution is also a good solution of the original optimization problem. Applying it to the industrial process shows the feasibility of the approach and it illustrates the benefit of using model-based optimization for improving the operation policy of emulsification.
1. Introduction Emulsification is an essential manufacturing technology in the food industry. Examples of emulsions^ are mayonnaise and all kind of dressings. The equipment as typically used for the production of o/w (oil-in-water) emulsions is shown in figure 1. It consists of a stirred vessel in combination with a colloid mill and a recirculation loop. The vessel is equipped with a scraper stirrer: a device that consists of several blades rotating at low speed at a small distance from the vessel wall in order to achieve mixing and breaking of the oil drops. The colloid mill consists of a stator and a rotor. In the narrow gap between these the intensity of the (hydrodynamic) forces acting on the drops is very high, which causes the breakage of the oil drops. The process is operated fed-batch wise and typical production times are in the order of 10-15 minutes. For profit maximization it is desirable to decrease the production time while maintaining the product quality specifications. In this paper it is investigated if this can be established by using modelbased optimization. To this end a model is developed that predicts the course of the DSD in time (the product quality is strongly influenced by the DSD). The model describes emulsification in a stirred vessel that is operated fed-batch wise. In this work it is used to calculate the stirrer speed as function of the time, such that a certain
^ An emulsion is a dispersion of drops of one liquid in another one with which it is incompletely miscible (Israelachvili, 1993).
974
Stirred vessel
Colloid
Figure I: Equipment for the production ofo/w-emulsions. predefined DSD is reached in minimal time. To the authors knowledge this is not addressed in the literature. A short description of the model is given in Section 2. The model contains continuous and discontinuous dynamics and therefore belongs to the class of so called hybrid discrete/ continuous models (Barton et al., 98). In Section 3 the optimization problem is presented and it is argued that "standard" gradient based optimization techniques are not suited for the solution of it. An approach to solve general hybrid discrete/ continuous optimization problems is discussed in Barton et al. (2000) and in Avraam et al. (1998); it is called the MIDO (mixed-integer dynamic optimization) approach. In Avraam et al. (1998) it is applied to some toy examples and they ended up with a Mixed Integer Nonlinear Program (MINLP). MINLP's are generally hard to solve, severely limiting the applicability of the MIDO approach (Barton et al., 2000). We also use this approach (Section 4), however due to our specific model structure and by using results of Bemporad and Morari (1999) we are able to reformulate the original minimum time optimization problem as a MILP instead of a MINLP. This enables to solve the optimization problem. An application is presented in Section 5 and concluding remarks are made in Section 6.
2. Outline of the model The hydrodynamic forces that are generated by the stirrer action cause the breakage of oil drops and because of this the DSD changes in time. The model describes the DSD(t) and it consists of two parts: a reactor model and a drop model. The reactor model describes the mixing in the vessel and the hydrodynamic forces that are generated by the stirrer action. This is modeled with compartment models and for each compartment a population balance equation PBE (Ramkrishna, D., 1985) is formulated to describe the DSD(t). It is assumed that drop breakage occurs in a small region round the stirrer with laminar flow and that in the bulk region the fluid is mixed. This is modeled with two
975
Time [s]
-9.5
8.5 Log of drop diameter [m]
Figure 2: Course of the normalized (number based) in the bulk compartment. compartments: the so-called laminar compartment and the bulk compartment. The drop model describes the phenomena occurring at the drop level. It describes the breakage condition, the breakup time and the number and sizes of the daughter drops. These relations are for the larger part based on the theory as currently available for the breakage of drops. Simulation results are included to illustrate that the DSD(t) is strongly affected by the stirrer speed. During the simulation the stirrer speed is kept constant for a period of 125 s, then its value is changed and kept constant for an other period of 125 s etc. The sequence as used is: 1, 3, 2 and 5 s ^ The corresponding normalized DSD(t)^ is shown in figure 2.
3. Features of the optimization problem We want to choose the stirrer speed as function of the time such that a certain predefined DSD is reached in minimum time. Next it is argued why "standard" gradient based optimization techniques are not suited for the solution of this optimization problem. The breakage phenomena (the breakage condition, the number and the sizes of the daughter drops) depend heavily on the stirrer speed and exhibit discrete events. A very small increase of the stirrer speed may already lead to the breakage of certain drop sizes that would not break with a slightly lower value of the stirrer speed. Comparable behavior is observed for the formation of certain drop sizes; until some stirrer speed they are not formed whereas they are formed rapidly at a stirrer speed that is only slightly higher. A further increase of the stirrer speed may suddenly lead to the nonformation or even breakage of these drop sizes. Gradient based optimization methods will fail because of this behavior. ^The normalized DSD equals the DSD divided by the total number of drops.
976
4. MILP approach Here an approach is suggested that approximates the original nonlinear optimization problem as a MILP. The method is derived as follows. First, by model analysis it can be shown that the strong non-linearity (discontinuity) is only in the dependence on the stirrer speed. Further, in small intervals of the stirrer speed the dynamics are bilinear (product of stirrer speed and states). These intervals form the modes of the system. At any given time, the system finds itself in exactly one mode; its mathematical behavior is then described by a given set of evolution ordinary differential equations. A transition from one mode to another is triggered when the stirrer speed passes a certain critical value. Hence, the stirrer speed determines completely in which mode the system is and when the transitions occur. This suggests that the model can be reformulated as a statetransition network (Avraam et al., 1998) where bilinear dynamics describe the behavior in a mode and where transitions between different modes are modeled using integer decision variables. However, this would result in a MINLP, which is undesirable. The dynamics are however linear if the stirrer speed is fixed at a constant value for the different modes. This enables to end up with a MILP. It is believed that this will not result in a loss of performance because the intervals are rather small (typically 0.05 s" ). The objective, being to reach a certain end-point condition in minimum time, can be enforced through the introduction of another set of integer decision variables. The precise formulation of the optimization problem as a MILP is discussed next. 4.1 System behavior We define a set of periods (k=l,...,N) of fixed duration. The following binary variable is introduced to characterize the system behavior: [l if the system is in mode s over period k [0 otherwise The system can be in only one mode at a point in time. This is expressed mathematically as V ' S[=\,
with n^ the number of modes of the system. For the
ease of implementation we use discrete time domain models. For each mode s we have the discrete time model:
x,^,=Alx,^Bl
(2)
The states x^ are the number of drops, with a certain volume, in the laminar and bulk compartment. While the system is in mode s, the corresponding equations characterizing the behavior of the system must hold in that mode. In Bemporad and Morari (1999) an approach is suggested for this. We follow the main idea, some modifications are however made to facilitate the solution process (details are omitted). A new continuous variable z[ is defined and with linear equality and inequality constraints it is enforced that this variable is equal to x^ as the system is in mode s and that it is zero as this is not the case. Mathematically:
977
zl < MlSl
(5)
z;>o
(6)
With Ml being a weighting matrix. Hence, if 5^ is zero then zl- is zero. Since 5^ is one for only one mode, zl = Xj^ for that mode. 4.2 End-point inequality constraints and the objective function We now turn our attention to handling the inequality end-point constraints and the formulation of the objective function J. The following binary variable is introduced to characterize the inequality end-point constraints: [l
if the inequality end - point constraints are met over period k
[0 otherwise The constraints are now expressed as:
Y,x^,-x,<0
(8)
^^-^max-a->^^)A^2<0
(9)
With M2 being a weighting matrix. Hence, if Xj^ is less than its lower bound x^^jn then Yj^ must be zero in order to satisfy inequality constraint (9). If x^ is larger than its upper bound Xn^x then Yj^ is also set to zero. K^ can be either one or zero if the bounds are met. Due to the formulation of the objective function 7, K^ will be set to one. The objective function is written as: N
max J = f^Y,
(10)
This way the number of periods, where the end-point constraints are satisfied, are maximized. Hence, the time needed to satisfy these constraints is minimized.
5. Optimization of the operation policy Here the results of an optimization study are presented. A sample interval of 50 s is used for the derivation of the discrete time domain models and the time horizon is set to 500 s. Upper and lower bounds are formulated for all 38 states. The system has 55 modes, so the resulting optimization problem consists of 561 binary variables, 810 equality and 46816 inequality constraints. The MILP is solved using GAMS/CPLEX. It took 304 nodes (less than a hour) to find the optimum. Its value is 3, so after 400 s the
978
Figure 3: Optimal state trajectories in the bulk compartment. desired DSD is reached. The corresponding trajectory for the stirrer speed is: 4.578, 4.578, 3.937, 4.578, 3.392, 3.837, 4.508, 3.392, 0.804 and 0.804 s"^ Some of the optimal state trajectories are depicted in figure 3. The states x5, x6 and xl are zero at the start of the process. With the stirrer speed of 4.578 s'^ they are formed from breakage of larger drops. The results of figure 3 might suggest that xl is not formed during the first 100 s. These drops are however formed, but they rapidly break down to x5 and x6. After 100 s the stirrer speed is lowered: now it is too low to cause breakage of xl, hence its increase. These results are non-trivial illustrating the benefit of using model-based optimization for improving the operation policy of emulsification.
6. Concluding remarks An approach is suggested that approximates the original minimum time optimization problem as a MILP. The solution of the MILP is also a good solution of the original optimization problem. Applying it to the industrial process shows the feasibility of the approach and it illustrates the benefit of using model-based optimization for improving the operation policy of emulsification. In order to establish what improvements can be reached the quality of the model has to be established; this is subject of current research. References Avraam, M. P., Shah, N. and Pantelides, C. C , 1998, Modelling and optimization general hybrid systems in the continuous time domain. Comp. Chem. Eng., 22: S221-S228. Barton, W. F. , Allgor, R. J., Feehery, W. F. and Galan S., 1998, Dynamic optimization in a discontiuous world. Ind. Eng. Chem. Res., 37: 966-981. Barton, W. F., Banga, J. R. and Galan, S., 2000, Comp. Chem. Eng., 24:2171-2182. Bemporad, A. and Morari, M. , 1999, Control of systems integrating logics, dynamics and constraints. Automatica, 35: 407-427. Israelachvili, J., 1993, The science and applications of emulsions- an overview. Colloids And Surfaces A: Physicochemical and Engineering Aspects, 91: 1-8. Ramkrishna, D., 1985, The status of population balances. Rev. Chem. Eng., 3: 49-95.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
979
The Effect of Modelling Assumptions on the Differential Index of Lumped Process Models Zs. Tuza, G. Szederkenyi, K. M. Hangos Computer and Automation Research Institute Hungarian Academy of Sciences Department of Computer Science, University of Veszprem Hungary
Abstract The effect of model simplification assumptions on the differential index of DAE process models is investigated in this paper. Efficient incremental graph-theoretical algorithms are proposed to follow the changes in the variable-equation assignments during the modelling process. Case studies are used to demonstrate the operation of the algorithms and the effect of steady-state assumptions on the differential index of a simple process model.
1. Introduction Lumped process models are in the form of differential-algebraic equations (Hangos, Cameron, 2001a) which are sometimes difficult to solve numerically, due to index and stiffness problems. The effect of some modelling decisions on the structural solvability has already been investigated (Moe, 1995) and it has been found that a change in the specification may transform an index-1 model to a higher-index one (Hangos, Cameron, 2001a). It is intuitively clear that simplification assumptions applied during the modelling process may also affect seriously the differential index of lumped process models. The aim of this paper is to propose a polynomial-time incremental algorithms for analysing the changes in the differential index during the modelling process. The algorithm warns the modeller about decisions causing index problems and advises him/her to change the specification to meet solvability requirements.
2. Computational Structure of Lumped Process Models For the purpose of the analysis, the structure of a lumped process model is described by a bipartite graph called the equation-variable graph. For this we use the fact that a lumped process model is given as a DAE system with the following special form. (D) (DS) (A) (AS)
u = dx/dt = f(x, Zs. z) = Ju dt + Xs 0 = g(x, Zs, z) Zs = spec (xs = spec)
X
980 where x is the vector of differential variables, u contains their derivatives and xs their initial values, z is the vector of algebraic variables, zs is that of the specified algebraic variables and spec stands for a general given constant (different for each variable). Note that the variables in the specification S = { Zs, xs ) appear in the structure in a special way. The equation-variable graph of the above model is a bipartite graph, where one vertex class represents the set of equations, ( (D), (DS), (A), (AS)) and the other one contains the variables, ( x, u, z, Zs, ^s )• A variable-vertex is adjacent to an equation-vertex if and only if the variable in question appears in the corresponding equation. Assume that the degrees-of-freedom requirement {DOF=0) is satisfied. Then the DAE model above is of index 1 if and only if there exists a perfect matching in the equationvariable graph (Hangos, Cameron, 2001a). 2.1 The effect of model simplification transformations A formal representation of assumptions in process is reported in the earlier paper (Hangos, Cameron, 2001b) where the modelling assumptions acted as formal transformations on process models. Similarly, model simplification transformations are described as graph transformations on the equation-variable graph. Assume that a full equation-variable assignment (a perfect matching) is given together with an equationvariable graph. Then a simplification transformation may be: 1. edge-changing transformation when only some non-matching edges are removed or added, 2. assignment-changing transformation when some of the matching edges are affected, 3. vertex-changing transformation when new equations and/or variables appear causing edge changes and change in the specification, too.
3. Algorithms for Finding Closest Maximum Assignment Let us have an equation-variable graph with a full equation-variable assignment. Moreover, let us consider a model simplification transformation of the 2"^^ or y^ type affecting the equation-variable assignment. A closest maximum assignment is a (not necessarily full) assignment in the transformed equation-variable graph, which has the largest possible number of edges and under this requirement the largest number of matching edges in common with the original full assignment. 3.1. The algorithms Two different cases will be distinguished from the algorithmic point of view. The first case (covered by Algorithm 1) occurs when a model simplification transformation deletes just one matching edge from the model, while in the second case (Algorithm 2) more than one edges are deleted. Algorithm 1. Let B = {X, K, E) be the equation-variable graph of the process model in question, with vertex classes X (variables) and Y (equations) and edge set E (dependence of equations on the variables). Let F c E be the assignment given (i.e. a matching selected in B ). We construct an auxiliary directed graph D as follows. The vertex set of D is F u {5, r}, where seX and te Y are the two vertices that have no
981 matching edge. An ordered pair (5, e) - where e = xy e F, x e X, ye K - is an edge in D if and only if (s, y) e E. Similarly, a pair {e, t) with e^xy e F is an edge in D if and only if (x, t) e E. Inside F, there is an edge from/;=.Yy>'7 to/2=X2>2 if {^i> ^2) ^ E - F. Now, we run the Breadth-First Search algorithm (see e.g. Gormen, Leiserson and Rivest (1990), Section 23.2) on D, starting from s. If t is reachable from s along any directed path, then BFS also fmds one shortest s~t path, say P(s, t). 1. If there is no directed path from s to t then the given assignment F is of maximum size in B. 2. If t is reachable from s and BFS fmds a shortest s~t path P(s, t) = sf\fi"-fkt then an assignment closest to F and of size |F| + 1 is (^ - {/b /2, •.., fk)) u {(5,>;;),(x7,3;2),...,(x^7,>'it),to.O}, where /;=x£y„ i=l,...X Since BFS can be implemented in linear time, the closest assignment is found in linear time, too. Algorithm 2. Let again B = (X, Y, E) be the equation-variable graph, and F c £ the assignment given. Assume |X| = |y| = ^z. We define an edge-weight function w as w(f) = n + 1 for all
/ G F,
w{e) = /i
for all ee
E-F.
We run a Maximum-Weight Bipartite Matching search algorithm on (B, w). If M Q E is the output of the algorithm, then the total weight of M is w(M) = n\M\ + \F n M\> (n + 1)(|A/| - 1). Thus, the algorithm first maximizes the possible number of edges in a matching of B, and under this condition it maximizes the possible number of edges selected from F. This means precisely a closest maximum assignment. Since a maximum-weight matching of a bipartite graph can be found in polynomial time (see e.g. Hopcroft and Karp (1973)), the same time bound is valid for the closest assignment, too. 3.2. The algorithmic complexity of the algorithms The following theoretical results show the efficiency of our algorithms above. Theorem 1. There is a linear-time algorithm to decide whether an assignment is of maximum size. In particular, if just one edge is deleted from a perfect matching, it can be decided in linear time whether the modified system still has differential index 1. Moreover, if the assignment is below the maximum size by 1, then a closest assignment can be found in polynomial time. Theorem 2. A closest assignment of maximum size can be found in polynomial time. There is a good reason for the different time complexities in the two theorems. For a closest maximum assignment, one cannot really expect a linear-time algorithm, because such an algorithm would solve the Bipartite Matching problem as a particular case where the initial „assignment" contains no edges. On the other hand, it remains an open question whether the removal of a bounded number k of edges admits an algorithm
982 finding a closest maximum assignment more efficiently than Bipartite Matching. Theorem 1 yields a positive answer for k = 1.
4. Case Studies The algorithm for finding the closest assignment is illustrated on the following example by examining the effect of steady state assumptions. The model used is a simplified version of the one described in (Hangos, Cameron, 2001b). The original model equations are the following (the labels of the equations are between parentheses): (dl) (d2) (dsl) (ds2)
mi^dML/dt= F-E-L UL = dUi/dt = = FhF-EhLv-LhL+Q+QE ML = fnii dt + Mu) UL = JUL dt + ULO
(al) (a2)
E = kLv(P*-P°)
(a3) (a4)
UL = ML CpL TL P*^HTL
(as4) (as5)
Mu)= spec ULO= spec
QE = ULV (TL - T^)
The specified variables are (asl) (as2) (as3)
Q = spec F= spec L= spec
The constant physico-chemical parameters are: h^, hiy, hi, C^L, P", f, and spec denotes a given specified value for the variable in question (different for each variable). This model has differential index 1. 4.1 Test case 1 We examine the effect of a steady-state assumption on the liquid mass Mi. For this, let us introduce the following new equation in the modified model: Add (a*)
mi = dMi/dt = 0
Since we have a new equation, we have to reduce the number of specified variables, i.e. the number of specification equations. This is done by changing the specifications (as])-(as4) in such a way that we delete equation (as4) from the new model. Del (as4)
Muo= spec
Using Algorithm 1 we find that the modified system still has differential index 1, and we can also determine the assignment which is closest to the original one. The algorithm is based on finding the shortest directed path in a directed graph assigned to the equation-variable graph. The results are shown in Figure 1 where the vertices s and t (see Algorithm 1) are {di) and Mjj) respectively. 4.2 Test case 2 Now we investigate the effect of changing the assignment in a less fortunate way together with the above steady state assumption f«*| For this, let us again introduce
983 fa*j as a new equation in the modified model but now we delete the specification equation (asl) from the new model. Del (asl)
Q = spec
Using Algorithm 1 we find that the modified system is no longer of differential index 1. 4.3 Test case 3 Let us investigate what happens if we put two steady state assumptions to our original model. For this, let us introduce the steady state assumptions for both the mass ML and the energy UL in the original model, that is: Add fa*j
niL = dMi/dt = 0
Add (bV
UL = dUi/dt = 0
but now we need to delete two specification equations (as4) and (as5) Del (as4)
Mu)= spec
Del (as5)
Uu)= spec
Using Algorithm 2 we find that the modified system is no longer of differential index 1. 4.4 Test case 4 Now we try to improve the situation in the previous test case with two steady state assumptions by selecting another, more fortunate specification equation to be deleted. For this, we again introduce the steady state assumptions ffl*j and (b"^) in the model, but now we delete two specification equations (as4) and (asl) from the new model. Del (as4)
M^o^ spec
Del (asl)
Q= spec
Using Algorithm 2 we find that now the modified system has differential index 1, and we can determine the assignment which is closest to the original one shown in Figure 2. Note that in this case we obtain the same result by applying Algorithm 1 in two steps: the first step is the same as in Test case 1. In the second step we delete (asl) and add the specification equation Q=spec, and then find a matching in the resulting graph.
References Cormen, T.H., Leiserson, C.E. and Rivest, R.L. (1990): Introduction to Algorithms. MIT Press. Hangos, K.M. and I.T. Cameron, 2001a, Process Modelling and Model Analysis. Academic Press. Hangos, K.M. and I.T. Cameron, 2001b, A Formal Representation of Assumptions in Process Modelling. Comput. Chem. Engng. 25, 237-255. Moe H.I., 1995, Dynamic Process Simulation, Studies on Modeling and Index Reduction. PhD Thesis, University of Trondheim Hopcroft, J. and Karp, R.M. (1973): An n^^^ algorithm for maximum matching in bipartite graphs. SIAM J. Comput. 2, 225-231.
984
(as5)
O-
O
U,,
unchanged matching edge edge removed from the original matching new matching edge other edge not belonging to any matching
Figure 1. Test case 1. a - Determining the closest assignment by finding the shortest path in Test Case 1. b - The closest assignment in the equation-variable graph in Test Case 1. The edges belonging to the original matching are denoted by solid and dotted lines. The new matching consists of the solid (unchanged) and the dashed (new) lines.
Figure 2. The resulting matching of Test Case 4. The edges belonging to the final matching are denoted by solid lines.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
985
Dynamic Simulation of Continuous Crystallizers by Moving Finite Elements with Piecewise Polynomials Zsoit Ulbert and Bela G. Lakatos Department of Process Engineering, University of Veszprem H-8200 Veszprem, Hungary
Abstract The moving finite element method is developed for numerical solution of the population balance equation of disperse systems. This approximation is obtained by using the Lagrange interpolation polynomials, spacing the interior nodes according to the orthogonal collocation technique. The method is used for solving the mixed set of nonlinear ordinary and partial integro-differential equations, forming a detailed dynamical model of the continuous cooling crystallizers.
1. Introduction The population balance model is the adequate mathematical description of crystallization processes. This is a mixed set of ordinary and partial differential equations and the crucial point of using this modelling approach is the numerical solution of the population balance equation. A number of different methods have been presented for solution of this equation (Subramanian and Ramkrishna,1971, Gelbard and Seinfeld, 1978, Chang and Wang,1984, Eyre et a/.,1988, Hounslow et a/., 1988, Steemson and White, 1988, Nicmanis and Hounslow, 1998), but it rarely has been considered in coupling with the nonlinear ordinary integro-differential equations describing the variations of concentrations and temperature in time (Lakatos et a/.,1984, Marchal et «/.,1988, Rawlings et al, 1992, Wulkow and Gerstlauer,1999, Mantzaris et a/. ,2001). In dynamic conditions, however, the solutions of the population balance equation often exhibits sharp profiles and steep moving fronts of the particle size distribution in certain regions of the particle size. In such cases it seems to be desirable to move and place the grid points into those regions where they are most efficient in approximating the exact population density function. The moving finite element method, developed by Miller and Miller (1971) and Miller (1971) has proved satisfactory for that purpose in computing processes occurring in finite intervals (Sereno et a/,1991, Coimbra et ai, 2001). The aim of the paper is to apply this method for solving the population balance equation, allowing computations also for such environmental conditions that produce sharp profiles and steep moving fronts in the particle size distribution.
2. Population Balance Model Consider a continuous cooling crystallizer, assuming that: (1) The crystal suspension is well mixed; (2) The volumetric feed and withdrawal rates of the crystallizer are constant and equal;
986 (3) There is a selective withdrawal of crystals characterised by the selection function (4) Crystal breakage and agglomeration are negligible; (5) All new crystals are formed at a nominal size L „ ^ , so that we take Ln=0. Then the mathematical model of the crystallizer takes the following form: the variation of the population density function n{L,t) with time is described by the population balance equation
dt dL dl} subject to the initial and boundary conditions
(1)
^
n(L,0) = no(L), L > 0 lim
G{L,CyC^q)n{Lj)-
limn(L,0 = 0 and
(2) a(Dc(L)n(L,0)l dL
B(c,c,^.fi^\
t>0
l i m ^ ^ ^ ^ ^ = 0, r > 0
(3) (4)
where the rate of nucleation B is given by the power law relation B(c,c,q,fi^) = ki,(c-c,q{T)fjul
(5)
and overall linear growth rate of crystals G is expressed as GiUc,c,^)
= kg(c-c,^)^(p(L).
(6)
Here: L - crystal size, DG - growth rate dispersion coefficient, q - volumetric rate, V volume, c -concentration of solute, c^,^ - solubility, k^, b, j , kg, g - constants, (p - empirical function, T - temperature, and /ij is the third order moment of the population density function, given as oo
fi^ = JL^n{Lj)dL
(7)
0
The terms on the left hand side of eq.(l) describe, respectively, the accumulation, growth and size dispersion of crystals, while the two terms on the right hand side describe the feed and removal rates of crystals. Boundary condition (3) expresses that the flow density of crystals at the size of nuclei L„=0 equals to the overall rate of nucleation of new crystals. In order to characterise the crystallizer entirely, we need also equations describing the variations with time of the concentrations of solute and solvent. The equation for the solute can be obtained from the overall mass balance
f=^r-,«c,„ - . . „ , c ; + ^ ( ( i - ^ , , p ) - ( i - ^ . „ , p ) ) - ^ ^ while the mass balance of the solvent can be written in the form
(8)
987
Finally, the heat balance can be written in the form
dt VeO 0 dt 0 where 0 = V£(C,,c,, +Cc) + V ( l - £ ) C p , O,, = V£,,(C,,c,w, + C c , J + V ( l - £ , J C p P ^out =y^out(CsvC,,+Cc)
+
V{l-e,,,,)C^p
(11) (Csv - concentration of solvent, C - heat capacity, p - density, ky - volume sorm factor, AHc - heat of crystallization, A - heat transfer surface, h_- heat transfer coefficient, TQ temperature of the cooling medium). Here, because of the selective withdrawal, the voidage in the crystallizer and that in the outlet stream are not equal, i.e. in turn, e^\-ky^^,
Si^^l-kyi^^i^,
e^^j=l-kyjs(L)L^n(Lj)dL,
(12)
0
Therefore, the state of the crystallizer is given by the quaternion [c(t),c,y{t),T{t)M'^t)] at time D>0, and its dynamics is described by the distributed parameter model formed by the mixed set of partial and ordinary integro-differential equations (1)-(12). The evolution in time of this system occurs in the state space ff^xN, which is the Descartes product of the vector space ff of concentrations and temperature and of the function space N of the population density functions. Projecting N into some finite (/C-)dimensional space spanned by appropriately chosen trial functions the problem is reduced to finding an approximating trajectory of the state of crystallizer in a A^+3 dimensional state space.
3. The Moving Finite Element Method In order to find a finite dimensional approximation of the system of equations (1)-(12), the moving finite element method is applied to the partial differential equation (1). Let us assume that the maximal size of crystals at time r>0 is L^axiO^ and let the interval [ 0 , L ^ be divided into M elements by M+1 separation nodes as it is shown in Fig.l. In the m-th element we define the local size variable T and Im local interpolation points, as well as the approximation /z'" of the population density function n by using an Im-l degree polynomial expressed through the Lagrange interpolation polynomials Ar as: r^L, A:'ir)=f]
(13)
ifL„
i = l,2,.:Im
n"(/'",/)=§nr(rK(^")
m = l,2,...M.
(14)
(15)
988 1
h
Hm
^
Local nodes
Global nodes I
i _
Li=0 L2
•^m+l
'-^m-k-2
LM-1
LM=^L^ 'M"'^max
Figure 1. Finite elements on the interval [0,LmiLx] cind the local nodes of the m-th element If we now take all the K nodes, either the separation nodes or the local ones inside the finite elements, and use the notation M^ (r) = n{L,^j) ~ n(L,^ J) at the ^-th global node, /:=1,2,.. .K, then the approximating function can be written in the form
fi{Lj)=Y^n,{f)F,{L)^n(L,t)
(16)
where W^, /:=1-^AT denotes the global interpolation functions identical with the appropriate local interpolation polynomials on the finite elements, respectively. In order to get the appropriate values of the time dependent coefficients n^ {t) and global moving nodes LJ^t) we minimise the functional ^max
Q{t)= J ResiUtfdL^Y^
dt
(17)
with respect to all time derivatives dniJdt and dLJdt, k-\-^K, m=\-^M as it was formulated by Miller (1981). Here, e^ and S^ are the so called internodal viscosity function and internodal spring function, respectively, regularising the movement of the global nodes during the process, and Res{L,t) stands for the residual of approximation of the population balance equation. The solution of the minimising problem (17) generates a set of ordinary differential equations for the time dependent coefficients AI^ (/) and global nodes L^(0 which together with the differential equations (8)-(10) can be solved by an ODE-solwcr. In order to reduce the problems of stiffness often arising in solving the model equations of crystallizers, the set of eqs (1)-(12) had been scaled before applying the finite element method and solving the resulted set of ordinary differential equations by means of the DA55L-solver (Petzold,1983). The details of the numerical solution will be presented elsewhere.
4. Simulation Results and Discussion In crystallizers, sharp profiles in population density function, describing the crystal size distribution, arise often when primary nucleation is the dominant mechanism of producing new crystals, described by the rate equation (1) withy=0 and b»l. This occurs
990 applying the present moving finite element solution, the predictions obtained by the moment equation model can be compared with the results given by the population balance model. In Fig.3.b, development of limit cycle oscillations is presented in isothermal case at parameter values for which the stability analysis based on the moment equation model predicted oscillatory instabilities.
6. Conclusions The moving finite element method presented for numerical solution of the population balance equation appears to be a good technique for solving the usually highly nonlinear integro-differential equations of crystallisation models. Since, however, the crystal size distribution may vary some orders of magnitude in the dynamic crystallisation processes, this method needs some adaptation crossing the boundaries of the significantly differing size regions. Changing the parameters of the movement regularising functions (17) what however requires some experience can do such adaptation.
7. References Chang, R.-Y. and M.-L. Wang, 1984, Computers chem. Engng, 8, 117. Coimbra, M.C., C Sereno and A. Rodrigues, 2001, Chem. Engng J., 84, 23. Eyre, D., C.J. Wright and 0. Reuter, 1988, J. Comp. Physics, 78, 288. Gelbard, F. and J.H. Seinfeld, 1978, J. Comp. Phys., 28, 357. Hounslow, M.J., R.L. Ryall and V.R. Marshall, 1988, AIChE J., 34, 1821. Kumar, S, and Ramkrishna, 1997, Chem. Eng. Sci., 50, 4659. Lakatos, B., E. Varga, S. Halasz and T. Blickle, 1984, Simulation of batch crystallizers. In: Jancic, S.J. and E.J. de Jong (Eds), Industrial Crystallization'84. Elsevier, Amster-dam, 185. Lakatos, B.C. and Zs. Ulbert, 2001, Proc. 8"*' Int. Workshop Ind.Crystallization. DUT, The Netherlands, 231. Lakatos, G.B. and T. Blickle, 1995, Computers chem.Engng, 19, S501. Mantzaris, N.V., P. Daoutidis and F. Srienc, 2001, Computers chem. Engng, 25, 1411. Marchal, P., R. David, J.P. Klein and J. Villermaux, 1988, Chem. Eng. Sci., 43, 59. Miller, K. and R.N. Miller, 1981, SIAM J. Numer. Anal., 18, 1019. Miller, K., 1981, SIAM J. Numer. Anal., 18, 1033. Nicmanis, M. and M.J. Hounslow, 1998, AIChE J., 44, 2258. Petzold, L.R., 1983, A description of DASSL. In: Stepleman, R.S et a/.(Eds), Scientific Computing, North Holland, Amsterdam, 65. Ramkrishna, D., 2000, Population Balances: Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego. Rawlings, J.B., W.R. Witkowski and J.W. Elton, 1992, Powder Technology., 69, 3. Sereno, C, A. Rodrigues and J. Villadsen, 1991,Computers chem. Engng, 15, 25. Steemson, M.L. and E.T. White, 1988, Computers chem. Engng, 12 81. Subramanian, G. and D. Ramkrishna, 1971, Math.Biosciences, 10, 1. Ulbert, Zs. and B.C. Lakatos, 1999, Computers chem.Engng, 23, S435. Wulkow, M. and A. Gerstlauer, 1999, Proc. 14''' Int. Symp. Ind. Cryst., IChemE, Rug-by, UK, no 7.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
991
An Object-Oriented Framework for Bill of Materials in Process Industries Marcela Vegettit, Gabriela Henningt, Horacio Leonet flNTEC, Guemes 3450, 3000 - Santa Fe, Argentina. [email protected] tINGAPOUTN, Avellaneda 3657, 3000 - Santa Fe, Argentina. [email protected], [email protected]
Abstract Products that were formerly quite standard turn now into custom-made ones, leading to an enormous increase in product variants. High degrees of redundancy can occur in data management when closely related product structures are treated as independent bills of materials. Since the bill of materials (BOM) is one of the most fundamental data in industrial enteq^rises a lot of work has been devoted to it in last years. In this contribution, a new object-oriented BOM has been proposed. It easily manages crucial aspects that should be taken into account in a product representation, such as the efficient handling of product families and variants concepts, composition and decomposition structures and the possibility of describing restrictions. Moreover, the model easily accommodates the needs of various organizational units within a company.
1. Introduction Product proliferation is uncontrollable nowadays. With customers demanding ever more customized products, manufacturers have responded with mass customization and even segment-of-one views of the market, as companies perceive each customer as an independent market segment. In addition to the enormous variety of product types, products life cycle has been shrinking dramatically. These phenomena, which have been referred as product flexibility by Van Veen and Wortmann (1992a), have become a new factor in competition. However, product flexibility has an enormous impact in supporting information systems, which are burdened by the growing number of products that must be managed. Many problems arise in defining and maintaining big amounts of product data, which affect the performance of information systems with regards to aspects such as storage requirements, data entry support, retrieval support as well as correctness and timeliness of provided information. Indeed, many of the modules that comprise overall company information systems make use of product information either in the form of a recipe or a BOM. However, the actual way of presenting a BOM depends on factors such as the point of view from which products are seen. For instance, the materials requirement planning (MRP) function uses it intensively to explode the requirements of end products into the needs of intermediate ones and raw materials, thus launching work orders for the various production departments and purchase orders for the different suppliers. Therefore, the BOM supporting MRP must be consistent with the way a product is manufactured. On the contrary, the Master Planning Scheduling (MPS) function does
992 not need those details and requires product information in terms of how products are sold instead of how they are manufactured, leading to different types of products models such as the modular and planning BOMs. Other functions within the production planning and control department like demand forecasting use different product information such as smoothing factors and previous forecast values. In addition, a variety of functional areas make use of the BOM (e.g., for calculating costs of final and intermediate products). These distinct needs have led in practice to a situation where many functional areas have developed their own product model, suitable for their particular requirements. As pointed out by Vollmann et al. (1997) a golden rule is that a company should have one, and only one BOM that should be maintained as an entity and be so designed that all legitimate company uses can be satisfied. Another problem faced by many industries in the fact that conventional BOM representations are only suitable for discrete manufacturing industries where products are always fabricated by putting parts together (composition) in assembly processes. In other words, they do not handle BOM representations where products are obtained by decomposing raw materials, like in some food industries (milk and meat ones) and in the petrochemical business where hybrid structures (combining composition and decomposition types of operations) may be associated to products. The problems described before reveal a demand for a new representation of BOMs, able to suit the needs of different functional areas, to efficiently deal with a growing number of product variants and to handle all types of production strategies. Associated with such representation, there is need for the corresponding bill of materials processor, the specific computer software that deals with data entry, maintenance and recovery. This contribution describes a conceptual representation of BOMs that tries to overcome the problems pointed out before. It has been organized as follows. First, a brief summary of former approaches is presented. Afterwards, the proposed model is introduced and then its application to a case-study is tackled. Finally, conclusions and future work are discussed.
2. Different types of bill of materials At the simplest conceptual level, the BOM is a list of all the parts needed to produce a finished product. Thus, the BOM is generally represented by a tree structure whose root is a final product, and the descendants of each node represent the components or necessary materials to produce it (multi-level BOM). Nevertheless, there are other BOMs like the single level BOM and the indented BOM. A typical BOM has been defined by Scheer (1998) using two types of entities: Part and BOM Relationship. A Part entity can be a finished product, an assembly, a component or a raw material. Each BOM Relationship entity defines a link between a product P (parent) and a product Q (component) if Q is one of the direct components required to produce P. In the traditional BOM, each variant is treated as an independent product. High degrees of redundancy can occur when closely related product structures are represented as independent bills of materials. This is true in the case of product variants, as well as in cases where the representation of a product is managed separately according to the different views of distinct functional areas of the organization. The creation of independent bills of materials for similar products can lead to an uncontrollable volume
993 of data, associated with high storage and management costs, and potential inconsistency problems. Thus, special representations are to be developed to manage similar bills of materials. To achieve an efficient representation of variants, Scheer (1998) outlines some modifications to this model called identical-part BOM, plus-minus BOM and multiple BOM. Van Veen and Wortmann (1992a, b) describe other approaches, which are a little more complex but more efficient than the previous ones: Modular BOM, Variant BOM and Generic BOM, where the last two belong to the group of "generative BOM" systems. Chung and Fisher (1994), as well as Usher (1993), present a BOM model based on an object-oriented representation.
3. An object-oriented model for a hybrid BOM The proposed model agrees with the "generative BOM" philosophy, where each variant structure is derived at the moment it is required by resorting to a valid specification, thus reducing the volume of stored data. Moreover, it is an object-oriented (00) model that minimizes the coupling among the products and their variants, allowing to carry out changes in products' structure with minimum consequences for the associated variants. The model, which is presented in Fig. 1 in a concise view (not showing specific attributes), considers the existence of similar products that have almost common structures and only differ in: (i) the presence (with quantity specification) or absence of the some components (at least one) in the structure, or (ii) the value of some of the characteristics that define the set of components. A group of similar products is referred as product family (named as Product in Fig. 1). The explicit representation of families allows the encapsulation of aggregated data usually stored in planning bills of materials. A particular member of this group is referred as a variant. A Product family represents a simple or a compound product. A Compound Product models at a generic level (i) an intermediate or a final product (intended for commercialization) resulting from the assembly or processing of raw materials and intermediates or (ii) a non-atomic raw material that can be decomposed into other products. These two situations are denoted in the model as Composed-by and Decomposed-into relationships and are described in the model by the CStructure and DStructure classes, which are subclasses of the Structure class. This class contains the number of units or the amount of the descendent product participating in the parent product, the classification of the structure and the restrictions that could exist for the required quantities of the component in its parent's composition. Regarding classification, a Structure relationship can be mandatory (when its links a component that is always present), optional (when the component might not participate) or mandatory-selective (when one out of a group of links should be present). It is also necessary to represent that in certain cases, either for technological or commercial reasons, not all the combinations of components are valid. Therefore, when defining a structure, a mechanism that allows expressing restrictions among parts is needed. It is possible to identify two groups of such constraints: obligatory (when a component family is included another is obliged to take part) and incompatible (when one component family is present another must not take part). These constraints are represented by the P_Restriction entity. According to Olsen et al. (1997), most restrictions should be included in generic structures in order to simplify the specification of product variants. Other constraints that may occur in a composition
994 structure are quantity restrictions (Q_Restriction), restraining the "amount" in which a given component participates (it can a maximum or a minimum bound, a range, etc.). Apart from the previous constraints that apply to a generic level, there might be restrictions among specific instances of variants (SV_Restriction), where also obligatory and incompatible constraints can be identified. «CV» Corrpound Variart i ^ n c l u d e : aclusion [] J ^include: inclusion []
«SP» Smple Product «V»
«E» Delusion
Variart
«P» Product
iB^variarts: variaitO
""^AsocFarnly
^
'
-
"••"
product "-A'
C^Variart: Variart jS^include= sinnple variart Q
Deoomposed-into
«cs»
CStructure
«CP» Corrpound Product il|i^Connixiner<s: Structure Q %Routing ^Processing Time
«PR» P_Re8triction ^jpiroducts: Product 4tH»=(0b(.lnc}
I
«S» Structure I^Connponenl: Product ^Quantity It^RestriclionB; Restriction |^Tipe = (Man,opt, manl-sel) ;
1
«DS» DStructure il^Pcart
«SV» Srrple Variart
^
! |
'
lp,Scpp(ier -String i
'Restriction-d
«R» Restriction
J «QR» Q_Restriction |^ti^=(mBxrTin)
«SMT» SV_Restriction jJijj^Veu'iants: Vev^iart
Figure 1: Class diagram of the proposed BOM model. Since a Variant is a particular member of a product family (Product), to represent its membership to such family, the Variant-of relationship is included in the model. As there are simple and compound products, simple and compound product variants must also be introduced in the model. The first situation is represented by resorting to the Simple Variant class, while the second one by the Compound Variant class. For each Compound Variant, the associated Compound Product family is specified, from which it is possible to obtain the variant's structure. Provided that certain components that are defined at the level of a family may not be present in a particular compound variant, the Exclusion relationship has been introduced in the model. On the other hand, a Compound Variant may be associated to a Simple Variant when there is a possibility of choosing among alternatives and a particular one is chosen; the selected variant is associated to the compound variant by means of an Inclusion relationship. This link may also contain information on the number of the simple components that are included in the compound variant.
4. Example The proposed model has been tested by representing product data of different industrial enterprises that are characterized by having complex structures (like in the meat industry, where composition and decomposition operations exist), high number of
995 variants, as well as needs for representing data both at the level of product families and instances. The case treated in this section and depicted in Fig. 2 corresponds to a candy production facility, described by Henning and Cerda, 2000, where an enormous amount of variants needs to be tackled. For instance, from a particular unwrapped candy (such as the strawberry candy depicted in Fig. 2) many variants originate (MPS Strawb. Price, MPS Straw Holid and MPS Strawb. Candy) due to the adoption of different wrappings (Strawb. candy, Holid. and Price, which are variants-of the entity wrapp). Similarly, the same wrapped candy can be packed in different quantities and formats (in bags, tubes and jars of distinct capacities), thus leading to different end products. Due to space limitations Fig. 2 depicts just one end product (named OOOXXX Strawb. Fil.), but it shows different alternative packagings that lead to a greater variety of them. The fact of representing packagings as separate intermediate assemblies (Packaging 000, 001, etc.) allows their "reuse" in the definition of other products, since many products having different characteristics (flavor, shape, filling, etc.) are packed in the same way.
5. Conclusions and Future Work Product data is used as the basis for many modules of an industrial information system. An object-oriented product model that tries to represent all the legitimate company uses of product information has been presented. It has been implemented using OODBMS technology that allows the creation of persistent objects, enabling the implementation of the object model as it was conceived, without transforming it into a relational outline. The chosen database administrator is VERS ANT, Release 5.2 with the Java interface JVI (Java-Versant Interface, VERSANT Corporation, 1999). Future work involves intensive model tesfing to assess its suitability in relation to the disfinct needs of the various functions of industrial organizations, easiness of data entry, modificafion, etc.
References Chung, Y. and G. Fischer, 1994, A Conceptual Structure and Issues for an Object Oriented Bill of Materials (BOM) Data Model. Computers Ind. Engng., 26, 321-339. Henning, G.P. and J. Cerda, 2000, Knowledge-based predictive and reactive scheduling in industrial environments, Computers and Chemical Engineering, 24, 2315-2338. Olsen, K.A., Saetre, P. and A. Thorstenson, 1997, A Procedure-Oriented Generic Bill of Materials. Computers Ind. Engng., 32, 29-45. Scheer, A.W., 1998, Business Process Engineering. Springer-Veriag, Beriin- Heidelberg. Usher, J.M., 1993, An Object-Oriented Approach to Product Modelling for Manufacturing Industries, Computers Ind. Engng., 25, 557. Van Veen, E.A. and J.C Wortmann, 1992a, Generative bill of materials processing systems. Production Planning & Control, 3, 314-326. Van Veen, E.A. and J.C. Wortmann, 1992b, New developments in generative BOM processing systems. Production Planning & Control, 3, 327-335. Versant Corporation, 1999, J/VERSANT Interface Release 2.4. Vollmann, T.E., Berry, W.L. and D.C. Whybark, 1997, Manufacturing Planning and Control Systems, Fourth Edition, Irwing McGraw-Hill, New York.
Figure 2: Partial view of the product model of a candy production facility
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European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
997
A Hybrid Global Optimization Approach for Solving MINLP Models in Product Design Yiping Wang and Luke E. K. Achenie* Department of Chemical Engineering, University of Connecticut, Storrs, CT 06269 [email protected] and [email protected] mailto:[email protected] * Phone: (860) 486 2756. Fax: (860) 486 2959.
Abstract This paper presents a hybrid global optimization approach for solving product design (specifically solvent design) problems modeled by mixed integer nonlinear programming (MINLP). The strategy incorporates a variant of the Outer Approximation mathematical programming algorithm and a soft computing global optimization approach (namely simulated annealing). The suggested approach is not provably globally optimal. However, computational experience with benchmark examples and solvent design MINLP models indicate strongly that the approach gives near globally optimal solutions. Keywords: Global optimization; Simulated annealing; MINLP; Computer Aided Product Design (CAPD); Outer approximation (OA).
1. Introduction In recent years, some computer-aided product design (CAPD) problems have been formulated as mixed integer nonlinear mathematical programming (MINLP) models in which performance requirements of the compounds are reflected in the objective and the constraints (Odele and Macchieto, 1993, Duvedi and Achenie, 1996, Churi and Achenie, 1996, Pistikopoulos and Stefanis, 1998, Sinha et al. 1999, Buxton et al. 1999, Wang and Achenie, 2001). Most of these CAPD models are multi-extremal and nonconvex from an optimization point of view. As a result, care needs to be taken when employing locally optimal techniques such as generalized benders decomposition (GBD, Geoffrion, 1972) and outer approximation (OA/ER/AP, Viswanathan and Grossmann, 1990). Developing efficient algorithms for identifying the global optima is not trivial and has been the target of research for some time. A recent book by Floudas (2000) presents an overview of the existing global algorithms for MINLP problems. Overall, the strategies in the open literature for global optimization of MINLP models can be classified as deterministic (Kocis and Grossmann, 1988; Floudas et al. 1989; Ryoo and Sahinidis, 1995), stochastic (Cardoso et al. 1997; Consta and Oliveria, 2001), or transformation techniques (Porn et al., 1999). Deterministic approaches tend to use gradient-based techniques to solve sub-problems. Stochastic methods often do not use gradient information, and can find a global minimum with an asymptotic convergence guarantee in probability (i.e. with probability 1, Dekkers and Aarts, 1991), such as simulated
998 annealing. Depending on the structure of the optimization model, Porn et al. (1999) proposed some convexification techniques to reformulate posynominal and negative binominal terms in the constraints so that the nonconvex problem is transformed into a convex problem. Some MINLP models can be reformulated or simplified to have the proper convexity assumptions such that existing algorithms can be employed. There are other types of MINLP models that do not lend themselves to such reformulations. Similarly, there are models for which approximation of physical property models may not be appropriate. For example, consider MINLP models that employ complicated activity models in phase equilibria. For these types of MINLP models we propose a hybrid method that incorporates stochastic and deterministic elements for global solutions. We initiate our investigation with the OA/ER/AP algorithm (Viswanathan and Grossmann, 1990). It should be emphasized that our aim is to provide a viable approach for finding a near global solution for nonconvex MINLP problems. The basic idea is to employ the simulated annealing method (Kirkpatrick et al. 1983) to solve the primal problem in OA/ER/AP algorithm to increase the possibility of getting a globally optimal solution. The paper is organized as follows. We start with an introduction to the simulated annealing method, followed by a discussion of the proposed hybrid global optimization method. To demonstrate the viability of the proposed approach, some benchmark MINLP examples and MINLP models of the solvent design problem are solved.
2. Simulated annealing (SA) approach Simulated annealing is a stochastic method, which can avoid getting trapped in a local minimum (Dekkers and Aarts, 1991). The SA method originates from an analogy between a physical annealing process and the problem of finding solutions to optimization problems. The SA algorithm involves the following steps (Zhu and Xu, 1999): Step 1: Initialization: set the initial and final values of the parameter T, namely TQ and Tf. Also provide an initial guess XQ for the vector of optimization variables. Set the iteration counter k <—0. Step 2: For the current value of T = T^ , perform a sequence (Markov chain loop with length Lo) of exploratory moves. (2a) Generate a neighbor of jc^ as x using a random number generator. {2b)LtiA=f(x)-f(x,). (2c) If ^ <0 or exp (-A/T) > random (0, J), set x^ = x. where random (0,1) is a random number (between 0 and 1) from a given probability distribution. We employed both a normal distribution and a uniform distribution. Step 3: if T
999
3. Proposed hybrid global optimization method (OA_Global) The outer approximation (OA) algorithm was developed by Duran and Grossmann (1986) to solve a MINLP problem without equality constraints. To accommodate equality constraints, Kocis and Grossmann (1987) provided a variant, namely outer approximation with equality relaxation (OA/ER). Both OA and OA/ER can identify the global solution if certain convexity requirements are satisfied (Floudas, 1995). To soften the convexity assumptions, Viswanathan and Grossmann (1990) proposed a variant of the OA/ER algorithm by adding a penalty, namely OA/ER/AP (outer approximation with equality relaxation and augmented penalty). However, as shown in (Floudas, 1995), the OA/ER/AP may still end up with a locally optimal solution. To obtain a global solution, we suggest a modification of OA/ER/AP, namely OA_Global. The premise for the proposed strategy is if a globally optimal solution for the nonconvex NLP primal problem is obtained in the OA/ER/AP, then a potentially global optimal solution will be obtained for the original MINLP problem. Simulated annealing is employed to solve the NLP primal problem to find a globally optimal solution. The OA_Global algorithm addresses MINLP problems of the general form z = minc^ v + /(jc) s.t Ay + /iU) = 0 By^g{x)<0 Cy-\-Dx
(^^
<x<x'']
yeY = mr In problem (1), all the binary variables appear linearly. For the problem involving nonlinear binary variables, pseudo-continuous variables can be employed (Kocis and Grossmann, 1988). The OA_Global solves the NLP primal of problem (1) without making convexity assumptions. The algorithmic steps are similar to the OA/ER/AP algorithm (Viswanathan and Grossmann, 1990). Since simulated annealing does not provide Lagrange multipliers, the formation of the relaxation matrix using Lagrange multipliers is not possible here. In this study, we have used the following heuristics to determine the elements of the relaxation matrix in the MELP master of problem (1).
T,={t^}
(2)
where 1 t'=i 0 -1
Ay' +
h,(x')>0
Ay' +/i,(jc') = 0 Ay'+h^(x')<0
^^^
1000 Theoretically at optimal solution jc*, the equality constraints should be satisfied, i.e. Ay' -^h(x' ) = 0 Simulated annealing satisfies the equality constraints to within a given tolerance thus \Ay' +h{x' )\<e To validate the OA_Global algorithm, a benchmark MINLP example from the open literature and a solvent design MINLP model are solved. Benchmark example Tmnf(x„X2,x^,y,,y,,y^,y,)
= (y, -1)^ i-iy, -ly
^{y^ -1)"
- ln(>;, + 1) + (X, -1)- + {X, - 2)- + ^3 - 3 ) ' sX. yx +y2+y^+x,+x^ +^3 <5 yl +;cf + x] +JC3 <5.5 y,+x,<\2 ^3 + JC3 < 2.5 y^ +JC, <1.2 yl +;C2-<1.64 >'3^+jC3^<4.25 yl + x] < 4.64 >'p)^2'>'3'>'4e{0,l}
(4)
This nonconvex MINLP model was solved by Floudas and Aggarwal (1989), Ryoo and Sahinidis (1995), and Cardoso et al. (1997). The best solution from the open literature is (•^7,-^2,^ij7,}'2,>'i,}^4.:/)=(0.2,0.8,1.908,l,l,0,l;4.576) (Floudas and Aggarwal, 1989). OA_Global obtained {xi,X2,X3yi,y2,y3,y4:f) = (0.206,0.804,1.910,1,1,0,1 ;4.540) (see Table 1). Table I: Computational results of illustrative example by OA_Global Optimal solution Simulated annealing parameters To a e Lo [0.206,0.804, 1.910,1,1,0,1] 1.0e6 0.98 l.Oe-3 100 [1,1,0,1,0.2,0.8,1.908] Floudas etal. (1989) [1,1,0,1,0.2,0.8,1.907878] Ryoo and Sahinidis (1995) [1,1,0,1,0.2,0.8,1.907878] Cardoso etal. (1997)
Objective function 4.540 4.576 4.579582 4.579582
4. Solvent Design Case Study Solvent design for extraction of acetic acid from water has been studied in the open literature (Odele and Macchietto, 1993; Pretel et al., 1994; Hostrup et al. 1999). In these studies, tetramethyl hexane, 1-nonanol, and diisobutyl ketones were identified as potential solvents. These solvents however were obtained based on infinite dilution
1001 activity coefficient models. When the concentration of the solute is very low, this simplification is reasonable; otherwise it can lead to unsuitable solvent choices. In this paper, we employ the rigorous UNIFAC model to calculate liquid-liquid equilibrium. We note that the highly non-linear nature of this rigorous model is likely to cause problems for many MINLP global solvers. We hope researchers in global MINLP will use this problem as a benchmark problem for their algorithms. We are genuinely interested in whether any one can find a solution better than what is presented here. Here we would like to design a globally optimal solvent for the extraction of acetic acid from water using the OA_Global algorithm. The objective is to maximize the distribution coefficient of acetic acid. Since one would like to extract as much acetic acid as possible, this is probably a better performance criterion than maximizing selectivity. For comparison purposes, we employ the same conditions as in the open literature (Hostrup et al. 1999). The selectivity of the candidate solvent should be greater than 7, and solvent loss should be less than 0.01. The boiling point of the solvent should be in the range 42IK to 54IK, and the melting point should be less than 31 OK. The results by OA_Global are given in Table 2. Table 2: Computational results of acetic acid extraction OA .Global
Hostrup etal. (1999)
Simulated annealing parameters Solvent
a e Lo To 60 1.0e4 0.85 0.01 Propanedioic acid, methyl dimethyl ester [609-02-9]
Optimal structure
(CH3COO)2(CH3)(CH)
(CH3)4(CH2)2(CH)2CO
Molecular Weight BP(K) MP(K) -log(LC5o) Selectivity of acetic acid
146 443.6 (449.7*) 224.9 0.98 12.43
142 462(441*) 227 (227*) 3.35
Distribution coefficient of acetic acid *Experimental value
0.66
0.28
Diisobutyl ketone [108-83-8]
20.3
As shown in Table 2, propanedioic acid, methyl-dimethyl ester (abbreviated as PAMDE) is the potentially optimal solvent for extracting acetic acid from water under the specified conditions. The global solution was partly verified by partially enumerating the MINLP using the binary variables and solving the individual NLP's. Compared with Hostrup's results (Hostrup et al. 1999), PAMDE shows a higher distribution coefficient (as dictated by the performance objective function), but a lower selectivity than diisobutyl ketone. If selectivity is more important, then selectivity should be used in the objective function. In addition, PAMDE has a lower value of log(LC5o) than diisobutyl ketone, which makes it environmentally benign.
1002
5. Conclusions This paper presents a hybrid global optimization strategy (for a given kind of MINLP model) based on the OA/ER/AP algorithm. The premise for OA_Global is if a global solution is obtained for NLP primal problem, then the final solution may be potentially global. Based on this, a near-global optimizer, namely simulated annealing, was employed to solve the NLP primal problem in OA/ER/AP. The OA_Global algorithm is demonstrated through the solution of a benchamrk example and a realistic solvent design problem. Although the suggested approach has not been mathematically proven to be globally optimal, our experience so far indicates very strongly that a global solution of the problem is obtained.
References: Buxton, A.; Linvingston, A. and Pistikopoulos, E. N., 1999, AlChEJ. 45(4), 817. Cardoso, M. F. Salcedo, R. L. de Azevedo, S. and Barbosa, D., 1997, Computers Chen Engng. 21 (12), 1349. Churi, N.; Achenie, L. E. K., 1996, Ind. Eng. Chem. Res.. 35(10), 3788. Costa, L. and Oliveira, P., 2001, Computers Chem. Engng. 25, 257. Dekkers, A. and Aarts, E., 1991, Mathematical Programming, 50, 367. Duran, M. A. and Grossmann, I. E., 1986, Mathematical Programming 36, 307. Duvedi, A. P. and Achenie, L. E. K., 1996, Chem. Eng. Science, 51, 3727. Floudas, C. A. Deterministic global optimisation: theory, algorithms and applications Kluwer Academic Publishers, 2000. Floudas, C. A. Nonlinear and mixed-integer optimisation. Oxford University Press, N \ 1995 Floudas, C. A.; Aggarwal, A. and Ciric, A. R., 1989, Comp. Chem. Engng. 13, 1117. Geoffrion, A. M., 1972, Journal Optimization and Theory Applications 10, 237. Hostrup M.; Harper, P. M.; and Gani, R., 1999, Computers Chem. Engng. 23, 1395. Kirkpatrick, S.; Gelatt, C. D., Jr; Vecchi, M. P., 1983, Science, 220, 671. Kocis, G. R. and Grossmann, I. E., 1987, Ind. Engng. Chem. Res. 26, 1869. Kocis, G. R. and Grossmann, I. E., 1988, Ind. Eng. Chem. Res. 27, 1407. Odele, O. and Machietto, S., 1993, Fluid Phase Equilibria. 82, 47. Pistikopoulos, E. N. and Stefanis, S. K., 1998, Computers Chem. Engng. 22, 717. Porn, R. Harjunkoski, I. and Westerlund, T., 1999, Computers Chem. Engng.23, 439. Pretel, E.J.; Lopez, P. A.; Bottini, S. B., 1994, AlChEJ., 40(8), 1349-1360 Ryoo, H. S. and Sahinidis, N. V. (1995). Computers Chem. Engng. 19(5), 551. Sinha, M.; Achenie, L.K.; Ostrovsky G., 1999, Computers Chem. Engng., 23, 1381. Viswanathan, J. and Grossmann, I. E., 1990, Computers Chem. Engng. 14(7), 769. Wang, Y. and Achenie, L. E. K., 2001, ESCAPE-ll, 585. Zhu, Y. and Xu, Z., 1999, Fluid Phase Equilibria, 154, 55.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
1003
Cluster Identification using a Parallel Co-ordinate System for Knowledge Discovery and Nonlinear Optimization K Wang\ A Salhi^ & E S Fraga^ * ^Centre for Process Systems Engineering, Department of Chemical Engineering, UCL (University College London), London WCIE 7JE, U.K. ^Department of Mathematics, University of Essex, Colchester C04 3SQ,U.K.
Abstract The visualization of multi-dimensional data using a Parallel Co-ordinate System is applied to process optimization. A Scan Circle algorithm is proposed for identifying clusters of "close-to-a-line" points in an ^-dimensional space by using the duality between the Cartesian co-ordinate and parallel co-ordinate systems. The lines identified by the algorithm are used to identify regions of interest in the domain of the constraints. These regions of interest can lead to the discovery of knowledge about the behaviour of the complex, nonlinear constraints and this knowledge is used as a basis for a genetic algorithm for efficient and robust nonlinear optimization.
1 Introduction Data visualization plays an indispensable role in uncovering hidden structures or patterns for information extraction tools (Lee and Ong, 1996). Most graphical representations display data in two-dimensional Euclidean space. For data having more than three dimensions, other than drawing all the pair-wise relationship individually, dynamic elements such as colors, shapes and/or motions are normally used to enhance visual effects (Keim, 1997). However, chemical process design problems are so large that direct visualization of entire space through such methods is difficult. The Parallel Co-ordinate System (PCS) representation (Inselberg and Dimsdale, 1990) transforms the search for relations among design variables into a two-dimensional pattern recognition problem, and design points become amenable to visualization. This paper describes the use of PCS as a basis for multi-dimensional data visualization and subsequent knowledge discovery. Moreover, PCS is explored further and a Scan Circle Algorithm (SCA) is proposed to find clusters of "close-to-a-line" points, to identify such lines and hence to locate regions of interest in the problem domain. Visualization and any subsequent knowledge discovery are useful in their own right. However, the knowledge discovered can provide useful information for optimization procedures, improving both reliability and efficiency. We therefore propose the combination of knowledge discovery with stochastic optimization for complex nonlinear process optimization. Genetic Algorithms (GA) (Goldberg, 1989) mimic natural evolution on an abstract level. In contrast to mathematical programming, the Author to whom correspondence should be addressed: e . f raga@ucl . a c . u k
1004 evaluation of a candidate process can be done by simulation, making GAs attractive for optimization in simulation-based chemical process design. Furthermore, nonlinear, multiple and complex objectives can be included. For these reasons, genetic algorithms have attracted much interest in process optimization (Caldas & Norford, 2002). The Scan Circle and genetic algorithms are combined in this work to provide a robust and efficient optimization procedure.
2 Parallel Co-ordinate Systems In a parallel co-ordinate system (PCS), n equally spaced vertical axes are used to represent n variables (Inselberg and Dimsdale, 1990). The distance between all the adjacent axes are assumed to equal one for simplicity of discussion. An ^-dimensional point in PCS is represented by points on the n axes joined by a polygonal curve connecting the n points sequentially. In the A2-dimensional space, a line in the Cartesian co-ordinate system (CCS) can be represented by nA linearly independent equations (Inselberg and Dimsdale, 1990):
^,+1=^,^,+*,
/ = 1,2,...,«-!
(1)
whereas the same line is represented by a set of nA indexed points in PCS with coordinate values ( i/ , '/ ), /• = 1,2, ... , A7-1. yl-my'/l-w/
3 Scan Circle Algorithm The basis of the new procedure is the correspondence between CCS and PCS. Visualization by PCS is used to identify a particular type of pattern, namely clusters of "close-to-a-line" points, in the optimization search space. A novel Scan Circle algorithm (SCA) is proposed here to identify such clusters and lines which represent these clusters with the aim of providing insight into possible solutions to the optimization problem and also to provide support for the development of new optimization algorithms. 3.1 Rationale of SCA As shown in Fig. 1(a), A^ lines in PCS (corresponding to A^ points in CCS) result in A^(A^l)/2 intersections within two adjacent axes. SCA uses the intersections as centres of circles of radius r, called scan-circles. The core of the novel algorithm is the identification of a set of circles which include a large number of intersection points and the selection, from this set, of the best circle. The rationale is that if a point and a line are close to each other in CCS, their counterparts in PCS are even closer (Chou et. al., 1999). If the best circle exists, its centre point will correspond to a line in CCS which represents a "close-to-a-line" cluster of points. To determine whether such a cluster exists in an ^-dimensional space, close proximity of the edges of polygonal lines within two adjacent axes being scanned alone is not sufficient. The entire set of polygonal lines containing these edges must also exhibit close proximity. The full polygonal lines spanning the n axes and which contain these edges must therefore be further analyzed.
1005 3.2 SCA implementation 1. Initialize S to the set of A^ data points. 2. Each pair-wise combination ofn variables is chosen to define a candidate region for analysis. In each region, e.g. the region ij shown as Fig. 1(a), the data points in S are represented by line segments with up to N(N'\)/2 intersections, where A^ is the size of S. Each intersection point defines a circle, of radius r, centred on the intersection point, and which contains a number of other intersection points. The value of r used is selected automatically as the minimum to ensure covering all feasible points. However, infeasible points are also covered, increasing the computational effort here and in the GA step described below. For each candidate region, a potentially interesting circle, the one with the most number of line segments, is selected. The circle with the largest number of line segments from the list of interesting circles is chosen and is known as the best circle. The region that contains the best circle is called the best region. The polygonal lines, whose line segments in the best region result in the intersections within the best circle (indicated by solid lines in Fig. 1(a)), are called the interesting polygonal lines and represent the first guess at a cluster of data points. 3. We now check to see if the interesting polygonal lines identified in step 2 represent clusters of points, with respect to the other variables, by analysing their situation in the other regions (i.e. all k /A regions except the best region). Assuming that region ij was identified as The best circle the best region in step 2 (as indicated in Fig. 1(a)), variable / cannot be selected as the left axis and j as the right axis to define Fig. I. SCA find the best circle automatically candidate regions. All other within candidate regions. combinations are allowed. For each region, we identify a new scan circle considering only the interesting polygonal lines defined in step 2. If the number of the lines that intersect this new circle is greater than a specified number (10% of total feasible points in the results presented below), the region is added to the best region and the lines which define the cluster is the intersection of the original interesting polygonal lines and those which intersect this circle. In the list of remaining regions, region 7/ and all regions with the same left (/) or right (/) variables are removed before selecting the next region to analyse. The termination criterion for this whole step is when the number of lines in the cluster falls below the minimum allowed (e.g. 10% of original data set size) or when n-\ candidate regions have been analysed. At the end of this whole step, a cluster of points and associated pairs of variables will have been identified. Output this cluster. 4. Let SbQ S minus the lines in the cluster found in step 3. If 5 is sufficiently large (greater than 10% of original data set size), go back to step 2. Otherwise, terminate. The result is a set of clusters. Each cluster is defined by a set of data points fi-om which we derive linear relationships between the variables, as described next.
1006 3.3 Interesting Region Identification by SCA As shown in Fig. 2, if three lines, LA, LB, ^O, in CCS are parallel, their corresponding points in PCS lie on a vertical line AOB (O is the center of the best circle). The converse is also true: a vertical line AOB in PCS represents a family of parallel lines in CCS. If the value of jc/ is in [a,b], then the family of parallel lines represents the region CDEF. As line AB slides horizontally from point O to the edge of the best circle, the width of the region covering the parallel lines decreases gradually to 0. CDEF defines a family of different lines transformed from the vertical lines in the best circle. If d represents the distance between point F and LQ in CCS, 1 < ^
< V2 if w < 0 (Chou
et. al., 1999). Therefore, CDEF is represented approximately by a<x.
(w,x, + 6,)-^^2(1 + m^) <x^ <{m,x, + ft,) + r^2(l + m , ' ) .
(2)
To meet with the assumption of w<0, the search of the best circle in each candidate region is restricted to the region between adjacent axes. If the input points for SCA are feasible points, PCS then the region CDEF is CCS identified as being Fig. 2. The feasible region defined by such lines. feasible. If, for a given optimization problem, several lines are found by SCA at the same time, each of these lines represents a linear feasible region. Their combination represents an approximation to the nonlinear feasible region.
4 Optimization By GA Using Feasible Region Knowledge The task in applying a genetic algorithm to a new problem is the definition of the genetic operators, crossover and mutation. In this paper, we focus on the mutation operator. In mutation, a linear feasible region is selected randomly from all of the feasible regions identified by the SCA. We then carry out a Gaussian mutation (Goldberg, 1989) for a randomly selected solution, emulating a toin coss for each variable in the genome. For example, in the feasible region defined by Fig. 2, the variable of JC, is mutated within [a,b]; x, is mutated within the corresponding dynamic feasible region: [m^x^ + b^ - r ^ 2 ( l + m}),m^x^ + b^ + r^2(l + mf)].\f
the coin toss for a
variable indicates that no mutation should be done, this is over-ridden if the variable is not within the feasible region. For the results presented below, the probability of mutation for individual variables, pmut, is 0.5. The total mutation rate (probability of
1007 mutation for members of the population) is 0.1. The crossover rate is 0.9. An overlapping population is used.
5 A Case Study M3 An oil stabilization process (OSP) (McCarthy et al., 2000) is used to demonstrate the new procedures. Fig. 3 shows the flow sheet structure in which, 1, 2, 3 and 4 represeni; flash vessels, M a mixer, and V a pressure; valve. The feed consists of 12 hydrocarbons There are n=5 continuous optimization Ml variables: the flash temperature as a fractiori relative to the bubble and dew points for the; I M2 I • feed to each flash, jci, JC2, ^3, X4, with initial Fig. 3. OSPflowsheetstructure. search region from 0 to 1, and the targei; pressure for the valve, xs, with initial search region of 1 to 30 bar. The objective is to minimize the vapour pressure (P\), at 310.8 K, of the liquid product such that Pv<0.817 bar (the liquid product specification). Jacaranda (Fraga et al., 2000) is used both for the generation of the initial data required by the SCA and visualization procedures and for the evaluation of the objective function. lO'^ random points are generated is > initially within the initial search la region. The feasible points are shown E in Fig. 4 which leads us to define the I reduced search region: Xie(0.5,0.9), JC2G(0.4,0.9), X3e(0.2,0.9), X4G(0.2,0.9)
E]4]
L,
I
and JC5G(1,30). SCA is subsequently
Fig. 4. Visualization of feasible points.
used to define two linear search regions, using r=0.08 (Fig. 5). To evaluate the effectiveness the linear feasible regions, we compare the results obtained by GAs with the mutation operator based on (A) the initial search region, (B) the reduced search region and (C) the linear search region. In all cases, the GA jC3e (0.27,0.60) JC4=-0.90 JC3+0.94 ±0.20 ;cie(0.61,0.71) JC2=-0.98 jci4-1.30 ±0.20 ;c'5=-2.42 ;c2+2.11 ±0.37 JC5= ;c'5(30-l)+l
Fig. 5. The linear
jC3e(0.50,0.80) JC4=-0.94 ;c3+1.00 ±0.20 ;cie(0.65,0.80) JC2=-1.08 jci+1.40 ±0.20 jc'5=-3.37 JC2+1.95 ±0.41 JC5= JC'5 (30-1)4-1
search regions.
population size is 100 and a convergence termination criterion is used. The GA is run 10 times in each case and the distribution of the resuUs and the standard deviation are shown in Fig. 6. The stopping criterion came into effect after approximately 20 generations in
1008 all cases. The results indicate that the consistency of the GA is improved through the incorporation of knowledge of the feasible region.
0.6
0.4 !
6 Conclusions 0.2
t
A scan circle algorithm has been proposed for identifying interesting regions of (C) (B) (A) feasibility based on parallel Case co-ordinate system multiFig. 6. Distribution of GA results for different dimensional data cases. visualization. The regions of interest provide some knowledge about the behaviour of the complex, nonlinear constraints and can be used as a basis for the definition of genetic operators for a genetic algorithm. We have shown how the mutation operator may be adapted to exploit such knowledge. One industrial case has demonstrated that the combination of the scan circle and genetic algorithms leads to more effective optimization.
Acknowledgments Funding provided by the EPSRC and valuable input from BP Amoco are gratefully acknowledged.
References Caldas L. G. & L.K. Norford (2002), A Design Optimization Tool Based on A Genetic Algorithm, Automation In Construction, 11 173-184. Chou S.Y., S.W. Lin & C.S. Yeh (1999), Cluster identification with parallel coordinates. Pattern Recognition Letters, 20 565-572. Fraga, E. S., M. A. Steffens, I. D. L. Bogle & A. K. Hind (2000), An object oriented framework for process synthesis and optimization, in "Foundations of Computer-Alded Process Design," M. F. Malone, J. A. Trainham, & B. Camahan (Editors), AIChE Symposium Series 323(96) 446-449. Goldberg, D.E. (1989), Genetic algorithms in search optimization and machine Learning, Addison Wesley, Reading. Inselberg A. & B. Dimsdale (1990), Parallel Coordinates: A tool for visualizing multidimensional geometry, in Proceedings of the First IEEE Conference on Visualization, 361-378. Keim D.A. (1997), Visual database exploration Techniques, in Proceedings of Tutorial Int. Conf On Knowledge Discovery and Data Mining (KDD'97), Newport Beach, CA, http://www.informatik.uni-halle.de/~keim/PS/KDD97.pdf Lee H.Y. & H.L. Ong (1996). Visualization support for data mining, IEEE Expert 11(5) 69-75. McCarthy E. C , E. S. Fraga & J. W. Ponton (1998), An automated procedure for multicomponent product synthesis. Computers chem. Engng 22(Suppl.) 877-884.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
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Application of CFD on a Catalytic Rotating Basket Reactor WarnaJ.^^ Ronnholm M.^\ Salmi T.^\ Keikko K.^\ ^^Laboratory of Industrial Chemistry, Process Chemistry Group, Abo Akademi University, Biskopsgatan 8, 20500 Turku Finland, Fax: +358-2-2154479, e-mail: [email protected] ^^Kemira Chemicals, Box 171, 90101 Oulu Finland Keywords Catalytic Rotating Basket Reactor, CFD
Abstract Using catalytic rotating basket reactors is often preferred instead of catalytic slurry reactors. This is especially true in the case of testing a specified catalyst particle size as the catalyst particles remain the same size during the experiment and they are not crushed as much as in a slurry reactor. However, in a rotating basket reactor, the liquidsolid mass transfer rate is poorer than in conventional slurry reactors The use of rotating catalyst basket reactors in experimental work was introduced by Carberry ( Carberry 1964). Different designs for spinning basket reactors for two- and three phase reactions have been presented. Turek and Winter (Turek and Winter 1990) made a comparison between two types of spinning basket reactors to determine the hydrogenation rate of butynediole with a nickel catalyst. Kawakami et. al. (Kawakami et. al.l976) studied a semibatch hydrogenation reaction of styrene in a basket reactor. The goal with the different designs of rotating catalyst basket reactors is to achieve a sufficiently high flow rate of the fluid over the catalyst particles. To build and test different reactor designs is costly and time consuming. With the use of modern CFD tools it is possible to investigate different reactor designs with a computer model and finally build the reactor that shows the best performance. In order to enhance the reactor performance, CFD calculations were used to estimate the liquid-solid mass transfer rates inside the rotating catalyst basket. Different catalyst basket designs were screened. The reactor design that showed the best performance was built and it was tested with catalytic esterification.
1. Rotating basket reactor The reactor consists of the reactor vessel and a rotating catalyst basket connected to the shaft. The catalyst basket is made of a metal net. The size of the rotating basket reactor was 100 mm in diameter and 150 mm in height. The liquid amount loaded into the reactor was 0.5 1 and the catalyst amount was 100 g. The rotating basket had four paddles on the side of the basket and four paddles at the bottom of the basket; the bottom and side paddles were tilted 45° (Fig. 1). The basket was rotating in anticlockwise direction with a rotation speed of 320 rpm. The reactor operates in batch mode.
1010
i
Figure 1. Top and side view of reactor 2. Two reactors were used in this work. Reactor I was the original design of the laboratory test reactor, with no paddles in the middle at the rotation axis, but instead this space was filled with catalyst particles. Reactor 2 was constructed according to the CFD calculations that showed better liquid-solid mass transfer rates, as paddles were inserted in the middle of the reactor as shown in Fig 1.
2. Liquid-solid mass transfer In order to compare the performance of the different reactor designs the liquid-solid mass transfer rates in the catalyst area were calculated. To estimate the limitations set by liquid-solid mass transfer, it is appropriate to use correlations for ks of the form (Beenackers, 1993): d k (1) S/l = = 2 + aRe"Sc^ D where Sh, Re and Sc denote the Sherwood, Reynolds and Schmidt numbers, respectively. The diffusion coefficients were calculated from the Wilke-Chang equation. (Reid et al, 1988) The mass transfer coefficient ks was obtained from eq. (1), Reynolds number, however, must be calculated with the actual velocity v' that the liquid passes the catalyst particles. The corrected velocity can be calculated from :
•=^f v ? + v ?
+ (^or-^c)
(2)
where Vrot is the rotation velocity of the basket in the calculation point and Vx,Vr and v^j are the velocities in the 3D coordinate system. The mass transfer rates kg are shown in Fig (2), and it can be seen that the mass transfer rates are higher in the catalyst area in reactor 2.
1011
0.01
0.02
0.03
0.04
0.05
Figure 2. Mass transfer rates in the catalyst basket as a function of radial position. Reactor 1 (squares), reactor 2 (triangles)
3. Pressure drop and physical properties The pressure drop in the catalyst area was calculated with a correlation equation. The friction factor was estimated from (Fried, 1989). r64
1.8 I, (3)
/ = 1.5 The flow resistance in porous areas (catalyst area) is accounted for with body forces in the CFD program code. The body force is obtained from:
5=
(4)
The body force caused by pressure drop is added to other body forces, i.e. centrifugal forces caused by rotation. The total body forces are included in the momentum transfer equation of the CFD program. The influence of the viscosity and density on the flow field was investigated by running simulations with viscosity and density values in the range 0.3 10'^ N s W - 2.22 10" N s W , 0.66 kg/dm"^ -1.0 kg/dm\ respectively. In this viscosity and density range only very small differences in the flow fields were observed.
4. CFD model The CFD calculations were performed with the software CFX-4.4. The turbulent k-E method was applied in the area outside the rotating catalyst basket, while laminar and porous flow methods were used to describe the flow in the catalyst area. Reynolds value in the catalyst area was in the range 100-1500. The flow profiles in the reactor were modelled using a 90° sector (Fig 3) of the cylindrical reactor and assuming a four-fold flow periodicity. The rotation of the catalyst basket was calculated with both the sliding grid and the multiple frames of reference (MFR) methods (AEA Technology, 2001). The MFR method was found to be much more rapid than the sliding grid method. The number of calculation elements in the 90° sector was 51450.
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Figure 3 Calculation domain Flow fields in a plane in the middle of the reactor show that the flow direction in reactor 2 is from the centre of the reactor through the catalyst towards the walls. In reactor 1 the flow direction for the liquid is mainly the same as the rotation direction as shown in Fig 4
Figure 4 Flow directions in reactor 2 (left) and reactor 1 (right) in the centre of the reactor.
5. Chemical reaction Chemical kinetics was also implemented in the CFD model. It showed out that, predicting the flow fields, catalyst basket rotation and the chemical reactions simultaneously required a lot of calculation time. Instead the flow profiles were firstly predicted in the absence of chemical reactions. Chemical reactions were then added and the obtained flow field was kept constant during the chemical reaction simulation. In this way, the calculation times became reasonable. A catalytic reversible reaction: esterification of acetic acid (Maki-Arvela et. al. 1999) over Amberlyst 15 catalyst, showed improved performance with the reactor 2 design shown in Fig. 6. By including the reaction rates in the CFD program the reaction in the reactor could also be simulated with the CFD program. The simulations showed a good
1013 agreement with experimental data presented in Fig. 5. Reaction zones could also be studied from the CFD calculations.
100
150
Time (min)
Figure 5 CFD calculation versus experimental data. Weight fraction as a function of time.
Figure 6 Experiment with reactor 1 (lower line) and reactor 2 (upper line). Weight % as a function of mass of catalyst * time.
6. Conclusions A rotating catalytic basket reactor was modelled with CFD in order to improve the performance of the reactor. The CFD calculations showed that liquid-solid mass transfer rates could be improved by a change of the design of the rotating catalyst basket. The new and old reactor designs were compared with experiments with a catalytic esterification reaction and it was shown that the new reactor design gave an improved performance.
Acknowledgements This work is part of the activities at the Abo Akademi Process Chemistry Group within the Finnish Centre of Excellence Programme (2000-2005) by the Academy of Finland. The financial support from TEKES (Technology Development Centre) is gratefully acknowledged.
Symbols ap
B dp
D f
K ksap
T V
Mass transfer area / volume Body force Catalyst particle diameter Diffusion coefficient Friction factor (Eq. 3) Liquid-solid mass transfer coefficient Mass transfer coefficient, ksap=ks6/dp Temperature Velocity
m-^ kg/m m m^/s m-^ m/s S-'
K m/s
1014 Greek letters e Porosity p Density
kg/m^
Subscripts X, r, (0 Indicating flow direction in axial, radial and rotation direction
References Carberry, J. J. Designing Laboratory Catalytic Reactors. Ind. Eng. Chem. 1964, 56, 39 Turek F., Winter H., Effectiveness Factor in a Three-Phase Spinning Basket Reactor: Hydrogenation of Butynediol. Ind. Eng. Chem. Res., Vol. 29, No. 7,pp 15461549, 1990 Kawakami K., Ura S., Kusonki K., The Effectiveness Factor of a Catalyst Pellet in the Liquid-Phase Hydrogenation of Styrene. Chem. Eng. Jpn. 9, 392, 1976 Beenackers, A. A. C. M. & van Swaaij W. P. M., Mass Transfer in Gas-Liquid Slurry Reactors, Chem. Eng. Sci. Vol. 48, pp. 3109-3193, 1993 Reid, R. C , Prausnitz, J. M., Poling, B. E., , The Properties of Gases and Liquids, 4* Ed., McGraw-Hill Book Company, New York, 1988. Fried, E. & Idelchik, I., Flow Resistance: A Design Guide For Engineers. Hemisphere Publishing Corporation., 1989 CFX-4.4: Solver, AEA Technology, 2001 Maki-Arvela, P., Salmi, T., Sundell, M., Ekman, K., Peltonen, R., Lehtonen, J., Comparison of polyvinylbenzene an polyolefm supported sulphonic acid catalyst in the esterification of acetic acid. Applied Catalysis A: General 184 (1999)25-32 Baldyga, J., & Bourne J., R,., Turbulent Mixing and Chemical Reactions, Wiley, 1999 Brucato, A. et. al. Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modelling approaches, Chem. Eng. Sci. Vol 53, No. 21, pp. 3653-3684, 1998 Harris , C. K., Roeckaerts, D., Rosendal, F. J. J., Computational Fluid Dynamics for Chemical Reactor Engineering, Chem. Eng. Sci. Vol. 51, pp. 1569-1594, 1996 Kluytmans, J. H. J., Markusse A. P., Kuster B. F. M., Marin G. B., and Schouten J. C , Engineering Aspects of the Aqueous Noble Metal Catalysed Alcohol Oxidation, Catalysis Today, vol. 57, pp. 143-155, 2000
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
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A post-graduate study in Process Design: An Innovative Model in the Netherlands Johan Grievink, Giljam Bierman, Cees P. Luteijn and Peter J.T. Verheijen Faculty of Applied Sciences, Department of Chemical Technology Delft University of Technology, 2600 GA, Delft, The Netherlands
Abstract The design of sustainable, cost effective processes, with perfect control of product quality, calls for strongly enhanced capabilities of designers. A thorough training in both fundamentals of design methods and CAPE tools as well as in the associated work processes is required to become an expert designer. Manufacturing and engineering companies in the Netherlands have indicated that a more advanced training in design is needed to meet the upper range of design capabilities. The Dutch chemical engineering departments responded by developing specialised two-year postgraduate studies in process systems design. As an example the structure and contents of a post-graduate study in process design at Delft University of Technology is highlighted and evaluated.
1. Introduction This paper describes and evaluates a higher level of engineering education in process design, as developed and practised by the chemical engineering departments of the technical universities in The Netherlands. It is a relatively new branch of the educational model for the engineering sciences with a clear focus on the integrative features of design. Furthermore, it offers a higher level of expertise to start a career in chemical engineering. This educational innovation must be placed against the background of the conventional academic engineering education on the European continent. About a fifteen years ago the manufacturing industry in the Netherlands pointed out that in the future a higher level of expertise in design of complex technological systems was needed, for reasons outlined in section 2. The technical universities have responded by developing post-graduate studies in the design of technological artefacts, as a follow-up to the graduate MSc level. The objectives of these two-year, full time studies are reviewed in section 3. Taking the post-graduate study developed by the Chemical Technology Department of Delft University of Technology as an example, the specifics of this study are explained with particular attention to CAPE elements (section 4). The experiences with this study are predominantly positive (section 5). Yet, the rapid pace of change in CAPE technology in particular and the process industry in general calls for continued developments in order to remain effective (section 6).
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2. Increased design capabilities: needs and means 2.1 Wider horizons in the Process Industry The drive for a sustainable society and for products with a high customer satisfaction with respect to cost and quality challenges scientists and engineers to think and act over much wider horizons than before. For example, the design and operation of chemical processes must integrate at least four different perspectives. • The product life span A chemical product (e.g. a polymer) passes through three processes during its life span: (1) manufacturing, (2) application and (3) recycling or recovery (of spent product). The design of each of these processes must acknowledge the functional requirements of the other two processes and contribute to an optimisation over the entire product life span. • Integration in a supply chain Many plants are imbedded in a supply chain on a site or within an enterprise. The main product is often an intermediate, to serve as a feed for other plants or to be sold as a finished product. The supply chain will exert its influence in the design and operation of such a plant by imposing additional constraints and economic trade-offs along the chain. • Optimisation over process life span The physical life of a plant can be much longer than the economic life span of a product (grade), implying that the plant must be flexible enough to handle different product grades. Furthermore, minimisation of life cycle costs forces one to strike a balance between investments in equipment and the resulting contributions to operational costs. • Integration over multiple scales within and outside a process. The concept of a process as just a connection of unit operations is not sufficient anymore. At the upper scale (in time, space) integration between a process and the site (utilities, supply chain) is required, in both design and operations. However, it is at the lower scales that new opportunities arise for better control of product properties and for process innovations. Especially, processes for the manufacture of products with an internal spatial structure (e.g. polymers, crystals, emulsions, and catalysts) demand a multi-scale approach to the simultaneous designs of product and process. In summary, the increasing interactions between a process and its business environment and the needs for a more finely tuned internal process structure call for enhanced design capabilities to integrate the multi-scale features. 2.2 CAPE related innovations This growth in demands from industrial practice has stimulated new developments in process engineering. In addition, the autonomous developments in computing and information technology have given rise to a wide array of new methods and tools for process design, control and optimisation of operations. Grossmann and Westerberg (2000) present a list of some major accomplishments in the PSE/CAPE area. However, the level of abstraction of many methods and the complexity of the computational schemes of these tools are high. To become a reliable, expert user of these tools one
1017 requires a thorough training in the underlying fundamentals. The graduate education in chemical engineering (to the MSc level) does not offer enough room for such training.
3. Innovation in design education at Dutch universities. 3.1 Structure of university education in design In late eighties the Netherlands industry were concerned about graduate education in engineering not being able to meet the upper range of the future demands for design capability. The main reason being the existing academic focus on developing capabilities for research rather than design. The education gives high priority to acquiring fundamental knowledge in the pertinent domains of natural sciences and engineering and to developing strong analytical skills. Although room is made for an introduction to design, it is not enough to confidently deal with the design complexity of modern technological systems. A doctorate in engineering research is not fully adequate for design either, although research on design methods and tools creates an excellent fundamental understanding in general. But a doctorate typically leads to highly trained specialists who can go in great detail, rather than pursuing the broader, integrative approach as needed for design. Furthermore, the organisational and social aspects of the work processes in design often fall beyond the scope of recognised research. To close this gap the technical universities established a specialised post-graduate study in design of complex technological systems. It is a two-year, full time study, starting from a MSc level in engineering. This study is a generic complement to a Doctorate in engineering. The study is (partially) funded by the government, who gave equal status to the design and research students. The completion of an post-graduate design study is rewarded with a certified degree 'Master in Technological Design'. The design objects can be technical products, processes and systems for control and logistics. The study has a focus on the integrative aspects of systems engineering. The key objective is to learn and master the design process, including its associated methods, tools and procedures. 3.2 Objectives The objectives of the advanced design study can be further specified by looking at aspects of the designer's conceptual environment and their mental outfit. • Interface between a designer and the business and society in general. To be capable of developing a consistent Basis of Design, including different performance criteria of economic, ecological and technological nature. • Design methods, tools and technical procedures. To get a thorough understanding and mastery of the technical aspects of design with a focus on the conceptual phase. • Work processes and project organisation. To understand and experiment with the social, organisational and management aspects of the work processes. • Knowledge of relevant disciplines of science and engineering. To extend and deepen knowledge of a relevant mix of disciplines. • Individual characteristics. To understand and train individual characteristics (creativity, social and management).
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4. Post-graduate chemical process design in Delft 4.1 Entrance level The preferred entrance level of expertise in design can be characterised by briefly eviewing the graduate education in Chemical Technology at Delft University as a typical example. Representative process engineering subjects in the core curriculum are: Chemical process technology Risk analysis and management; Process dynamics and control; Computer laboratory in process simulations; Process systems design (based on Douglas, 1988) or, chemical product design. The integration and application of the acquired knowledge takes place at the end of the fourth year by means of the: • Conceptual Process Design project (12 weeks effort in a team of four students) The courses and projects in process design are accredited by the British Institution of Chemical Engineering and ABET. While at this stage of education the students obtain the experience of making a design of a (conventional) continuous process with simple products (e.g. base chemicals), they do not yet master the design process in general. Students admitted to the post-graduate study end coming from other universities have a similar profile of initial design expertise. 4.2 Post-graduate study in Process and Equipment design The post-graduate study aims for higher levels of abstraction, understanding and creativity, i.e. to master the design process. The focus remains on the conceptual design stage. A post-graduate design student learns: (a) To think in life span and supply cliain dimensions (as outlined in section 2.1); (b) To master methods and CAPE tools for the design and integration of a process, involving synthesis, analysis, evaluation and optimisation steps for the process and its control, (c) Deal with the non-technical elements of the design process. These latter two points reflect the generic objectives outlines in section 3.2. A more complete overview of the study and its organisation can be found at the web-site: http://www.dct.tudelft.nl, looking under Delft Ingenious Design. 4.3 Post-graduate study: first year courses and design project The major part of the first year (28 weeks) is spent on knowledge expansion, both on engineering and non-technical topics. This expansion will be tailored to the needs of the individual student, depending on the profile acquired in the preceding education. The non-technical topics involve, among others, project organisation, economic evaluations in the process industry, presentation and writing skills. Some CAPE topics are: • Advanced process design (e.g. Biegler et.al., 1997; Seider et.al. 1999) • Process & heat integration (e.g. Smith, 1995) • Loss prevention and sustainable technology. • Thermodynamics for designers • Process modelling and model analysis (e.g. Hangos & Cameron, 2001)
1019 • Design of plant wide control (e.g. Luyben et.al., 1998) • Optimisation of Chemical Processes (e.g. Edgar, et.al., 2001); All teaching is in English, using international text books. For each course the students are also trained in using the associated CAPE tools, e.g. process dynamic simulators, heat integration design, control design. In addition to these topics students can take courses in catalysis and reactor engineering, and in separation technology. The remainder of the first year (15 weeks effort) is devoted to a group design project, in which project management and teamwork can be practiced. A manufacturing or an engineering contracting firm usually commissions the design problem; e.g. design of a Gas-to-Liquid plant. The basis of design has to be extracted from the problem owners in a negotiation process. Design alternatives are to be generated, evaluated and compared. In addition to writing a design report, the results must be orally presented and defended to some experienced process designers of the commissioning firm. 4.4 Post-graduate study: second year design project The full second year is spent on an individual design project (42 weeks) in an industrial setting on a contract basis. Here, the challenge is to demarcate a tractable design problem in a real life environment and solve it adequately in time and with limited, available means. The design is often combined with some experimental process development work to generate data for design or with optimisation of plant operations. This project can be carried out on the site of the commissioning firm or in a university laboratory. The design student is responsible for managing his own project. Consultants to the project advise the student, which will be academic staff and a process engineer of the commissioning firm. Some examples of recent design projects are: • Degassing process for plastics; • New process for IPA production; • ETBE recovery using permeative distillation; • Debottlenecking of a desulphurisation unit; • Processes for food products (e.g., Yadhav et al., 2002).
5. Experiences and evaluation 5.1 Recruitment of students Since the start in 1991 the number of applicants have exceeded the number of available places for many years. However, the number of admitted students with suitable qualifications was lower than the maximum capacity (-15 students per year). This screening for admittance has kept the attrition rate below the ten-percent. Since 1998 it is difficult to attract enough qualified students from the Netherlands: • Low number of chemical engineering students (~ 60 % drop in 5 years); • The short term prospect of making more money in industry as junior engineer; • Concerns about international recognition of this type of study, since it deviates from the international education pattern BSc, MSc and PhD. Yet, recruiting on an international scale appears to be effective; most design students are now from abroad. Although this study also offers a nice opportunity for process design engineers to take a break from an industrial job and upgrade their knowledge, this has hardly happened so far. It is an option to be developed for the future.
1020 5.2 Starting positions of designers About 40 % ends up in the process industry, 30 % with engineering-contracting firms, 7 % goes to consultancy firms and 7 % goes to work for (semi-) governmental institutes of technology. Less than 7 % stays on at a university (for a PhD degree). Industry recognises the added value of these designers by the fact that they are found to be effective almost immediately after their start. This is reflected in starting salaries and rate of rise. 5.3 Quality control with industrial feed back Every five years a committee of the National Accreditation Board reviews the quality of the post-graduate design studies. Each committee includes design experts from the industry. For the Delft course these experts have given recommendations with respect to its contents; i.e. putting more emphasis on process dynamics and plant wide control design and on project management skills. The appreciation for these design study is also reflected by companies volunteering to offer design projects.
6. Future developments This post-graduate design study has proven effective from an educational point of view. The process industry recognise the advanced skills of these designers. Attracting enough talented students and finding more international recognition remain challenges. The technical contents of the courses will be adapted to anticipate the developments in the process industry. More attention need to be given to processes for structured products and for multiple modes of production. Last but not least, the advances in computer aided product and process synthesis need to be captured in the teaching.
7. References Biegler, L.T., I.E.Grossman, A.W.Westerberg, 1997, Systematic methods of process design. Prentice Hall Douglas, J.M., 1988, Conceptual Design of Chemical Processes, McGraw-Hill. Edgar, Th.F., D.M.Himmelblau, L.S.Lasdon, 2001, Optimization of chemical processes, McGraw-Hill Grossmann, I.E., A.W.Westerberg, 2000, AIChE J., p. 1700, Research challenges in process systems engineering. Vol. 46. Hangos, K., I.Cameron, 2001, Process modelling and model analysis, PSE Volume 4, Academic Press. Yadhav, N., M.L.M.vander Stappen, R.Boom, G.Bierman and J.Grievink, 2002, Conceptual design of processes for structured food products, submitted to ESCAPE-12. Luyben, W.L., B.D.Tyreus, M.L.Luyben, 1998, Plantwide process control, McGrawHill Seider, W.D., J.D.Seader, D.R.Lewin, 1999, Process design principles. Synthesis, analysis, and evaluation, Wiley & Sons, Inc.. Smith, R., 1995, Chemical process design, McGraw-Hill.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
1021
Preconceptions and Typical Problems of Teaching with Process Simulators Laureano Jimenez and Rene Banares-Alcantara Chem. Eng. Dept, University Rovira i Virgili, Av. PaTsos Catalans 26, 43007 Tarragona, Spain. Tel.: +34-977-559617; Fax: x9667/21; e-mail: Ijimenez, [email protected]
Abstract The basic input to any process simulator is the result of a set of decisions. For this reason, it is convenient to provide students with a road map to assist them in the solution of the typical dilemmas. In particular, continuous effort to foster off-line work (conceptual design, process synthesis) is required. Overall, the use of accurate physical properties and a correct selection of models for property estimation are key factors to succeed in process modelling, although they are frequently neglected.
1. What is Process Modelling? Commercially available process simulators (HYSYS®, ASPENPLUS®, CHEMCAD®, PRO-II®...) help to solve, once all degrees of freedom are fixed, the mass and energy balance of a process with rigorous physical property calculations, kinetic considerations and detailed unit operation models. The simulator calculates the flowrate and compositions of the process streams, size/rates equipment, predict the operation variables and the dynamic behaviour. The use of process simulators should result in the increase of the engineer's efficiency, and improve the opportunities to discern between configurations, to optimise variables and to explore the dynamic behaviour of the process. Traditionally, the faculty staff stress the emphasis in the design of individual units, perceived as the core characteristics of the design, while the considered ancillary issues (such as conceptual design and process synthesis) are left aside, despite the fact that the capital costs are fixed when the P&ID is established, prior to the final modelling.
2. Methodological Aspects Computer aided process design plays a critical role in bridging the gap between theory and practice, as students face practical problems (operability, dynamic behaviour) and connect scatter units as a whole (separation units, reactors, and controllers). Technical competence is no longer sufficient to fulfil these objectives if it is not combined with soft skills (decision-making, problem solving, teamwork, communication and management abilities). To increase the retention and improve the integration of concepts, the design is organised as a 'stop, think and go' procedure. This frame forces students to analyse, interpret and extract information to establish or re-direct decisions. The traditional teacher and student roles change. Students assume increasing responsibility thus enhancing motivation. In turn, the faculty staff role is to guide
1022 students to prevent misconceptions, rather than to transmit formal knowledge to passive students (they freely operate the simulator without interference unless they fall in a dead-end situation). As the course advances, Socratic questions are posed to enhance critical thinking. In this way self-confidence is improved and autonomy is encouraged.
3, Decision-Making The process simulator should not be used as a black box, because it requires a sequence of decisions: selection of a thermodynamic model, setting of the flowsheet, fixing the operating conditions, sizing and rating equipment or setting the control strategy. Conflict resolution has to be used, as often several criteria point in irreconcilable directions and their relative importance has to be balanced. In addition, it is not clear when/if real behaviour has to be introduced (e.g. subcooling, efficiency, leakage or entrainment). For a beginner, a too agile mouse operation may foster uncertainty (students are used to solve close-ended problems and they are suddenly required to solve ill-defined problems), insecurity (students have to surf over an avalanche of results of varying relevance) and helplessness (students may choke with the huge amount of values). As a general rule, we can state that each mouse click is equivalent to a decision that must be supported by arguments.
4. What Does a Physical Property Model Imply? Students perceive the physical property package selection as a wild territory. We should transmit students the idea that absolutely all properties (for pure components and mixtures) are computed based on these predictions. For this reason we suggest to validate each model in a two level approach, with respect to physical properties (pure component, VLB, LLE) and to plant data. In addition, and depending on the unit operation to model, the stress is switched to different properties (distillation: VLB; extraction: LLB; pumping: viscosity and density; reaction: kinetics and enthalpy; heat exchangers: specific heat and latent heat, etc.). A list of the typical issues to take into account follows: • How to model the vapour phase? Systems with volatile organic acids (e.g. acetic acid), aldehydes or HCN the Hayden-O'Connell or Nothnagel models must be selected due to vapour phase association. HF requires a specific model due to the formation of hexamers. • Number of phases. Computing efficiency and convergence are improved if the presence of two/three phases is provided a priori. • Are there components at trace level? In this case, special emphasis must be taken in the validation. In addition, tolerance can be relaxed to improve convergence. • What to do with supercritical components? If supercritical gases are present, Henry's law in conjunction with an activity coefficient model should be used to calculate the solubility. Most equations of state can predict this behaviour without further consideration. • Modelling oil. Petroleum and refinery products are systems with hundreds of non-polar components and thus it is convenient to estimate their properties in
1023
•
•
•
•
terms of pseudo-components. The critical step is the oil characterisation from the assay data curves (TBP and ASTM distillation, density, viscosity, sulphur content), and it is imperative to focus on the discretisation methods. Presence of electrolytes. Ions do not participate directly in the phase equilibrium, but influence the activity coefficients of the other species because of their possibility to combine and precipitate. Typical electrolyte models are Pitzer (P<10 atm and Ci< 6M), NRTL-modified and Bromley-Pitzer (predictive). Simulation with solids. Each industrial sector describes the solids with specific properties (e.g. length of the cellulose fiber in the paper industry). Models are based on these attributes to characterise the solid and estimate any other property. Transport properties are important, as solid products require both the chemical composition and particle size distribution to be defined. The component is not in the database! Different alternatives are possible (e.g. use predictive models or simplify the mixture) depending on the importance of the missing component in the system. If a predictive model is used, it is strongly recommended to provide the simulator at least with accurate TB, MW and critical constants since all models are very sensitive to these parameters. What can I do if there is a lack of properties? We can not assume that all parameters are available just because no error message appeared. If additional properties are needed it is recommended to: a) search in databases revised by editors (DIPPR, DECHEMA); b) search in non-revised databanks (Reid, Perry, CRC); c) obtain experimental data, or d) use predictive models (Carlson, 1996).
4.1. Criteria to select physical property estimation methods Due to the great casuistry (some methods are not recommended with certain components, while others are incompatible), selecting a method is a complex task (Ballinger et al, 1994; Carlson, 1996; Agarwall et at, 2001; Satyro et al, 2001). Methods can be clustered in equations of state and activity coefficient models, their applicability is compared in Table 1. The commercial process simulators provide a vague selection procedure (HYSYS , ASPENPLUS®, Pro-II®). Only CHEMCAD® has a selection algorithm implemented within the software. However there are several applications (5, 6) that can assist in these tasks: 1. Select the components and requests information about their importance. 2. Search for the nature of the properties of interest. 3. Assess the range of pressure, temperature and composition. 4. Check the available parameters and perform a model discrimination procedure. The parameters that are not available are estimated by a purely predictive method, e.g. group contribution methods. 4.2. When are purely predictive models recommended? The purely predictive models are the unique option when qualitative results are sufficient or when no data is available. These models are based on the structural information of the components, and the interactions between functional groups are computed based on average values. They should be used with caution and their predictions must be validated.
1024 Table 1. - Comparison between equations of state and activity coefficient models. Equations of state Activity coefficient models T . . J . i..,Valid for highly non-ideal mixtures Limited in ability in representing non/T^ i A^ . ^ T? r» m . i . , , , . , , . / ^ ,^ (P<10 atm). If P > 10 atm use purely ideal binary liquidsparameters (polar components) i- binary • .i Few required Many parameters required ' ^ r / predictive models Parameters extrapolate reasonably with Binary parameters are highly temperature temperature dependent Consistent in the critical region Inconsistent in the critical region 4.3. Models for special systems Some systems (e.g. absorption of acidic gases with amines, water and steam) present deviations so particular that it has been necessary develop specific models valid in a certain range of pressure, temperature and composition. 4.4. Why select different thermodynamic models in a case? The operation of the chemical plants can be divided in different sections (e.g. reaction, separation and utilities), each one with certain particularities. Accordingly, it is frequent to use different thermodynamic models in each subsection, due to the disparity of objectives, components and operating conditions.
5. Convergence For vendor companies the mathematical and thermodynamic methods are public, and the access to databases is obtained through commercial agreements. Thereby, one of the key factors in simulation is the robustness of the calculation methods. Some aspects to observe in order to improve convergence are to verify the information introduced, check the number of phases, provide estimated values, partition and stream tearing or modify the integration interval in a dynamic simulation. A common problem is to request for an unfeasible separation (e.g. disregard an azeotrope in a distillation column), usually pointing the finger to a convergence problem. These situations fosters the importance of the off-line work.
6. Preconceptions It is common to start modelling just by sitting in front of a computer, without previously applying any systematic off-line work (e.g. search for information, planning and conceptual design). In this situation, students may feel that approaching process simulation relies purely on experience or is a question of luck, and a trial and error approach is reinforced. On the contrary, it is necessary to use a systematic procedure to detect preconceptions (P&ID ^ PFD, recycle ^ tear stream, unit operation ^ icon). Students often fail in their first approximation to process simulation because their notion of degrees of freedom is not clear, and they fail to conclude that the simulator requires exactly the same information than with the 'by hand" approach. For instance, students have problems in identifying why each type of reactor (e.g. PFR, equilibrium) only accepts certain type of reactions (e.g kinetics, equilibrium). Counterexamples are used to help students to detect and correct erroneous preconceptions, in particular the
1025 implication of the flow-pressure relationship in dynamic modelling and that if a case had converged that means the results are feasible. In addition, the simulator is perceived as a labyrinth and a road map is provided (Figure 1). 6.1. Good practices in process simulation • Build the PFD in several phases, make modifications step by step and approach problems by first isolating them. • Classify components in high, medium and low priority; explore the properties prediction of pairs of high/high priority components and their combinations. Estimate the non-available parameters. • Divide the process and product specifications into fixed, flexible and modifiable. • Discern among different process alternatives (sensitivity analysis, optimisation).
7. Size Matters! All units must be rated according to technical criteria (velocity for a pipe, flooding percentage for the column diameter, volume for a CSTR/PFR). Many problems that appear during dynamic simulation are due to incorrect sizing (flooding, reverse flow, uncontrollable valves).
8. Control Strategy There is a growing recognition of the need to consider the controllability of a chemical process during its design. Setting the appropriate control strategy according to each objective (P, PI, PID, cascade, ratio) is performed off-line. It is a common mistake to consider all control schemes with the same importance (e.g. the reflux flowrate is controlled by the accumulator liquid level, thus indicating a lack of understanding of distillation; or reactor feeds that are controlled independently, without any ratio consideration).
jng) ^^^ Setup: units, reports...
1
Property package: EOSorVi?
1_
Components: news or from the datat)ase?
1
^9 Unit sizing and rating
Stream definition: q, T, P & X, or q„ T & P
Unit definition: degrees of freedom?
Set the PFD: connect units
^ '.
Set the control strategy: P, PI, PID or cascade?
Figure 1. Road map for process simulation.
:9_
Robust analysis: ±10% process upsets
1026 The controller parameters can be tuned with different models. The auto tuning value method has a good performance for simple controllers (level, pressure), while for complex configurations the open or closed Ziegler-Nichols strategy is commonly used.
9. Steady State is Not Enough Anomalous, ambiguous and/or contradictory situations in steady state are usually detected during dynamic simulation, such as multiple stationary states (reactive distillation), or reverse flow. Also, dynamic analyses expose the need for extra equipment (for start-up or shut-down), exhibit typical asymmetric behaviour and help to select appropriate control variables. Among all the advantages of dynamic simulation, the main one is the possibility to analyse the system robustness for typical process upsets (±10%). In addition, the model can check if upstream or downstream units bottleneck the process.
10. The Analysis of Results It is fundamental to foster in students the use of graphical analysis techniques (trends, behaviours and profiles). In this way, results are found more quickly, and in a fashion that may allow small problems to slip-in but big mistakes to be detected. In particular, beginners are treading a thin line separating fact from fiction, where the reality is the interpretation of the simulation results, based on knowledge and rigor.
11. Conclusions Process simulation is not an objective in itself, but a tool that avoids the most repetitive, routine and heavy tasks. Its main problems in education are that the necessary computer requirements are expensive and that it is difficult to dedicate enough teaching effort for groups with many students. The use of a simulator is not exempt of risks, since there is the possibility to act as a video game player, without knowing the underlying assumptions, applying the necessary criteria and without carrying out critical analysis of the results (reality is the interpretation of the simulation results). To achieve good results in process simulation it is necessary to foster off-line work and emphasise the appropriate selection of physical property models.
12. References Agarwall, R., Y.-K. Li, O. Santollani, M. A. Satyro and A. Vieler, 2001, Chem. Eng. Prog. 97(5), 42. Ballinger, G.H., R. Bafiares-Alcantara, D. Costello, E.S. Fraga, J. Krabbe, H. Labadibi, D.M. Laing, R.C. McKinnel, J.W. Ponton, N. Skilling and M.W. Spencely, 1994, Comp. Chem. Eng. 18. Carlson, E.C., 1996, Chem. Eng. Prog. 10, 35. http://www.ip-sol.com/TMS.htm. Accessed on Feb 2002. http://www.virtualmaterials.com/products.html. Accessed on Feb 2002. Satyro, M. A., R. Agarwall, Y.-K. Li, O. Santollani and A. Vieler, 2001, Chem. Eng. Prog. 97(6), 64.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
1027
A Novel Course on Integrated Batch-Plant Management Zofia Verwater-Lukszo and Petra Heijnen Delft University of Technology The Netherlands
Abstract The course "Integrated Plant Management", developed at the Department of Technology, Policy and Management at the Delft University of Technology, is aimed to provide knowledge and understanding of the plant operation in such a way that the challenges imposed by the economic, environmental and social sustainability are made more transparent (Verwater-Lukszo e.a., 2001). The course focuses on the batch processing industry, but the most concepts are applicable to continuous and discrete industry as well. The integration of the enterprise functions as strategic and tactical management, forecasting, planning, scheduling, recipe management, process execution, optimisation and control are central to the main course aim. To realise this integrated manner of plant management the modern concepts of manufacturing execution systems (MES), plant modelling according to the ISA-S88 and ISA-S95 standards, total quality management and system thinking are very useful. Those issues constitute the main focus of the course. The course is concluded by an emphasis on the importance of the integration programs for quality and environment, which can be realised and maintained according to the principles of the new ISO quality (ISO 9001:2000, 2000) and environmental (ISO14001: 1996, 1996) standards. Modelling enterprise activities and production processes as well as experimental and model-based process optimisation are the enablers of the intended improvements. Monitoring business and process performance form the next step. The principles of Statistical Quality Control form on one hand a basis for a sound assurance of quality and enterprise performance. On the other hand, they create a framework for continual performance improvement.
1. Introduction In the last few decades, developments in the global economic structure have changed the environment in which process industries operate. They have to cope with the following problems: more short-term dynamics in supply and end-products markets as well as more unpredictable and turbulent demand patterns more complicated processes which may be more difficult to operate short series of the manufactured products stricter requirements on product quality greater emphasis on shorter and more reliable production time a growing number of product grades and brands
1028 a need for improved customer service level. A flexible batch-wise mode of operation is a good answer for these trends. Batch processing of higher added-value specialities has been a fast growing segment of the process industry (i.e. food & beverages, chemical, pharmaceutical, metal industry etc.) in most industrialised countries. However, the flexibility of a batch plant poses a difficult problem of the allocation to available equipment for producing the desired products and setting up a production plan to decide if, when, and in what amounts, products should be produced. Moreover, the dynamic character of processing steps, which do not operate in a steady-state mode, complicates further the operation and control of a batch plant. Despite their complexity, the attractiveness of batch processing plants should be mentioned, too. It is found in the flexibility they offer to produce different (types of) products with the same equipment and to use the same pieces of equipment for different processing operations. This feature makes batch plants eminently suitable for production situations with a large number of product grades and short series of (tailor made) products. Chemical engineering research into methods to support an eco-efficient design of batch processes is gaining momentum since a few years. It is striking that, in comparison with batch processes design, the operation of batch processes is hardly explored as a chemical engineering field. The course "Integrated Plant Management", developed at the Department of Technology, Policy and Management at the Delft University of Technology, is aimed to provide knowledge and understanding of the batch-wise plant operation in such a way that the challenges imposed by the economic, environmental and social sustainability are made more transparent. The course is delivered to the students of the Systems Engineering, Policy Analysis and Management (SEPA) educational program at the Delft University of Technology as part of the Master's curriculum. The students from other faculties, i.e. Chemical Engineering and Mechanical Engineering, participate in the course, too. The development of this course has gone through a number of stages. It started as part of the module Cleaner Production and Process Design in an educational European Socrates program ELCE (Environmental Life-Cycle Engineering) in which various universities developed four advanced modules in the area of environmental engineering (Verwater-Lukszo e.a., 1999). The original course on Cleaner Production and Process Design is still given to international students as a distance-learning module on the Internet, in particular via the World Wide Web. The two main application areas in this course are plant design and plant operation (Herder e.a., 2(X)1). The course described in this paper, given for the regular SEPA students, is an extension of the ELCE module in the direction of more sustainable operation of an industrial plant, and in particular a batch plant. The students learn to see the "whole picture" of the plant operation with economic, ecological and technical aspects discussed together.
2. How to operate a plant in a changing environment? Industrial companies operating in a rapidly changing world of global economy and more short-term dynamics are continuously searching for opportunities to improve their competitive position. This involves improvements on the one hand in the production processes that produce products more efficiently and on the other hand in the internal
1029 methods of operation that enable companies to be more effective. Surprisingly, improvements in internal methods related to integrated plant management paying attention to both economic as environmental issues have often received relatively litde attention. Mostly, the companies concentrate on improving one task without taking sufficiently into account interactions with other activities and with the surroundings. The goal of the course it to create an understanding of bringing the batch plants operation in agreement with the strategic goals related to developments on the market. The leading thread running through the course is the integration of planning, scheduling, recipe, and quality management and process control according to operational objectives defined in compliance with strategic goals. The question that rises firsdy, is: How could strategic and tactical objectives (as those related to growth, profitability, sustainability) be translated to operational objectives (as those related to production results, personnel satisfaction, environmental impact of production activities etc.), which are measurable and achievable? In the course the students learn to structure the enterprise objectives in a hierarchical way by using the so-called objective-tree technique. An objective tree gives a good structure for the hierarchy and connection between objectives. Firstly, the overall objective should be determined - it characterises the reason for interest and defines the breadth of concern. Next, other objectives, which contribute to this main objective, should be specified. The procedure continues till the main objectives (at the strategic level) are translated into the operational objectives. The general framework for the objective tree will be the same for all sites of the company. Further, to support an integrated way of plant management a clear representation of activities performed in the company in relation to the specified objectives is desirable. In the course the students learn to model the key business activities by using so-called IDEFO techniques, which represent the structure of activities in a hierarchical way (IDEFO, 1993). The activities in the system - in this case an industrial plant - are analysed independently of the objects (persons, departments) that perform them. Such a purely functional perspective allows for a clear separation of the issues of meaning from the issues of implementation. The acfivity defined at the highest level is decomposed in a number of activifies, which collectively should achieve the main objective of the plant. In the decomposed model an output from one activity can be an input to another activity. In this way activities can be combined in a chain or network. The decomposition stops at that level, which makes it possible to associate the objectives at the lower level of the objective tree with the decomposed activities. Modelling enterprise activities and their interdependencies, and relating them to the operational objectives makes it possible to visualise how improvements or changes in one activity interact with other activities and which results could be expected. This visualisafion supports the decision, which activities at the operational level contribute mostly to the overall objective of the plant, in other words which ones are most effective. Improvement of the efficiency of those activities will be most effective. To improve manufacturing processes, i.e. to improve quality, yields, environmental performance, statistical thinking plays an important role. In general, statistical thinking
1030 can be used for the identification of problems or issues having common sources, in contract with those having unique causes. The following aspects could characterise it: the processes (activities) and not the products (outputs) are emphasised; identification, quantification, control and reduction of variation are the central issues; the problems are tackled on the basis of gathered data. The learnt techniques for modelling objectives and activities as well as improving them are also useful in the implementation of quality and environmental management systems according to the ISO standards: respectively (ISO 9001:2000, 2000) and (ISO 14001:1996, 1996). The quality standards, published in 2000, make the integration of a quality management system with an environmental one much easier. Both standards address: identification and satisfaction of the expectation of all involved parties (employees, owners, suppliers, society), aimed to achieve chosen objectives and to do this in an effective and efficient manner attain, maintain and improve overall organizational performance and capabilities. The mentioned standards encourage the adoption of the so-called process approach (process model), in which continual improvements play a central role.
3. The program of the course The course is given during two educational periods of six weeks each. Table 1 presents the main subject of each weekly part and the corresponding individual assignment. The course ends with an oral examination.
Table 1. The main subjects of the course "Integrated Plant Management"
NR. 1.
2
Subjects and aims Introduction to the course and to the process industry Economic and ecological potential of the process industry in the Netherlands and in Europe Various ways of processing (continue, batch, discrete) and various operating regimes (start-up, switch-over, shut down) Mutually related enterprise functions Concepts of activity modelling according to the SADT technique, including IDEFO diagrams Planning Examples of planning problems Short term planning, capacity planning Customer Order Uncouple Point Main technologies for planning (simulation, programming)
Assignment Read the paper "Planning, scheduling and control systems: why can they not work together" of D.E. Shobry and D.C White (Shobry e.a., 2000). Make a short description of forecasting, planning, scheduling and control activities in an industrial company Make an IDEFO model of a manufacturing site with the software tool BPWin. Zoom in on the four functions: Forecast, Planning, Scheduling and Control.
_ _ _ _ ^ Extend the IDEFO model with the planning activities
1031 NR.
Subjects and aims Scheduling Examples of scheduling problems Industrial practice Recent academic developments (disturbance management) Quahty control for process operation Quahty as reducing variations Statistical thinking Introduction Statistical Process Control (SPC) Quahty improvement: Statistical process Control Introduction to control charts Development of control charts Interpretation of control charts Process judgement Experimental design for process optimisation Introduction to the basis principles of design of experiments Introduction to factorial design Experimental and model-based process optimisation Discussion of various ways of process modelhng First principle models Black-box models (regression and statistical test) Quality and environmental management systems The new and old ISO 90(X) quahty management systems Environmental management system 14001 Introduction to process control Control configurations Control algorithms Control Hierarchy Formulafion of a control problem
10.
11
12
Introduction of batch plant modelling standards ISA-S88 (process cell level) ISA-S95 (integration of MES level with ERP level) Optimisation of the plant operation (Non) Linear programming Dynamic programming Integer programming HeurisUc methods
Assignment Extend the IDEFO model with the scheduhng activities
In a plant there are three multi-purpose batch reactors. Monitor the manufacturing process with Comerstone - a statistical package for industrial data analysis. The progress of the reaction is tested by continuous temperature measurements and by taking samples from the reactor. Develop a control chart for the production process of epoxy resin. Calculate the process capabihty indices. Develop a cause-and -effect diagram for the cases with a wrong WPE-number (quality parameter) Defme designed experiments to find an optimised recipe for a fermentation process, for which a new recipe has to be found. Comerstone is again the supporting tool. Use Comerstone to model a chemical process. Develop a way for integration of scheduling and processing activities by using process models as a part of a master recipe
Discussion of common pitfalls on the path to registration, false expectations and focal points in external audits and of the role of statistical techniques in quality management systems
Identify goals and the design space for a control problem described for a batch process producing small polymer particles. Translate the goals to objectives and constraints, and develop the tests necessary for mapping the control variables onto objectives and constraints. Demonstration of the software package proCX for recipe management, scheduling, execution of recipes, material flow control, tracking & tracing
Create a master recipe for the production of an alkyd resin, taking into account the S88 standard terminology for recipe elements. Make a schematic representation of the plant and distinguish process cells, process classes and transfer classes Final discussion on Integrated Plant Management MES (Manufacturing Execution Systems) as a integration framework
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4. Final remarks The ICT-supported course is given in an interactive way. In addition to the conventional class, which is given once a week, the students make every week an individual assignment with the business-modelling tool BPWin (BPWin, 2001) and with Cornerstone (Cornerstone, 2001), an exploratory tool for analysing manufacturing and engineering data. They submit them via Blackboard© (Blackboard, 2002), the general course supporting-tool, which gives an excellent environment for communication between teacher and students, for working in groups and for discussion on the topics from class. By the evaluation of assignments and during the oral exams it becomes clear that the students have assimilated the integrated view on plant management in the right manner. From the other side, it is also clear that the application of the mentioned ideas in a real plant is still difficult to operationalise. However, we are sure that the students understand very well that the industrial systems should be managed in an integrative way and that there is no way to escape from this responsibility. To do so, we still need to improve the scientific basis of the process operation discipline. Hopefully this course has made a small contribution towards this goal.
References Blackboard, 2002, http://www.blackboard.com/. Blackboard Inc. BPWin, 2001, http://www.cai.com/products/alm/bpwin.htm. Computer Associates International, Inc. Cornerstone, 2001, http://www.brooks.com/products/POBU/cstone/index.htm, Brooks Automation Inc. Foster S.T., 2001, Managing Quality. An Integrative Approach, Prentice Hall, New Jersey. Herder, P.M., Z. Verwater-Lukszo, M.P.C. Weijnen, 2001, A Novel Perspective on Chemical Engineering Education - Experiences with a Broad, ICT-Supported Course on the Integrated Design of Industrial Systems, 6* World Congress On Chemical Engineering, Melbourne. ISO 9001:2000, 2000, Quality Management Systems - Requirements, ISO, Geneva. ISO 14001:1996, 1996, Environmental Management Systems, specification with guidance for use, ISO, Geneva. Rao, A.,et al., 1996, Total Quality Management: A Cross Functional Perspective, John Wiley & Sons, New York. Shobry, D.E., D.C. White, 2000, Planning, scheduling and control systems: why can they not work together. Computers and Chem. Eng, 24 Standard Integration Definition for Function Modelling (IDEFO), 1993, Publication 183, FIPS PUBS. Verwater-Lukszo, Z., P.M.Herder, M.P.C. Weijnen, 1999, Cleaner Production and Process Design. Experiences with a distance-learning module on Internet, Proceedings of ENTREE'99, Tampere. Verwater-Lukszo, Z., P. Heijnen, 2001, Integrated Plant Management for better economic and ecological business performance, Proceedings of ENTREE'2001, Florence.
European Symposium on Computer Aided Process Engineering - 12 J. Grievink and J. van Schijndel (Editors) ® 2002 Elsevier Science B.V. All rights reserved.
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Designing a Multi-user Web-based Distributed Simulator for Process Control eLearning S. H. Yang and J. L. Alty Computer Science Department, Loughborough University, Loughborough, Leicestershire, LEI 1 3TU, United Kingdom
Abstract Although web-based elearning has been used for various disciplines, web-based experiments are still unusual. A web-based distributed simulator can be a powerful tool for elearning and a good alternative for web-based experiments. This paper discusses the design issues of a web-based distributed simulator for elearning in process control, including architecture selection, communication protocol, interface design, and process modelling. An industrial catalytic reactor is used as a case study to illustrate the methods described here.
1. Introduction Engineering and Science courses such as control theory and process control courses, are strongly founded on mathematics on one hand, but also need to develop the student's intuition for bridging the gap between theory and practice on the other hand. Obviously, providing opportunities for students to experience the theory which they have learned should form an essential part for the web-based courses (Copinga et. al., 2000). Unfortunately, web-based experiments (in contrast with the use of the web as a simple source of information) are still uncommon in engineering and science courses (Cartwright, 1998). Web-based dynamic simulators designed to support elearning are currently becoming available on the web (Granlund et. al., 2000). The advantages of learning, supported by dynamic simulations and the availability of the web, form a powerful combination. This combination in terms of speed, potential sophistication and wide availability seems certain to make them widespread and in frequent use within a few years. Developing web-based dynamic simulators for elearning presents special problems in software, networking, and interfacing. This paper will address these problems by reference to an industrial case study. The contents of the rest of this paper are arranged as follows: Section 2 describes the potential architectures of web-based dynamic simulators for elearning Then communication protocol is explored in section 3 for the distributed architecture chosen in section 2. User interface design and process modelling for the web-based dynamic simulator have raised some new issues which are briefly studied in sections 4 and 5. The industrial catalytic reactor is used in section 6 to illustrate the discussion given in pre-sections as case study. Conclusions are presented in section 7.
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2. Architectures of the Web-based Simulator One of the goals of the Internet is the Stuirt Student Instmcta global accessibility, which is a strong Irterfeoe Interfaoe Interfaoe reason for designing web-based simulators for elearning. The problems that may occur are slow and unpredictable network performance, and uncertainty as to who the users are. Therefore some mechanism is required to solve time delay and control conflict problems between multi-users. Figure 1 shows the perspective view of the web-based simulator. It consists of a web server running the server application, a webbased instructor interface and several Figure 1, Overview of the web-based distributed simulator. identical web-based student interfaces. The instructor interface is designed for the tutor to monitor and assess the operation of the students. The process model runs on the same machine as the instructor interface in order to reduce the working load of the server. The instructor interface acts as a userinterface front end to the process model. The web server acts as a post office and transfer data and control commands between the student interfaces and the process model through a HTTP/TCP connection. At the remote site (student site and instructor site), the HTML browsers display information about the process, including current states, alarm, dynamic trends, et al., and allow the students and the tutor to control the process. These functions are implemented in Java applet. When a student with a Javacapable browser loads the student interface (a web page) that includes applets, the browser downloads the applets from the web server and runs them on the student's own system. The instructor interface and the process model must be loaded and run before the student interface can really work. No specific software is required to be installed at the remote site. At the local site, the server establishes and controls communication between students/tutor and the process model by using the socket techniques. A hash table is built on the server to remain the update registration of the student interfaces. This configuration enables the students as many as possible to operate the same simulator simultaneously to develop their teamwork skill, but requires the students to stay on-line during the session. When no teamwork is required, the simulator can be simplified into a stand-alone application in the Java applet form. In that case once they load the interface front end from the web-server the students can work off-line. Only the distributed architecture will be discussed in the rest of this paper.
3. Communication Protocol As described above the server acts as a messenger for the instructor interface and the student interfaces, passing message from one to the other according to a special protocol. Two protocols were considered for the student interfaces and the instructor interface connections to the server. The first of these was UDP (Universal Datagram Protocol). UDP is a connectionless protocol and communicates fast. It is however unreliable, and cannot guarantee the delivery of the data in the right order. TCP
1035 however, whilst being a slightly slower protocol, guarantees that data delivered will be correct and in the right order. Therefore TCP is chosen as the protocol for all communication in this work. The server, the student interfaces, and the instructor interface can be a sender and/or a receiver. It enables information pass between them. The protocol defines a strict set of instructions which allow the students to give instructions to the process model to manipulate its operation, allow the student interfaces to regularly update the data they are displaying, and allow the instructor to monitor the operations carried out on the student interfaces as well. The student interface and the instructor interface need to identify themselves to the server program by sending the command '"CONNECT INT' and "CONNECT SIAf' respectively. From then on, the server program recognizes the client as a student interface or an instructor interface and treats it as such. The UPDATE commands from the instructor interface and the server are not of fixed length however, as the command may have multiple <device>
4. Web-based User Interface Design The central design objective for a user interface in the web-based distributed simulator is to enable the students (operators) to appreciate more rapidly what is happening in the process model and to provide a more stimulating problem-solving environment over the Internet. It should be born in mind that media available in the Internet environment will be very limited comparing with those in the central simulation room. Nevertheless, in the Internet environment no specific software and no special type of computer can be provided. Technologies from the areas of "multimedia" and "Virtual Reality" show considerable potential for improving yet further the human-computer interface used in process control technology (Alty, 1999). Using the audio information for warnings and
1036 alarms and using animation for displaying information are essential in the user interface design for the web-based simulators. Minimizing the amount of irrelevant information in the interface is another key issue because the irrelevant information may obscure important information by attracting the attention of the user.
5. Requirements of Process Modelling for Web-based Simulators One of the cores of the web-based simulator is the real-time dynamic model of the actual process. There are several different requirements that must be satisfied in order for the web-based simulator to be used for on-line experiments. A prerequisite is that the process model must be able to reproduce the process dynamic behaviour over a very wide range of operational modes including not just normal operation, but fault and emergency operation, start-up and shutdown as well. In addition, it should be able to handle logical procedures related to actions such as switching on or off valves, purges and so on. Also, main equipment failures should be considered in the process model so that the equipment failure can be invoked for special experiment task. Finally, the computation of the process model must not be time-consuming and complex in order to run the model on-line, and any iterative calculation should be avoided as much as possible. These requirements make web-based simulators distinctive from process simulators, which are concerned with improving optimisation and control. In this paper only a subset of a chemical process plant is simulated and it incorporates a mathematic model that integrates normal operation, start-up operation, and emergency handling operation together using several transition switches. For more details see our recent publication (Yang et al., 2001).
6. Case Study - a Web-based Distributed Simulator for an Industrial Catalytic Reactor The purpose of the web-based distributed simulator for an industrial catalytic reactor which we implemented here is twofold: (1) to provide students a virtual experiment environment to practice the basics of plant operation, and to build up the teamwork skill, and (2) to provide researchers a test bed for the design of Internet-based process control. The second aspect of the purpose will be addressed in our further publication. The simulator allows students situated in geographically diverse locations to experiment with some aspects of operating an industrial catalytic reactor. The main scenarios include that desired values of the process variables need to be updated and/or malfunctions occur during normal operation of the reactor. The task of the students is to act in response to these requirements and failures and to bring the reactor to a safe and steady state. Figure 2 shows the student interface running in a remote site. The process consists of a heat exchanger E201, a catalytic reactor R201, and four hand valves for Nitrogen inlet, liquid outlet, gas outlet, and emergency liquid outlet. The inlet temperature of the reactor is controlled by the hot stream flowrate of the heat exchanger E201. The buttons shown at the top of Figure 2 start various web instruments (pop windows) for displaying dynamic trends, alarm panel, controller panel and evaluation panel. These web
1037 instruments can be activated as well by clicking a mouse left button over the corresponding components in the process flowchart. E(ft
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Figure 2. Student interface running in a remote site. Before developing the web-based simulator several standard web instruments have been implemented for the Internet-based process control system. These web instruments include PID (Proportional Integral, and Differential) controller, manipulator, bar graph alarm panel, dynamic trend panel, and are pure Java applet pop windows. Figure 3 shows the web PID controller for the heat exchanger E201 that reappears the appearance of the front panel of a physical digital PID controller. The group radio buttons are used to choose automatic or manual mode for the controller. The bar charts at the top show the current input, output, and setpoint values. The scroll bars at the bottom allow a user to adjust
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1038 output value while in the manual mode and setpoint while in the automatic mode. Pressing the setting button allows an authorized user to tune P, I and D parameters and change the output range for the controller. All the web instruments here include security measures in the form of password and user name to identify the authorized users. Another important part of the distributed simulator is the instructor interface. Its main purpose is to provide a tutor a platform to set various malfunctions and operating circumstances for the remote site students, and monitor their operations. The process model is running on the same machine with the instructor interface.
7. Conclusions This paper has described the design issues of a web-based distributed simulator, such as distributed architecture, communication protocol, interface design and process modelling. A case study is implemented to show the design issues. One of the most important advantages for developing web-based distributed simulators is the availability of the system on the web. The initial motivation of developing this distributed simulator was to provide a test bed for an Internet-based process control system. In this paper we demonstrate that it is straightforward for this system to be used as an online experiment environment for elearning students in their process control practice, especially when teamwork skill is required. We believe that web-based distributed simulators will play an important role in future elearning.
Acknowledgement The contribution is part of the work of the EPSRC funded project "design of Internetbased process control". The authors would like to thank the support given by the EPSRC, Grant No. GR/R13371/01. Appreciation should be also to our students, R Baxter, C Childs, C George, S Nichols, and P Kay for their valuable programming.
References Alty, J.L., 1999, Multimedia and Process Control Interfaces: Signals or Noise? Transaction of Inst MC, 21(4/5), 181-190. Cartwright, H. M., 1998, Remote control: How science students can learn using Internet-based experiments. New Network-based Media in Education; Proceedings of the International CoLoS Conference, Maribor, Slovenia, September, pp. 51-59. Copinga, G. J. C , Verhaegen, W. H. G., and van de Ven, M. J. J. M., 2000, Toward a web-based study support environment for teaching automatic control. IEEE Control Systems Magazine, August, 8-19. Granlund, R., Berglund, E., and Eriksson, H., 2000, Designing web-based simulation for learning. Future Generation Computer Systems, 17, 171-185. Yang, S. H., Yang, L., and He, C. H., 2001, Improve safety of industrial processes using dynamic operator training simulators. Transactions of IchemE, Process Safety and Environmental Protection, 79(B6), 329-338.
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Author Index Abbas, A. 409 Abe, H. 739 Achenie, L.E.K. 883,997 Agachi, P.S. 463 Aguer, M. 625 Alexander, B. 817 Alexandrisis, A. 949 Alhammadi, H. 817 Alle,A 613 Almeida-Rivera, C.P. 151 Alonso, A.A. 415 Alty,J.L. 601,1033 Anoprienko, A. 583 Arellano-Garcia, H. 619 Arpomwichanop, A. 421 Arrieta, J.J.C. 547 Arruda, L.V.R. 715,805 Attarakih, M.M. 823 Avramenko, Y. 157 Azzaro-Pantel, C. 127 Backx, T. 643 Badell, M. 625 Balliu, N. 427 Bafiares-Alcantara, R.889, 1021 Banga, J.R. 415 Bansal,V. 163 Barbosa-Povoa, A.P.F.D. 649 Barnes, R.J. 631 Barolo, M. 505 Bart, H.J. 823 Barton, G.W. 607 Basaf, G. 949 Baton, G.W. 817 Batzias, F.A. 433, 829 Batzias, A.F. 433 Becks, A. 835 Belaud, J.P. 847 Bender, N. 937 Benqlilou, C. 853 Berg van den, J. 511 Betlem, B.H.L. 637 Bierman, G. 85,1015 Bigaran, C. 763 Bildea, C.S. 229, 439 Biller, N.C.T. 445
Biscaiajr, E.G. 865,967 Biscans, B. 127 Bitzer, M. 451 Bliek,A. 175 Blomberg, A. 133 Bogle, I.D.L. 271 Bok, J.K. 799 Boom, R. 85 Bosgra,O.H. 511,811,973 Braunschweig, B. 859 Brempt van, W. 643 Briesen, H. 205 Broglio, M.I. 313 Caballero, J.A. 169 Cabassud, M. 475 Cameron, I. 427,925 Cao, Yi 457 Cappi, A. 91 Camieri, R. 751 Castro, P. 649 Cerda,J. 721 Cervantes, A. 739 Cezac, P. 325 Cezerac, J. 475 Chen,X. 601 Cheung, K.Y. 655 Christofides, P.D. 877 Continillo, G. 403 Corvalan, S.M. 145 Costa jr., E.F. 865 Crescitelli, S. 403,535 Cristea, V.M. 463 Cruz, S.C. 439 Davin, A. 127 Demicoli, D. 385 Didriksen, H. 469 Dimian,A.C. 175,229,439 Domenech, S. 127 Drinkenburg, B. 1 Druart,F. 181 Dua, V. 541 Dumont, M.N. 685 Eden, M.R. 79 Edwards, D.W. 601 Eggersmann, M. 871 El-Farra, N.H. 877 Elgue, S. 475
El-Halwagi, M. 79 Eliceche, A.M. 145 Ender, L. 481 Engell, S. 703,775 Espinosa, J. 187 Espuna, A. 757,853 Fabro, J.A. 805 Faqir, N.M. 823 Feord, D. 661 Fletcher, N.M. 487 Flores, A.T. 547 Floudas, C.A. 97 Folestad, S. 133 Fonyo,Z. 121,205,361 Fraga,E.S. 271,1003 Franco, T.T. 91 Eraser, D.M. 361 Frausto-Hemandez, S. 193 Frouzakis, C. 415 Gabbar, H.A. 793 Galli, S. 307 Gani, R. 79,289 Garcia, C. 553 Garg, S. 883 Gelete, S.S. 889 Georgiadis, M.C. 667 Giannelos, N.F. 667 Gilles.E.D. 919,937 Glasser, D. 211,217 Goel,H.D. 199 Gomes, V.G. 607 Gonzalez, R. 769 Goossens, L.H.J. 337 Govatsmark, M.S. 499 Graells. M. 787,853 Granados-Aguilar, S.A. 301 Greaves, M.A. 505 Grievink,J. 85,151,199, 259,529,589,955,1015 Grosch, R. 205 Grossmann, I.E. 169 Gruner, S. 919 Guevara, V. 409 Hakagawa, K. 739 Hale, A.R. 337 Hamataka, H. 739
1040 979 Hangos, K.M. 103 Harmsen, G.J. Hausberger, B. 217 643 Hayot, P. Heijnen, P. 673,1027 Heinonen, J. 679 Heinz, D. 283 Henning, G. 991 Herder, P.M. 199 Hernandez-Castro, S. 193 Hessem van, D.VI. 511 Heyen, G. 685 Hildebrandt, D. 211,217 Hirata, K. 655 Holland, S.T. 211 Homeman, D.A. 691 Hui, C.W. 655 Huitzil-Melendez, P. 301 Hurme, M. 265,343 Hussain, M.A. 505 Huziwara, W.K. 943 ledema, P.D. 229,439 Iribarren, O.A. 139 Irons, K 859 Jadhav, N.Y. 85 Jakemann, C. 661 Jesus, S.S. 91 Jimenez, A. 193 Jimenez, L. 889,1021 Johansson, M. 133 Jones, W.E. 577 Jorgensen, S.B.79,289,901 Josefsson, L. 133 Joulia, X. 907 Julka, N. 895 Kadam, J.V. 511 Kaibel, G. 9 Kakalis, N.M.P. 541 Kalitventzeff, B. 685 Karhela, T. 265 Karimi, I. 895 Kauchali, S. 217 Keikko, K. 1009 Kevrekidis, I. 415 Kienle, A. 583,919 Kim, J.K. 223 Kiparissides, C. 961 Kiss, A.A. 229 Kittisupakom, P. 421 Kjelstrup, S. 235
Klepeis, J.L. 97 Koeijer de, G.M. 235 Kokossis,A. 115,247,367, 631 Konecsny, H. 763 Korevaar, G. 103 Kosmidis, V.D. 697 Kraslawski, A. 157,241, 319 Kravanja, Z. 349 Krishna, R. 109,277,355 Kristensen, N.R. 901 Kussi, J. 23 Lakatos, B.G. 841,985 Le Lann, J.M. 475,907 Leineweber, D. 283 Lelkes,Z. 361 Lemkowitz, S.M. 103,337 Leone, H. 991 Leung, D. 763 Li,P 619 Li, P. 493 Li, X.N. 241 Liao, J.C. 877 Ligero, E.L. 253 Lim, Y.I. 907 Lima,R.M. 913 Linke, P. 115,247,367,631 Lintomen, L. 313 Lohl, T. 703 Lopes, J.A. 709 Lu,Z. 517 Ludlage, J. 643 Luteijn, C.P. 1015 Luyben, M.L. 31 LuzJr, L.F.L. 313 Maciel Filho, R. 91,313, 481,565,781 Madsen, H. 901 Maffettone, P.L. 535 Magatao, L. 715 Manca, D. 307 Mancusi, E. 403,535 Manczinger, J. 121 Mangione, M. 307 Mangold, M. 919 Marcoulaki, E.G. 829 Marquardt, W.42,205,283, 511,871 Martin, E. 517,523
Martin, E.B. 487,727 Martini, W. 619 Matos, H. 649 Mazarakis, S. 949 McGahey, S. 925 McPherson, L. 523 Meeuse, P.M. 259,529 Meier, H.F. 943 Meireiles, A.J.A. 313 Mendez, C.A. 721 Menezes, J.C. 709 Mercer, E. 727 Meyer, M. 181 Michel, A. 769 Mitova, E. 217 Mizsey, P. 121 Mogk,G. 931 Mohl,K.D. 919 Monnigmann, M. 205 Montastruc, L. 127 Morais de, E.R. 565 Mori, M. 943 487,727 Morris, A.J. Morris, J. 517,523 Motz, S. 937 Mrziglod. Th 931 Msiza,A.K. 361 Mujtaba,I.M. 379,421,505 Muller, A. 595 Nakagawa, K. 733 Namatama, K. 739 Nappa, M. 265 Neves jr,F. 751,805 Newell, R. 427 Nielsen, J.S. 469 Niklasson Bjom, I. 133 Nougues, A. 733 Nystrom, L. 157 Oddone, S. 139 O'Grady, A.R.F. 271 Ojeda, J.A. 277 Oldenburg, J. 283 Omota, F. 175 Oostvogels, L. 643 Ortiz, I. 145 Osipova, T. 583 Ota, Y. 739 Ottens, M. 691 Overschee van, P. 643 O'Young, L. 655
1041 Pajula, E. Palanki, S. Papaeconomou, I. Papageorgiou, L.G.
343 589 289 295, 613 Park, H. 799 Park, S. 799 Park, S.W. 553 Pasman, H.J. 337 Patsiatzis, D.I. 295 Peres, A.P. 943 Perez-Cisneros, E.S. 301 Perkins, J.D. 163,331,541, 697 Peme, R. 23 Pettersson, F. 679 Pibouleau, L. 127 Pierucci, S. 307 Pinto, J.M. 373,613 Pinto, R.T.P. 313 Pistikopoulos, E.N 163, 331,541,697 Pons, M 847 Powers, G. 559 Prat, L. 475 Puigjaner, L. 625,757,787, 853 Raab,A. 121 Rahman, S. 643 Ramirez, J. 277 Ravagnani, T.M.K. 253 Reneaume, J.M. 181 Rev,E. 361 Rico-Ramirez, V. 193 Rodera, H. 745 Rodrigues, L.C.A.751,805 Roffel, B. 637 Romagnoli, J.A. 409,607, 763,817 Romero, J. 757 Rong,B.G. 241,319 Ronnholm, M. 1009 Roques, M. 325 Roquet, D. 325 Rosmalen van, S. 637 Ross, R. 163 Rouzineau, D. 181 Russo, L. 403,535 Sakamoto, H. 655 Sakizlis,V. 331,541
913 Salcedo, R.L. 1003 Salhi, A. 1009 Salmi, T. 529 Samyudia, Y. Sanchez, A. 769 763 Sanchez, M. Sand, S. 775 Santos, G. 625 781 Santos, M.M. 949 Sarimveis, H. 481 Scheffer, R. Schenk, M. 331 Schlegel, M. 511 Schmal, J.P. 955 Schneider, P.A. 571 Schneider, R. 871 Schoenmakers, H. 9 Schupp, B.A. 337 Schuppert, A. 23,931 Secchi, A.R. 865 961 Seferlis, P. 787 Sequeira, S.E. 343 Seuranen, T. 661 Shah, N. 745 Shethna, H.K. Shimada, Y. 793 Shimono, F.N. 91 Silva, A.B. 547 967 Silva, CM. Skogestad, S. 57,499 Smeets, J.F.C. 595 223,367 Smith, R. Snoeren, R. 733 799 Song, J. Sorensen, E. 445 349 Sor§ak, A. Sotomayor, 0.A.2:. 553 Sousa, R.R. 91 Springer, P.A.M. 355 Srinivasan, R. 895 Stappen, M.L.M. vander 85 805 Stebel, S.L. 91 Stein, G.C. Stichlmair, J. 385 70 Stockill, D. 973 Stork, M. Suzuki, K. 793 397 Syamlal, M. Szanyi, A. 121
Szederkenyi, G. 979 Szitkai, Z. 361 Tapp,M. 211 Tesson, M. 691 Thonus, P.B. 637 Tjoa, LB. 739 Toebermann, J.C. 835 Tousain, R.L. 511,811,973 Trotta, A. 505 Turk, A.L. 559 Tuza, Zs. 979 Ulbert, Z. 985 Uppaluri, R. 367 Vadnais, P. 733 Valli, F. 739 Vasco de Toledo, E.G. 565 Vasquez-Alvarez, E. 373 Vegetti,M. 991 Verheijen, P.J.T. 595,955, 1015 Verwater-Lukszo, Z. 673, 1027 Villafafila, A. 379 Viveros-Garcia, T. 301 Vu,T.T.L. 571 Wahl, T. 577 Wang,K. 1003 Wang, Y. 997 Wama,J. 1009 Warter, M. 385 Waschler, R. 583 Weijnen, M.P.C. 199 Wendt,M. 619 Wielen van der, L.A.M. 691 Wilson, J.A. 577 Wissen van, M.E. 589,595 Wolfde, S. 163 Wolf-Maciel, M.R. 313 Wozny,G. 493,619 Yang, S.H. 601,1033 Yang, Zhijia 457 Zeaiter, J. 607 Zeitz, M. 451 Zhelev,T.K. 391 Zitney, S.E. 397
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