Robust Power System Frequency Control (Power Electronics and Power Systems)

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Robust Power System Frequency Control

Power Electronics and Power Systems Series Editors:

M.A. Pai University of Illinois at Urbana-Champaign Urbana, Illinois

Alex Stankovic Northeastern University Boston, Massachusetts

Robust Power System Frequency Control Hassan Bevrani ISBN 978-0-387-84877-8 Synchronized Phasor Measurements and Their Applications A.G. Phadke and J.S. Thorp ISBN 978-0-387-76535-8 Digital Control of Electical Drives Slobodan N. Vukosavi´c ISBN 978-0-387-48598-0 Three-Phase Diode Rectifiers with Low Harmonics Predrag Pejovi´c ISBN 978-0-387-29310-3 Computational Techniques for Voltage Stability Assessment and Control Venkataramana Ajjarapu ISBN 978-0-387-26080-8 Real-Time Stability in Power Systems: Techniques for Early Detection of the Risk of Blackout Savu C. Savulesco, ed. ISBN 978-0-387-25626-9 Robust Control in Power Systems Bikash Pal and Balarko Chaudhuri ISBN 978-0-387-25949-9 Applied Mathematics for Restructured Electric Power Systems: Optimization, Control, and Computational Intelligence Joe H. Chow, Felix F. Wu, and James A. Momoh, eds. ISBN 978-0-387-23470-0 HVDC and FACTS Controllers: Applications of Static Converters in Power Systems Vijay K. Sood ISBN 978-1-4020-7890-3 Power Quality Enhancement Using Custom Power Devices Arindam Ghosh and Gerard Ledwich ISBN 978-1-4020-7180-5 Computational Methods for Large Sparse Power Systems Analysis: An Object Oriented Approach S.A. Soman, S.A. Khaparde, and Shubha Pandit ISBN 978-0-7923-7591-3 Operation of Restructured Power Systems Kankar Bhattacharya, Math H.J. Bollen, Jaap E. Daalder ISBN 978-0-7923-7397-1 Transient Stability of Power Systems: A Unified Approach to Assessment and Control Mania Pavella, Damien Ernst, and Daniel Ruiz-Vega ISBN 978-0-7923-7963-8 Continued after index

Hassan Bevrani

Robust Power System Frequency Control

ABC

Hassan Bevrani University of Kurdistan Sanandaj, Kurdistan Iran [email protected] Queensland University of Technology Brisbane, QLD Australia [email protected]

ISBN: 978-0-387-84877-8 e-ISBN: 978-0-387-84878-5 DOI: 10.1007/978-0-387-84878-5 Library of Congress Control Number: 2008934165 c 2009 Springer Science+Business Media, LLC ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper springer.com

Dedicated to my parents

Preface

Frequency control as a major function of automatic generation control is one of the important control problems in electric power system design and operation, and is becoming more significant today because of the increasing size, changing structure, emerging new uncertainties, environmental constraints and the complexity of power systems. In the last two decades, many studies have focused on damping control and voltage stability and the related issues, but there has been much less work on the power system frequency control analysis and synthesis. While some aspects of frequency control have been illustrated along with individual chapters, many conferences and technical papers, a comprehensive and sensible practical explanation of robust frequency control in a book form is necessary. This book provides a thorough understanding of the basic principles of power system frequency behaviour in wide range of operating conditions. It uses simple frequency response models, control structures and mathematical algorithms to adapt modern robust control theorems with frequency control issue and conceptual explanations. Most developed control strategies are examined by real-time simulations. Practical methods for computer analysis and design are emphasized. This book emphasizes the physical and engineering aspects of the power system frequency control design problem, providing a conceptual understanding of frequency regulation, and application of robust control techniques. The main aim is to develop an appropriate intuition relative to the robust load frequency regulation problem in real-world power systems, rather than to describe sophisticated mathematical analytical methods. This book could be useful for engineers and operators in power system planning and operation, as well as for academic researchers. It could be useful as a supplementary text for university students in electrical engineering at both undergraduate and postgraduate levels in standard courses of power system dynamics, power system analysis and power system stability and control.

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Outlines The book is divided into ten chapters and two appendices. Chapter 1 provides an introduction on the general aspects of power system controls. Fundamental concepts and definitions of stability and existing controls are emphasized. The timescales and characteristics of various power system controls are described and the importance of frequency stability and control is explained. Chapter 2 introduces the subject of real power frequency control, providing definitions and basic concepts. The load–frequency control (LFC) mechanism of a single control area is first described and then extended to a multi-area control system. Frequency operating standards, tie-line bias and its application to a multi-area frequency control system are presented. Past achievements in the frequency control literature are briefly reviewed. Chapter 3 describes LFC characteristics and dynamic performances. Static and dynamic performances are explained, and the effects of physical constraints (generation rate, dead band, time delays and uncertainties) on power system frequency control performance are emphasized. The impacts of power system restructuring on frequency regulation are discussed, and a dynamical model to adapt a well-tested classical LFC model to the changing environment of power system operation is simulated. Chapter 4 describes a systematic H∞ control technique using a fundamental control theorem and an iterative linear matrix inequalities (ILMI) algorithm for proportional–integral (PI)-based LFC design. In the proposed synthesis approach, the frequency control synthesis is reduced to static output feedback control problem. The closed loop performance is compared with conventional control design. Chapter 5 provides an H∞ control method to the design of robust PI-based LFC system in the pretense of communication delays. A laboratory environment for doing real-time simulations to evaluate the developed power system frequency control framework is described. A simplified model of a real power system is used to perform a comparison study on the proposed control strategy. Chapter 6 is organized into two main parts. First, the mixed H2 /H∞ control technique is used to synthesize simple robust PI controllers in a multi-area power system. Then, the LFC problem, considering multiple delays/uncertainties, is formulated as a multi-objective control problem. The advantages of the proposed method are examined by an experimental study and real-time non-linear simulations. Chapter 7 presents an agent-based control strategy for the designing of the decentralized LFC system. A two-agent control system measures/receives the required signals/data and estimates the total power imbalance, generator participation factors and produces control action signal through an H∞ -based PI controller. The results are examined using an analog power simulator. Chapter 8 presents the application of structured singular value theory ( μ ) for robust decentralized LFC design. System uncertainties and practical constraints are

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properly considered during a synthesis procedure. The robust performance is formulated in terms of the structured singular value for the measuring of control performance within a systematic approach. Chapter 9 describes the power system frequency behaviour in emergency conditions. The conventional frequency response model is generalized by considering the dynamics of emergency control/protection schemes such as under-frequency load shedding (UFLS) and under-frequency/over-frequency generation trips. A method for UFLS by using the regional frequency decline rate is proposed. Chapter 10 presents an overview of the key issues and the new challenges on frequency regulation, concerning the integration of renewable energy units into the power systems. The impact of power fluctuation produced by variable wind and solar renewable sources on system frequency performance via a simulation study is analysed. An updated LFC model is introduced, and the need for the revising of frequency performance standards is emphasized. Finally, a brief survey on the recent studies on the frequency regulation in the presence of renewable energy resources (RESs) and associated issues is presented. Appendices include mathematical descriptions and simulation data.

Acknowledgements Much of the information, outcomes and insight presented in this book were achieved through a long-term research conducted by the author on robust control and power system frequency regulation over the last 15 years in Iran (1993–2002: K.N. Toosi University of Technology, West Regional Electric Company, and University of Kurdistan), Japan (2002–2006: Osaka University, Kumamoto University and Research laboratory of Kyushu Electric Power Company) and Australia (2007–2008: Queensland University of Technology). It is pleasure to acknowledge the scholarships, awards and support the author received from various sources: The Ministry of Education, Culture, Sports, Science and Technology, Government of Japan (Monbukagakusho); Japan Society for the Promotion of Science (JSPS); West Regional Electric Company (WREC); Research Office at University of Kurdistan (UOK) and the Australian Research Council (ARC). The author thanks Prof. T. Hiyama (Kumamoto University), Prof. Y. Mitani (Kyushu Institute of Technology), Prof. K. Tsuji (Osaka University), Prof. G. Ledwich and Prof. Arindam Ghosh (Queensland University of Technology) for their continuous support and valuable comments. The assistance provided by Dr. J.J. Ford, Dr. J. Banks and Ms J. Stanbrook (Queensland University of Technology) is appreciated. Special thanks go to Prof. M.A. Pai (University of Illinois) for his support to provide this book. Finally, the author offers his deepest personal gratitude to his family for their support and patience during his work on this book.

Contents

1

Power System Control: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 A Brief Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Instability Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Controls Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Controls at Different Operating States . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Dynamics and Control Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 Power System Frequency Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.1 Load–Frequency Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.2 Why Robust Power System Frequency Control? . . . . . . . . . . 9 1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2

Real Power Compensation and Frequency Control . . . . . . . . . . . . . . . . . 2.1 Fundamental Frequency Control Loops . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Frequency Response Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Frequency Control in an Interconnected Power System . . . . . . . . . . . 2.4 LFC Participation Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Frequency Operating Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 A Literature Review on LFC Synthesis/Analysis . . . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 20 25 26 28 30 31

3

Frequency Response Characteristics and Dynamic Performance . . . . . 3.1 Frequency Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 State-Space Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Frequency Control in a Deregulated Environment . . . . . . . . . . . . . . . . 3.4 LFC Dynamics and Bilateral Contacts . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 43 44 47 47 50

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Physical Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Generation Rate and Dead Band . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 56 58 60 60

4

PI-Based Frequency Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 H∞ -SOF Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Static Output Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 H∞ -SOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Problem Formulation and Control Framework . . . . . . . . . . . . . . . . . . . 4.2.1 Transformation from PI to SOF Control Problem . . . . . . . . . . 4.2.2 Control Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Iterative LMI Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Developed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Weights Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Using a Modified Controlled Output Vector . . . . . . . . . . . . . . . . . . . . . 4.6 Considering Bilateral Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 64 64 66 66 67 69 70 71 73 73 75 77 81 81 82

5

Frequency Regulation with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1.1 H∞ Control for Time-Delay Systems . . . . . . . . . . . . . . . . . . . . 86 5.1.2 LFC with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2 Proposed Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.2 H∞ -SOF-Based LFC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.2.3 Application to a Three-Control Area . . . . . . . . . . . . . . . . . . . . 92 5.3 Real-Time Laboratory Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.1 Analog Power System Simulator . . . . . . . . . . . . . . . . . . . . . . . 92 5.3.2 Configuration of Study System . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.3 H∞ -SOF-Based PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6

Multi-Objective Control-Based Frequency Regulation . . . . . . . . . . . . . . 103 6.1 Mixed H2 /H∞ : Technical Background . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Proposed Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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6.2.1 Multi-Objective PI-Based LFC Design . . . . . . . . . . . . . . . . . . 106 6.2.2 Modelling of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.3 Developed ILMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.2.4 Weights Selection (μi ,Wi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.5 Application to Three-Control Area . . . . . . . . . . . . . . . . . . . . . . 110 6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4 Real-Time Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.1 Configuration of Study System . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.4.2 PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7

Agent-Based Robust Frequency Regulation . . . . . . . . . . . . . . . . . . . . . . . 123 7.1 Frequency Response Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.1.1 Frequency Response Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 7.1.2 Total Power Imbalance Estimation . . . . . . . . . . . . . . . . . . . . . . 125 7.2 Proposed Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.2.1 Overall LFC Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.2.2 Computing of Participation Factors . . . . . . . . . . . . . . . . . . . . . 127 7.2.3 Structure of Two-Agent System . . . . . . . . . . . . . . . . . . . . . . . . 129 7.3 Tuning of PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.4 Real-Time Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4.1 Configuration of Study System . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.4.2 PI Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.4.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 7.5 Laboratory Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8

Application of Structured Singular Values in LFC Design . . . . . . . . . . . 141 8.1 Sequential Decentralized LFC Design . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.1.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8.1.2 Synthesis Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.1.3 Synthesis Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.1.4 Application Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.1.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.2 A Decentralized LFC Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.2.1 Synthesis Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.2.2 Application Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.2.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 8.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

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Frequency Control in Emergency Conditions . . . . . . . . . . . . . . . . . . . . . . 165 9.1 Frequency Response Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9.1.1 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 9.1.2 Considering of Emergency Control/Protection Dynamics . . . 168 9.1.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 9.2 Under-Frequency Load Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.2.1 Why Load Shedding? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 9.2.2 A Brief Literature Review on UFLS . . . . . . . . . . . . . . . . . . . . . 175 9.3 UFLS in Multi-Area Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3.1 On Targeted Load Shedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.3.2 A Centralized UFLS Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 179 9.3.3 Targeted Load Shedding Using Frequency Rate Change . . . . 180 9.3.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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Renewable Energy Options and Frequency Regulation . . . . . . . . . . . . . 191 10.1 RESs and New Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 10.2 Impact on Frequency Regulation: A Simulation Study . . . . . . . . . . . . 194 10.3 Considering RESs Effect in LFC Model . . . . . . . . . . . . . . . . . . . . . . . . 198 10.3.1 LFC Model with RESs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10.3.2 Required Supplementary LFC Reserve . . . . . . . . . . . . . . . . . . 200 10.3.3 RESs and Frequency Performance Standards . . . . . . . . . . . . . 201 10.4 A Survey on Recent Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Chapter 1

Power System Control: An Overview

This introductory chapter provides a general description of power system control. Fundamental concepts/definitions of power system stability and existing controls are emphasized. The role of power system controls (using automatic processing and human operating) is to preserve system integrity and restore the normal operation subjected to a physical (small or large) disturbance [1]. In other words, power system control means maintaining the desired performance and stabilizing of the system following a disturbance, such as a short circuit and loss of generation or load. From the viewpoint of control engineering, a power system is a highly non-linear and large-scale multi-input multi-output (MIMO) dynamical system with numerous variables, protection devices and control loops, with different dynamic responses and characteristics. The term power system control is used to define the application of control theory and technology, optimization methodologies and expert and intelligent systems to improve the performance and functions of power systems during normal and abnormal operations. Power system controls keep the power system in a secure state and protect it from dangerous phenomena [1, 2].

1.1 A Brief Historical Review Power system stability and control was first recognized as an important problem in the 1920s [3, 4]. Until recently, most engineering efforts and interests have been concentrated on rotor angle (transient and steady state) stability. For this purpose, many powerful modelling and simulation programs, and various control and protection schemes have been developed. A survey on the basics of power system controls, literature and past achievements is given in [5, 6]. Frequency stability problems, related control solutions and long-term dynamic simulation programs have been emphasized in the 1970s and 1980s following some major system events [7–10]. Useful guidelines were developed by an IEEE working group for enhancing power plant response during major frequency disturbances [11]. H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 1, c Springer Science+Business Media LLC 2009

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Since the 1990s, supplementary control of generator excitation systems, static VAR compensator (SVC) and high voltage direct current (HVDC) converters are increasingly being used to solve power system oscillation problems [5]. There has also been a general interest in the application of power-electronics-based controllers known as flexible AC transmission system (FACTS) controllers for the damping of system oscillations [12]. Following several power system collapses worldwide [13–15], in the 1990s, voltage stability attracted more research interests. Powerful analytical tools and synthesis methodologies have been developed. Since the 1980s, several integrated control design approaches have been developed for power system oscillation damping and voltage regulation [16–19]. Recently, following the development of synchronized phasor measurement units (PMUs), communication channels and digital processing, wide-area power system stabilization and control have become areas of interest [20,21]. Attempts to improve data exchange and coordination between the different existing control systems [22], as a wide-area control solution, are considered as an important control trend. In a modern power system, the generation, transmission and distribution of electric energy can only be met by the use of robust/optimal control methodologies, infrastructure communication and information technology (IT) services in the designing of control units and supervisory control and data acquisition system (SCADA) centres. Some important issues for power system control solutions in a new environment are appropriate lines of defence [21], uncertainties consideration and more effective dynamic modelling [23], assessments/predictions and optimal allocations and processing of synchronized devices [24], appropriate visualizations of disturbance evaluations, proper consideration of distributed generation units [25] and robust control design for stabilizing power systems against danger phenomena [26]. Considerable developments have recently been made on renewable energy sources (RESs) technologies. The increasing penetration of RESs has many technical implications and raises important questions, as to whether the conventional power system control approaches to operate in the new environment are still adequate. Recently, there has been a strong interest in the area of RESs and their impacts on power systems dynamics and stability, and possible control solutions [27–31].

1.2 Instability Phenomena The most recent proposed definition of power system stability is [32] “the ability of an electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact”. As the electric power industry has evolved over the last century, different forms of instability have emerged as being important during different periods. Similarly, depending on the developments in control theory, power system control technology and computational tools, different control syntheses/analyses have been developed.

1.2 Instability Phenomena

3

Fig. 1.1 Different phenomena that lead to power system instability

Power system control can take different forms and is influenced by the instabilizing phenomena. Conceptually, definitions and classifications are well founded in [32]. As shown in Fig. 1.1, important phenomena that lead to power system instability are rotor angle instability, voltage instability and frequency instability. Rotor angle instability is the inability of the power system to maintain synchronization after being subjected to a disturbance. In case of transient (large disturbance) angle instability, a severe disturbance does not allow a generator to deliver its output electricity power into the network. Small signal (steady state) angle instability is the inability of the power system to maintain synchronization under small disturbances. The considered disturbances must be small enough that the assumption of system dynamics being linear remains valid for analysis purposes [1,32–34]. The rotor angle instability problem has been fairly well solved by power system stabilizers (PSSs), thyristor exciters, fast fault clearing and other stability controllers and protection actions such as generator tripping. Voltage instability is the inability of a power system to maintain steady acceptance voltages at all system’s buses after being subjected to a disturbance from an assumed initial equilibrium point. A system enters a state of voltage instability when a disturbance changes the system’s condition to make a progressive fall or rise of voltages of some buses. Loss of load in an area, tripping of transmission lines and other protected equipments are possible results of voltage instability. Frequency instability is the inability of a power system to maintain system frequency within the specified operating limits. Generally, frequency instability is a result of a significant imbalance between load and generation, and it is associated with poor coordination of control and protection equipment, insufficient generation reserves and inadequacies in equipment responses [35, 36]. The size of disturbance, physical nature of the resulting instability, the dynamic structure and the time span are important factors to determine the instability form [1]. The above instability classification is mainly based on dominant initiating phenomena. Each instability form does not always occur in its pure form. One may lead to the other, and the distinction may not be clear.

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Fig. 1.2 Progressive power system response to a serious disturbance

As shown in Fig. 1.2, a fault on a critical element (serious disturbance) may influence much of the control loops and the equipments through different channels, and finally, may affect the power system performance and even stability [1]. Therefore, during frequency excursions following a major disturbance, voltage magnitudes and power flow may change significantly, especially for islanding conditions with under-frequency load shedding that unloads the system [3]. In real power systems, there is clearly some overlap between the different forms of instability, since as systems fail, more than one form of instability may ultimately emerge [5]. However, distinguishing between different instability forms is important in understanding the underlying causes of the problem in order to develop appropriate design and operating procedures.

1.3 Controls Configuration

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Fig. 1.3 General structure for power system controls

1.3 Controls Configuration Power system controls are of many types including [1, 21, 37] generation excitation controls, prime mover controls, generator/load tripping, fast fault clearing, highspeed re-closing, dynamic braking, reactive power compensation, load–frequency control, current injection, fast phase angle control and HVDC special controls. From the point of view of operations, all controls can be classified into continuous and discontinuous controls. A general structure for a power system with the main required control loops in a closed-loop scheme is shown in Fig. 1.3. Most of continuous control loops such as prime mover and excitation controls operate directly on generator units and are located at power plants. The continuous controls include generator excitation controls (PSS and automatic voltage regulator (AVR)), prime mover controls, reactive power controls and HVDC controls. All these controls are usually linear, continuously active and use local measurements. In a power plant, the governor voltage and reactive power output are regulated by excitation control, while energy supply system parameters (temperatures, flows and pressures) and speed regulation are performed by prime mover controls. Automatic generation control balances the total generation and load (plus losses) to reach the nominal system frequency (commonly 50 or 60 Hz) and scheduled power interchange with neighbouring systems.

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The discontinuous controls generally stabilize the system after severe disturbances and are usually applicable for highly stressed operating conditions. They perform actions such as generator/load tripping, capacitor/reactor switching and other protection plans. These power system controls may be local at power plants and substations, or over a wide area. These kinds of controls usually ensure a post-disturbance equilibrium with sufficient region of attraction [21]. Discontinuous controls evolve discrete supplementary controls [38], special stability controls [39] and emergency control/protection schemes [40–42]. Furthermore, there are many controls and protections systems on transmission and distribution sides, such as switching capacitor/reactors, tap-changing/phase shifting transformers, HVDC controls, synchronous condensers and static VAR compensators. Despite numerous existing nested control loops that control different quantities in the system, working in a secure attraction region with a desired performance is the objective of an overall power system control strategy. It means generating and delivering power in an interconnected system is as economical and reliable manner as possible while maintaining the frequency and the voltage within permissible limits.

1.4 Controls at Different Operating States Power system controls attempt to return the system in off-normal operating states to a normal state. Classifying the power system operating states to normal, alert, emergency, in extremis and restorative is conceptually useful to designing appropriate control systems [1, 43]. In the normal state, all system variables (such as voltage and frequency) are within the normal range. In the alert state, all system variables are still within the acceptable range. However, the system may be ready to move into the emergency state following disturbance. In the emergency state, some system variables are outside of the acceptable range and the system is ready to fall into the in extremis state. Partial or system wide blackout could occur in the in extremis state. Finally, energizing of the system or its parts and reconnecting/resynchronizing of system parts occurs during the restorative state. Based on the above classification, power system controls can be divided into the main two different categories (1) normal/preventive controls, which are applied in the normal and alert states to stay in or return into normal condition and (2) emergency controls, which are applied in emergency or in extremis state to stop the further progress of the failure and return the system to a normal or alert state. Automatic frequency and voltage controls are part of the normal and the preventive controls, while some of the other control schemes such as under-frequency load shedding, under-voltage load shedding and special system protection plans can be considered under emergency controls. Control command signals for normal/preventive controls usually include active power generation set points, flow controlling reference points (FACTS), voltage set point of generators, SVC, reactor/capacitor switching, etc. Emergency control mea-

1.5 Dynamics and Control Timescales

7

sures are some control commands such as tripping of generators, shedding of load blocks, opening of interconnection to neighbouring systems and blocking of transformers’ tap changer.

1.5 Dynamics and Control Timescales For the purpose of dynamic analysis, it is noteworthy that the timescale of interest for rotor angle stability in transient (large disturbance) stability studies is usually limited to 3–10 s, and in steady state (small signal) studies is of the order of 10–20 s. The rotor angle stability is known as a short-term stability problem, while a voltage stability problem can be either a short-term or a long-term stability problem. The time frame of interest for voltage stability problems may vary from a few seconds to several minutes. Although power system frequency stability is impacted by fast as well as slow dynamics, the time frame will range from a few seconds to several minutes [5]. Therefore, it is known as a long-term stability problem. For the purpose of power system control designs, generally the control loops at lower system levels (locally in a generator) are characterized by smaller time constants than the control loops active at a higher system level. For example, the AVR, which regulates the voltage of the generator terminals to the reference value, responds typically in a timescale of a second or less. The secondary voltage control, which determines the reference values of the voltage controlling devices, among which the generators, operates in a timescale of several seconds or minutes. That means these two control loops are virtually de-coupled. On the other hand, since the excitation system time constant is much smaller than the prime mover time constant and its transient decay is much faster and does not affect the LFC system dynamic, the cross-coupling between the LFC loop and the AVR loop is negligible. This is also generally true for the other control loops. As a result, for the purpose of system protection, turbine control, frequency and voltage control, a number of de-coupled control loops are operating in a power system operating in different timescales. The overall control system is complex. However, due to the de-coupling, in most cases it is possible to study each control loop individually. Depending on the loop nature, the required model, important variables, uncertainties and objectives, different control strategies may be applicable. A schematic diagram showing the important different timescales for the power system controls and the dynamics is shown in Fig. 1.4.

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Frequency stability

Dynamics

Trans. S. S. angle angle stability stability Voltage stability Generator/excitation dynamics

Milliseconds

Boiler dynamics

Seconds

Time

Slow protective relaying

Fast protective relaying

Controls

Minutes

SVC

LFC Prime mover control

Mechanical capacitor/reactor switching Under-frequency/under-voltage load shedding FACTS, AVR, PSS, HVDC

Fig. 1.4 Schematic diagram of different timescales of power system dynamics and controls

1.6 Power System Frequency Control 1.6.1 Load–Frequency Control A severe system stress resulting in an imbalance between generation and load seriously degrades the power system performance (and even stability), which cannot be described in conventional transient stability and voltage stability studies. This type of usually slow phenomena must be considered in relation with power system frequency control issue. Power system frequency regulation entitled load–frequency control (LFC), as a major function of automatic generation control (AGC), has been one of the important control problems in electric power system design and operation. Off-normal frequency can directly impact on power system operation and system reliability [1]. A large frequency deviation can damage equipment, degrade load performance, cause the transmission lines to be overloaded and can interfere with system protection schemes, ultimately leading to an unstable condition for the power system [44]. Maintaining frequency and power interchanges with neighbouring control areas at the scheduled values are the two main primary objectives of a power system LFC. These objectives are met by measuring a control error signal, called the area control error (ACE), which represents the real power imbalance between generation and load, and is a linear combination of net interchange and frequency deviations. After filtering, the ACE is used to perform an input control signal for a usually proportional integral (PI) controller. Depending on the control area characteristics, the resulting output control signal is conditioned by limiters, delays and gain constants. This control signal is then distributed among the LFC participant generator units in accordance with their participation factors to provide appropriate control

1.6 Power System Frequency Control

9

commands for set points of specified plants. The probable accumulated errors in frequency and net interchange due to used integral control have to be corrected by tuning the controller settings according to procedures agreed upon by the whole interconnection. Tuning of the dynamic controller is an important factor to obtain optimal LFC performance. Proper tuning of controller parameters is needed to obtain good control without excessive movement of units [45]. The frequency control is becoming more significant today due to the increasing size, the changing structure and the complexity of interconnected power systems. Increasing economic pressures for power system efficiency and reliability have led to a requirement for maintaining system frequency and tie-line flows closer to scheduled values as much as possible. Therefore, in a modern power system, LFC plays a fundamental role, as an ancillary service, in supporting power exchanges and providing better conditions for the electricity trading.

1.6.2 Why Robust Power System Frequency Control? As mentioned, the power systems are being operated under increasingly stressed conditions due to the prevailing trend to make the most of existing facilities. Increased competition, open transmission access and construction and environmental constraints are shaping the operation of electric power systems in new ways that present greater challenges for secure system operation [5]. Frequently changing power transfer patterns causes new stability problems. Different ownership of generation, transmission and distribution makes power system control more difficult. A main complication brought on by the separation of ownership of generation and transmission is lack of coordination in long-term system expansion planning. This results in the much-reduced predictability (increased uncertainty) of the utilization of transmission assets and correct allocation of controls. The increasing number of major power grid blackouts that have been experienced in recent years [46–49], for example, the Brazil blackout of March 1999, Iran blackout of Spring 2001 and Spring 2002, Northeast USA-Canada blackout of August 2003, Southern Sweden and Eastern Denmark blackout of September 2003, the Italian blackout of September 2003 and the Russia blackout of May 2005 shows that today’s power system operations require more careful consideration of all forms of system instability and control problems. The network blackouts show that to improve the overall power system control response, it is important to provide more effective and robust control strategies in order to achieve a new trade-off between system security, efficiency and dynamic robustness. Significant interconnection frequency deviations can cause under-/overfrequency relaying and disconnect some loads and generations. Under unfavourable conditions, this may result in a cascading failure and system collapse [48]. In the last two decades, many studies have focused on damping control and voltage stability and related issues. However, there has been much less work on power sys-

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tem frequency control analysis and synthesis, while violation of frequency control requirements was known as a main reason for numerous power grid blackouts [46]. Most published research in this area neglects new uncertainties [23] and practical constraints [50], and furthermore, suggest complex control structures with impractical frameworks, which may have some difficulties while implementing in real-time applications [51, 52]. Operating the power system in the new environment will certainly be more complex than in the past, due to the considerable degree of interconnection, and due to the presence of technical and economic constraints (deriving by the open market) to be considered, together with the traditional requirements of system reliability and security. In addition to various market policies, the sitting of numerous generators units and RESs in distribution areas and the growing number of independent players is likely to have an impact on the operation and control of the power system, which is already designed to operate with large, central generating facilities. At present, the power system utilities participate in the LFC task with simple and classical tuned controllers. Most of the parameters adjustments are usually made in the field using heuristic procedures. Existing LFC systems’ parameters are usually tuned based on experiences, classical methods and trial and error approaches, and they are incapable of providing good dynamical performance over a wide range of operating conditions and various load scenarios. Therefore, the novel modelling and control approaches are strongly required, to obtain a new trade-off between market outcome (efficiency) and market dynamics (robustness). In response to the above challenge, recent development in robust linear control theory has provided powerful tools such as μ synthesis/analysis, optimal H2 , H∞ and mixed H2 /H∞ techniques for power system load–frequency control design. The resulting robust controls will play an important role in system security and reliable operation. Robust power system frequency control means the control must provide adequate minimization on a system’s frequency and tie-line power deviation, and expend the security margin to cover all operating conditions and possible system configurations. The main goal of robust LFC designs in the present monograph is to develop new load–frequency control synthesis methodologies for multi-area power systems based on the fundamental LFC concepts, together with the powerful robust control theory and tools. The LFC objectives are satisfied, i.e., frequency regulation and maintaining the tie-line power interchanges to specified values in the presence of physical constraints and model uncertainties. The proposed control techniques meet all or a combination of the following specifications: • Robustness. Guarantee robust stability and robust performance for a wide range of operating conditions. For this purpose, robust control techniques are to be used in synthesis and analysis procedures. • Decentralized property. In a new power system environment, centralized design is difficult to numerically/practically implement for a large-scale multi-area frequency control synthesis. Because of the practical advantages it provides, the decentralized LFC design is emphasized in the proposed design procedures for real-world power system applications.

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• Simplicity of structure. In order to meet the practical merits, the robust decentralized LFC design problem is reduced to a synthesis of low-order or a proportional integral control problem, which is used usually in a real LFC system. • Formulation of uncertainties and constraints. The LFC synthesis procedure must be flexible enough to include generation rate constraints, time delays and uncertainties in the power system model and control synthesis procedure. The proposed approaches advocate the use of a physical understanding of the system for robust LFC synthesis. The presented techniques and algorithms in this monograph address systematic, fast and flexible design methodologies for robust power system frequency regulation. The developed control strategies attempt to invoke the well-known strict conditions and bridge the gap between the power of robust/optimal control theory and practical power system frequency control synthesis.

1.7 Summary This chapter provides an introduction on the general aspects of power system controls with a brief historical review. Fundamental concepts and definitions of stability and existing controls are emphasized. The timescales and characteristics of various power system controls are described, and the importance of frequency stability and control and the main goal of robust frequency control designs in the next chapters are explained.

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36. P. Kundur, D. C. Lee, J. P. Bayne, and P. L. Dandeno, Impact of turbine generator controls on unit performance under system disturbance conditions, IEEE Trans. Power App. Syst., 104(6), 1262–1267, 1985. 37. T. Torizuka and H. Tanaka, An outline of power system technologies in Japan, Electr. Power Syst. Res., 44, 1–5, 1998. 38. IEEE Discrete Supplementary Control Task Force, A description of discrete supplementary controls for stability, IEEE Trans. Power App. Syst., 97(1), 149–165, 1978. 39. IEEE Special Stability Controls Working Group, Annotated bibliography on power system stability controls: 1986–1994, IEEE Trans. Power Syst., 11(2), 794–800, 1996. 40. P. Pourbeik, P.S. Kundur, and C.W. Taylor, The anatomy of a power grid blackout, IEEE Power Energy Mag., 4(5), 22–29, 2006. 41. P.M. Anderson and M. Mirheydar, An adaptive method for setting underfrequency load shedding relays, IEEE Trans. Power Syst., 7(2), 647–655, 1992. 42. V.V. Terzija, Adaptive underfrequency load shedding based on the magnitude of the disturbance estimation, IEEE Trans. Power Syst., 21(3), 1260–1266, 2006. 43. L. H. Fink and K. Carlsen, Operating under stress and strain, IEEE Spectr., 15, 48–53, 1978. 44. B. J. Kirby, J. Dyer, C. Martinez, et al., Frequency control concerns in the North American electric power system, Available on-line at: http://www.ornl.gov/sci/btc/apps/ Restructuring/ORNLTM200341.pdf, 2002. 45. N. K. Stanton, J. C. Giri, and A. Bose, Energy management, Power System Stability and Control, Chapter 17. Boca Raton, FL: CRC, 2007. 46. UCTE, Final report of the investigation committee on the 28 September 2003 blackout in Italy, Available online at: http://www.ucte.org, 2004. 47. G. Andersson, P. Donalek, R. Farmer, et al., Causes of the 2003 major grid blackouts in North America and Europe and recommended means to improve system dynamic performance, IEEE Trans. Power Syst., 20(4), 1922–1928, 2005. 48. Y. V. Makarov, V. I. Reshetov, V. A. Stroev, et al., Blackout prevention in the United States, Europe and Russia, Proc. IEEE, 93(11), 1942–1955, 2005. 49. M. Sanaye-Pasand, Security of the Iranian national grid, IEEE Power Energy Mag., 5(1), 31–39, 2007. 50. N. Jaleeli, D. N. Ewart, and L. H. Fink, Understanding automatic generation control, IEEE Trans. Power Syst., 7(3), 1106–1112, 1992. 51. H. Bevrani and T. Hiyama, Robust load–frequency regulation: A real-time laboratory experiment, Optimal Control Appl. Methods, 28(6), 419–433, 2007. 52. H. Bevrani and T. Hiyama, On load–frequency regulation with time delays: Design and realtime implementation,” IEEE Trans. Energy Convers., in press.

Chapter 2

Real Power Compensation and Frequency Control

This chapter introduces the subject of real power and frequency control, providing definitions and basic concepts. The load–frequency control mechanism of a single control area is first described and then extended to a multi-area control system. Tie-line bias control and its application to a multi-area frequency control system are presented. Past achievements in the frequency control literature are briefly reviewed.

2.1 Fundamental Frequency Control Loops The frequency of a power system is dependent on real power balance. A change in real power demand at one point of a network is reflected throughout the system by a change in frequency. Therefore, system frequency provides a useful index to indicate system generation and load imbalance. Any short-term energy imbalance will result in an instantaneous change in system frequency as the disturbance is initially offset by the kinetic energy of the rotating plant. Significant loss in the generation without an adequate system response can produce extreme frequency excursions outside the working range of the plant. The control of frequency and power generation is commonly referred to as load–frequency control (LFC) which is a major function of automatic generation control (AGC) systems. Depending on the type of generation, the real power delivered by a generator is controlled by the mechanical power output of a prime mover such as a steam turbine, gas turbine, hydro-turbine or diesel engine. In the case of a steam or hydro-turbine, mechanical power is controlled by the opening or closing of valves regulating the input of steam or water flow into the turbine. Steam (or water) input to generators must be continuously regulated to match real power demand, failing which the machine speed will vary with consequent change in frequency. For satisfactory operation of a power system, the frequency should remain nearly constant [1, 2]. In addition to a primary frequency control, most large synchronous generators are equipped with a supplementary frequency control loop. A schematic block diagram of a synchronous generator equipped with frequency control loops is shown in Fig. 2.1. H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 2, c Springer Science+Business Media LLC 2009

15

16

2 Real Power Compensation and Frequency Control

Fig. 2.1 Schematic block diagram of a synchronous generator with basic frequency control loops

In Fig. 2.1, the speed governor senses the change in speed (frequency) via the primary and supplementary control loops. The hydraulic amplifier provides the necessary mechanical forces to position the main valve against the high-steam (or hydro-) pressure, and the speed changer provides a steady-state power output setting for the turbine. The speed governor on each generating unit provides the primary speed control function, and all generating units contribute to the overall change in generation, irrespective of the location of the load change, using their speed governing. However, primary control action is not usually sufficient to restore the system frequency, especially in an interconnected power system and the supplementary control loop is required to adjust the load reference set point through the speed-changer motor. The supplementary loop performs a feedback via the frequency deviation and adds it to the primary control loop through a dynamic controller. The resulting signal (ΔPC ) is used to regulate the system frequency. In real-world power systems, the dynamic controller is usually a simple integral or proportional integral (PI) controller. According to Fig. 2.1, the frequency experiences a transient change (Δ f ) following a change in load (ΔPL ). Thus, the feedback mechanism comes into play and generates an appropriate signal for the turbine to make generation (ΔPm ) track the load and restore the system frequency.

2.2 Frequency Response Modelling Power systems have a highly non-linear and time-varying nature. However, for the purpose of frequency control synthesis and analysis in the presence of load disturbances, a simple low-order linearized model is used. In comparison with voltage and rotor angle dynamics, the dynamics affecting frequency response are relatively slow, in the range of seconds to minutes.

2.2 Frequency Response Modelling

17

To include both the fast and the slow power system dynamics [3], by considering generation and load dynamics in detail, complex numerical methods are needed to permit varying the simulation time step with the amount of fluctuation of system variables [4]. Neglecting the fast (voltage and angle) dynamics reduces the complexity of modelling, computation and data requirements. Analysis of the results is also simplified. In this section, a simplified frequency response model for the described schematic block diagram in Fig. 2.1 with one generator unit is described, and then the resulting model is generalized for an interconnected multi-machine power system in Sect. 2.3. The overall generator–load dynamic relationship between the incremental mismatch power (ΔPm − ΔPL ) and the frequency deviation (Δ f ) can be expressed as d Δ f (t) + DΔ f (t), (2.1) ΔPm (t) − ΔPL (t) = 2H dt where Δ f is the frequency deviation, ΔPm the mechanical power change, ΔPL the load change, H the inertia constant and D is the load damping coefficient. The damping coefficient is usually expressed as a percent change in load for a 1% change in frequency. For example, a typical value of 1.5 for D means that a 1% change in frequency would cause a 1.5% change in load. Using the Laplace transform, (2.1) can be written as ΔPm (s) − ΔPL (s) = 2HsΔ f (s) + DΔ f (s).

(2.2)

Equation (2.2) can be represented in a block diagram as in Fig. 2.2. This generator– load model can simply reduce the schematic block diagram of a closed-loop synchronous generator (Fig. 2.1) as shown in Fig. 2.3. Several low order models for representation of turbine and generator dynamics (Gt and Gg ) have been proposed for use in power system frequency analysis and control design [5]. Slow system dynamics of the boiler and the fast generator dynamics are usually ignored in these models. The block diagram representation of the speed governor and the turbine for steam and hydraulic governor units appropriate for LFC synthesis/analysis is shown in Fig. 2.4 [1]. Here, R (and Rh ) is the speeddroop characteristic and shows the speed regulation due to governor action. Tg , Tt , Tr , Ttr , Tgh and Tth are generator–turbine time constants. Figure 2.5 shows a combination of the block diagrams in Figs. 2.3 and 2.4a. That is, a block diagram representation for a non-reheat steam generator unit with

Fig. 2.2 Block diagram representation of generator–load model

18

2 Real Power Compensation and Frequency Control

Fig. 2.3 Reduced block diagram of Fig. 2.1

a

b

c

Fig. 2.4 Block diagram of turbine–governor system; (a) non-reheat steam unit, (b) reheat steam unit and (c) hydraulic unit

2.2 Frequency Response Modelling

19

Δ f (Hz)

Fig. 2.5 Block diagram model of governor with frequency control loops for a non-reheat steam generator unit 0.02 0 −0.02 −0.04 −0.06 −0.08

0

5

10

15

0

5

10

15

0

5

10

15

Δ Pm (pu)

0.04 0.02 0

Δ PC (pu)

0.03 0.02 0.01 0 −0.01

Time (sec)

Fig. 2.6 Dynamic response of the closed-loop system with (solid) and without (dotted) supplementary control

associated frequency control loops (LFC system) comprising turbine, generator, governor, supplementary control and load. The dynamic response of the closed-loop system for a step load disturbance of 0.02 pu is plotted in Fig. 2.6. For the sake of comparison, the frequency deviation of system without a supplementary control is also plotted on the same figure. The system parameters for the performed simulation are given in Table 2.1.

20

2 Real Power Compensation and Frequency Control

Table 2.1 Simulation data K (s) −0.3/s

D (pu/Hz)

2H (pu s)

R (Hz/pu)

Tg (s)

Tt (s)

0.015

0.1667

3.00

0.08

0.40

2.3 Frequency Control in an Interconnected Power System In order to conduct a frequency response analysis for an isolated power system in the presence of sudden load changes, it is usual to model a multi-machine dynamic behaviour by an equivalent single machine as shown in Fig. 2.5. In this case, the proposed model can be used as an equivalent frequency response model for the whole multi-machine power system. The equivalent model lumps the effects of system loads and generators into a single damping constant; the equivalent inertia constant is assumed to equal the sum of the inertia constant of all the generating units. Furthermore, it is assumed that the individual control loops and turbine–generators have the same regulation parameter and response characteristics. It should be reminded that the equivalent model is only useful to simplify the frequency response analysis of an isolated power system. In an isolated power system, regulation of interchange power is not a control issue, and the LFC task is limited to restore the system frequency to the specified nominal value. In order to generalize the described model for interconnected power systems, the control area concept needs to be used, as it is a coherent area consisting of a group of generators and loads, where all the generators respond to changes in load or speed-changer settings, in unison. The frequency is assumed to be the same in all points of a control area. A multi-area power system comprises areas that are interconnected by highvoltage transmission lines or tie-lines. The trend of frequency measured in each control area is an indicator of the trend of the mismatch power in the interconnection and not in the control area alone. The LFC system in each control area of an interconnected (multi-area) power system should control the interchange power with the other control areas as well as its local frequency. Therefore, the described dynamic LFC system model (Fig. 2.5) must be modified by taking into account the tie-line power signal. For this purpose, consider Fig. 2.7, which shows a power system with N-control areas. The power flow on the tie-line from area 1 to area 2 is Ptie,12 =

V1V2 sin(δ1 − δ2 ), X12

(2.3)

where X12 is the tie-line reactance between areas 1 and 2; δ1 , δ2 the power angles of equivalent machines of the areas 1 and 2 and V1 ,V2 are the voltages at equivalent machine’s terminals of the areas 1 and 2.

2.3 Frequency Control in an Interconnected Power System

21

Fig. 2.7 N-control areas power system

By linearizing (2.3) about an equilibrium point (δ10 , δ20 ) ΔPtie,12 = T12 (Δδ1 − Δδ2 ),

(2.4)

where T12 is the synchronizing torque coefficient given by T12 =

|V1 | |V2 | cos δ10 − δ20 . X12

(2.5)

Considering the relationship between area power angle and frequency, (2.4) can be written as ΔPtie,12 = 2π T12

Δ f1 −

Δ f2 ,

(2.6)

where Δ f1 and Δ f2 are frequency deviations in areas 1 and 2, respectively. The Laplace transform of (2.6) means that ΔPtie,12 (s) is obtained ΔPtie,12 (s) =

2π T12 (Δ f1 (s) − Δ f2 (s)). s

(2.7)

Similarly, the tie-line power change between areas 1 and 3 is given by ΔPtie,13 (s) =

2π T13 (Δ f1 (s) − Δ f3 (s)). s

(2.8)

Considering (2.7) and (2.8), the total tie-line power change between area 1 and the other two areas (2) and (3) can be calculated as 2π (2.9) ΔPtie,1 = ΔPtie,12 + ΔPtie,13 = ∑ T1 j Δ f1 − ∑ T1 j Δ f j . s j=2,3 j=2,3 Similarly, for N control areas (Fig. 2.7), the total tie-line power change between area 1 and other areas is

22

2 Real Power Compensation and Frequency Control

Fig. 2.8 Block diagram representation for tie-line power change of control area i in N-control area power system

⎡ ΔPtie,i =

N

∑ ΔPtie,i j =

j=1 j=i

⎤

N 2π ⎢ ⎢ ∑ Ti j Δ fi − s ⎣ j=1 j=i

N

⎥

∑ Ti j Δ f j ⎥⎦ .

(2.10)

j=1 j=i

Equation (2.10) is represented in the form of a block diagram in Fig. 2.8. The effect of changing the tie-line power for an area is equivalent to changing the load of that area. Therefore, the ΔPtie,i must be added to the mechanical power change (ΔPm ) and area load change (ΔPL ) using an appropriate sign. A combination of block diagrams Figs. 2.5 and 2.8 creates a simplified block diagram for control area i in an N-control area power system (Fig. 2.9). The next point to consider is the supplementary control loop in the presence of a tie-line. In the case of an isolated control area, this loop is performed by a feedback from a control area frequency deviation through a simple dynamic controller (Fig. 2.5). As shown in Fig. 2.6, this structure provides a sufficient supplementary control action to force the steady-state frequency deviation to zero. In a multi-area power system, in addition to regulating area frequency, the supplementary control should maintain the net interchange power with neighbouring areas at scheduled values. This is generally accomplished by adding a tie-line flow deviation to the frequency deviation in the supplementary feedback loop. A suitable linear combination of frequency and tie-line power changes for area i, is known as the area control error (ACE), ACEi = ΔPtie,i + βi Δ fi ,

(2.11)

2.3 Frequency Control in an Interconnected Power System

23

Fig. 2.9 Block diagram representation of control area i

Fig. 2.10 Control area i with complete supplementary control

where βi is a bias factor and its suitable value can be computed as follows [1]

βi =

1 + Di . Ri

(2.12)

The block diagram shown in Fig. 2.10 illustrates how supplementary control is implemented using (2.11) and the block diagram shown in Fig. 2.9.

24

2 Real Power Compensation and Frequency Control

The effects of local load changes and interface with other areas are properly considered as two input signals. Each control area monitors its own tie-line power flow and frequency at the area control centre. The ACE signal is computed and allocated to the controller K(s). Finally, the resulting control action signal is applied to the turbine–governor unit. Therefore, it is expected that the supplementary control shown in Fig. 2.10 can ideally meet the basic LFC objectives and maintain area frequency and tie-line interchange at scheduled values. To illustrate LFC system behaviour in a multi-area power system, consider three identical interconnected control areas as shown in Fig. 2.11. The simulation parameters are given in Table 2.2. The system dynamic response following a simultaneous 0.02-pu load step disturbance in control areas 1 and 3 is shown in Fig. 2.12. This figure shows that the power to compensate the tie-line power change initially comes from all areas to respond to the step load increase in areas 1 and 3, and results in a frequency drop sensed by the speed governors of all generators. However, after a few seconds (at steady state), the additional powers against the local load changes come only from areas 1 and 3.

Fig. 2.11 Three-control area power system Table 2.2 Simulation parameters for Fig. 2.11 Area

K (s) D (pu/Hz) 2H (pu s) R (Hz/pu)

Tg (s)

Tt (s)

β (pu/Hz) Ti j (pu/Hz)

Area 1 −0.3/s

0.015

0.1667

3.00

0.08

0.40

0.3483

T12 = 0.20 T13 = 0.25

Area 2 −0.2/s

0.016

0.2017

2.73

0.06

0.44

0.3827

T21 = 0.20 T23 = 0.12

Area 3 −0.4/s

0.015

0.1247

2.82

0.07

0.30

0.3692

T31 = 0.25 T32 = 0.12

2.4 LFC Participation Factor

25

0.02 Δ fi (Hz)

0 −0.02 −0.04 −0.06 0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10

12

14

16

18

20

0

2

4

6

8

10 12 Time (sec)

14

16

18

20

Δ Ptie, i (pu)

0.02 0

−0.02

Δ Pmi (pu)

0.03 0.02 0.01 0 −0.01

Fig. 2.12 Dynamic response following simultaneous 0.02 pu load step disturbance in areas 1 and 3; Area 1 (solid), area 2 (dotted-line) and area 3 (dotted)

2.4 LFC Participation Factor There are many generators in each control area with different turbine–governor parameters and generation types. Furthermore, in the new environment, generators may or may not participate in the LFC task and participation rates are not the same for all participant generators. In order to consider the variety of generation dynamics and their sharing rate in the supplementary control action, the dynamic model of control area i in Fig. 2.10, can be modified to that shown in Fig. 2.13. Here, Mki (s) and αki are the governor–turbine model and LFC participation factor for generator unit k, respectively. Following a load disturbance within the control area, the produced appropriate supplementary control signal is distributed among generator units in proportion to their participation, to make generation follow the load. In a given control area, the sum of participation factors is equal to 1 n

∑ αki = 1,

k=1

0 ≤ αki ≤ 1.

(2.13)

26

2 Real Power Compensation and Frequency Control Primary Control

Supplementary Control

Bi

α 1i

+ Ki (s)

+ ACEi

α 2i

+

1/R1i 1/R2i … 1/Rni

Governor-Turbine

− Mli(s)

− +

M2i(s)

ΔPLi ΔPm1i

+ ΔPm2i

ΔPCi

α ni

− +

+

+

− −

1 Di+2Hi s

Δfi

Rotating mass and load N Tij j=1 j≠1

Σ

Mni(s)

ΔPmni

ΔPtie, i

+

2π/s

− N

Σ

Tij Δf j=1 j≠1

Fig. 2.13 LFC system with different generation units and participation factors in area i

In a competitive environment, LFC participation factors are actually time-dependent variables and must be computed dynamically by an independent organization based on bid prices, availability, congestion problems, costs and other related issues (see Chap. 7). Since the 1970s, the LFC scheme described in Fig. 2.10 has been widely used by researchers for LFC analysis and synthesis. Most research attempts in LFC synthesis have focused on designing a more effective supplementary control. A brief review on previous works is given in Sect. 2.6. The far-reaching deregulation of the power system industry and concomitant new concepts of operation requires an evaluation and re-examination of this scheme, which is already designed to operate with large and central generating facilities. Some issues are discussed in Chap. 3.

2.5 Frequency Operating Standards Following a large generation loss disturbance, a power system’s frequency may drop quickly if the remaining generation no longer matches the load demand. As discussed above, frequency provides a useful index to indicate the system generation and load imbalance. Frequency changes in large-scale power systems are a direct result of the imbalance between the electrical load and the power supplied by system connected generators. Any short-term energy imbalance will result in an instantaneous change in system frequency as the disturbance is initially offset by the kinetic energy of a rotating plant. Significant loss in the generating plant without an adequate system response can produce extreme frequency excursions outside the working range of the plant. As highlighted in Chap. 1, off-normal frequency deviations can directly impact on a power system operation, system reliability and

2.5 Frequency Operating Standards

27

Table 2.3 Frequency operating and control actions Frequency deviation range

Condition

Control action

Δ f1 Δ f2 Δ f3 Δ f4

No contingency or load event Generation/load or network event Separation event Multiple contingency event

Normal operating LFC operating Emergency operating Emergency operating

efficiency. Large frequency deviations can damage equipments, degrade load performance, overload transmission lines and interfere with system protection schemes. These large-frequency deviation events can ultimately lead to a system collapse. Depending on the size of the frequency deviation experienced, supplementary control (LFC), natural governor response and emergency control may all be required to maintain power system frequency. One method of characterizing frequency deviations experienced by a power system is in terms of the frequency deviation ranges and related control actions as shown in Table 2.3. The frequency variation ranges Δ f1 , Δ f2 , Δ f3 and Δ f4 are identified in terms of different power system operating conditions (perhaps specified in terms of local regulations). Under normal operation, frequency is maintained near to nominal frequency by balancing generation and load. That is, for small frequency deviations up to Δ f1 , these deviations can be attenuated by the governor natural autonomous response (primary control). The supplementary control can be used to restore area frequency if it deviates more than Δ f1 Hz. In particular, any LFC system must be designed to maintain the system frequency and time deviations within the limits of specified frequency operating standards. The value of Δ f2 is mainly determined by the available amount of operating reserved power in the system. The LFC system is designed to operate during a relatively small and slow change in real power load and frequency. For large imbalances in real power associated with rapid frequency changes that occur during a fault condition, the LFC system is unable to control frequency. There is a risk that these large frequency deviation events might be followed by additional generation events, load/network events, separation events or multiple contingency events. For such large frequency deviations and in a more complex condition (such as Δ f3 and Δ f4 frequency deviation events), the emergency control and protection schemes must be used to restore the system frequency. This issue is emphasized in Chap. 9. Frequency operating standards could be different from network to network. For example in Australia, for the mainland regions, the Δ f1 , Δ f2 , Δ f3 and Δ f4 are specified as 0.3 Hz, 1 Hz, 2 Hz and 5 Hz, respectively [6]. In the Australian network, the frequency threshold used to start emergency control plans such as under-frequency load shedding is 49 Hz (50 Hz is the nominal system frequency). In all available standards, the acceptable frequency deviation for normal operation is small (about 1%). Larger deviations may activate the protection relays to trip generators and interrupt power system supply. Of course, the relay settings are such

28

2 Real Power Compensation and Frequency Control

Fig. 2.14 Trip conditions for generation units

that the system frequency can deviate for short time periods to allow for transients following a fault. A typical generation trip area according to frequency deviation limits is schematically shown in Fig. 2.14 [7]. Over the years, many frequency control criteria and standards have been established for how well each control area must balance its aggregate generation and load. For instance, the North American Electric Reliability Council (NERC) developed two criteria called control performance standards 1 and 2 (CPS 1 and CPS 2) to obtain the minimum load frequency control performance [8–11]. CPS 1 measures the relationship between the control area ACE and the frequency on a 1-min basis, while CPS 2 is based on a monthly standard and set limits on the maximum average ACE for each 10-min period [11].

2.6 A Literature Review on LFC Synthesis/Analysis LFC synthesis and analysis in power systems has a long history and its literature is voluminous. The LFC scheme described in previous sections has evolved over the past decades, and interest continues in proposing new frequency control approaches with an improved ability to maintain tie-line power flow and system frequency close to nominal values. A survey and exhaustive bibliography on the LFC is given in [12]. The first attempts in the area of power system frequency control are given in [13–17]. Then the standard definitions of the terms associated with power systems frequency control were provided by the IEEE working group [18]. The first optimal control concept for frequency control design of interconnected power systems was addressed by Elgerd and Fosha [19, 20]. A two-area power system consisting of two identical power plants of non-reheat thermal turbines was considered for LFC synthesis.

2.6 A Literature Review on LFC Synthesis/Analysis

29

According to physical constraints and/or to cope with the changed system environment, suggestions for dynamic modelling and modifications to the LFC definitions were given from time to time [21–26]. System non-linearities and dynamic behaviours such as governor dead-band and generation rate constraint have been considered in [22, 27–32]. Some research considered load characteristics [33–36] and the interaction between the frequency (real power) and the voltage (reactive power) control loops [37–39] during the LFC design procedure. Furthermore, LFC analysis/modelling, special applications, constraints formulation, frequency bias estimation, model identification and performance standards have lead to the publishing of numerous reports [8–10, 40–51]. Since Elgerd and Fosha’s work, extensive research has been done on the application of modern control theory to design more effective supplementary controllers. References [52–67] have suggested several LFC synthesis approaches using optimal control techniques. The efforts were usually directed towards the application of suitable linear state feedback controllers to the LFC problem. They have mainly optimized a constructed cost function to meet LFC objectives by well-known optimization techniques. Since an optimal LFC scheme needs the availability of all state variables, some developed strategies have used state estimation using an observer [53–58]. Due to the technical limitations in the design of LFC using all state variables, sub-optimal LFC systems were introduced [68–71]. Apart from optimal/sub-optimal control strategies, the concept of variable-structure systems has also been used to design LFC regulators for power systems [72–79]. These approaches enhance the insensitivity of an LFC system to parameter variations. Since, parametric uncertainty is an important issue in LFC design, the application of robust control theory to the LFC problem in multi-area power systems has been extensively studied during the last two decades [80–97]. The main goal is to maintain robust stability and robust performance against system uncertainties and disturbances. For this purpose, various robust control techniques such as H∞ , linear matrix inequalities (LMI) and Riccati-equation approaches, Kharitonov’s theorem, structured singular value (μ ) theory, quantitative feedback theory, Lyapunove stability theory, pole placement technique and Q-parameterization, have been used. Apart from these design methodologies, adaptive and self-tuning control techniques have been widely used for power system frequency control design during the last three decades [98–103]. The major part of the work reported so far has been performed by considering continuous time power system models. The digital and discrete-type frequency regulator is also reported in some work [32, 62, 101–109]. In the light of recent advances in artificial intelligent control, various intelligentbased control methodologies have been proposed to solve the power system frequency regulation problem [110–130]. Artificial neural networks have been applied to the LFC problem [110–114]. The application of fuzzy logic and genetic algorithms in power system frequency control is witnessed in [115–127]. Fuzzy logic is mainly applied based on fuzzy scheduling of PI-based load–frequency controller parameters. A combination of the intelligent methods has also been applied to the LFC problem [128–130].

30

2 Real Power Compensation and Frequency Control

Few publications have appeared on the application (or in the presence) of special devices such as superconductivity magnetic energy storage (SMES) and solid-state phase shifter [131–133]. The increasing need for electrical energy in the twenty-first century, and in the other hand limited fossil fuel reserves and the increasing concerns to environmental issues call for a fast development in the area of renewable energy sources (RES). Some recent studies analyse the impacts of battery energy storage (BES), photovoltaic (PV) power generation, capacitive energy and wind turbine on the performance of the LFC system or their application in power system frequency control [7, 134–143]. Considerable research on the LFC incorporating an HVDC link is contained in [144–148]. The above-mentioned work has been conducted on power systems under vertically integrated organizations. The classical LFC scheme is difficult to implement in a deregulated power system environment, which includes separate generation companies, independent power producers (IPPs) and distribution and transmission companies with an open access policy. Control strategies for a new structure with a few LFC participators may not be as straight forward as for vertically integrated utility structures. Novel control strategies based on modified dynamical models are needed to maintain reliability and eliminate frequency error. Under a new organization, considerable scenarios on LFC modelling, control and structure description are contained in [149–169]. There are various schemes and organizations for the provision of ancillary services in countries with a restructured electric industry. The type of LFC scheme in a restructured power system is differentiated by how free the market is, who controls the generator units and who has the obligation to execute the LFC [149]. Several modelling and control strategies have reported to adapt well-tested classical LFC schemes to the changing environment of power system operation under deregulation [150, 151, 154, 156, 157, 161]. The effects of deregulation of the power industry on LFC and several general LFC scenarios for power systems after deregulation have been addressed in [149, 152, 153, 155, 159–161].

2.7 Summary The subject of real power frequency control providing definitions and basic concepts is addressed. The load–frequency control mechanism of a single control area is first described and then extended to a multi-area control system. Frequency operating standards, tie-line bias and its application to a multi-area frequency control system are presented. Past achievements in the frequency control literature are briefly reviewed.

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146. M. Sanpei, A. Kakehi, H. Takeda, Application of multi-variable control for automatic frequency controller of HVDC transmission system, IEEE Trans. Power Deliv., vol. 9, no. 2, pp. 1063–1068, 1994. 147. N. Rostamkolai, C. A.Wengner, R. J. Piwko, H. Elahi, M. A. Eitzmann, G. Garzi, P. Taetz, Control design of Santo Tome back-to back HVDC link, IEEE Trans. Power Syst., vol. 8, no. 3, pp. 1250–1256, 1993. 148. K. Y. Lim, Y. Wang, R. Zhou, Decentralised robust load–frequency control in coordination with frequency-controllable HVDC links, Int. J. Electr. Power Energy Syst., vol. 19, no. 7, pp. 423–431, 1997. 149. R. D. Chritie and A. Bose, Load frequency control issues in power system operation after deregulation, IEEE Trans. Power Syst., vol. 11, no. 3, pp. 1191–1200, 1996. 150. J. Kumar, N. G. K. Hoe, G. B. Sheble, AGC simulator for price-based operation, Part I: A model, IEEE Trans. Power Syst., vol. 2, no. 12, pp. 527–532, 1997. 151. J. Kumar, N. G. K. Hoe, G. B. Sheble, AGC simulator for price-based operation, Part II: Case study results, IEEE Trans. Power Syst., vol. 2, no. 12, pp. 533–538, 1997. 152. B. H. Bakken and O. S. Grande, Automatic generation control in a deregulated power system, IEEE Trans. Power Syst., vol. 13, no. 4, pp. 1401–1406, 1998. 153. A. P. S. Meliopoulos, G. J. Cokkinides, A. G. Bakirtzis, Load–frequency control service in a deregulated environment, Decis. Support Syst., vol. 24, pp. 243–250, 1999. 154. V. Donde, M. A. Pai, I. A. Hiskens, Simulation and optimization in a AGC system after deregulation, IEEE Trans. Power Syst., vol. 16, no. 3, pp. 481–489, 2001. 155. J. M. Arroyo and A. J. Conejo, Optimal response of a power generator to energy, AGC, and reserve pool-based markets, IEEE Trans. Power Syst., vol. 17, no. 2, pp. 404–410, 2002. 156. B. Delfino, F. Fornari, S. Massucco, Load–frequency control and inadvertent interchange evaluation in restructured power systems, IEE Proc. Gener. Transm. Distrib., vol. 149, no. 5, pp. 607–614, 2002. 157. H. Bevrani, Y. Mitani, K. Tsuji, Robust AGC: Traditional structure versus restructured scheme, IEE J. Trans. Power Energy, vol. 124-B, no. 5, pp. 751–761, 2004. 158. H. Bevrani, Y. Mitani, K. Tsuji, H. Bevrani Bilateral-based robust load–frequency control, Energy Convers. Manage., vol. 46, pp. 1129–1146, 2005. 159. F. Liu, Y. H. Song, J. Ma, S. Mei, Q. Lu, Optimal load–frequency control in restructured power systems, IEE Proc. Gener. Transm. Distrib., vol. 150, no. 1, pp. 377–386, 2003. 160. S. Bhowmik, K. Tomsovic, A. Bose, Communication models for third party load frequency control, IEEE Trans. Power Syst., vol. 19, no. 1, pp. 543–548, 2004. 161. H. Bevrani, Decentralized Robust Load–Frequency Control Synthesis in Restructured Power Systems. PhD dissertation, Osaka University, 2004. 162. L. Vanslyck, N. Jaleeli, W. R. Kelley, Implications of frequency bias settings on interconnected system operation and inadvertent energy accounting, IEEE Trans. Power Syst., vol. 4, no. 2, pp. 712–723, 1989. 163. H. Singh and A. Papalexopoulos, Competitive procurement of ancillary services by an independent system operator, IEEE Trans. Power Syst., vol. 14, no. 2, pp. 498–504, 1999. 164. K. W. Cheung, P. Shamsollahi, D. Sun, J. Milligan, M. Potishanak, Energy and ancillary service dispatch for the interim ISO New England electricity market, IEEE Trans. Power Syst., vol. 15, no. 3, pp. 968–974, 2000. 165. X. S. Zhao, F. S. Wen, D. Q. Gan, M. X. Huang, C. W. Yu, C. Y. Chung, Determination of AGC capacity requirement and dispatch considering performance penalties, Electr. Power Syst. Res., vol. 70, no. 2, pp. 93–98, 2004. 166. H. Bevrani and T. Hiyama, Robust decentralized PI based LFC design for time-delay power systems, Energy Convers. Manage., vol. 49, pp. 193–204, 2007. 167. H. Bevrani and T. Hiyama, Robust load–frequency regulation: A real-time laboratory experiment, Optimal Control Appl. Methods, vol. 28, no. 6, pp. 419–433, 2007. 168. G. Dellolio, M. Sforna, C. Bruno, M. Pozzi, A pluralistic LFC scheme for online resolution of power congestions between market zones, IEEE Trans. Power Syst., vol. 20, no. 4, pp. 2070–2077, 2005. 169. B. Tyagi and S. C. Srivastava, A decentralized automatic generation control scheme for competitive electricity market, IEEE Trans. Power Syst., vol. 21, no. 1, pp. 312–320, 2006.

Chapter 3

Frequency Response Characteristics and Dynamic Performance

This chapter describes load–frequency control characteristics and dynamic performance. Static and dynamic performances are explained, and the effects of physical constraints (generation rate, dead band, time delays, and uncertainties) on power system frequency control performance are emphasized. The impacts of power system restructuring on frequency regulation are simulated, and a dynamical model to adapt a well-tested classical load–frequency control model to the changing environment of power system operation is presented.

3.1 Frequency Response Analysis A linear dynamical model useful for LFC analysis and synthesis was described in Chap. 2. Figure 3.1 shows the block diagram of typical control area i with n generator units in an N-multiarea power system. The blocks and parameters are defined as follows: Δ f : Frequency deviation ΔPm : Governor valve position ΔPC : Supplementary control action ΔPP : Primary control action ΔPtie : Net tie-line power flow H: Equivalent inertia constant D: Equivalent damping coefficient Ti j : Tie-line synchronizing coefficient with area j B: Frequency bias v: Area interface R: Droop characteristic ACE: Area control error α : Participation factors M(s): Governor–turbine dynamic model K(s): Dynamic controller H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 3, c Springer Science+Business Media LLC 2009

39

40

3 Frequency Response Characteristics and Dynamic Performance Primary Control

Supplementary Control

Bi

α1i

+

1/R1i 1/R2i … 1/Rni

+ ACEi

Mli(s)

ΔPC1i

− +

+ Ki (s)

Governor-Turbine

− ΔPP1i

α2i

ΔPCi

ΔPP2i M2i(s)

ΔPC2i

− +

αni

ΔPLi ΔPm1i

+ ΔPm2i

+

+

− −

1 2Hi s+Di

Δfi

Rotating mass and load N Tij j=1 j≠1

Σ

ΔPPni Mni(s)

ΔPCni

ΔPmni

ΔPtie, i

+

2π/s

− ni

Fig. 3.1 LFC system model

Several low-order models for representing turbine–governor dynamics Mi (s) for use in power system frequency analysis and control design are introduced in Chap. 2. Here, it is assumed that all generators are non-reheat steam units, therefore Mki (s) =

1 1 · , (1 + Tg k s) (1 + Tt k s)

(3.1)

where Tg k and Tt k are governor and turbine time constants, respectively. The balance between connected control areas is achieved by detecting the frequency and the tieline power deviations to generate the ACE signal, which is in turn utilized in a dynamic controller. In Fig. 3.1, the vi is the area interface and can be defined as follows N

vi =

∑ Ti j Δ f j .

(3.2)

j=1 j=i

Considering the effect of primary and supplementary controls, the system frequency can be obtained as n 1 (3.3) Δ fi (s) = ∑ ΔPmki (s) − ΔPtie,i (s) − ΔPL i (s) , 2Hi s + Di k=1 where

ΔPmki (s) = Mki (s) ΔPCki (s) − ΔPPki (s)

and ΔPPki (s) =

Δ fi (s) . Rki

(3.4) (3.5)

3.1 Frequency Response Analysis

41

Here ΔPP and ΔPC are primary (governor natural response) and supplementary (LFC) control actions. The expressions (3.4) and (3.5) can be substituted into (3.3) with the result n 1 1 Δ fi (s) = ∑ Mki (s) ΔPCki (s) − Rki Δ fi (s) − ΔPtie,i (s) − ΔPL i (s) 2Hi s + Di k=1 (3.6) For the sake of load disturbance analysis, we are usually interested in ΔPL i (s) in the form of a step function, i.e., ΔPL i . (3.7) ΔPL i (s) = s Substituting ΔPL i (s) in (3.6) and summarizing the result yields n 1 1 (3.8) Δ fi (s) = ∑ Mki (s)ΔPCki (s) − ΔPtie,i (s) − sgi (s) ΔPL i , gi (s) k=1 where

n

Mki (s) . k=1 Rki

gi (s) = 2Hi s + Di + ∑

(3.9)

Substituting Mki (s) from (3.1) in (3.8) and (3.9), and using the final value theorem, the frequency deviation in steady state Δ fss,i can be obtained from (3.8). Δ fss,i = Lim s Δ fi (s) = s→0

1 1 ΔPC i − ΔPL i . gi (0) gi (0)

(3.10)

It is assumed that ΔPtie,i approach zero at steady state, and n

∑ Mki (s)ΔPCki (s),

(3.11)

1 1 = Di + . R R sys,i ki k=1

(3.12)

ΔPC i = Lim s s→0

k=1 n

gi (0) = Di + ∑

Here, Rsys,i is the equivalent system drooping characteristic, and 1 Rsys,i

n

=

1

∑ Rki .

(3.13)

k=1

By definition (2.12), gi (0) is equivalent to the system’s frequency response characteristic (βi ). 1 βi = Di + . (3.14) Rsys,i Using (3.12), (3.10) can be rewritten into the following form Δ fss,i =

ΔPC i − ΔPL i . Di + 1/Rsys,i

(3.15)

42

3 Frequency Response Characteristics and Dynamic Performance

Equation (3.15) shows that if the disturbance magnitude matches with the available power reserve (supplementary control) ΔPC i = ΔPL i , the frequency deviation converges to zero in steady state. Since the value of a droop characteristic Rki is bounded between about 0.05 and 0.1 for most generator units (0.05 ≤ Rki ≤ 0.1) [1], for a given control system according to (3.13) we can write Rsys,i ≤ Rmin . For a small enough, DRsys,i , (3.15) can be reduced to Δ fss,i =

Rsys,i (ΔPC i − ΔPL i ) ∼ = Rsys,i (ΔPC i − ΔPL i ). (Di Rsys,i + 1)

(3.16)

Without a supplementary control signal (ΔPC i = 0), the steady-state frequency deviation will be proportional to disturbance magnitude as follows Δ fss,i = −

Rsys,i ΔPL i . (Di Rsys,i + 1)

(3.17)

It is noteworthy that the above result is obtained by assuming no supplementary control and tie-line variations. However, practically tie-line deviation is not zero, and hence to achieve an exact result, it should be properly reflected in the steadystate frequency deviation (3.17). In an interconnected power system, the frequency deviation following a load variation becomes zero, after all tie-line flow changes (and ACE signals) have been zeroed. Without the intervention of the supplementary control, the steady-state frequency deviation would depend on the equivalent drooping characteristic of all the system, namely of all generators in all areas, as well as from the damping factors of all areas. The time constants of governor–turbine units are smaller than the time constant of an overall power system (rotating mass and load) [2]. Hence, for the purpose of simplification in dynamic frequency analysis, it is reasonable to assume that Tg k = 0 and Tt k = 0. With this assumption, the frequency response (3.8) for a generator unit with primary control loop (ΔPC i (s) = 0, ΔPtie,i (s) = 0) can be reduced to Δ fi (s) ∼ =

ΔPL i 1 − 2Hi s + Di + (1/Rsys,i ) s

(3.18)

Simplification of (3.18) and resolving into partial fractions yields ⎛ ⎞ ΔP R 1 sys,i Li ⎝ 1 ⎠. Δ fi (s) ∼ − =− (Di Rsys,i + 1) s s + Di Rsys,i +1 2Hi Rsys,i

(3.19)

3.2 State-Space Dynamic Model

43

Considering (3.17), (3.19) becomes Δ fi (s) ∼ = Δ fss,i

1 1 − s s + τi

,

(3.20)

where τi is time constant of the closed-loop system

τi =

Di Rsys,i + 1 2Hi Rsys,i

(3.21)

and inverse Laplace transformation of (3.20) gives Δ fi (t) ∼ = Δ fss,i (1 − e−τi t ).

(3.22)

As shown in the next few chapters, the frequency dynamic behaviour of an LFC system with a supplementary control loop is more complex than the described dynamic given in (3.22).

3.2 State-Space Dynamic Model State-space model of a LFC dynamical system is a useful representation for the application of the modern/robust control theory. Using appropriate definitions and state variables, as given in (3.24)–(3.28), the state-space realization of control area i shown in Fig. 3.1 can be easily obtained as (3.23) [3]. x˙i = Ai xi + B1i wi + B2i ui (3.23) yi = Cyi xi , xiT = Δ fi xm i = ΔPm 1i

ΔPm 2i

ΔPtie−i

xm i

xg i ,

· · · ΔPm ni , xg i = ΔPg 1i

(3.24) ΔPg 2i

· · · ΔPg ni , (3.25)

ui = ΔPC i ,

(3.26)

yi = ACEi = βi Δ fi + ΔPtie,i ,

(3.27)

wTi = ΔPL i

vi ,

where ΔPg i denotes the governor valve position change, and

(3.28)

44

3 Frequency Response Characteristics and Dynamic Performance

⎡

Ai12 Ai22 Ai32

Ai11 Ai = ⎣ Ai21 Ai31 ⎡

⎤ ⎤ ⎤ ⎡ ⎡ Ai13 B1i1 B2i1 Ai23 ⎦, B1i = ⎣ B1i2 ⎦, B2i = ⎣ B2i2 ⎦ , Ai33 B1i3 B2i3 −1/2Hi

−Di /2Hi

N ⎢ Ai11 = ⎣ 2π ∑ Ti j

0

⎤ ⎥ ⎦, Ai12 =

j=1 j=i

Ai22 = −Ai23 = diag −1/Tt1i Ai33 = diag

−1/Tg1i

−1/Tg2i

⎡

−1/(Tg ni Rni )

B1i1 = ⎣

−1/Tt2i

, 2×n

· · · −1/Tt ni ,

· · · −1/Tg ni

,

⎤

0

0

−2π

⎦, B1i2 = B1i3 = 0n×2 ,

B2i1 = 02×1 , B2i2 = 0n×1 , BT2i3 = α1i /Tg 1i 1 01×n

0

−1/2Hi

Cyi = βi

· · · 1/2Hi ··· 0

⎤ 0 .. ⎥, A = AT = 0 , A = 0 , n×n 2×n i32 i21 . ⎦ i13

−1/(Tg 1i R1i ) ⎢ .. Ai31 = ⎣ . ⎡

1/2Hi 0

α2i /Tg 2i

· · · αni /Tg ni ,

01×n .

According to Fig. 3.1, in each control area the control input is performed by the ACE signal. (3.29) ui = ΔPC i = f (ACEi ). where f (·) is a function, which identifies the dynamics of the controller.

3.3 Frequency Control in a Deregulated Environment Towards the end of the twentieth century many countries sought to reduce direct government involvement in, and to increase the economic efficiency of, their electricity industries through a change in industry managements, often described as electricity industry deregulation. The power system frequency control becomes more difficult to predict with the restructuring occurring in many parts of the electric industry. This area has recently received increased attention by various technical

3.3 Frequency Control in a Deregulated Environment

45

committees [4, 5]. Part of the reason for this attention is the need to identify and quantify frequency control as an ancillary service that must be provided by generation companies. In a traditional power system, generation, transmission, and distribution are owned by a single entity called a vertically integrated utility (VIU), which supplies power to the customers at regulated rates. Usually, the definition of a control area is determined by the physical boundaries of a VIU. All such control areas are interconnected by tie-lines. In an open energy market, generation companies (Gencos) may or may not participate in the LFC task. On the other hand, a distribution company (Disco) may contract individually with Gencos or independent power producers (IPPs) for power in different areas. The Gencos submit their ramp rates (MW/min) and bids to the market operator. After a bidding evaluation, those Gencos selected to provide regulation services must perform their functions according to the ramp rates approved by the responsible organization [6]. Therefore, in a new environment, control is highly decentralized. Each loadmatching contract requires a separate control process, yet this process must cooperatively interact to maintain system frequency and tie-line power interchange. In these structures, a separate control process exists for each control area. The boundary of the control area encloses the Gencos and the Discos associated with the performed contracts. The Disco is responsible for buying power from Gencos and getting it directly or through transmission companies (Transcos) to its load. Such a configuration is conceptually shown in Fig. 3.2. The control areas are interconnected to each other, either through Transco or through Gencos. Each control area has its own LFC and is responsible for tracking its own load and honouring tie-line power exchange contracts with its neighbours. Currently, these transactions are conducted under the supervision of an independent system operator (ISO), independent contract administrator (ICA), transmission system operator (TSO) or another responsible organization. The Gencos send the bid regulating reserves F($, T, Q) to an area/local control centre through a secure network service. These bids are sorted by pre-specified time period and price. Then, the sorted regulating reserves with the demanded load from Discos, and the measured tie-line flow and area frequency, are used to provide control decision. The bids are checked and re-sorted according to the congestion condition and screening of available capacity. Then, the control centre provides the ACE signal and computes the participation factors αi in order to load following by the available Gencos, to cover the total contracted load demand ∑ ΔPL (t) and probable disturbances. Since, ultimately, the Genco must adjust the governor setpoint(s) of its generator(s) for the LFC, the control algorithm for each control area is executed at the Genco end (which is equipped with an appropriate controller). With enough fast communication channels, the ACE signal is computed and the raise/lower control signals are transmitted to the generator units once every 1–3 s, while the results of computing participation factors and load generation scheduling by market operator (economic dispatch centre) are executed daily or every few hours.

46

3 Frequency Response Characteristics and Dynamic Performance

Fig. 3.2 A conceptual control area in a deregulated environment

There are several schemes and different structures for the provision of LFC services in countries with a restructured electric industry, differentiated by how free the market is, who controls generator units and who has the obligation to execute LFC. Some possible LFC structures are introduced in [6–12]. For example, in Europe, three different types of control are defined by the Union for the Co-ordination of Transmission of Electricity (UCTE): centralized network control, decentralized pluralistic network control and decentralized hierarchical network control [10]. The countries with a central electricity supply system use the central network control, where LFC is operated through a single secondary controller. The other two decentralized methods consider some separate control areas, and each control area has individual controller. One or more control areas operating together for what concerns LFC can establish a control block, and in this case a block co-ordinator is defined as the overall control centre for the LFC and for the accounting of the whole control block. In a decentralized pluralistic network, each control area regulates the frequency by its own controller. If some control areas perform a control block, a separate controller (block coordinator) coordinates the whole block towards its neighbour’s blocks/control area by means of its own controller and regulating capacity [13].

3.4 LFC Dynamics and Bilateral Contacts

47

3.4 LFC Dynamics and Bilateral Contacts Technically, the basic concepts of conventional LFC structure are not changed, and therefore it is possible to adapt well-tested conventional LFC scheme to the changing environment of power system operation under deregulation. This section addresses a modified dynamical model for analysis and synthesis of a bilateral-based LFC scheme, following the ideas presented in [14]. The proposed LFC model uses all information required in a vertically operated utility industry plus the contract data information. Based on the bilateral transactions, a Disco has the freedom to contract with any available Genco in its own or another control area.

3.4.1 Modelling Analogous to the traditional LFC [15, 16], the physical control area boundaries are assumed for each Disco, its distribution area and the local Gencos as before. However, the Disco may have a contract with a Genco outside its distribution area boundaries, in another control area. There can be various combinations of contracts between each Disco and available Gencos. On the other hand, each Genco can contract with various Discos. Similar to the Disco participation matrix in [14], the generation participation matrix (GPM) concept is defined to visualize these bilateral contracts conveniently in the generalized model. The GPM shows the participation factor of each Genco in the considered control areas, and each control area is determined by a Disco. The rows of a GPM correspond to Gencos and the columns to control areas that contract power. For example, for a large-scale power system with m control areas (Discos) and n Gencos, the GPM will have the following structure ⎡

gpf11 ⎢ gpf21 ⎢ ⎢ .. GPM = ⎢ . ⎢ ⎣ gpf(n−1)1 gpfn1

··· gpf1(m−1) ··· gpf2(m−1) .. .. . . · · · gpf(n−1)(m−1) ··· gpfn(m−1)

gpf12 gpf22 .. . gpf(n−1)2 gpfn2

gpf1m gpf2m .. . gpf(n−1)m gpfnm

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(3.30)

where, gpfki refers to generation participation factor and shows the participation factor of Genco k in the load following of area i (based on a specified bilateral contract). The sum of all the entries in a column in this matrix is unity, i.e., n

∑ gpfki = 1.

(3.31)

k=1

Any entry in a GPM that corresponds to a contracted load by a Disco, demanded from the corresponding Genco, must be reflected to the control area system. This

48

3 Frequency Response Characteristics and Dynamic Performance

Fig. 3.3 LFC structure with bilateral contracts

introduces new information signals that were absent in the traditional LFC structure. These signals identify which Genco has to follow a load demanded by a specified Disco. The scheduled flow over the tie-lines must be adjusted by demand signals of those distribution control areas having a contract with Gencos outside its boundaries. The difference between scheduled and current (actual) tie-line power flows gives a tie-line power error, which is used to compose an ACE signal. Based on the above explanations, a modified LFC block diagram for control area i in a contract-based environment can be obtained as shown in Fig. 3.3. New information signals due to various possible contracts between Disco i and other Discos and Gencos are shown as dashed line inputs. The v1i includes the sum of local contracted demand and area load disturbances. The v2i includes the interface effects between each control area and other areas v1i = ΔPL i + ΔPd i , v2i =

N

∑ Ti j Δ f j

(3.32)

j=1 j=i

The v3i is the scheduled tie-line power change. Using the given idea in [14], the scheduled tie-line power v3i for a N control area can be generalized as follows v3i = ∑ (Total export power − Total import power) = ⎛

⎞

N

n

j=1 j=i

k=1

∑ ∑ gpfk j

ΔPL j

n ⎜ N ⎟ ⎟ ΔPL i −∑⎜ gpf ∑ jk ⎝ ⎠ k=1

j=1 j=i

(3.33)

3.4 LFC Dynamics and Bilateral Contacts

49

and according to Fig. 3.3 ΔPtie−i, error = ΔPtie−i, actual − v3i .

(3.34)

The input signal v4i shows a vector that includes contracted demands of other Discos from Gencos of area i, (3.35) v4i = v4i−1 v4i−2 · · · v4i−n , where

N

v4i−1 = ∑ gpf1 j ΔPL j j=1 j=i

(3.36)

.. . N

v4i−n = ∑ gpfn j ΔPL j , j=1

N is the number of control areas, ΔPL i is the contracted demand of area i, ΔPd i is the disturbance and uncontracted demand in area i and ΔPtie−i actual is the actual ΔPtie−i . The generation of each Genco must track the contracted demands of Discos in steady state. The desired total power generation of a Genco i in terms of GPM entries can be calculated as N (3.37) ΔPm i = ∑ gpfi j ΔPL j . j=1

In order to take the contract violation cases into account, as given in [14] and [17], the excess demand by a distribution area (Disco) is not contracted out by any Genco, and the load change in the area appears only in terms of its ACE and is shared by all the Gencos of the area (in which the contract violation occurs). The associated expressions and the place of new input signals in the bilateralbased LFC model were selected in such a way that the model covers all possible contract combinations given by GPM, and the calculation results from (3.33) and (3.37) are completely matched to the corresponding simulation results for a given set of bilateral contracts. Assuming the state and control variables given in (3.24)–(3.27), the state-space dynamical representation for the described LFC model can be easily obtained [18]. The vi in (3.28) must be rewritten as (3.38) (3.38) wTi = v1i v2i v3i v4i , vT4i = v4i−1 v4i−2 · · · v4i−n Therefore, B1i in (3.23) is changed as follows: ⎤ ⎡ ⎡ B1i11 B1i12 −1/2Hi 0 B1i = ⎣ B1i21 B1i22 ⎦ , B1i11 = ⎣ B1i31 B1i32 0

0 −2π 0

B1i21 = B1i31 = 0n×3 , B1i12 = 03×n , B1i22 = 0n×n , B1i32 = diag 1/Tg 1i 1/Tg 2i · · · 1/Tg ni .

⎤ 0 0 ⎦, −1

50

3 Frequency Response Characteristics and Dynamic Performance

The other coefficient matrices and vectors can be defined the same as those given in Sect. 3.2. The above bilateral based LFC model has been recently used in several reported LFC design approaches [18–20].

3.4.2 Simulation Example A three-control area power system shown in Fig. 3.4 is considered as a test system. It is assumed that each control area includes two Gencos and one Disco. The power system parameters are tabulated in Tables 3.1 and 3.2. Case 1. It is assumed that a step increase in demand as ΔPL1 = 100 MW, ΔPL2 = 70 MW and ΔPL3 = 60 MW are applied to the control areas and each Disco demand is sent to its local Gencos only, based on the following GPM

Fig. 3.4 Three-control area power system Table 3.1 Applied data for Gencos Quantity Rating (MW) R (Hz/pu) Tt (s) Tg (s) α

Genco 1

Genco 2

Genco 3

Genco 4

Genco 5

Genco 6

800 2.4 0.36 0.06 0.5

1,000 3.3 0.42 0.07 0.5

1,100 2.5 0.44 0.06 0.5

1,200 2.4 0.4 0.08 0.5

1,000 3 0. 36 0.07 0.5

1,000 2.4 0.4 0.08 0.5

3.4 LFC Dynamics and Bilateral Contacts

51

Table 3.2 Applied control area parameters Quantity

Δ Ptie,2-3 (pu) Δ Ptie,1-2 (pu) Δ f3 (Hz)

Δ f2 (Hz)

Δ f1 (Hz)

D (pu/Hz) 2H(pu s) B (pu/Hz) K (s), [19] Ti j (pu/Hz)

Area 1

Area 2

Area 3

0.0084 0.1667 0.8675 −0.2695 − 0.3788/s

0.014 0.2 0.795 −0.0418 − 0.1806/s 0.545

0.011 0.1667 0.870 −0.2319 − 0.3796/s

0.05 0 −0.05 0 0.05

5

10

15

5

10

15

5

10

15

5

10

15

10

15

0 −0.05 0 0.02 0 −0.02 0 0.01 0 −0.01 0 0.01 0 −0.01 0

5 Time (sec)

Fig. 3.5 System response for simulation case 1; frequency deviation and tie-line power changes

⎡ ⎢ ⎢ ⎢ GPM = ⎢ ⎢ ⎢ ⎣

0.5 0 0 0.5 0 0 0 0.5 0 0 0.5 0 0 0 0.5 0 0 0.5

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

The frequency deviation (Δ f ), tie-line power flow (ΔPtie ) and mechanical power changes (ΔPm ) are shown in Figs. 3.5 and 3.6 for the closed-loop system. Since there are no contracts between areas, the scheduled steady-state power that flows over the tie-lines is zero. The actual tie-line powers are shown in Fig. 3.5. As shown in Fig. 3.6, the actual generated powers of the Gencos, according to (3.37), reach the desired values in the steady state ΔPm1 = gpf11 ΔPL1 + gpf12 ΔPL2 + gpf13 ΔPL3 = 0.5(0.1) + 0 + 0 = 0.05 pu

3 Frequency Response Characteristics and Dynamic Performance Δ Pm6 (pu) Δ Pm5 (pu) Δ Pm4 (pu) Δ Pm3 (pu) Δ Pm2 (pu) Δ Pm1 (pu)

52 0.1 0.05 0 0 0.1

5

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Fig. 3.6 System response for simulation case 1; mechanical power changes

and ΔPm2 = 0.05 pu, ΔPm3 = ΔPm4 = 0.035 pu, ΔPm5 = ΔPm6 = 0.03 pu. Case 2. Consider the following larger demands by Disco 2 and Disco 3, i.e., ΔPL1 = 100 MW, ΔPL2 = 100 MW and ΔPL3 = 100 MW, and assume the Discos contract with the available Gencos in other areas, according to the following GPM ⎡ ⎤ 0.25 0.25 0 ⎢ 0.5 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ 0.25 0.75 ⎢ ⎥. GPM = ⎢ ⎥ 0.25 0.25 0 ⎢ ⎥ ⎣ 0 0.25 0 ⎦ 0 0 0.25 The closed-loop response is shown in Figs. 3.7 and 3.8. According to (3.37), the actual generated powers of the Gencos for this scenario can be obtained as follows. ΔPm1 = 0.25(0.1) + 0.25(0.1) + 0 = 0.05 pu, ΔPm2 = 0.05 pu, ΔPm3 = 0.1 pu, ΔPm4 = 0.05 pu, ΔPm5 = ΔPm6 = 0.025 pu.

53

0.05 0 −0.05 0

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Δ Ptie,2-3 (pu)

Δ Ptie,1-2 (pu) Δ f3 (Hz)

Δ f2 (Hz)

Δ f1 (Hz)

3.4 LFC Dynamics and Bilateral Contacts

0.1 0.05 0

Time (sec)

Δ Pm6 (pu) Δ Pm5 (pu)

Δ Pm4 (pu)

Δ Pm3 (pu)

Δ Pm2 (pu)

Δ Pm1 (pu)

Fig. 3.7 System response for simulation case 2; frequency deviation and tie-line power changes

0.1 0.05 0 0 0.1

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Fig. 3.8 System response for simulation case 1; mechanical power changes

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3 Frequency Response Characteristics and Dynamic Performance

The simulation results show the same values in steady state. The scheduled tie-line powers in the directions from area 1 to area 2, and area 2 to area 3, using (3.33) are obtained as follows. Figure 3.7 shows actual tie-line powers, and they reach to the below values at steady state. ΔPtie,1−2 = (gpf12 + gpf22 )ΔPL2 − (gpf31 + gpf41 )ΔPL1 = (0.25 + 0)0.1 − (0 + 0.25)0.1 = 0 pu and ΔPtie,2−3 = (0.75 + 0)0.1 − (0.25 + 0)0.1 = 0.05 pu.

Δ Pm6 (pu) Δ Pm5 (pu)

Δ Pm4 (pu)

Δ Pm3 (pu)

Δ Pm2 (pu)

Δ Pm1 (pu)

Also, the effect of the contract violation problem is simulated. Assume Disco 1 demands 50 MW more power than that specified in the contract. This excess power must be reflected as an uncontracted local demand of area 1 and must be supplied by local Gencos only. The simulation result is shown in Fig. 3.9, which shows that the excess load is only taken up by Genco 1 and Genco 2, according to their LFC participation factors, and Gencos in other distribution areas do not participate to compensate it. Since GPM is the same as in case 2, the generated power of Gencos in area 2 and area 3 is the same as in case 2 in steady state.

0.1 0.05 0

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Time (sec)

Fig. 3.9 Generated power in responses to contract violation

3.5 Physical Constraints

55

The simulation results for various cases demonstrate the effectiveness of the described model as a suitable dynamical model for LFC analysis and synthesis, in a bilateral-based large-scale power system.

3.5 Physical Constraints The performed studies in the previous sections on LFC dynamic performance have been made based upon a linearized analysis. The described LFC model so far does not consider the effects of the physical constraints. Although considering all dynamics in frequency control synthesis and analysis may be difficult and not useful [6], it should be noted that to get an accurate perception of the LFC subject it is necessary to consider the important inherent requirement and the basic constraints imposed by the physical system dynamics, and model them for the sake of performance evaluation.

3.5.1 Generation Rate and Dead Band An important physical constraint is on the rate of change of power generation due to the limitation of thermal and mechanical movements. LFC studies that do not take into account the delays caused by the crossover elements in a thermal unit, or the behaviour of the penstocks in a hydraulic installation, in addition to the sampling interval of the data acquisition system, results in a situation where frequency and tie-line power could be returned to their scheduled value within 1 s. In a real LFC system, rapidly varying components of system signals are almost unobservable due to various filters involved in the process. Hence, the performance of a designed LFC system is dependent on how generation units respond to the control signal. A very fast response for an LFC system is neither possible nor desirable [21]. A useful control strategy must be able to maintain sufficient levels of reserved control range and control rate. The generation rate for non-reheat thermal units is usually higher than the generation rate for reheat units [22, 23]. The reheat units have a generation rate about of 3–10% pu MW/min. For hydro-units, the rate is of the order of 100% continuous maximum rating (CMR) per minute [24]. Results of investigations of the impacts of generation rate constraint (GRC) on the performance of LFC systems are reported in [25–27]. Speed governor dead band is known as another important issue in power system performance. By changing the input signal, the speed governor may not immediately react until the input reaches a specified value. All governors have a dead band in response, which is important for power system frequency control in the presence of disturbances. Governor dead band is defined as the total magnitude of a sustained speed change, within which there is no resulting change in valve position.

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3 Frequency Response Characteristics and Dynamic Performance

Fig. 3.10 Non-reheat generator unit model with GRC and dead band

The effect of the governor dead band is to increase the apparent steady-state speed regulation [28]. The maximum value of dead band for governors of large steam turbines is specified as 0.06% (0.036 Hz) [29]. The effects of the governor dead band on power system dynamics and frequency control performance were studied in the last five decades [21, 28, 30–33]. The results indicate that for a wide dead band the frequency control performance can be significantly degraded. Several methods have been developed to consider the GRC and speed governor dead band for the analysis/synthesis of LFC systems. For the frequency response analysis and simulations, these non-linear dynamics can be usually considered by adding a limiter and a hysteresis pattern to the governor–turbine system model, as shown for a non-reheat steam turbine in Fig. 3.10. The VU and VL are the maximum and minimum limits that restrict the rate of valve (gate) closing (opening) speeds.

3.5.2 Time Delays In LFC practice, rapid responses and varying components of frequency are almost unobservable due to various filters and delays involved in the LFC process. Any signal processing and filtering introduces delays that should be considered. Typical filters on tie-line metering and ACE signal (with the response characteristics of generator units) use about 2 s or more for the data acquisition and decision cycles of the LFC systems. In a new environment, the communication delays in the LFC synthesis/analysis are becoming a more significant challenge due to the restructuring, expanding of physical setups, functionality and complexity of power systems. Most published research works on the LFC design during the last decades have neglected problems associated with the communication network. Although, under the traditional dedicated communication links, this was a valid assumption, the use of an open communication infrastructure to support the ancillary services in deregulated environments raises concerns about problems that may arise in the communication system. In the control systems, it is well known that time delays can degrade a system’s performance and even cause system instability [34–36].

3.5 Physical Constraints

57

Fig. 3.11 Time delays representation in a LFC system

The time delays in a LFC system mainly exist on the communication channels between the control centre and operating stations; specifically on the measured frequency and power tie-line flow from remote terminal units (RTUs) to the control centre and the delay on the produced rise/lower signal from control centre to individual generation units [37, 38]. These delays are schematically shown in Fig. 3.11. Here, the delay is expressed by an exponential function e−sτ , where τ gives the communication delay time. The introduction of time delays in the supplementary control loop reduces the effectiveness of LFC system performance. It is shown that the frequency control performance declines when the delay time increases [33]. In order to satisfy the desired performance for a multi-area power system, the design of a controller should take into account these delays. Recently, several papers have been published to address the LFC modelling/synthesis in the presence of communication delays [33, 37–44]. The effects of signal delays on the load following task have been discussed. Reference [37] is focused on the network delay models and communication network requirement for a third party LFC service. A compensation method for communication time delay in the LFC systems is addressed in [39], and some control design methods based on linear matrix inequalities (LMI) are proposed for the LFC system with communication delays in [38, 40–42, 44]. Dynamic analysis of LFC performance and standards are given in [33, 43]. In LFC practice, to remove the fast changes and probable added noises, system frequency gradient and ACE signals must be filtered before being used. Considering the total effects of generating rate, dead band, filters and delays for LFC performance analysis gives an appropriate model, which is shown in Fig. 3.12. This model is useful for digital simulations. The ACE signal is filtered and if exceeds a threshold at an interval TW , it will be applied to a proportional integral (PI) control block. The controller can be activated to send higher/lower pulses to generating plants if its input ACE signal exceeds a standard limit. Delays, ramping rate and range limits are different for various plants.

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3 Frequency Response Characteristics and Dynamic Performance

Fig. 3.12 An LFC model for dynamic performance analysis

In the presence of GRC, dead band and time delays, the LFC system model becomes highly non-linear; hence, it will be difficult to use the linear control theory for performance optimization and control design. In the next chapters, it is shown that the above constraints can be easily considered through the robust synthesis procedure using appropriate fictitious weights on the controlled signals.

3.5.3 Uncertainties Investigation of power system behaviour typically involves numerous uncertainties. With ongoing system restructuring, continuous change of dynamics/load and operating conditions, the uncertainty issue in power system operation and control has increasingly become a challenge.

3.5 Physical Constraints

59

The uncertainty reflects the lack of complete knowledge of the exact value of parameters, components, and quantities being measured. Generally, continuous parameters variation, unmodelled dynamics, inexact definition/measurement and consequent approximations are the main sources of power system uncertainties. Recently, in order to address uncertainties and their formulation in a power system control, various approaches have been proposed [45–50]. The linearized frequency control system model may only be valid for a narrow band around a particular operating condition. To deal with this problem, for frequency control design it is important to formulate/model the effects of uncertainties on the system dynamics and performance [6]. For the application of the robust control theory in LFC synthesis, the control area uncertainties can be represented using appropriate modelling techniques. For instance, the uncertainties due to unmodelled dynamics and parameter variations can be modelled by an unstructured multiplicative uncertainty block as shown in Fig. 3.13. Let Gˆ i (s) denotes the transfer function from the control input ui (ΔPC i ) to the control output yi (ACEi ) at operating points other than nominal point. Then, following a practice common in robust control, the uncertainties transfer function is represented as (3.39) Δi (s) = [Gˆ i (s) − Gi (s)]Gi (s)−1 , Gi (s) = 0, where Δi (s) shows the uncertainty block corresponding to the uncertainties and Gi (s) is the nominal transfer function model. As an example, consider the power system example described in Sect. 3.4.2, and assume that the rotating mass and load pattern parameters have uncertain values in each control area. The variation range for Di and Hi parameters in each control area is assumed to be ±20%. The resulting uncertainty in each control area can be modelled as a multiplicative uncertainty. Using (3.39), some sample uncertainties corresponding to different values of Di and Mi for control area 1 are obtained, as shown in Fig. 3.14. Since the frequency responses of the above uncertainties are close to each other, using a single norm bounded transfer function to cover all possible perturbed plants reduces the complexity of the frequency control synthesis procedure [51].

Fig. 3.13 Block diagram representation of multiplicative uncertainty

60

3 Frequency Response Characteristics and Dynamic Performance 100

Magnitude

10−1

10−2

10−3

10−4

10−5

0

1

2

3

4

5

6

7

8

9

10

Frequency (rad/sec)

Fig. 3.14 Uncertainty plots due to parameters changes in area 1; Di (dotted) and Mi (dash-dotted)

3.6 Summary The load–frequency control characteristics and dynamic performance are described. The effects of physical constraints (generation rate, dead band, time delays and uncertainties) on power system frequency control performance are emphasized. The impacts of power system restructuring on frequency regulation are discussed, and a dynamical model to adapt a well-tested classical load–frequency control model to the changing environment of power system operation is simulated.

References 1. P. M. Anderson and M. Mirheydar, A low-order system frequency response model, IEEE Trans. Power Syst., 5(3), 720–729, 1990. 2. T. K. Nagsarkar and M. S. Sukhija, Power System Analysis. New Delhi: Oxford University Press, 2007. 3. H. Bevrani, Y. Mitani and K. Tsuji, Robust decentralized load–frequency control using an iterative linear matrix inequalities algorithm, IEE Proc. Gener. Transm. Distrib., 150(3), 347–354, 2004. 4. IEEE System Dynamics Performance Committee Panel Session, Frequency control requirement, trends and challenges in new utility environment, Proc. IEEE PES Winter Meeting, New York, NY, 1999. 5. CIGRE SCTF 38.02.14, Analysis and modelling needs of power systems under major frequency disturbances, CIGRE Technical Brochure, No. 148, 1999. 6. H. Bevrani, Decentralized Robust Load–Frequency Control Synthesis in Restructured Power Systems. PhD dissertation, Osaka University, Japan, 2004.

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7. R. D. Chritie and A. Bose, Load frequency control issues in power system operation after deregulation, IEEE Trans. Power Syst., 11(3), 1191–1200, 1996. 8. B. Delfino, F. Fornari and S. Massucco, Load–frequency control and inadvertent interchange evaluation in restructured power systems, IEE Proc. Gener. Transm. Distrib., 149(5), 607–614, 2002. 9. J. Kumar, NG. K. Hoe and G. B. Sheble, AGC simulator for price-based operation, Part I: A model, IEEE Trans. Power Syst., 2(12), 527–532, 1997. 10. UCPTE Doc. UCPTE rules for the co-ordination of the accounting and the organization of the load–frequency control, 1999. 11. A. P. S. Meliopouls, G. J. Cokkinides and A. G. Bakirtzis, Load–frequency control service in a deregulated environment, Decision Support Syst., 24, 243–250, 1999. 12. B. Roffel and W. W. deBoer, Analysis of power and frequency control requirements in view of increased decentralized production and market liberalization, Control Eng. Pract., 11, 367–375, 2003. 13. H. Bevrani, Y. Mitani and K. Tsuji, On robust load–frequency regulation in a restructured power system, IEEJ Trans. Power Energy, 124-B(2), 190–198, 2004. 14. V. Donde, M. A. Pai and I. A. Hiskens, Simulation and optimization in a AGC system after deregulation, IEEE Trans. Power Syst., 16(3), 481–489, 2001. 15. O. I. Elgerd and C. Fosha, Optimum megawatt-frequency control of multiarea electric energy systems, IEEE Trans. Power App. Syst., PAS-89(4), 556–563, 1970. 16. C. Fosha and O. I. Elgerd, The megawatt-frequency control problem: A new approach via optimal control, IEEE Trans. Power App. Syst., 89(4), 563–577, 1970. 17. J. Kumar, NG. K. Hoe and G. B. Sheble, AGC simulator for price-based operation, Part II: Case study results, IEEE Trans. Power Syst., 2(12), 533–538, 1997. 18. H. Bevrani, Y. Mitani and K. Tsuji, Robust decentralized AGC in a restructured power system, Energy Convers. Manage., 45, 2297–2312, 2004. 19. H. Bevrani, Y. Mitani and K. Tsuji, Robust AGC: Traditional structure versus restructured scheme, IEEJ Trans. Power Energy, 124-B(5), 751–761, 2004. 20. H. Bevrani, Y. Mitani, K. Tsuji and H. Bevrani, Bilateral-based robust load–frequency control, Energy Convers. Manage., 46, 1129–1146, 2005. 21. N. Jaleeli, D. N. Ewart and L. H. Fink, Understanding automatic generation control, IEEE Trans. Power Syst., 7(3), 1106–1112, 1992. 22. IEEE Committee Report, Power plant response, IEEE Trans. Power App. Syst., 86, 484–399, 1967. 23. IEEE Committee Report, Dynamic models for steam and hydro turbines in power system studies, IEEE Trans. Power App. Syst., 92, 1904–1915, 1973. 24. P. Kundur, Power System Stability and Control. New York, NY: McGraw-Hill, 1994. 25. J. Nanda, M. L. Kothari and P. S. Satsangi, Automatic generation control of an interconnected hydro-thermal system in continuous and discrete modes considering generation rate constraints, IEE Proc., Pt D, 130(1), 455–460, 1983. 26. T. Hiyama, Optimisation of discrete-type load–frequency regulators considering generationrate constraints, IEE Proc., Pt C, 129(6), 285–289, 1982. 27. M. L. Kothari, P. S. Satsangi and J. Nanda, Sampled data automatic generation control of interconnected reheat thermal systems considering generation rate constraints, IEEE Trans. Power App. Syst., 100, 2334–2342, 1981. 28. C. Concordia, L. K. Kirchmayer and E. A. Szymanski, Effect of speed governor dead-band on tie-line power and frequency control performance, Am. Inst. Electr. Eng (AIEE). Trans., 76, 429–435, 1957. 29. IEEE Standard 122–1991, Recommended practice for functional and performance characteristics of control systems for steam turbine–generator units, 1992. 30. C. W. Taylor, K. Y. Lee and D. P. Dave, Automatic generation control analysis with governor dead band effects, IEEE Trans. Power App. Syst., 98, 2030–2036, 1979. 31. S. C. Tripathy, G. S. Hope and O. P. Malik, Optimization of load frequency control parameters with reheat steam turbines and governors dead-band nonlinearity, IEE Proc. Gener. Transm. Distrib., 129(1), 10–16, 1982.

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32. S. C. Tripathy, T. S. Bhatti, C. S. Jha, O. P. Malik and G. S. Hope, Sampled data automatic generation control analysis with reheat steam turbines and governor dead-band effects, IEEE Trans. Power App. Syst., 103(5), 1045–1051, 1984. 33. T. Sasaki and K. Enomoto, Dynamic analysis of generation control performance standards, IEEE Trans. Power Syst., 17(3), 806–811, 2002. 34. M. S. Mahmoud, Robust control and filtering for time-delay systems. New York, NY: Marcel Dekker, 2000. 35. J. Aweya, D. Y. Montuno and M. Ouellette, Effects of control loop delay on the stability of a rate control algorithm, Int. J. Commun. Syst., 17, 833–850, 2004. 36. S. I. Niculescu, Delay effects on stability: A robust control approach. Berlin: Springer, 2001. 37. S. Bhowmik, K. Tomosovic and A. Bose, Communication models for third party load frequency control, IEEE Trans. Power Syst., 19(1), 543–548, 2004. 38. H. Bevrani and T. Hiyama, Robust load–frequency regulation: A real-time laboratory experiment, Optimal Control Appl. Methods, 28(6), 419–433, 2007. 39. T. Hiyama, T. Nagata and T. Funabashi, Multi-agent based automatic generation control of isolated stand alone power system, Proc. Int. Conf. Power Syst. Technol., 1, 139–143, 2004. 40. H. Bevrani and T. Hiyama A robust solution for PI-based LFC problem with communication delays, IEEJ Trans. Power Energy, 25(12), 1188–1193, 2005. 41. X. Yu and K. Tomosovic, Application of linear matrix inequalities for load frequency control with communication delays, IEEE Trans. Power Syst., 19(3), 1508–1515, 2004. 42. H. Bevrani and T. Hiyama, Robust decentralized PI based LFC design for time-delay power systems, Energy Convers. Manage., 49, 193–204, 2007. 43. S. Fukushima, T. Sasaki, S. Ihara, et al., Dynamic analysis of power system frequency control, Proc. CIGRE 2000 Session, No. 38–240, Paris, 2000. 44. H. Bevrani and T. Hiyama, On load–frequency regulation with time delays: Design and realtime implementation, IEEE Trans. Energy Convers., in press. 45. I. A. Hiskens and Jassim Alseddiqui, Sensitivity, approximation, and uncertainty in power system dynamic simulation, IEEE Trans. Power Syst., 21(4), 1808–1820, 2006. 46. A. K. Al-Othman and M. R. Irving, A comparative study of two methods for uncertainty analysis in power system state estimation, IEEE Trans. Power Syst., 20(2), 1181–1182, 2005. 47. J. R. Hockenberry and B. C. Lesieutre, Evaluation of uncertainty in dynamic simulations of power system models: The probabilistic collocation method, IEEE Trans. Power Syst., 19(3), 1483–1491, 2004. 48. A. K. Al-Othman and M. R. Irving, Uncertainty modeling in power system state estimation, IEE Proc. Gener. Transm. Distrib., 152(2), 233–239, 2005. 49. V. A. Maslennikov, S.M. Ustinov and J. V. Milanovic, Method for considering uncertainties for robust tuning of PSS and evaluation of stability limits, IEE Proc. Gener. Transm. Distrib., 149(3), 295–299, 2002. 50. A. T. Saric and A. M. Stankovic, Model uncertainty in security assessment of power systems, IEEE Trans. Power Syst., 20(3), 1398–1407, 2005. 51. H. Bevrani, Y. Mitani and K. Tsuji, On robust load–frequency regulation in a restructured power system, IEEJ Trans. Power Energy, 124-B(2), 190–198, 2004.

Chapter 4

PI-Based Frequency Control Design

Most robust and optimal load–frequency control methods published in the last two decades suggest complex state-feedback or high-order dynamic controllers [1–8], which are impractical for industry practices. Furthermore, some researchers have used new and untested control frameworks, which may have some difficulties in being implemented in real-world power systems. In practice, LFC systems use simple proportional–integral (PI) controllers. However, since the PI controller parameters are usually tuned based on experiences, classical or trial-and-error approaches, they are incapable of obtaining good dynamical performance for a wide range of operating conditions and various load changes scenarios in a multi-area power system. Recently, some control methods have been applied to the design of decentralized robust PI or low-order controllers to solve the LFC problem [9–12]. A PI control design method has been reported in [9], which used a combination of H∞ control and genetic algorithm techniques for tuning the PI parameters. The sequential decentralized method based on μ -synthesis and analysis has been used to obtain a set of low-order robust controllers [10]. The decentralized LFC method has been used with the structured singular values [11]. The Kharitonov’s theorem and its results have been used to solve the same problem [12]. In this chapter, the decentralized PI-based LFC synthesis in the multi-area power systems is formulated as an H∞ -based static output feedback (SOF) control problem, and is solved using an iterative linear matrix inequalities (ILMI) algorithm. This chapter is organized as follows. Technical background on H∞ -based SOF control design is given in Sect. 4.1. Section 4.2 presents the transformation from PI to SOF control problem and the overall control framework. An iterative LMI approach to solve the mentioned LFC problem is introduced in Sect. 4.3. The proposed methodology is applied to a multi-area power system example in Sect. 4.4. A modified approach and the effects of deregulation are discussed in Sects. 4.5 and 4.6.

H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 4, c Springer Science+Business Media LLC 2009

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4.1 H∞ -SOF Control Design 4.1.1 Static Output Feedback Control The SOF control problem is one of the most important research areas in control engineering [13–15]. One reason why SOF has received so much attention is that it represents the simplest control structure that can be realized in the real-world systems. Another reason is that many existing dynamic control synthesis problems can be transferred to an SOF control problem by well-known system augmentation techniques [15, 16]. A comprehensive survey on SOF control is given in [15]. It is generally known that because of using simple constant gains, pertaining to the SOF synthesis for dynamical systems in the presence of strong constraints and tight objectives are few and restrictive. Usually, the design of a full-order output feedback controller reduces to the solution of two convex problems, a state feedback, and a Kalman filter; however, the design of an SOF gain is more difficult. The reason is that the separation principle does not hold in such cases [17]. The existence of the SOF controller is shown to be equivalent to the existence of a positive definite matrix simultaneously satisfying two Lyapunov inequalities [18], where the determination of such a matrix leads to solving a non-convex optimization problem [16, 18–20]. Approaching a solution can be a difficult task, demanding to the great computational effort. Necessary and sufficient conditions for SOF design, as mentioned above, can be obtained in terms of two LMIs couple through a bilinear matrix equation [16, 19, 21]. Particularly, the problem of finding an SOF controller can be restated as a linear algebra problem, which involves two LMIs. For example, an LMI on a positive definite matrix variable P, an LMI on a positive definite matrix variable Q and a coupling bilinear matrix equation of the form PQ = I. But, finding such positive definite matrices is a difficult task, since the bilinear matrix equation implies Q = P−1 . Thus, the two LMIs are not convex in P [16]. A variety of SOF problems were studied by many researchers with many analytical and numerical methods to approach a local/global solution [13–15, 19, 22, 23]. In this paper, to solve the resulted SOF problem from the LFC synthesis, an iterative LMI is used based on the given necessary and sufficient condition for SOF stabilization in [23], via the H∞ control technique.

4.1.2 H∞ -SOF This section gives a brief overview of H∞ -based SOF control design. Consider a linear time invariant system G(s) with the following state-space realization: x˙ = Ax + B1 w + B2 u z = C1 x + D12 u y = C2 x,

(4.1)

4.1 H∞ -SOF Control Design

65

Fig. 4.1 Closed-loop system via H∞ control

where x is the state variable vector, w is the disturbance and other external input vector, z is the controlled output vector and y is the measured output vector. The H∞ -based SOF control problem for the linear time invariant system G(s) with the state-space realization of (4.1) is to find a given matrix K (static output feedback law u = Ky), as shown in Fig. 4.1, such that the resulted closed-loop system is internally stable, and the H∞ norm from w to z is smaller than γ , a specified positive number, i.e., (4.2) ||Tzw (s)||∞ < γ . Under certain assumptions on system matrices, Theorem 4.1 can be extendable to H∞ -SOF control problem. Theorem 4.1. It is assumed that (A, B2 ,C2 ) is stabilizable and detectable. The matrix K is a dynamic H∞ controller, if and only if there exists a symmetric matrix X > 0 such that ⎤ ⎡ T Acl X + XAcl XBcl CclT ⎥ ⎢ T ⎢ Bcl X −γ I DTcl ⎥ < 0, (4.3) ⎦ ⎣ Ccl Dcl −γ I where Acl = A + B2 KC2 , Bcl = B1 , Ccl = C1 + D12 KC2 , Dcl = 0. Proof. The proof is given in [19] and [24].

We can rewrite (4.3) in the following matrix inequality form [23] T

XBKC + (XBKC)T + A X + XA < 0, where ⎡

⎤ ⎤ ⎡ ⎡ 0 A B1 B2 X A = ⎣ 0 −γ I/2 0 ⎦ , B = ⎣ 0 ⎦ , C = C2 0 0 , X = ⎣ 0 0 −γ I/2 C1 D12 0

(4.4)

0 I 0

⎤ 0 0 ⎦. I (4.5)

Hence, the H∞ -based SOF control problem is reduced to find X > 0 and K such that matrix inequality (4.4) holds. It is a generalized SOF stabilization problem of the system (A, B,C) which can be solved via Theorem 4.2.

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Theorem 4.2. The system (A, B, C) that may also be identified by the following representation x˙ = Ax + Bu (4.6) y = Cx is stabilizable via SOF if and only if there exist P > 0, X > 0 and K satisfying the following quadratic matrix inequality T A X + XA − PBBT X − XBBT P + PBBT P (BT X + KC)T < 0. (4.7) BT X + KC −I Proof. According to the Schur complement, the quadratic matrix inequality (4.7) is equivalent to the following matrix inequality AT X + XA − PBBT X − XBBT P + PBBT P + (BT X + KC)T (BT X + KC) < 0. (4.8) Here, the matrices A, B and C are constant and have appropriate dimensions. The X and P are symmetric and positive-definite matrices. For this new inequality notation (4.8), the sufficiency and necessity of theorem are already proven [23].

4.2 Problem Formulation and Control Framework 4.2.1 Transformation from PI to SOF Control Problem In this section, the PI-based LFC problem is transferred to an SOF control problem. The main merit of this transformation is to use the well-known SOF control techniques to calculate the fixed gains, and once the SOF gain vector is obtained, the PI gains are ready in hand and no additional computation is needed. In control area i, the ACE performs the input signal of the PI controller to be used by the LFC system, and we can write ui = ΔPC i = kP i ACEi + kI i

ACEi ,

(4.9)

where kP i and kI i are constant real numbers. Therefore, by augmenting the system description to include the ACE and its integral as a measured output vector, the PI-based LFC problem becomes one of finding a SOF that satisfies the prescribed performance requirements. In order to change (4.9) to a simple SOF control as ui = Ki yi .

(4.10)

We can rewrite (4.9) as follows

ui = kP i

ACEi kI i . ACEi

(4.11)

4.2 Problem Formulation and Control Framework

67

b a

Fig. 4.2 Transformation from PI to SOF control; (a) PI control and (b) SOF control

Therefore, yi in (4.10) can be augmented as given in (4.12). Figure 4.2 shows this transformation (4.12) yTi = ACEi ACEi .

4.2.2 Control Framework Consider Gi (s) as a linear time invariant model for the given control area i with the following state space model:

Gi (s) :

x˙i = Ai xi + B1i wi + B2i ui zi = C1i xi + D12i ui ,

(4.13)

yi = C2i xi where xi is the state variable vector, wi is the disturbance and area interface vector, zi is the controlled output vector and yi is the measured output vector performed by ACE signal. It is expected the robust H∞ –SOF controller (Ki ) to be able to minimize the fictitious output (zi ) in the presence of disturbance and external input (wi ). Therefore, the vector zi must properly cover all signals and should be minimized to meet the LFC goals, e.g., frequency regulation and tracking the load changes, maintaining the tie-line power interchanges close to specified values in the presence of generation constraints, and minimizing the ACE signal. A useful fictitious output vector can be considered as follows (4.14) zTi = η1i Δ fi η2i ACEi η3i ui , where η1i , η2i and η3i are constant weighting coefficients. The fictitious controlled outputs η1i Δ fi and η2i ACEi are used to minimize the effects of input disturbances on area frequency and ACE (and tie-line power flow) signals. Furthermore, the fictitious output η3i ui sets a limit on the allowed control signal with regard to corresponding practical constraint on power generation by generator units. The main control framework to formulate the PI-based LFC via an H∞ -based SOF control design problem, for control area i, is shown in Fig. 4.3a. Using the LFC

68

4 PI-Based Frequency Control Design

a

b

Fig. 4.3 (a) Proposed control framework and (b) Control area i

structured described in Fig. 3.1, a simplified model for control area i with n nonreheat steam units can be represented, as shown in Fig. 4.3b. All parameters have been already defined in Sect. 3.1. The state variables and coefficients of the statespace model (4.13) for the block diagram of Fig. 4.3b can be obtained as follows: T (4.15) ACEi xm i xg i , xi = Δ fi ΔPtie−i xm i = [ΔPm1i ΔPm2i · · · ΔPm ni ] , xg i = ΔPg1i ΔPg2i · · · ΔPg ni , T (4.16) yi = ACEi ACEi , ui = ΔPC i , wTi = [w1i

w2i ] , w1i = ΔPdi , w2i =

N

∑ Ti j Δ f j

j=1 j=i

(4.17)

4.3 Iterative LMI Algorithm

and

69

⎤ ⎤ ⎤ ⎡ ⎡ Ai13 B1i1 B2i1 Ai23 ⎦ , B1i = ⎣ B1i2 ⎦ , B2i = ⎣ B2i2 ⎦ , Ai33 B1i3 B2i3 ⎤ ⎤ ⎡ ⎡ η1i 0 0 0 C1i = c1i 03×n 03×n , c1i = ⎣ 0 0 η2i ⎦ , D12i = ⎣ 0 ⎦ , 0 0 0 η3i Bi 1 0 , C2i = c2i 02×n 02×n , c2i = 0 0 1 ⎡ ⎤ −Di /2Hi −1/2Hi 0 ⎡ ⎤ N 1/2Hi · · · 1/2Hi ⎢ ⎥ ⎢ 0 0 ⎥, A = ⎣ 0 ∑ Ti j ··· 0 ⎦ , Ai11 = ⎢ 2π j=1 ⎥ i12 ⎣ ⎦ j=i 0 ··· 0 3×n Bi 1 0 Ai22 = −Ai23 = diag −1/Tt 1i −1/Tt 2i · · · −1/Tt ni , Ai33 = diag −1/Tg 1i −1/Tg 2i · · · −1/Tg ni , ⎡ ⎤ −1/(Tg 1i R1i ) 0 0 ⎢ .. .. .. ⎥ , A = AT = 0 , A = 0 , Ai31 = ⎣ n×n 3×n i32 i21 . . . ⎦ i13 ⎡

Ai11 Ai = ⎣ Ai21 Ai31

⎡

Ai12 Ai22 Ai32

−1/(Tg ni Rni ) −1/2Hi 0 0

0 0 ⎤

0 −2π ⎦ , B1i2 = B1i3 = 0n×2 , 0 B2i1 = 03×1 , B2i2 = 0n×1 , BT2i3 = α1i /Tg 1i α2i /Tg 2i B1i1 = ⎣

· · · αni /Tg ni .

4.3 Iterative LMI Algorithm Figure 4.4 summarizes the main steps of the developed LFC design. In the proposed methodology, the PI-based LFC design is reduced to SOF control synthesis through the H∞ control technique, and then the optimal constant gains can be carried out using an appropriate solution algorithm. But, it is notable that the H∞ -SOF reformulation generally leads to bilinear matrix inequalities (BMI) which are non-convex. This kind of problem is usually solved by an iterative algorithm that may not converge to an optimal solution.

70

4 PI-Based Frequency Control Design

Fig. 4.4 Design logic

4.3.1 Developed Algorithm Here, the quadratic matrix inequality (4.7) shows a non-convex optimization problem, and its solution cannot be directly achieved by using general LMI technique. In order to solve the mentioned H∞ -SOF, an iterative LMI algorithm has been used. The algorithm is mainly based on the idea given in [23]. The key point is to formulate the H∞ problem via a generalized static output stabilization feedback, such that all eigenvalues of (A − BKiC) shift towards the left half plane through the reduction of a, a real number, to close to feasibility of (4.7). Theorem 4.2 gives a family of internally stabilizing SOF gains is defined as KSOF . A desirable solution can be obtained using the following optimization problem. Optimization problem. Given an optimal performance index γ (4.2), simply obtained from the application of H∞ dynamic output feedback control to the control area i (for example, using hinflmi function in MATLAB LMI Control Toolbox), determine an admissible SOF law

such that

ui = Ki yi , Ki ∈ KSOF

(4.18)

||Tzi wi (s)||∞ < γ ∗ , |γ − γ ∗ | < ε ,

(4.19)

where ε is a small positive number. The performance index γ ∗ indicates a lower bound such that the closed-loop system is H∞ stabilizable. The following algorithm gives an iterative LMI solution for the above optimization problem:

4.3 Iterative LMI Algorithm

71

Step 1. Set initial values and compute the generalized system (Ai , Bi ,Ci ) as shown in (4.5). Step 2. Set i = 1, Δγ = Δγ0 and let γi = γ0 > γ . Δγ0 and γ0 are positive real numbers. Step 3. Select Q > 0, and solve X from the following algebraic Riccati equation T

T

Ai X + XAi − XBi Bi X + Q = 0.

(4.20)

Set P1 = X. Step 4. Solve the following optimization problem for X i , Ki and ai . Minimize ai subject to the LMI constraints T T T T T Ai X i + X i Ai − Pi Bi Bi X i − X i Bi Bi Pi + Pi Bi Bi Pi − ai X i (Bi X i + KiCi )T < 0, T Bi X i + KiC −I (4.21) T

X i = X i > 0.

(4.22)

Denote a∗i as the minimized value of ai . Step 5. If a∗i ≤ 0, go to Step 8. Step 6. For i > 1, if a∗i−1 ≤ 0, Ki−1 ∈ KSOF is an H∞ controller and γ ∗ = γi + Δγ indicates a lower bound such that the above system is H∞ stabilizable via SOF control, go to Step 10. Step 7. If i = 1, solve the following optimization problem for X i and Ki : Minimize trace (X i ) subject to the above LMI constraints (4.21) and (4.22) with ∗ ai = a∗i . Denote X i as the X i that minimized trace (X i ). Go to Step 9. Step 8. Set γi = γi − Δγ , i = i + 1. Then do Steps 3–5. ∗ Step 9. Set i = i + 1 and Pi = X i−1 , then go to Step 4. Step 10. If the obtained solution (Ki−1 ) satisfies the gain constraint, it is desirable, otherwise change constant weights (ηi ) and go to Step 1. The proposed iterative LMI algorithm, which is summarized in the flowchart of Fig. 4.5, shows that if one simply perturbs Ai to Ai − (a/2)I for some a > 0, a solution of the matrix inequality (4.7) can be obtained for the performed generalized plant. That is, there exists a real number (a > 0) and a matrix P > 0 to satisfy inequality (4.21). Consequently, the closed-loop system matrix Ai − Bi KCi has eigenvalues on the left-hand side of the line ℜ(s) = a in the complex s-plane. Based on the idea that all eigenvalues of Ai − Bi KCi are shifted progressively towards the left half plane through the reduction of a. The given generalized eigenvalue minimization in the proposed iterative LMI algorithm guarantees this progressive reduction.

4.3.2 Weights Selection The vector ηi = [η1i η2i η3i ] is a constant weight vector that must be chosen by the designer to get the desired closed-loop performance. The selection of these weights

72

Fig. 4.5 Iterative LMI algorithm

4 PI-Based Frequency Control Design

4.4 Application Example

73

is dependent on specified performance objectives. In fact, an important issue with regards to the selection of these weights is the degree to which they can guarantee the satisfaction of design performance objectives. The weights η1i and η2i at controlled outputs set the performance goals; e.g., tracking the load variation and disturbance attenuation. The η3i sets a limit on the allowed control signal to penalize fast changes, large overshoot with a reasonable control gain to meet the physical constraints. Therefore, the selection of constant weights entails a compromise among several performance requirements. Here, for the sake of weight selection, the following steps are simply considered through the proposed ILMI algorithm: Step 1. Set initial values, e.g., [1 1 1]. Step 2. Run the ILMI algorithm (summarized in Fig. 4.5). Step 3. If the ILMI algorithm gives a feasible solution, such that satisfies the robust H∞ performance and the exist constraint, the assigned weights vector is acceptable. Otherwise, retune ηi and go to Step 2. In Sect. 4.4, two types of robust controllers are developed for a power system example including three control areas. The first one is dynamic controller based on general robust LMI-H∞ design and the second controller is based on the iterative LMI (ILMI) static output H∞ approach (described in Sect. 4.3) with the same assumed objectives to achieve robust performance.

4.4 Application Example 4.4.1 Case Study To illustrate the effectiveness of the proposed control strategy, a three-control area power system, shown in Fig. 4.6, is considered as a test system. It is assumed that each control area includes three Gencos. The power system parameters are considered to be the same as in [9], and given in Appendix A. For the sake of comparison, in addition to the proposed control strategy to obtain the robust PI controller, a robust H∞ dynamic output feedback controller using an LMI control toolbox is designed for each control area. Specifically, based on general LMI, first the control design is reduced to an LMI formulation, and then the H∞ control problem is solved using the function hinflmi, provided by the MATLAB LMI control toolbox [25]. This function gives an optimal H∞ controller through minimizing the guaranteed robust performance index (γ ) subject to the constraint given by the matrix inequality (4.3), and returns the controller K(s) with optimal robust performance index. The resulted controllers using the hinflmi function are of a dynamic type and have the following state-space form, whose orders are the same as the size of the plant model (9th order in the present example)

74

4 PI-Based Frequency Control Design

Fig. 4.6 Three-control area power system

Table 4.1 Control parameters (ILMI design) Parameter a∗ kP i kI i

Area 1

Area 2

Area 3

−0.3285 −0.0371 −0.2339

−0.2472 −0.0465 −0.2672

−0.3864 0.0380 −0.3092

x˙ki = Aki xki + Bki yi ui = Cki xki + Dki yi .

(4.23)

At the next step, according to the described synthesis methodology summarized in Fig. 4.5, a set of three decentralized robust PI controllers are designed. As already mentioned, this control strategy is entirely suitable for LFC applications which usually employ the PI control, while most other robust and optimal control designs (such as the LMI approach) yield complex controllers whose size can be larger than real-world LFC systems. Using the ILMI approach, the controllers are obtained following several iterations. The control parameters are shown in Table 4.1. A set of suitable values for constant weights [η1i , η2i , η3i ] is chosen as [0.1, 5, 500], respectively. These weights are very important to realize the designed controller for the real-world power systems. The large coefficient 500 for η3i results in a smooth control action signal with reasonable changes in amplitude.

4.4 Application Example

75

Table 4.2 Robust performance index Control design H∞ ILMI

Control structure

Performance index

Area 1

Area 2

Area 3

Ninth order PI

γ γ∗

500.0103 500.0183

500.0045 500.0140

500.0065 500.0105

It is notable that the robust performance index given by the standard dynamic H∞ control design can be used as a useful reference performance index to analyse the robustness of the closed-loop system for the proposed control design. The resulting robust performance indices (γ ∗ ) of both synthesis methods are close to each other and shown in Table 4.2. It shows that although the proposed ILMI approach gives a set of much simpler controllers (PI) than the LMI-based dynamic H∞ design, they also give a robust performance like the dynamic H∞ controllers.

4.4.2 Simulation Results The proposed controllers were applied to the three-control area power system described in Fig. 4.6. In this section, the performance of the closed-loop system using the robust PI controllers compared to the designed dynamic H∞ controllers will be tested for the various load disturbances. Case 1. As the first test case, the following load disturbances (step increase in demand) are applied to three areas: ΔPd1 = 0.1, ΔPd2 = 0.08 and ΔPd3 = 0.05 pu. The frequency deviation (Δ f ), area control error (ACE) and control action (ΔPC ) signals of the closed-loop system are shown in Fig. 4.7. Using the proposed method (ILMI), the area control error and frequency deviation of all areas are quickly driven back to zero as well as dynamic H∞ control (LMI). Case 2. Consider larger demands by area 2 and area 3, i.e., ΔPd1 = 0.1 pu, ΔPd2 = 0.1 pu, ΔPd3 = 0.1 pu. The closed-loop response for each control area is shown in Fig. 4.8. Case 3. As another severe condition, assume a bounded random load change shown in Fig. 4.9 is applied to all control areas simultaneously, where −0.05 pu ≤ ΔPd ≤ +0.05 pu. The purpose of this scenario is to test the robustness of the proposed controllers against random large load disturbances. The control area responses are shown in Fig. 4.9b–d. This figure demonstrates that the designed controllers track the load fluctuations effectively. The simulation results show the proposed PI controllers perform as robustly as robust dynamic H∞ controllers (with complex structures) for a wide range of load disturbances.

76

4 PI-Based Frequency Control Design

Δ f1 (Hz)

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Fig. 4.7 System response in case 1; (a) Area 1, (b) Area 2 and (c) Area 3. Solid (ILMI-based PI controller) and dotted (dynamic H∞ controller)

4.5 Using a Modified Controlled Output Vector b

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Fig. 4.8 System response in case 2; (a) Area 1, (b) Area 2 and (c) Area 3. Solid (ILMI-based PI controller) and dotted (dynamic H∞ controller)

4.5 Using a Modified Controlled Output Vector In the proposed control framework (Fig. 4.3), it is expected the robust controller Ki to be able to minimize the fictitious output (zi ) in the presence of disturbance and external input (wi ). Therefore, the vector zi must properly cover all signals which must be minimized to meet the LFC goals, e.g., frequency regulation, tracking the load changes, maintaining the tie-line power interchanges to specified values in the presence of generation constraints and minimizing the ACE signal. By considering the tie-line power flow changes in the proposed fictitious output vector, we can rewrite (4.14) as follows: (4.24) zTi = η1i Δ f η2i ACEi η3i ΔPtie−i η4i ui . The new fictitious output η3i ΔPtie−i is used to minimize the effects of input disturbances on tie-line power flow signal. Referring to (4.13), the related coefficients to the fictitious output vector (zi ) in the proposed state-space model can be obtained as

78

4 PI-Based Frequency Control Design b 0.1 0.08

ACE1 (pu)

Δ Pd (pu)

0.04 0.02 0 −0.02

Δ Pc1 (pu)

−0.06 −0.08 20

30 40 50 Time (sec)

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a

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Fig. 4.9 System response in case 3; (a) Random load demand signal, (b) Area 1, (c) Area 2 and (d) Area 3. Solid (ILMI-based PI controller) and dotted (dynamic H∞ controller)

⎡

C1i = c1i

04×n

04×n

η1i ⎢0 , c1i = ⎢ ⎣0 0

0 0 0 η3i

⎡ ⎤ ⎤ 0 0 ⎢ ⎥ η2i ⎥ ⎥,D = ⎢ 0 ⎥. 0 ⎦ 12i ⎣ η4i ⎦ 0 0

A set of suitable values for constant weights according to the new control framework for the present power system example is considered as follows:

η1i = 0.4, η2i = 1.075, η3i = 0.39, η4i = 333 Using ILMI approach, the controllers are obtained following several iterations. For example, for control area 3, the final result is obtained after 29 iterations. Some iterations are listed in Table 4.3. The control parameters for three control areas are shown in Table 4.4. The resulting robust performance indices of both synthesis methods are shown in Table 4.5. The proposed controllers are applied to the three-control area power system described in Fig. 4.6. The performance of the closed-loop system using the robust PI controllers compared to the designed dynamic H∞ controllers and proposed control design in [9] is tested for some serious load disturbances.

4.5 Using a Modified Controlled Output Vector

79

Table 4.3 ILMI algorithm result for design of K3

γ

kP3

kI3

449.3934 419.1064 352.6694 340.2224 333.0816 333.0332 333.0306 333.0270 333.0265 333.0238

−0.0043 −0.0009 0.1022 −0.0006 −0.0071 0.0847 0.0879 0.0956 0.0958 −0.0038

−0.0036 −0.0042 −0.2812 −0.0154 −0.1459 −0.2285 −0.2382 −0.2537 −0.2560 −0.2700

Iteration 1 5 11 14 19 22 24 26 28 29

Table 4.4 Control parameters (ILMI design) Parameter a∗ kP i kI i

Area 1

Area 2

Area 3

−0.0246 −9.8 × 10−03 −0.5945

−0.3909 −2.6 × 10−03 −0.3432

−0.2615 −3.8 × 10−03 −0.2700

Table 4.5 Robust performance index Control design H∞ ILMI

Control structure

Performance index

Area 1

Area 2

Area 3

Ninth order PI

γ γ∗

333.0084 333.0261

333.0083 333.0147

333.0080 333.0238

For the first test scenario, the following large load disturbances (step increase in demand) are applied to the three areas. The system response is shown in Fig. 4.10. ΔPd1 = 0.15 pu, ΔPd2 = 0.15 pu, ΔPd3 = 0.15 pu. Figure 4.11 compares the frequency deviation (Δ f ) and governor load set-point (ΔPC ) signals for the proposed method and recent published design technique [9], following 0.1-pu step load increase in each control area. A combination of genetic algorithm (GA) and LMI-based H∞ control (GALMI) has been used in [9]. As seen from Fig. 4.11, the proposed controllers track the load changes and meet the robust performance, as well as reported results for the same simulation case in [9]. Consider the tie-line power change as the fictitious controlled output in the H∞ control framework adds enough flexibility to set the desired level of performance. Moreover, in comparison of [9], the proposed control design uses a simpler algorithm that takes a short time (few seconds) for tuning of controller parameters. More simulation tests are given in [26].

80

4 PI-Based Frequency Control Design Δ f1 (Hz)

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Fig. 4.10 Frequency deviation and ACE signals following a large step load demand (105 MW) in each area. Solid (ILMI-based PI controller) and dotted (dynamic H∞ controller)

b

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Fig. 4.11 (a) Frequency deviation and (b) control action signals, following a 100-MW step load increase in each area. Solid (ILMI) and dotted [9]

4.7 Summary

81

Fig. 4.12 Proposed control framework for the bilateral based LFC scheme

4.6 Considering Bilateral Contracts The proposed control framework to the design of PI controller, via the H∞ -based SOF control problem for a given control area in a deregulated environment, is shown in Fig. 4.12. Gi (s) denotes the dynamic model corresponds to the control area which is represented in Fig. 3.3. Assume the same variables as given in (4.14), (4.15), and (4.16). According to (4.13), we can write (4.25) wTi = v1i v2i v3i v4i , vT4i = v4i−1 v4i−2 · · · v4i−n ⎡

and

B1i11 B1i = ⎣ B1i21 B1i31

⎤ ⎡ B1i12 −1/2Hi B1i22 ⎦ , B1i11 = ⎣ 0 B1i32 0

0 −2π 0

⎤ 0 0 ⎦, −1

B1i21 = B1i31 = 0n×3 , B1i12 = 03×n , B1i22 = 0n×n , B1i32 = diag 1/Tg1i 1/Tg2i · · · 1/Tg ni . The other coefficient matrices and vectors can be defined the same as those given in Sect. 4.2.2. A detailed LFC design example is given in [27].

4.7 Summary In this chapter, a new decentralized method to design robust LFC using a developed ILMI algorithm has been provided for a large-scale power system. The proposed design control strategy gives a set of simple PI controllers via the H∞ -based SOF control design, which is commonly useful in real-world power systems. The proposed method was applied to multi-area power system examples with different LFC

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schemes, and the closed-loop system is tested under serious load change scenarios. The results were compared with the results of applied full dynamic H∞ controllers. It was shown the designed controllers can guarantee the robust performance under a wide range of area–load disturbances.

References 1. T. Hiyama, Design of decentralised load–frequency regulators for interconnected power systems, IEE Proc., Pt. C, 129, 17–23, 1982. 2. A. Feliachi, Optimal decentralized load frequency control, IEEE Trans. Power Syst., PWRS-2, 379–384, 1987. 3. C. M. Liaw and K. H. Chao, On the design of an optimal automatic generation controller for interconnected power systems, Int. J. Control, 58, 113–127, 1993. 4. Y. Wang, R. Zhou and C. wen, Robust load–frequency controller design for power systems, IEE Proc., Pt. C, 140(1), 11–16, 1993. 5. K. Y. Lim, Y. Wang and R. Zhou, Robust decentralized load–frequency control of multi-area power systems, IEE Proc. Gener. Transm. Distrib., 143(5), 377–386, 1996. 6. T. Ishi, G. Shirai and G. Fujita, Decentralized load frequency based on H∞ control, Electr. Eng. Jpn, 136(3), 28–38, 2001. 7. M. H. Kazemi, M. Karrari and M. B. Menhaj, Decentralized robust adaptive-output feedback controller for power system load frequency control, Electr. Eng. J., 84, 75–83, 2002. 8. M. K. El-Sherbiny, G. El-Saady and A. M. Yousef, Efficient fuzzy logic load–frequency controller, Energy Convers. Manage., 43, 1853–1863, 2002. 9. D. Rerkpreedapong, A. Hasanovic and A. Feliachi, Robust load frequency control using genetic algorithms and linear matrix inequalities, IEEE Trans. Power Syst., 18(2), 855–861, 2003. 10. H. Bevrani, Y. Mitani and K. Tsuji, Sequential design of decentralized load–frequency controllers using μ -synthesis and analysis, Energy Convers. Manage., 45(6), 865–881, 2004. 11. T. C. Yang, H. Cimen and Q. M. ZHU, Decentralised load frequency controller design based on structured singular values, IEE Proc. Gener. Transm. Distrib., 145(1), 7–14, 1998. 12. H. Bevrani, Application of Kharitonov’s theorem and its results in load–frequency control design, J. Electr. Sci. Technol.-BARGH (in Persian), 24, 82–95, 1998. 13. D. Bernstein, Some open problems in matrix theory arising in linear systems and control, Linear Algebra Appl., 162–164, 409–432, 1992. 14. V. Blondel, M. Gevers and A. Lindquist, Survey on the state of systems and control, Eur. J. Control, 1, 5–23, 1995. 15. V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback: A survey, Automatica, 33(2), 125–137, 1997. 16. F. Leibfritz, Static Output Feedback Design Problems, PhD dissertation, Trier University, 1998. 17. E. Prempain and I. Postlethwaite, Static output feedback stabilization with H∞ performance for a class of plants, Syst. Control Lett., 43, 159–166, 2001. 18. T. Iwasaki, R. E. Skelton and J. C. Geromel, Linear quadratic suboptimal control with static output feedback, Syst. Control Lett., 23, 421–430, 1994. 19. P. Gahinet and P. Apkarian, A linear matrix inequality approach to H∞ control, Int. J. Robust Nonlinear Control, 4, 421–448, 1994. 20. J. C. Geromel, C. C. Souza and R. E. Skeltox, Static output feedback controllers: Stability and convexity, IEEE Trans. Autom. Control, 42, 988–992, 1997. 21. T. Iwasaki and R. E. Skelton, All controllers for the general H∞ control problem: LMI existence conditions and state space formulas, Automatica, 30, 1307–1317, 1994.

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22. K. C. Goh, M. G. Safanov and G. P. Papavassilopoulos, A global optimization approach for the BMI problem, Proc IEEE CDC Conf., 2009–2014, 1994. 23. Y. Y. Cao, Y. X. Sun and W. J. Mao, Static output feedback stabilization: An ILMI approach, Automatica, 34(12), 1641–1645, 1998. 24. R. E. Skelton, J. Stoustrup and T. Iwasaki, The H∞ control problem using static output feedback, Int. J. Robust Nonlinear Control, 4, 449–455, 1994. 25. P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI Control Toolbox, The MathWorks, Natick, MA, 1995. 26. H. Bevrani, T. Hiyama, Y. Mitani and K. Tsuji, Automatic generation control: A decentralized robust approach, Intell. Autom. Soft Comput. J., 13(3), 273–287, 2007. 27. H. Bevrani, Y. Mitani and K. Tsuji, Robust AGC: Traditional structure versus restructured scheme, IEE J Trans. Power Energy, 124-B(5), 751–761, 2004.

Chapter 5

Frequency Regulation with Time Delays

An effective power system market needs an open communication infrastructure to support the increasing decentralized property of control processes, and a major challenge in a new environment is to integrate computing, communication and control into appropriate levels of real-world power system operation and control. With rapid advancements in wide-area measurement (WAM) systems technology such as phasor measurement units (PMU), the transmission of measured signals to remote frequency control centre has become relatively simpler. However, there is an unavoidable time delay involved before these signals are received at the control centre. For large interconnected power systems, because of the distance involved in WAM, communication delay cannot be ignored. Unlike the small time delay in a local signal control/measurement, in wide-area power systems, the time delay can vary in a wide range. This chapter focuses on robust PI-based frequency control synthesis taking into account the communication time delays. As described in Sect. 3.5, it is well known that time delays can degrade a control system’s performance and even cause system instability [1–3]. In light of this fact, in the near future, the communication delays, as one of the important uncertainties in the LFC synthesis/analysis due to expanding physical setups, functionality and complexity of power systems, may become a significant problem. To ensure satisfactory frequency regulation performance, these delays need to be taken into account in the frequency control synthesis stage. The real-world LFC systems use the proportional–integral (PI) type controllers. A H∞ control solution for the PI-based LFC problem is described in Chap. 4. However, the communication time delays have not been considered during the control synthesis procedure. Recently, several reports have been published to address the LFC analysis/synthesis in the presence of communication delays [4–10]. These references clearly addressed the effects of signal delays on the power system frequency control issue. Reference [4] introduces the network delay models and communication network requirement for a third party LFC service in a deregulated environment. A compensation method for communication time delay in the LFC systems is presented in [5], and a control synthesis technique based on linear matrix H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 5, c Springer Science+Business Media LLC 2009

85

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inequalities (LMI) is proposed for the LFC system with communication delays in [6]. References [7] and [8] address frequency control design methods using LMI techniques, and dynamic analysis of LFC performance, considering time delays given in [9, 10]. Most published research works on the PI-based LFC have neglected problems associated with the communication network. Although, under the traditional dedicated communication links, this was a valid assumption; however, the use of an open communication infrastructure in deregulated environments raises concerns about problems that may arise in the communication system. It should be noted, for a variety of reasons, the optimal setting of the PI parameters is difficult and as a result, most robust and optimal approaches suggest complex state-feedback or high-order dynamic controllers. In this chapter, the PI-based multi-delayed LFC problem is converted into a static output feedback (SOF) control design problem, and to tune the PI parameters, the optimal H∞ control is used via a multi-constraint minimization problem. The problem formulation is based on expressing the constraints as LMI, which can be easily solved using available semi-definite programming methods [11, 12]. Simplicity of control structure, keeping the fundamental LFC concepts, using a multi-delaybased LFC system and no need for an additional controller can be considered as advantages of the proposed LFC design methodology. In order to demonstrate the efficiency of the proposed control method, some real-time simulations have been performed on an analog power system simulator (APSS) system. This chapter is organized as follows: some preliminary explanations on timedelay-based H∞ control and LFC scheme are given in Sect. 5.1. The proposed control strategy and real-time laboratory experiment are described in Sects. 5.2 and 5.3, respectively. Section 5.4 demonstrates simulation results, and a summary is given in Sect. 5.5.

5.1 Preliminaries 5.1.1 H∞ Control for Time-Delay Systems Consider a class of time-delay systems in the following form [1]: x(t) ˙ = Ax(t) + Bu(t) + Ad x(t − d) + Bh u(t − h) + Fw(t) z(t) = C1 x(t) y(t) = C2 x(t),

(5.1) x(t) ∈ ψ (t) ∀ t ∈ [−max(d, h), 0].

Here x ∈ ℜn is the state, u ∈ ℜn is the control input, w ∈ ℜn is the input disturbance, z ∈ ℜn is the controlled output, y ∈ ℜn is the measured output and C2 ∈ ℜn is the constant matrix such that the pair (A, C2 ) is detectable. d and h represent the delay amounts in the state and the input, respectively. A ∈ ℜn×n and B ∈ ℜn×m represent

5.1 Preliminaries

87

the nominal system without delay such that the pair (A, B) is stabilizable. Ad ∈ ℜn×n , Bh ∈ ℜn×m and F ∈ ℜn×q are known matrices and ψ (t) is a continuous vectorvalued initial function. Theorem 5.1 adapts the H∞ theory in the control synthesis for time-delay systems (using LMI description) and establishes the conditions under which the state feedback control law u(t) = Kx(t) (5.2) stabilizes (5.1) and guarantees the H∞ norm bound γ of the closed-loop transfer function Tzw , namely Tzw ∞ < γ ; γ > 0. Theorem 5.1. The state feedback controller K asymptotically stabilizes the timedelay system (5.1) and Tzw ∞ < γ for d, h ≥ 0 if there exists matrices 0 < PT = P ∈ ℜn×n , 0 < QT1 = Q1 ∈ ℜn×n , 0 < QT2 = Q2 ∈ ℜn×n satisfying the LMI ⎡ ⎤ PAc + ATc P + Q1 + Q2 ATd P K T BTh P C1 F T P ⎢ −Q1 0 0 0 ⎥ PAd ⎢ ⎥ ⎢ 0 −Q2 0 0 ⎥ PBh K (5.3) ⎢ ⎥ < 0, ⎣ 0 0 I 0 ⎦ C1T PF 0 0 0 −γ 2 I where Ac = A + BK.

(5.4)

Proof. According to the Schur complement method [11], LMI (5.3) is equivalent to the following matrix inequality T −1 T T T −2 T PAc + ATc P + Q1 + Q2 + PAd Q−1 1 Ad P + PBh KQ2 K Bh P +C1 C1 + γ PFF P < 0. (5.5) The sufficiency of theorem for the inequality notation (5.5) is given in [1].

5.1.2 LFC with Time Delays The subject of delay and its analysis in a wide range of engineering applications has already received much attention. Delay is often ignored in the power system control designs. However, it becomes a pertinent topic in recent years with the proposal of wide area power system control design [13–17]. The delay involved between the instant of measurement and that of the signal being available to the controller is an important problem. This delay for a signal feedback in an interconnected power system is usually considered to be in the range of 0.1–1 s [13, 14]. Unlike the small time delay in a local control area, in a multi-area power system, the communication delay can vary from tens to several hundred milliseconds or more [15]. If routing delay and the natural response delay in thermal and hydrogenerator dynamics are included, there is a potential of experiencing long variability in these delays, in various power system controls. The introduction of delay in a

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Fig. 5.1 A general control area with time delays

power system feedback control loop has a destabilizing effect and reduces the effectiveness of control system damping. In some cases, the power system synchronism may be lost [13, 14]. The time-delayed LFC system is discussed in Sect. 3.5.2. The communication delays can be mainly considered on the control input and the control output of the LFC system: The delays on the measured frequency and power tie-line flow from RTUs to the control centre, which can be reflected into the ACE and the produced control command signal from control centre to individual generation units. A simplified time-delayed LFC system is shown in Fig. 5.1. The communication delay is expressed by an exponential function e−sτ where τ indicates the communication delay time. Following a load disturbance within the control area, the frequency of the area experiences a transient change and the feedback mechanism comes into work and generates the appropriate control signal, to make the generation readjust to meet the load demand. The balance between connected control areas is achieved by detecting the frequency and tie-line power deviation via communication line, to generate the ACE signal used by PI controller. The control signal is submitted to the participating generation companies (Gencos) via other links, based on their participation factors. w1i and w2i demonstrate the area load disturbance and interconnection effects (area interface), respectively w1i = ΔPd i , w2i =

N

∑ Ti j Δ f j .

(5.6)

j=1 j=i

All variables and parameters are described in Sect. 3.1. In order to satisfy the LFC performance specifications in a multi-area power system, the design of frequency controllers should take into account communication delays. In other words, the LFC system should tolerate not only the range of operating conditions desired, but also the uncertainty in delay.

5.2 Proposed Control Strategy

89

5.2 Proposed Control Strategy 5.2.1 Problem Formulation As described in Sect. 4.2.1, the PI-based LFC problem can be transferred to a SOF control problem by augmenting the measured output signal to include the area control error and its integral.

u(t) = ky(t) = kP ACE + kI

ACE = [kP kI ] ACE

T ACE ,

(5.7)

where kP and kI are constant real numbers (PI parameters). ACE is the area control error signal for which each control area can be expressed as a linear combination of tie-line power change and frequency deviation. The overall control framework to formulate the time-delayed LFC problem via a H∞ -based SOF (H∞ − SOF) control design is shown in Fig. 5.2. The output channel z∞i is associated with the H∞ performance, while the yi is the augmented measured output vector (performed by ACE and its integral). Here, μ1i , μ2i and μ3i are constant weights that must be chosen by the designer to get the desired closed-loop performance. Experience suggests that one can fix the weights μ1i , μ2i and μ3i to unity and use the method with the regional pole placement technique for performance tuning [18]. The first two terms of z∞i output are used to minimize the effects of disturbances on area frequency and ACE by introducing appropriate fictitious controlled outputs. Furthermore, fictitious output μ3i ΔPCi sets a limit on the allowed control signal to penalize fast changes and large overshoot in the governor load set point, with regards to practical constraint on power generation by generator units [19, 20]. Gi (s) is the nominal dynamic model of the given control area, ui is the control input and wi includes the perturbed and disturbance signals in the given control area.

Fig. 5.2 H∞ -SOF control framework

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According to (5.1), the open-loop state space model (Gi (s)) for the LFC system of control area “i” can be obtained as follows x˙i (t) = Ai xi (t) + Bi ui (t) + Adi xi (t − d) + Bhi ui (t − h) + Fi wi (t) zi (t) = C1i xi (t)

(5.8)

yi (t) = C2i xi (t). Using the standard simplified LFC model for the prime mover and governor in Fig. 5.1, the state variables can be considered as follows: T (5.9) ACEi xm i xg i , xi = Δ fi ΔPtie−i where xm i = [ΔPm1i and

ΔPm2i

···

ΔPm ni ] , xgi = ΔPg1i

ΔPg2i

= ACEi ACEi , ui = ΔPCi , T zi = μ1i Δ fi μ2i ACEi μ3i ΔPC i ,

yTi

wTi = [w1i

w2i ] ,

···

ΔPgni

(5.10)

(5.11) (5.12) (5.13)

where Δ fi is the frequency deviation, ΔPgi is the governor valve position, ΔPCi is the governor load setpoint and ΔPmi is the turbine mechanical power.

5.2.2 H∞ -SOF-Based LFC Design Using the described transformation from PI to SOF control design, the time-delayed LFC problem is reduced to synthesis of SOF control (u(t) = ky(t)) for the time-delay system given in (5.1). A variety of SOF problems were studied by many researchers with many analytical and numerical methods to approach a feasible solution [21–24]; however, only a few references have addressed the time-delayed systems. Here, in order to achieve an optimal LMI-based H∞ solution for the mentioned SOF problem from the delay-based LFC synthesis, Theorem 5.2 is used. Theorem 5.2. The SOF controller k asymptotically stabilizes system (5.1) and Tzw ∞ < γ for d, h ≥ 0 if there exists matrices 0 < Y T = Y ∈ ℜn×n , 0 < QTt = Qt ∈ ℜn×n and 0 < QTs = Qs ∈ ℜn×n satisfying the following matrix inequality

5.2 Proposed Control Strategy

⎡ ⎢ ⎢ ⎢ ⎢ Ws = ⎢ ⎢ ⎢ ⎢ ⎣

91

⎤

AY +YAT + Qt + Qs (BkC2 )T Y YATd (Bh kC2Y )T C1Y F T BkC2 −In 0 0 0 0 0 ⎥ ⎥ 0 0 0 ⎥ Y 0 −In 0 ⎥ 0 0 −Qt 0 0 0 ⎥ AdY ⎥ < 0. 0 0 0 −Qs 0 0 ⎥ Bh kC2Y ⎥ 0 0 0 0 −Ip 0 ⎦ YC1T F 0 0 0 0 0 −γ 2 Iq (5.14)

Proof. The control law can be considered as a replica of the state-feedback controller (5.2) (5.15) u(t) = ky(t) = kC2 x(t). Based on Theorem 5.1, there exists a memory-less feedback controller with constant gain (5.16) K = kC2 such that the closed-loop system is asymptotically stable and the Tzw ∞ < γ is satisfied for d, h ≥ 0. According to (5.4), for the closed-loop system we have Ac = A + BkC2 .

(5.17)

The stabilizing controller satisfies inequality (5.5). Therefore, using (5.17) one can write T P(A + BkC2 ) + (A + BkC2 )T P + Q1 + Q2 + PAd Q−1 1 Ad P (5.18) T T T −2 T + PBh kC2 Q−1 2 (kC2 ) Bh P +C1 C1 + γ PFF P < 0. Pre-multiplying and post-multiplying (5.18) by P−1 and letting P−1 = Y , inequality (5.19) can be obtained T AY +YAT +Y Q1Y +Y Q2Y + BkC2Y +Y (kC2 )T BT + Ad Q−1 1 Ad T T T −2 T + Bh kC2 Q−1 2 (kC2 ) Bh +YC1 C1Y + γ FF < 0.

(5.19)

Now, assuming Y Q1Y = Qt , Y Q1Y = Qs and using the following inequality [11] ∀Ω1 , Ω2 ∈ ℜ :

Ω1T Ω2 + Ω2T Ω1 ≤ αΩ1T Ω1 + α −1 Ω2T Ω2 , α > 0

inequality (5.19) can be reduced to T AY +YAT + Qt + Qs + BkC2 (BkC2 )T +Y TY + AdY Q−1 t (AdY ) T T −2 T + Bh kC2Y Q−1 s (Bh kC2Y ) +YC1 C1Y + γ FF < 0.

(5.20)

(5.21)

Using the Schur complement method, (5.21) can be arranged conveniently to yield the block form (5.14) as desired. An equivalent theorem with a different configuration using some relaxation parameters is given in [1]. Since the mentioned theorem and its result are not straightly applicable to the PI-based LFC design, the above-modified theorem is proposed.

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Theorem 5.2 shows that to determine the SOF controller k, one has to solve the following minimization problem: min γ subject to −Y < 0, −Qt < 0, −Qs < 0, −Ws < 0.

Qt ,Qs ,Y,k

(5.22)

The matrix inequality (5.14) points to an iterative approach to solve k, Qt and Qs ; namely, if Y is fixed, then it reduces to an LMI problem in the unknown k, Qt and Qs . The LMI problem is convex and can be solved efficiently using the LMI Control Toolbox [12], if a feasible solution exists. Remarks • It is shown that the necessary condition for the existence of a solution is that the nominal transfer function T (s) = kC2 [sI − A]−1 B

(5.23)

is strictly positive real (SPR) [25]. To approach the solution for some positive real cases, it is possible to use a reasonable approximation to close those systems to SPR ones. • It is significant to note that because of using simple constant gains, pertaining to SOF synthesis for dynamical systems in the presence of strong constraints and tight objectives are few and restrictive. Under such conditions, the minimization problem (5.22) may not lead to a strictly feasible solution.

5.2.3 Application to a Three-Control Area The proposed control technique is applied to the three-control area power system shown in Fig. 4.6, with the same simulation data used in Chap. 4. The application details are given in [26]. The results are compared with the delay-less H∞ -SOF control design (described in Chap. 4). It is shown that the communication delays seriously degrade the performance of delay-less LFC design, while the proposed methodology in Sect. 4.2 provides a desirable system performance [26].

5.3 Real-Time Laboratory Experiment 5.3.1 Analog Power System Simulator For many years, real-time analysis and the factory test of the designed power system controllers and protection systems are carried out effectively by using real-time APSSs [27–29]. A high-performance APSS uses more detailed equipment models,

5.3 Real-Time Laboratory Experiment

93

Fig. 5.3 Overview of APSS

implemented by operational amplifiers and electronic circuits, with electrical characteristics closer to real-world power system devices. Today analog simulators are highly precise, easy to use in time domain analysis, and suited to give an understanding of power system dynamics. To demonstrate the effectiveness of the developed control methodology, the designed PI-based controller on a personal computer (PC) was set on APSS at the Research Laboratory of Kyushu Electric Power Co., in Japan. For the sake of realtime operations, an appropriate digital signal processing (DSP) board is installed in the personal computer. The APSS has the capability to simulate various electro-mechanical transients and steady-state dynamics for system stability and performance analysis subject to various types of small and even large disturbances, without any risks associated with the site tests in actual power systems. The mentioned APSS shown in Fig. 5.3 was manufactured at Toshiba Company. The simulator is composed of numerous modules for generators, transformers, substations, transmission lines, loads, circuit breakers, static condensers, shunt reactors, and power system equipments. The modules can be operated/monitored by a host computer and two monitoring terminals.

5.3.2 Configuration of Study System For the purpose of the designed LFC in the present chapter, a longitudinal threemachine infinite bus system is considered as a test system. The study system is

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Fig. 5.4 Study power system

Fig. 5.5 SIMULINK-based control loop

shown in Fig. 5.4. All generator units are of thermal type, with separately conventional excitation control systems. A set of three generators represent a control area (Area I), and, the infinite bus is considered as other connected systems (Area II). The whole power system has been implemented in the mentioned laboratory. The proposed controller, ACE computing unit and participation factors, which are build in SIMULINK environment (shown in Fig. 5.5), have been connected to the power system using a DSP board equipped with analog to digital (A/D) and digital to analog (D/A) converters as the physical interface between the personal computer and the APSS. Figure 5.6 shows the overview of the applied laboratory experiment

5.3 Real-Time Laboratory Experiment

95

Fig. 5.6 Overview of the control/monitoring desk for performed laboratory experiment

Fig. 5.7 The block diagram of generator unit

devices. The block diagram given in Fig. 5.5 has been implemented in a personal computer. The digital oscilloscope and the notebook computer (shown in the left side of Fig. 5.6) are used for monitoring purposes. The detailed block diagram of each generator unit and its associated turbine system (including the high-pressure, intermediate-pressure, and low-pressure parts) is illustrated in Fig. 5.7. The power system parameters are given in Appendix A (Table A.2).

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5.3.3 H∞ -SOF-Based PI Controller To adapt the represented state-space dynamic model (5.8) with the shown timedelayed LFC system in Fig. 5.1, the Adi can be easily computed by transferring the ACE delay (τd ) through its components (Δ fi and ΔPtie−i as states). Therefore, the delay is considered in both the states and the control input. Based on a simple stability condition [30], the open-loop system (5.8) with real matrices is stable if

μ (Ai ) + Adi < 0,

(5.24)

where

1 max λ j ATi + Ai . (5.25) 2 j Here, λ j denotes the jth eigenvalue of ATi + Ai . Using the above stability rule, we note that for the example at hand, the control area is unstable

μ (Ai ) =

μ (Ai ) + Adi = 12.9714 > 0.

(5.26)

According to the described synthesis methodology in Sect. 5.2, the PI parameters are obtained as in (5.27). For the system at hand, the total time delay of communication channels is considered near to the LFC cycle rate of the power system, and suitable values for constant weights μ1i , μ2i , and μ3i are considered as 0.5, 1 and 25, respectively. (5.27) kP = −0.0611, kI = −0.1369. Based on Theorem 5.2, since a solution for the time-delayed LFC problem will be obtained through minimizing the guaranteed the H∞ performance index γ (as a valid performance measure) subject to the given constraints in (5.22), the designed PI controllers satisfy the robustness of the closed-loop system. In other words, the basis of designing the SOF controllers (5.7) is to simultaneously stabilize (5.8) and guarantee the H∞ -norm bound γ of the closed-loop transfer function Tzw ; namely Tzw ∞ < γ ; γ > 0.

(5.28)

5.4 Simulation Results In the performed non-linear real-time laboratory simulations, the proposed PI controller was applied to the control area power system described in Fig. 5.4. The performance of the closed-loop system is tested in the presence of load disturbances and time delays. Two types of communication delays, fixed and random, are simulated. To simplify the presentation and because of space limitation, case studies of fixed delays are used. The nominal area load demands that PL1 , PL2 and PL3 (in Fig. 5.4) during simulation tests are considered as 0.3, 0.6 and 0.6 pu, respectively.

5.4 Simulation Results

97

ΔPtie (pu)

Δω (rad/s)

For the first test scenario, the power system is examined with and without delays, following a 5% step load increase at 5 s in control area. The total communication delay is assumed as 10 s. The closed-loop system response including frequency deviation (Δω ), tie-line power change (ΔPtie ), control action signals (ui ) and area control error (ACE) are shown in Fig. 5.8. The designed PI controller acts to return the frequency, tie-line power and ACE signals to the scheduled values properly. Figure 5.8 shows the changes in control signals applied to the generator units are provided according to their participation factors (α ) listed in Table 5.1. Figure 5.9 shows the closed-loop response in the presence of a 6 s total communication delay, following a 10% step load increase in the control area. System

0.05 0 −0.05

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Time(s)

Fig. 5.8 System response with 10-s delay (solid) and without delay (dotted), following a 5% step load increase Table 5.1 Participation factors Generators

α

Unit 1

Unit 2

Unit 3

0.4

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0.2

5 Frequency Regulation with Time Delays

u3 (pu)

u2 (pu)

u1 (pu)

ΔPtie (pu)

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Time(s)

Fig. 5.9 System response with 6-s delay (solid) and without delay (dotted), following a 10% step load increase

response for 10 s delay with the same step load change is shown in Fig. 5.10. The figures show that the frequency deviation and ACE of control area are properly maintained within a narrow band using smooth control efforts. Further simulation results show that by using the time delay-less H∞ approach given in Chap. 4, the resulting closed-loop system will be unstable for the abovementioned scenarios; while the designed controller can ensure good performance despite load disturbance and delays in the communication network. The proposed real-time non-linear simulation demonstrates that the robust PI controller acts to maintain area frequency and total exchange power closed to the scheduled values, by sending corrective smooth signals to the generator units in proportion to their participation in the LFC task.

Δω (rad/s)

0.05

ΔPtie (pu)

5.5 Summary

0.2

0

u1 (pu)

− 0.05

u2 (pu)

0

50

100

150

200

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50

100

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200

0

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0

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0 − 0.2 0.2 0 − 0.2 0.2 0 − 0.2

u3 (pu)

99

0.1 0 − 0.1

ACE (pu)

1 0 −1

Time(s)

Fig. 5.10 System response in the presence of 10-s delay, following a 10% step load increase

5.5 Summary For large interconnected power systems, because of the distance involved in WAM, communication delay cannot be ignored. Unlike the small time delay in a local signal control/measurement, in wide-area power systems the time delay can vary in a wide range. This chapter focuses on robust PI-based frequency control synthesis taking into account the communication time delays. The PI-based LFC problem with communication delays in a multi-area power system is formulated as a robust SOF optimization control problem. To obtain the constant gains, an LMI-based H∞ methodology has been proposed. Simplicity of control structure, keeping the fundamental LFC concepts, using multi-delay-based LFC system and no need for an additional controller can be considered as advantages of the proposed methodology. The proposed method was applied to a control area power system using a laboratory real-time non-linear simulator.

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References 1. M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, Marcel Dekker, New York, NY, 2000. 2. J. Aweya, D. Y. Montuno and M. Ouellette, Effects of control loop delay on the stability of a rate control algorithm, Int. J. Commun. Syst., 17, 833–850, 2004. 3. S. I. Niculescu, Delay Effects on Stability – A Robust Control Approach, Springer, London, 2001. 4. S. Bhowmik, K. Tomosovic and A. Bose, Communication models for third party load frequency control, IEEE Trans. Power Syst., 19(1), 543–548, 2004. 5. T. Hiyama, T. Nagata and T. Funabashi, Multi-agent based automatic generation control of isolated stand alone power system, Proc. Int. Conf. Power Syst. Technol., 1, 139–143, 2004. 6. X. Yu and K. Tomosovic, Application of linear matrix inequalities for load frequency control with communication delays, IEEE Trans. Power Syst., 19(3), 1508–1515, 2004. 7. H. Bevrani and T. Hiyama, Multiobjective PI/PID control design using an iterative matrix inequalities algorithm, Int. J. Control Autom. Syst., 5(2), 117–127, 2007. 8. H. Bevrani and T. Hiyama, Robust decentralized PI based LFC design for time-delay power systems, Energy Convers. Manage., 49, 193–204, 2008. 9. S. Fukushima, T. Sasaki, S. Ihara, et al., Dynamic analysis of power system frequency control, Proc. CIGRE 2000 Session, No. 38–240, Paris, 2000. 10. T. Sasaki and K. Enomoto, Dynamic analysis of generation control performance standards, IEEE Trans. Power Syst., 17(3), 806–811, 2002. 11. S. Boyd, L. El. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, vol. 15. SIAM Books, Philadelphia, PA, 1994. 12. P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI Control Toolbox, The MathWorks, Natick, MA, 1995. 13. B. Chaudhuri, R. Majumder and B. C. Pal, Wide-area measurement based stabilizing control of power system considering signal transmission delay, IEEE Trans. Power Syst., 19(4), 1971–1979, 2004. 14. A. F. Snyder, et al., Delay-input wide-area stability control with synchronized phasor measurements, Proc. IEEE PES Summer Meeting, 2, 1009–1014, 2000. 15. H. Wu, K. S. Tsakalis and G. T. Heydt, Evaluation of time delay effects to wide-area power system stabilizer design, IEEE Trans. Power Syst., 19(4), 1971–1979, 2004. 16. M. Amin, Evolving energy enterprise: Possible road ahead and D&D ‘grand challenge’, Presented at IEEE PES Winter Meeting Panel Session on Grand Challenges in Electric Power Engineering. Available online at: http://www.ee.mtu.edu/faculty/ljbohman/peec/panels/ grand challenges W02/Amin.pdf 17. B. Pal and B. Chaudhuri, Robust Control in Power systems, Springer, New York, NY, 2005. 18. P. Gahinet and M. Chilali, H∞-design with pole placement constraints, IEEE Trans. Autom. Control, 41(3), 358–367, 1996. 19. N. Jaleeli, D. N. Ewart and L. H. Fink, Understanding automatic generation control, IEEE Trans. Power Syst., 7(3), 1106–1112, 1992. 20. T. Hiyama, Optimization of discrete-type load–frequency regulators considering generationrate constraints, IEE Proc. – C Gener. Transm. Distrib., 129, 285–289, 1982. 21. D. Bernstein, Some open problems in matrix theory arising in linear systems and control, Linear Algebra Appl., 162–164, 409–432, 1992. 22. V. Blondel, M. Gevers and A. Lindquist, Survey on the state of systems and control, Eur. J. Control, 1, 5–23, 1995. 23. V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback: A survey, Automatica, 33, 125–137, 1997. 24. F. Leibfritz, Static Output Feedback Design Problems, PhD Dissertation, Trier University, Trier, 1998. 25. K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.

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26. H. Bevrani and T. Hiyama, A robust solution for PI-based LFC problem with communication delays, IEEJ Trans. Power Energy, 25(12), 1188–1193, 2005. 27. Y. Kokai, I. Mtori and J. Kawakamt, Multiprocessor based generator module for a real-time power system simulator, IEEE Trans. Power Syst., 3(4), 1633–1639, 1988. 28. Y. Tamura, E. Dan, I. Horie, Y. Nakanishi and S. Yokokawa, Development of power system simulator for research and education, IEEE Trans. Power Syst., 5(2), 492–498, 1990. 29. J. Arai and Y. Noro, An induction machine model for an analog power system simulator, IEEE Trans. Power Syst., 8(4), 1478–1482, 1993. 30. T. Mori and H. Kokame, Stability of x(t) ˙ = Ax(t) + Bx(t − τ ), IEEE Trans. Autom. Control, 34, 460–462, 1989.

Chapter 6

Multi-Objective Control-Based Frequency Regulation

Load–frequency regulation systems are faced by new uncertainties in the liberalized electricity markets, and modelling of these uncertainties and dynamic behaviour is important to designing suitable controllers and providing better conditions for electricity trading. The communication delay as a significant uncertainty in the LFC synthesis/analysis can degrade the system’s performance and even cause system instability. In Chap. 5, a robust decentralized H∞ control strategy for the designing of PI-based LFC system in the presence of time delays has been developed. In order to tune the PI parameters, the optimal H∞ control is used via a multi-constraint minimization problem. The problem formulation is based on expressing the constraints as LMI which can be easily solved using available semi-definite programming methods. Unfortunately, in the presence of strong constraints and tight objective conditions, because of the following reasons the addressed optimization theorem in Chap. 5 may not approach a strictly feasible solution for a given time-delayed LFC system, and a more comprehensive/flexible control design algorithm is needed: 1. Naturally, LFC is a multi-objective control problem [1]. LFC goals, i.e. frequency regulation and tracking load changes, maintaining tie-line power interchanges to specified values in the presence of generation constraints, and time delays determine the LFC synthesis as a multi-objective control problem. Therefore, it is expected that an appropriate multi-objective control strategy (such as the mixed H2 /H∞ control technique [2]) would be able to give a better solution for this problem than a single norm control method (for example H∞ control). 2. Although the proposed H∞ –PI-based LFC design in Chap. 5 gives a simple design procedure, as shown in Sect. 5.2.2, the necessary condition for the existence of a solution is that the nominal system transfer function should be strictly positive real (SPR). This condition limits the application of the addressed control strategy to a class of dynamical power systems. 3. It is significant to note that because of using simple constant gains, pertaining to optimal SOF synthesis for dynamical systems in the presence of strong constraints and tight objectives are few and restrictive. Under such conditions, the H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 6, c Springer Science+Business Media LLC 2009

103

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minimization problem (5.22) given in Chap. 5 may not approach an optimal solution for all of the assumed dynamic LFC systems (such as the considered case study in the present work). In this chapter, a more relaxed control strategy is introduced to invoke the strict positive realness condition, and to cover all of the LFC performance targets. The PIbased multi-delayed LFC problem is transferred to a static output feedback (SOF) control design. The time delay is considered as a model uncertainty and the mixed H2 /H∞ control is used via an iterative LMI algorithm to approach a sub-optimal solution for the specified LFC design objectives. To demonstrate the efficiency of the proposed control method, some real-time non-linear laboratory tests have been performed on the analog power system simulator (APSS) system.

6.1 Mixed H2 /H∞ : Technical Background In many real-world control problems, it is desirable to follow several objectives such as stability, disturbance attenuation and reference tracking, and consider the practical constraints, simultaneously. Pure H∞ synthesis cannot adequately capture all design specifications. For instance, H∞ synthesis mainly enforces closed-loop stability and meets some constraints and limitations, while noise attenuation or regulation against random disturbances is more naturally expressed in LQG terms (H2 synthesis). The mixed H2 /H∞ control synthesis gives a powerful multi-objective control design addressed by the LMI techniques. This section gives a brief overview of the mixed H2 /H∞ output feedback control design. A general synthesis control scheme using a mixed H2 /H∞ control technique is shown in Fig. 6.1. G(s) is a linear time invariant system with the following statespace realization x˙ = Ax + B1 w + B2 u z∞ = C∞ x + D∞1 w + D∞2 u z2 = C2 x + D21 w + D22 u y = Cy x + Dy1 w,

Fig. 6.1 Closed-loop system via mixed H2 /H∞ control

(6.1)

6.2 Proposed Control Strategy

105

where x is the state variable vector, w is the disturbance and other external input vector and y is the measured output vector. The output channel z2 is associated with the LQG aspects (H2 performance), while the output channel z∞ is associated with the H∞ performance. Let T∞ (s) and T2 (s) be the transfer functions from w to z∞ and z2 , respectively, and consider the following state space realization for the closedloop system x˙cl = Acl xcl + Bcl w z∞ = Ccl1 xcl + Dcl1 w (6.2) z2 = Ccl2 xcl + Dcl2 w y = Ccl xcl + Dcl w. Theorems 6.1 and 6.2 express the design objectives in terms of linear matrix inequalities. Interested readers can find more details and proof in [2–4]. Theorem 6.1. (H∞ performance) The closed-loop RMS gain for T∞ (s) does not exceed γ∞ if and only if there exists a symmetric matrix X∞ > 0, such that ⎡ ⎢ ⎢ ⎣

Acl X∞ + X∞ ATcl

Bcl

BTcl

−I

Ccl1 X∞

Dcl1

T X∞Ccl1

⎤

⎥ DTcl1 ⎥ ⎦ < 0. −γ∞2 I

(6.3)

Theorem 6.2. (H2 performance) The H2 norm of T2 (s) does not exceed γ2 if and only if Dcl2 = 0, and there exist two symmetric matrices X2 and Q, such that

Acl X2 + X2 ATcl BTcl

Q Bcl < 0, T X2Ccl2 −I

Ccl2 X2 > 0, Trace(Q) < γ22 . X2

(6.4)

The mixed H2 /H∞ control design method uses both Theorems 6.1 and 6.2 and gives us an output feedback controller K(s) that minimizes the following trade-off criterion (6.5) k1 T∞ (s)2∞ + k2 T2 (s)22 , (k1 ≥ 0, k2 ≥ 0). An efficient algorithm to solve this problem is available in function hinfmix of the LMI control toolbox for Matlab [5].

6.2 Proposed Control Strategy For the purpose of this work, similar to Fig. 5.1, the communication delays are considered on the control input and the control output of the LFC system. The delays on the measured frequency and power tie-line flow from RTUs to the control centre can be easily transferred to the ACE signal side.

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This chapter addresses a more flexible methodology based on mixed H2 /H∞ control technique. The time delay is considered as model uncertainty, and the stability and performance objectives are formulated via H∞ and H2 norms. Finally, a suboptimal solution is obtained using a developed ILMI algorithm.

6.2.1 Multi-Objective PI-Based LFC Design Here, the LFC synthesis problem with time delay is formulated as a mixed H2 /H∞ SOF control problem to obtain the appropriate PI controller. Specifically, the H∞ performance is used to meet the robustness requirement of the closed-loop system against communication delays (as uncertainties), modelled as uncertainties. The H2 performance is used to satisfy the other LFC performance objectives, e.g. minimizing the effects of load disturbances on area frequency, ACE and penalizing fast changes and large overshoot on the governor load set point. The overall control framework to formulate the time-delayed LFC problem via a mixed H2 /H∞ control design is shown in Fig. 6.2. Using the standard simplified LFC model [6] for the prime mover and the governor, it is easy to find the state-space realization of each control area in the following form:

Fig. 6.2 H2 /H∞ SOF control framework

6.2 Proposed Control Strategy

107

x˙i = Ai xi + B1i wi + B2i ui z∞i = C∞i xi + D∞1i wi + D∞2i ui z2i = C2i xi + D21i wi + D22i ui yi = Cyi xi + Dy1i wi ,

(6.6)

where xi is the state variable vector, wi is the disturbance or other external input vector and yi is the measured output vector. For a power system control area, the state variables and input–output signals can be considered as (5.9)–(5.12), and T (6.7) z2i = μ1i Δ fi μ2i ACEi μ3i ΔPCi , z∞i = (Wi (s)Tz∞i v1i (s)) v1i ,

(6.8)

wTi = [v1i

(6.9)

v2i ] ,

where vT2i = [w1i

w2i ] .

(6.10)

The Tz∞i v1i is the transfer functions from the v1i to the z∞i . w1i and w2i are defined as input disturbances due to local load change and the area interface (5.6), respectively. The output channel z∞i is associated with the H∞ performance while the fictitious output vector z2i is associated with LQG aspects of H2 performance. μ1i , μ2i and μ3i are constant weights that must be chosen by the designer. Gi (s) is the nominal dynamic model of the given control area, yi is the augmented measured output vector (performed by ACE and its integral), ui is the control input and wi includes the perturbed and disturbance signals in the given control area. Δi shows the uncertainty block corresponding to delayed terms and Wi (s) is the associated weighting function. Here, time delays are considered as uncertainty; and unlike (5.8), the delay elements indirectly appeared in (6.6) and the LMIs in the coming pages. The H2 performance is used to minimize the effect of disturbances on area frequency and ACE by introducing appropriate fictitious controlled outputs, and fictitious output μ3i ΔPCi satisfies the physical constraint on governor load set point. The H∞ performance is used to meet the robustness requirement of the closed-loop system against specified uncertainties, due to communication delays and reduction of its impact on the system performance. The PI-based LFC problem as a multi-objective SOF control design can be expressed to determine an admissible SOF law Ki = [kPi kIi ], belonging to a family of internally stabilizing SOF gains KSOF , ui = Ki yi , Ki ∈ KSOF such that

where Tz2i v2i

inf Tz2i v2i 2 subject to Tz∞i v1i ∞ < 1, Ki ∈K SOF are the transfer functions from v2i to z2i .

(6.11) (6.12)

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6 Multi-Objective Control-Based Frequency Regulation

Fig. 6.3 Modelling the time delays as multiplicative uncertainty

6.2.2 Modelling of Uncertainties For a given power system, the uncertainties due to time delays can be modelled as an unstructured multiplicative uncertainty block that contains all possible variation in the assumed delays range. Some methods to model the uncertainties in power systems are presented in [7, 8]. Similar to the described method in Sect. 3.5.3, Gˆ i (s) denotes the transfer function from the control input ui to the control output yi at operating points other than the nominal point. According to Fig. 6.3, following a practice common in robust control, we can represent this transfer function as |Δi (s)Wi (s)| = [Gˆ i (s) − Gi (s)]Gi (s)−1 ,

(6.13)

Δi (s)∞ = supω |Δi (s)| ≤ 1; Gi (s) = 0

(6.14)

where such that Δi (s) shows the uncertainty block corresponding to delayed terms and Gi (s) is the nominal transfer function model. Wi (s) is the associated weighting function such that its respective magnitude bode plot covers all possible time-delayed structures. Figure 6.3 shows the simplified open-loop system after modelling the time delays as a multiplicative uncertainty.

6.2.3 Developed ILMI The optimization problem given in (6.12) defines a robust performance synthesis problem, where the H2 norm is chosen as the performance measure. Here, an ILMI algorithm is introduced to get a sub-optimal solution for the above optimization problem. Specifically, the proposed algorithm formulates the H2 /H∞ SOF control through a general SOF stabilization problem. The proposed algorithm searches the desired sub-optimal H2 /H∞ SOF controller Ki within a family of H2 stabilizing controllers KSOF , such that |γ2∗ − γ2 | < ε , γ∞ = Tz∞i v1i ∞ < 1, (6.15)

6.2 Proposed Control Strategy

109

where ε is a small real positive number, γ2∗ is H2 performance corresponding to H2 /H∞ SOF controller Ki , and γ2 is optimal H2 performance index that can result from the application of standard H2 /H∞ dynamic output feedback control. In the proposed strategy, based on the generalized static output stabilization feedback (Chap. 4), first the stability domain of PI parameters space is specified, which guarantees the stability of closed-loop system. In the second step, the subset of the stability domain in the PI parameter space is specified to minimize the H2 tracking performance. Finally, the design problem becomes the point with the closest H2 performance index to an optimal one which meets the H∞ constraint. The main effort is to formulate the H2 /H∞ problem via the generalized static output stabilization feedback lemma such that all eigenvalues of (A-BKC) shift towards the left halfplane through the reduction of ai , a real number, to close to feasibility of (6.12). The proposed algorithm includes the following steps: Step 1. Compute the state-space model (6.6) for the given control system. Step 2. Tune the constant weights and compute the optimal guaranteed H2 performance index γ2 using function hinfmix in the MATLAB-based LMI control toolbox [5], to design standard H2 /H∞ dynamic output controller for the performed system in Step 1. Step 3. Set i = 1, Δγ2 = Δγ0 , and let γ2i = γ0 > γ2 . Δγ0 and γ0 are positive real numbers. Select Q = Q0 > 0, and solve X from the following algebraic Riccati equation Ai X + XATi − XCyiT Cyi X + Q = 0,

X > 0.

(6.16)

Set P1 = X. Step 4. Solve the following optimization problem for Xi , Ki , and ai : minimize ai subject to the LMI constraints Ai Xi + Xi ATi + B1i BT1i + Fi B2i Ki + XiCyiT T < 0, (6.17) B2i Ki + XiCyiT −I trace(C2ic XiC2ic T ) < γ2i ,

(6.18)

Xi = XiT

(6.19)

> 0,

where Fi = −PiCyiT Cyi Xi − XiCyiT Cyi Pi + PiCyiT Cyi Pi − ai Xi . Step 5. Step 6. Step 7.

Step 8.

(6.20)

Denote a∗i as the minimized value of ai . If a∗i ≤ 0, go to Step 9. For i > 1 if a∗i−1 ≤ 0, Ki−1 ∈ KSOF and go to Step 10. Otherwise go to Step 7. Solve the following optimization problem for Xi and Ki : Minimize trace(Xi ) subject to LMI constraints (6.17), (6.18) and (6.19) with ai = a∗i . Denote Xi∗ as the Xi that minimized trace(Xi ). ∗ , then go to Step 4. Set i = i + 1 and Pi = Xi−1

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Step 9. Set γ2i = γ2i − Δγ2 , i= i + 1. Then complete Steps 3–5. Step 10. If γ∞,i−1 = Tz∞i v1i ∞ ≤ 1, Ki−1 is a sub-optimal H2 /H∞ SOF controller and γ2∗ = γ2i − Δγ2 indicates a lower H2 bound such that the obtained controller satisfies (6.15). Otherwise, go to Step 7. The proposed algorithm is schematically described in Fig. 6.4.

6.2.4 Weights Selection (μi , Wi ) As mentioned, μi = [μ1i μ2i μ3i ] is a constant weight vector that must be chosen by the designer to get a desired closed-loop performance. The selection of these performance weights is dependent on specified LFC performance objectives. The selection of weights entails a trade off among several performance requirements. Similar to weight vector ηi in Sect. 4.3.2, the elements of μ1i and μ2i at controlled outputs set the performance goals (tracking the load variation and disturbance attenuation), and μ3i sets a limit on the allowed control action to penalize fast change and large overshoot in the governor load set-point signal. Here, another alternative to select the mentioned weights, the designer can fix the weights μ1i , μ2i and μ3i to unity and use the method with regional pole placement technique for performance tuning [9]. The weight function Wi in each control area to be computed as described in Sect. 6.2.2. The Wi (s) must be considered such that its respective magnitude bode plot covers the bode plots of all possible time-delayed structures.

6.2.5 Application to Three-Control Area The proposed control technique is applied to the three-control area power system shown in Fig. 4.6, with the same simulation data used in Chap. 4. The application details are given in [10]. The results are compared with the delay-less and delayed H∞ -SOF control design (described in Chaps. 4 and 5). It is shown that the communication delays seriously degrade the performance of delay-less LFC design, while the proposed methodologies in Sect. 5.2 and the present chapter provides a desirable system performance [10].

6.3 Discussion • As has been mentioned, the complex and high-order dynamic controllers are inapplicable for real-world LFC systems. Usually, the load–frequency controllers used in the industry are PI type. Since the PI controller parameters are commonly tuned online based on experiences and trial and error approaches, they are incapable of obtaining good dynamical performance for a variety of load sce-

6.3 Discussion

111

Fig. 6.4 Iterative LMI algorithm

narios and operating conditions. There are hardly any results in PI-based LFC design literature with time-delay consideration. The design of PI-based load– frequency controllers is, in most cases, performed using classical tuning rules

112

•

•

•

•

6 Multi-Objective Control-Based Frequency Regulation

without consideration of delay impacts. On the other hand, the modern and postmodern control theory including H2 and H∞ optimal control cannot be directly applied to the PI-based LFC problem. Indeed, until recently, it was not known how to even determine whether stabilization of a nominal system was possible using PI/PID controllers [11]. Therefore, in comparison of previous works, the appropriate formulation of time delay in the PI-based LFC design through an optimal minimization problem can be considered as a significant contribution. The stability margin and performance specifications could be evaluated using classical analysis tools such as gain and phase margin, as well as modern ones such as H2 and H∞ norms of closed-loop transfer functions. In the proposed LFC solutions (Chaps. 5 and 6), robust performance indices, resulting from solution of the optimal H∞ and H2 /H∞ control synthesis, that provide strong criteria and powerful tools have been used as robust performance measures for the sake of closed-loop stability and performance analysis. Since the main theme in both SOF control designs is to stabilize the overall system and guarantee the H∞ and H2 norms of the closed-loop transfer functions, the designed load–frequency controllers meet the robust specifications. For example, in the resulting PI solution from the H∞ -based LFC design (described in Chap. 5), since the solution for the time-delayed LFC problem is obtained through minimizing the H∞ performance index γ subject to the given constraints in (5.22), the designed PI controllers satisfy the robustness of the closed-loop system. In other words, the basis of designing the SOF controllers is to stabilize simultaneously (5.8) and guarantee the H∞ norm bound γ of the closed-loop transfer function Tzw , namely, Tzw ∞ < γ ; γ > 0. Although the H∞ -based LFC design (Chap. 5) gives a simple design procedure, the proposed PI-based LFC design strategy (H2 /H∞ ) in the present chapter provides a more flexible control strategy, and could be applicable for a wider range of control area power systems. The necessary condition for the existence of a solution is that the nominal transfer function given in (5.23) should be strictly positive real, but satisfying this condition for dynamical systems in the presence of strong constraints and tight objectives are few and restrictive, and the minimization problem (5.22) may not approach a strictly feasible solution. The stability area for any controlled system is limited by a border, such that for the outside operating points, the system may go to an unstable condition. In the proposed LFC designs, the stability area is dependent on the considered range of time delay in the related LFC loop during the synthesis procedures. Therefore, in the assumed delay range, robust stability and robust performance are guaranteed for the power system control areas. However, to get a larger margin of stability, for example, in the mixed H2 /H∞ PI-based LFC design, a wider range of delays can be considered by choosing a larger delay terms τd and τh (Fig. 5.1). As a result, it provides a new upper bound for the modelled uncertainty without any change in the design procedure. From the view point of stability and performance analysis, it is shown that the impact of delay on the dynamic behaviour of a control system is the same as the effect of a perturbation and system uncertainty [12]. Similar to unmodelled

6.4 Real-Time Laboratory Experiments

113

dynamic uncertainties, time delays can degrade a system’s performance and stability [13–15]. Hence, it could be reasonable to consider the time delay as a model uncertainty. Here, with regards to the communication issue, the general theme is based on the premise that the necessary communication software/hardware facilities are available in the power system network to receive/transmit the measurements and control signals via appropriate secure links. • As described in Chap. 2, a power system is an inherently non-linear and complex system. However, since considering all dynamics in LFC synthesis and analysis may not be useful and is difficult, a simplified linear model is usually used by researchers, but it should be noted that to get an accurate perception of the LFC subject, it is necessary to consider the important inherent requirement and the basic constraints (described in Sect. 3.5) imposed by the physical system dynamics and model them for the sake of performance evaluation. A useful control strategy must be able to maintain sufficient levels of reserved control range and control rate. Here, the effect of delays and generation rate constraint is properly considered in the synthesis procedure to produce a smooth set point behaviour. The proposed H2 /H∞ control strategy includes enough flexibility to set a desired level of performance to cover the practical constraint on the control action signal. It is easily performed by the tuning of μ3i (or η3i in Chap. 5) in the fictitious controlled output. Hence, it is expected the designed controllers could be useful to perform the LFC task in a real-world power system. • In the proposed LFC methods in the present chapter and Chap. 5, an important goal was to keep the simplicity of control algorithms (as well as control structure) for computing the PI parameters among the well-known LFC scheme. For the reasons of simplicity, flexibility and straightforwardness of the control algorithms, this work acts as a catalyst to bridge the gap between robust H∞ and the mixed H2 /H∞ control theory and real-world LFC synthesis as well as the gap between classical and modern LFC tuning methods.

6.4 Real-Time Laboratory Experiments 6.4.1 Configuration of Study System To illustrate the effectiveness of the proposed control strategy, a real-time experiment has been performed on the large-scale APSS described in Sect. 5.3. For the purpose of this study, a longitudinal four-machine infinite bus system is considered as the test system. A single line representation of the study system is shown in Fig. 6.5. All generator units are thermal type, with separately conventional excitation control systems. The set of four generators represents a control area (Area I), and, the infinite bus is considered as other connected systems (Area II).

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6 Multi-Objective Control-Based Frequency Regulation

Fig. 6.5 Study power system

Table 6.1 Oscillation modes in actual and laboratory systems Power system Actual WJPS Laboratory system

Global oscillation mode (Hz)

Local oscillation mode (Hz)

0.30 0.30

0.10–2.50 0.15–2.40

The detailed block diagram of each generator unit and its associated turbine system (including the high-pressure, intermediate-pressure and low-pressure parts) is illustrated in Fig. 5.7. The power system parameters are given in Appendix A (Table A.3). Although, in the given model the number of generators is reduced to four, it closely represents the dynamic behaviour of the West Japan Power System (WJPS) at the time of experiment. As shown in Table 6.1, the most important global and local oscillation modes of the actual system are included. The essential global oscillation mode of actual WJPS is around 0.3 Hz. Depending on the individual generators, the local oscillation modes are varied in the frequency range of 0.1–2.5 Hz. There also exists an inter-area oscillation mode around 0.7 Hz.

6.4.2 PI Controller The proposed control loop including robust PI controller, ACE computing unit and participation factors which have been built in a personal computer were connected

6.4 Real-Time Laboratory Experiments

115

to the power system using a digital signal processing (DSP) board equipped with analog to digital (A/D) and digital to analog (D/A) converters, as the physical interfaces between the personal computer and the analog power system hardware. Figure 6.6a shows the applied laboratory devices for the experiment including the control/monitoring desks. The PC-based control loop is shown in Fig. 6.6b. A digital oscilloscope and a notebook computer are used for monitoring purposes. Using the stability test rule described in Sect. 5.3.3, the control area in the example at hand is shown to be unstable in the presence of the specified time delays.

μ (A) + Ad = 11.0519 > 0.

(6.21)

Fig. 6.6 Performed laboratory experiment; (a) the control/monitoring desks and (b) PC-based implemented control loop

116

6 Multi-Objective Control-Based Frequency Regulation 102

101

W(s)

Magnitude

100 10−1 10−2 10−3 10−4

0

0.2

0.4

0.6

0.8 1 1.2 1.4 frequency (rad/sec)

1.6

1.8

2

Fig. 6.7 Uncertainty plots (dotted) due to communication delays and the upper bound (solid)

It was also shown that the described H∞ -based technique in Chap. 5 is unable to obtain a feasible optimal solution. In order to get a sub-optimal solution, the proposed control strategy given in Sect. 6.2 is applied. Some sample uncertainties due to delay variations, within the following range which is close to the real LFC cycle, are shown in Fig. 6.7.

τ = τd + τh ∈ [0 8]s.

(6.22)

Specifically, to obtain the uncertainty curves, (6.13) should be solved for some different points in the assumed delay range (6.21). To keep the complexity of the design procedure low, we can model the uncertainties from both delayed channels by using a low-order norm bonded multiplicative uncertainty to cover all possible plants as follows 2.1012s + 0.2130 . (6.23) W (s) = s + 0.5201 Figure 6.7 clearly shows that the chosen first-order weight W (s) provides a little conservative design at low frequencies; however, it provides a good trade-off between robustness and design complexity. It is notable that using a high-order weighting function to find a tighter upper bound may result in a failure to obtain feasible optimal PI parameters. On the other end, the determined low-order W (s) must cover all the uncertainty curves. Otherwise, for the obtained PI parameters, the robustness cannot be guaranteed for all the specified delay changes. In the present example, the time delay of communication channels is considered near the LFC cycle rate. However, as mentioned, one can consider a wider range of delays by choosing a larger τ . As a result, it provides a new upper bound for the modelled uncertainty without any change in the design procedure.

6.5 Simulation Results

117

Considering the existing limits on the rate and range of generation change and the fact that the generation units (for example steam units) need time to fully respond, the proposed control strategy includes enough flexibility to set a desired level of performance and to cover practical constraint on the control action signal. This can be easily done by tuning the constant weights μi associated with the fictitious controlled outputs in Fig. 6.2. Based on the given explanation on selection of constant weights for the present LFC system, values 0.5, 1 and 5, are chosen for the weights μ1i , μ2i and μ3i , respectively. Finally, according to the synthesis methodology described in Sect. 6.2, the parameters of PI controller are obtained as kPi = −0.3509, kIi = −0.2104.

(6.24)

6.5 Simulation Results In the performed non-linear real-time laboratory experiment, the proposed PI controller was applied to the control area power system described in Fig. 6.5. The performance of the closed-loop system is tested in the presence of load disturbances and the time delays. The nominal area load demands PL1 , PL2 and PL3 (Fig. 6.5) during test scenarios are considered as 0.3 pu, 0.6 pu and 0.6 pu, respectively. For scenario 1, the power system is examined with delays, following a 0.1 pu step load increase in control area. The total communication delay is assumed as 5 s. The closed-loop system response including frequency deviation (Δω ), tie-line power change (ΔPtie ), ACE and control action signals (ui ) are shown in Fig. 6.8. The system performance is compared with a designed robust H∞ –PI controller based on the given methodology in Chap. 4 and [16] for the delay-less LFC systems. As shown in Fig. 6.8, using delay-less H∞ design the system falls in a critical condition and leads to an unstable operating point, while the proposed H2 /H∞ –PI controller acts to return the frequency, tie-line power and ACE signals to the scheduled values properly. Figure 6.8b shows the changes in control signals applied to the generator units that are provided according to their participation factors listed in Table 6.2. In scenario 2, the power system was tested for a longer time delay. Figure 6.9 shows the closed-loop response in the presence of 8 s total communication delay, following a 0.1 pu step load increase in the control area. In scenario 3, the system response was tested for a sequence of step load changes as shown in Fig. 6.10. The total delay was fixed at 6 s. Figures show that the frequency deviation and the area

Table 6.2 Participation factors in scenarios 1 and 2 Gen. unit

α

1

2

3

4

0.4

0.4

0.2

0.0

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6 Multi-Objective Control-Based Frequency Regulation

a

Δω (rad/s)

Load (pu)

0.1 0.05 0 0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

0.04 0.02 0 −0.02 −0.04

ΔPtie (pu)

0.5 0 −0.5

ACE (pu)

1

u1(pu)

b

0 −1 Time(s)

0.2 0 −0.2

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

0

20

40

60

80

100

u2(pu)

0.2 0 −0.2

u3(pu)

0.1 0 −0.1

u4(pu)

0.1 0 −0.1

Time(s)

Fig. 6.8 System response for scenario 1 (5-s delay following a 0.1 pu step load increase), using the proposed method (solid) and design technique given in Chap. 5

6.5 Simulation Results

a

119

Load (pu)

0.1 0.05 0 0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40

50

60

70

80

0

10

20

30

40 50 Time(s)

60

70

80

Δω (rad/s)

0.05 0

ΔPtie (pu)

−0.05

0.2 0 −0.2

ACE (pu)

0.5 0 −0.5 −1

u1(pu)

b

40 Time(s)

0.2 0.1 0 −0.1

u2(pu)

0.2 0.1 0 −0.1

u3(pu)

0.2 0.1 0 −0.1

u4(pu)

0.2 0.1 0 −0.1

Fig. 6.9 System response for scenario 2: with 8-s delay (solid) and without delay (dotted), following a 0.1 pu step load increase

120

6 Multi-Objective Control-Based Frequency Regulation

a Load (pu)

0.1 0.05

Δω (rad/s)

0 0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

0

50

100

150

200

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350

0

50

100

150 200 Time(s)

250

300

350

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

0

50

100

150 200 Time(s)

250

300

350

0.05 0

ΔPtie (pu)

−0.05

0.2 0 −0.2

ACE (pu)

1 0 −1

u1(pu)

b

0.1 0 −0.1

u2(pu)

0.1 0 −0.1

u3(pu)

0.05 0 −0.05

u4(pu)

0.05 0 −0.05

Fig. 6.10 System response for scenario 3: with 6-s delay (solid) and without delay (dotted), following a sequence of step load changes

References

121

Table 6.3 Participation factors in scenario 3 Gen. unit

1

2

3

4

α

0.40

0.25

0.20

0.15

control error of control area are properly maintained within a narrow band using smooth control efforts. The participation factors for the recent experiment are given in Table 6.3. The results obtained show that the designed controller can ensure good performance despite load disturbance and indeterminate delays in the communication network. The proposed real-time simulation shows the robust PI controller acts properly to maintain area frequency and total exchange power close to the scheduled values by sending corrective smooth signal to the generators in proportion to their participation in the LFC task. Considering the time delays as structured uncertainties, the proposed method provides a conservative design, but it gives a good trade-off among the specified LFC objectives using the H2 and H∞ performances. The experiment results show that this controller performs well for a wide range of operating conditions considering the load fluctuation and the communication delays.

6.6 Summary The PI-based LFC problem with communication delays in a multi-area power system is formulated as a robust SOF optimization control problem. To obtain the constant gains, a flexible methodology is developed to invoke the existing strictness. The time delay is considered as a model uncertainty and the H2 /H∞ control is used via an iterative LMI algorithm to approach a sub-optimal solution for the assumed design objectives. The proposed method was applied to a control area power system through a laboratory real-time experiment. In the proposed LFC method, an important goal was to keep the simplicity of control algorithms (as well as control structure) for computing the PI parameters among the well-known LFC scheme.

References 1. H. Bevrani, Decentralized Robust Load–Frequency Control Synthesis in Restructured Power Systems, PhD Dissertation, Osaka University, 2004. 2. P. P. Khargonekar and M. A. Rotea, Mixed H2 /H∞ control: A convex optimization approach, IEEE Trans. Autom. Control, 39, 824–837, 1991. 3. C. W. Scherer, Multiobjective H2 /H∞ control, IEEE Trans. Autom. Control, 40, 1054–1062, 1995.

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4. C. W. Scherer, P. Gahinet and M. Chilali, Multiobjective output-feedback control via LMI optimization, IEEE Trans. Autom. Control, 42(7), 896–911, 1997. 5. P. Gahinet, A. Nemirovski, A. J. Laub and M. Chilali, LMI Control Toolbox, The MathWorks, Natick, MA, 1995. 6. A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control, Wiley, New York, NY, 1984. 7. M. Djukanovic, M. Khammash and V. Vittal, Sequential synthesis of structured singular value based decentralized controllers in power systems, IEEE Trans. Power Syst., 4(2), 635–641, 1999. 8. M. Rios, N. Hadjsaid, R. Feuillet and A. Torres, Power system stability robustness evaluation by μ analysis, IEEE Trans. Power Syst., 14(2), 648–653, 1999. 9. P. Gahinet and M. Chilali, H∞ -design with pole placement constraints, IEEE Trans. Autom. Control, 41(3), 358–367, 1996. 10. H. Bevrani and T. Hiyama, Robust decentralized PI-based LFC design for time-delay power systems, Energy Convers. Manage., 49, 193–204, 2008. 11. G. J. Silva, A. Datta and S. P. Bhattacharyya, PID Controllers for Time-Delay Systems, Birkhauser, Boston, MA, 2005. 12. K. Gu, V. L. Kharitonov and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, MA, 2003. 13. M. S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, Marcel Dekker, New York, NY, 2000. 14. J. Aweya, D. Y. Montuno and M. Ouellette, Effects of control loop delay on the stability of a rate control algorithm, Int. J. Commun. Syst., 17, 833–850, 2004. 15. S. I. Niculescu, Delay Effects on Stability – A Robust Control Approach, Springer, Berlin, 2001. 16. H. Bevrani, Y. Mitani and K. Tsuji, Robust decentralized load–frequency control using an iterative linear matrix inequalities algorithm, IEE Proc. Gener. Transm. Distrib., 150(3), 347–354, 2004.

Chapter 7

Agent-Based Robust Frequency Regulation

As mentioned in the previous chapters, serious frequency deviations can directly impact on power system operation, system reliability and efficiency. Frequency changes in large-scale power systems are a direct result of the imbalance between the electrical load and the power supplied by system connected generators [1]. In a deregulated environment, load–frequency control (LFC), as an ancillary service, plays a fundamental role in supporting power exchanges and providing better conditions for system reliability and electricity trading. In the new environment, generation companies (Gencos) submit their ramp rates (Megawatts per minute) and bids to the market operator. After a bidding evaluation, those Gencos selected to provide the regulation service must perform their functions according to the ramp rates approved by the responsible organization. Recently, several scenarios have been proposed to adapt the well-known conventional LFC scheme to the changing environment of power system operation under deregulation [2–6]. As mentioned in Chap. 3, there are several schemes and organizations for the provision of regulation services in countries with a restructured electric industry, differentiated by how free the market is, who controls generator units, and who has the obligation to execute LFC. This chapter introduces an agent-based scenario to follow LFC objectives in a deregulated environment. The multi-agent systems philosophy and its potential value to the power industry are discussed in [7,8]. In the present chapter, a two-agent control system is used to cover a minimum number of required processing/activities to the LFC objectives in a control area. Based on the proposed control strategy, a decision and control agent uses the received data from a data acquisition and monitoring agent to provide the generation participation factors and appropriate control action signal, through an H∞ robust proportional–integral (PI) controller. The system frequency is analytically described, and to demonstrate the efficiency of the proposed control method, real-time non-linear laboratory tests have been performed on the APSS laboratory system. The results show that the proposed control scheme guarantees optimal performance for a wide range of operating conditions. The LFC model and power imbalance estimation is explained in Sect. 7.1. An overview of the proposed agent-based control framework is given in Sect. 7.2. H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 7, c Springer Science+Business Media LLC 2009

123

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7 Agent-Based Robust Frequency Regulation

Sect. 7.3 addresses the problem formulation for the robust tuning of dynamic PI controller, and Sect. 7.4 presents a real-time implementation of the developed control scheme. Some laboratory results for different load disturbance scenarios are given in Sect. 7.5. Finally, a discussion and chapter summary are given in Sect. 7.6 and 7.7, respectively.

7.1 Frequency Response Analysis 7.1.1 Frequency Response Model The preliminary concepts on LFC mechanism are discussed in Chaps. 2 and 3. A multi-area power system is comprised of areas that are interconnected by highvoltage transmission lines or tie-lines. The trend of frequency measured in each control area is an indicator of the trend of mismatch power in the interconnection and not in the control area alone. The primary control provides a fast control action made by the speed governor due to the frequency deviations in the control area, to keep the instantaneous balance between system production and consumption. Generators are required to participate in this control by setting the droop (R) according to specifications by the system (market) operator. The supplementary control is a relatively slow action to recover the nominal system frequency and scheduled power interchanges after a load/generation disturbance. Here, for the sake of dynamic frequency analysis of a power system in the presence of load disturbances, the frequency response model for the LFC system is used, which is described in Sect. 3.1 (Fig. 3.1). According to Fig. 3.1, in a control area, the ACE performs the input signal for the LFC system, and a simple PI control law is used to provide the supplementary control action signal KI ACE(s). (7.1) ΔPC i (s) = αi KP + s It is shown in Chap. 3 that without supplementary control signal and tie-line power deviation (ΔPC i = 0, ΔPtie,i = 0), the steady-state frequency deviation, which is proportional to disturbance magnitude, can be calculated as follows (3.17): Δ fss = −

Rsys ΔPL , (DRsys + 1)

(7.2)

where Δ fss is the steady state frequency deviation; ΔPL is the total load disturbance; D is the equivalent damping coefficient and Rsys is the equivalent droop characteristic. For a sufficient small DRsys and considering the remark made in Sect. 3.1, the above equation can be approximated as follows Δ fss ∼ = −Rsys ΔPL .

(7.3)

7.1 Frequency Response Analysis

125

7.1.2 Total Power Imbalance Estimation In order to predict the amount of necessary power reserve for the recovery of system frequency, the total load change (due to the local load demands and disturbances) should be estimated by the DC agent periodically. As previously mentioned, frequency changes can induce a corresponding change in load demand. This effect is called load relief and should be taken into account when calculating the amount of required ancillary service [9]. The change in demand is always in a direction that tends to alleviate the frequency deviation, i.e., for a reduction in frequency, the load relief is negative (decrease in demand), which tends to alleviate the falling frequency. Consider the ith generator swing equation for a control area with N generators (i =, . . ., N) 2Hi

dΔ fi (t) + Di Δ fi (t) = ΔPm i (t) − ΔPL i (t) = ΔPd i , dt

(7.4)

where ΔPm i is the mechanical turbine power, ΔPL i is the load demand (electrical power), Hi is the inertia constant, Di is load damping, fi is frequency and ΔPd i is the load–generation imbalance. By adding N generators within the control area, one obtains the following expression for the total load–generation imbalance N

ΔPD (t) = ∑ ΔPd i (t) = 2H i=1

dΔ f (t) + DΔ f (t). dt

(7.5)

Equation (7.5) shows the multi-machine dynamic behaviour by an equivalent single machine [10]. Using the concept of an equivalent single machine (Chap. 2), a control area can be represented by a lumped load generation model using an equivalent frequency Δ f , system inertia H and system load damping D [11–13] N

Δ f = Δ fsys = ∑ (Hi Δ fi ) i=1 N

N

∑ Hi ,

(7.6)

i=1

N

H = Hsys = ∑ Hi , D = Dsys = ∑ Di . i=1

(7.7)

i=1

The magnitude of total load–generation imbalance (or total system load reduction) ΔPD , after a while, can be obtained from (7.5) ΔPD = DΔ f ,

(7.8)

ΔPD = ΔPm − ΔPL ,

(7.9)

where N

ΔPm = ∑ Pm i ,

(7.10)

ΔPL = ∑ PL i .

(7.11)

i=1 N i=1

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7 Agent-Based Robust Frequency Regulation

The total mechanical power change indicates the total power generation change due to governor action, which is in proportion to the system frequency deviation [14]: ΔPm ∼ = ΔPg = −

1 Δf. Rsys

The equations (7.8), (7.9) and (7.12) give 1 +D Δf. ΔPL = − Rsys

(7.12)

(7.13)

Thus, the total load change in a control area is proportional to the system frequency deviation. This result agrees with the result obtained in Chap. 3 (3.17).

7.2 Proposed Control Strategy 7.2.1 Overall LFC Framework The power system industry is already designed to operate with large and central generating facilities. The far reaching deregulation and concomitant new concepts of operation require an evaluation and re-examination of the conventional schemes, to find ways to maintain and to improve system efficiency and reliability. In the present chapter, a multi-agent-based scenario is proposed to perform the LFC task in a deregulated environment. A survey on the potential benefits of multiagent technology and its use in power system engineering application can be found in [7, 8, 15]. The overall control framework is conceptually shown in Fig. 7.1. The boundary of control area encloses the generation companies (Gencos) and the distribution companies (Discos) associated with the performed contracts. The operating centre for each area includes two agents:data acquisition and monitoring (DAM) agent and decision and control (DC) agent. The Gencos send the bid regulating reserves F($, T, Q) to the DAM agent through a secure network service. The DAM agent sorts these bids by a pre-specified time period, quantity and price. Then, it sends the sorted regulating reserves with the demanded load from Discos and the measured tie-line flow and area frequency to the DC agent, continuously. DC agent checks and re-sorts the bids according the congestion condition and screening of available capacity. Then, DC agent provides the ACE signal and computes the participation factors αi (t) in order to load tracking by the available Gencos to cover the total contracted load demand ∑ ΔPL (t) and local load disturbance ΔPd (t). Limits on the unit ramp rates should be considered in evaluation of the participation factors. Here, it is assumed that in a given control area, the necessary hardware and communication facilities to enable reception of data and control signals are available, and Gencos can bid up and down regulations by price and MW-volume for each predetermined time period T to the regulating market. Also, the control centre can distribute load demand signals to available generating units on a real-time basis.

7.2 Proposed Control Strategy

127

7.2.2 Computing of Participation Factors The participation factors, which are actually time-dependent variables, must be computed dynamically by DC (Decision and Control) agent based on the received bid prices, availability, congestion problem, and other related costs in case of the used each applicant (Genco). Assume that the DC agent determines a regulating price RP i for each applicant based on the mentioned aspects and the all yielding costs, and (7.14) RP1 < RP2 < · · · < RP n . The DC agent prepares a priority list sorted by regulating price as given in Table 7.1. Then, the DC agent tries to fully utilize the capacity of one regulating object in the table before calling on the next, which is more expensive. In each T step, the DC agent is comparing the total load demand and available unit capacity. First, the capacity of the cheapest unit is checked. If this is sufficient to cover the demand, the participation factor of the unit is set equal to 1; otherwise, to cover the total load demand, the second cheapest item is considered and so forth. For three (or more) units, the participation factors are calculated to set all but the most expensive unit in use at the minimum capacity. In a given control area, the sum of participation factors will always be equal to 1. n

∑ αi (t) = 1.

i=1

Fig. 7.1 Proposed two-agent-based LFC scheme

(7.15)

128

7 Agent-Based Robust Frequency Regulation

Table 7.1 The sorted list for Ti < t < Ti+1 Genco 1

Genco 2

...

Genco n

ΔPR1 RP1

ΔPR2 RP2

... ...

ΔPR n RP n

Regulating power (MW) Regulating price ($/MW)

Based on the above explanation for each time step T , the area participation factors to be computed are as follows:

α1 (t) =

! ΔP

R 1 (t) ΔPD (t) ;

1

;

ΔPD (t) > ΔPR 1 (t)

(7.16)

ΔPD (t) ≤ ΔPR 1 (t)

and

αk (t) =

⎧ ΔPD (t)−ΔPR1 (t)−ΔPR2 (t)···−ΔPR (k−1) (t) ⎪ ⎪ ⎪ ; ⎨ ΔPD (t) ⎪ ⎪ ⎪ ⎩0

k−1

∑ αi < 1

i=1

k−1

;

,

(7.17)

∑ αi = 1

i=1

where ΔPD is the total load change in the area ΔPD (t) = ∑ ΔPL (t) + ΔPd (t).

(7.18)

Therefore, each Genco tracks the area load change according to its participation factor αi (t) ΔPC i (t) = αi (t)ΔPD (t) (7.19) and

n

∑ ΔPC i (t) = ΔPC (t) ≈ ΔPD (t),

(7.20)

i=1

where ΔPD (t) is satisfied by the supplementary control signal ΔPC (t). Since both of the number and the order of LFC participant Gencos may change, a smooth transition must be carried out from one set of units (at Ti ) to another (at Ti+1 ). A suitable scenario is introduced in [16]. Each participating unit will receive its share of the demand ΔPC i (t) according to its participation factor, through a dynamic Controller which usually includes a simple PI structure in a real-world power system. Technically, as described in the previous chapters, this unit has a vital role to guarantee desired LFC performance [14]. An effective design ensures a smooth coordination between generator set-point signals and the scheduled operating points αi (t)ΔPC i (t).

7.2 Proposed Control Strategy

129

7.2.3 Structure of Two-Agent System In the design of real-time multi-agent system, at least the following steps must be performed: denote the number and the type of agents in the system, define the internal structure of every agent, design the communication protocol among agents and finally the schedulability of the different agents of the system from the point of view of real-time system [17]. For a real-time LFC system, the structural flexibility and having a degree of intelligence are highly important. In such systems, agents require real-time responses and must eliminate the possibility of massive communication among agents. In this work, the proposed control structure presents two agents. The two-agent schema entailed the minimum number of measurement/monitoring and control activities in a control area to track the LFC tasks. Each agent of the LFC system is implemented on a software platform that supports the general component of this special kind of agent. The software platform must provide a communication environment among the agents and supports a standard interaction protocol [18]. The proposed two-agent structure with the related tasks is summarized in Fig. 7.2. In the first agent (DAM agent), special care was taken regarding the

Fig. 7.2 Two-agent structure

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7 Agent-Based Robust Frequency Regulation

provision of appropriate signals following a filtering and signal conditioning process on the measured signals and the received data from the input channels. The sorted information and wash out signals will be passed to the second agent (DC agent). The DC agent evaluates and reorders the bids and performs the ACE signal using the measured frequency and tie-line power signals. The agent software estimates the total power imbalance and determines LFC participation factors by assigned algorithms, considering the ramp rate limits and based on the explained methods in the present section. It is noteworthy that the market described in the present section is simple. In most markets, a Genco’s bid price will increase in the higher part of its range.

7.3 Tuning of PI Controller The tuning of PI controller ensures a smooth coordination between generator setpoint signals and the scheduled operating points αi (t)ΔPC i (t). As described in Chaps. 5 and 6, and as shown in recent research works [19, 20], time delays can seriously degrade the LFC system’s performance, and should be considered during the dynamic PI-based frequency control design. Three robust control methodologies have been developed for the tuning of PIbased LFC parameters in the last three chapters. In this section, the control strategy described in Chap. 5 is reused for PI control design, following appropriate problem simplification. Consider the open-loop state-space model of a control area in the following timedelayed system form: x(t) ˙ = Ax(t) + Bu(t) + Ad x(t − d) + Bh u(t − h) + Fw(t) z(t) = C1 x(t) y(t) = C2 x(t).

(7.21)

Here x ∈ ℜn is the state vector, C2 ∈ ℜn is the constant matrix such that the pair (A,C2 ) is detectable. d and h represent the delay amounts in the state and the input, respectively. A ∈ ℜn×n and B ∈ ℜn×m represent the nominal system without delay such that the pair (A, B) is stabilizable. Ad ∈ ℜn×n , Bh ∈ ℜn×m and F ∈ ℜn×q are known matrices and ψ (t) is a continuous vector-valued initial function. Similar to Fig. 5.1, the communication delays are considered on the control input and the control output of the LFC system. The PI control design problem can be transferred to a static output feedback (SOF) control problem by augmenting the measured output signal to include the ACE and its integral

u(t) = ky(t),

u(t) = KP ACE + KI ACE = [KP

T ACE . KI ] ACE

(7.22) (7.23)

7.3 Tuning of PI Controller

131

Fig. 7.3 H∞ -SOF control framework

The overall control framework to formulate the time-delayed PI control problem via H∞ -based SOF (H∞ -SOF) control design is redrawn in Fig. 7.3. Here, G(s) is the nominal dynamic model of the given control area, u is the control input and w includes the perturbed and disturbance signals in the given control area. The output channel z is associated with the H∞ performance while y is the augmented measured output vector (performed by ACE and its integral). Also, μ1 , μ2 and μ3 are constant weights that must be chosen by the designer to get the desired closed-loop performance. Using the LFC block diagram (Fig. 3.1) and remaining consistent with the standard dynamic model for the prime mover and governor in a control area [21], the state variables of G(s) can be determined as follows: ACE xm xg , (7.24) xT = Δ f ΔPtie where xm = ΔPm1

ΔPm2

· · · ΔPm n , xg = ΔPg1

ΔPg2

· · · ΔPg n

and yT = [ ACE z T = μ1 Δ f

ACE], u = ΔPC , wT = [ΔPD μ2 ACE μ3 ΔPC ,

v]

(7.25) (7.26)

where the ΔPg , ΔPm , and v indicates the governor valve position, turbine mechanical power changes and area interface effect, respectively. Using the described transformation from PI to SOF control design, the timedelayed PI control problem is reduced to the synthesis of an SOF controller for the time-delay system (7.21) of the form of (7.22). Here, k is a static gain vector to be determined. In order to obtain an optimal LMI-based H∞ solution for the problem at hand, one can use the Theorem 5.2 (Sect. 5.2). This theorem adapts the H∞ theory with time-delayed systems (using LMI description) and establishes the conditions under which the SOF control law u(t) = ky(t) stabilizes (7.21) and

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guarantees the H∞ norm bound γ of the closed-loop transfer function Tzw , namely Tzw ∞ < γ ; γ > 0. Based on the mentioned theorem, the SOF controller k asymptotically stabilizes the system (7.21) and Tzw ∞ < γ for d, h ≥ 0 if there exist matrices 0 < Y T = Y ∈ ℜn×n , 0 < QTt = Qt ∈ ℜn×n and 0 < QTs = Qs ∈ ℜn×n satisfying the matrix inequality (7.27) T AY +YAT + Qt + Qs ] + [BkC2 (BkC2 )T +Y TY + AdY Q−1 t (AdY ) (7.27) T T −2 T + Bh kC2Y Q−1 s (Bh kC2Y ) +YC1 C1Y + γ FF < 0.

7.4 Real-Time Implementation 7.4.1 Configuration of Study System To illustrate the effectiveness of the proposed control strategy, a real-time experiment has been performed on the APSS laboratory system introduced in Chap. 5. For the purpose of this study, the longitudinal four-machine infinite bus system, described in Chap. 6, is considered as a test system. A single line representation of the study system is redrawn in Fig. 7.4. All generator units are thermal type, with separately conventional excitation control systems. The set of four generators represents a control area (Area I), and the

Fig. 7.4 Study power system

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Fig. 7.5 (a) Conventional speed governing system and (b) detailed turbine system

infinite bus is considered as other connected systems (Area II). Each unit has a full governor–turbine system (governor, steam valve servo-system, high-pressure turbine, intermediate-pressure turbine and low-pressure turbine), as shown in Fig. 7.5. The power system parameters are given in Table 6.1.

7.4.2 PI Parameters To adapt (7.21) with the structure of time-delayed LFC system [22], Ad and Bh can easily be computed by transferring the delays on ACE signal (τd ) and ΔPc signal (τh ) through their components (Δ f , ΔPtie and u). For the study system at hand, the total time delays of communication channels is considered near to the LFC cycle rate of the power system, and suitable values for the weights μ1 , μ2 and μ3 are chosen as 0.5, 1 and 5, respectively. According to the synthesis methodology described in Sect. 7.3, and following the suitable assumption to the relaxation of the control problem, the parameters of PI controller are obtained as follows KP = −0.4095, KI = −0.0241.

(7.28)

7.4.3 Implementation The whole power system has been implemented using the APSS laboratory system. Figure 7.6a shows the block diagram representation of the proposed controlled system. Figure 7.6b shows the overview of the applied laboratory experiment devices including the DAM and DC agents. In the performed DAM agent, a PC screen

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Fig. 7.6 Performed laboratory experiment; (a) block diagram representation, (b) physical configuration

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with a digital oscilloscope (shown in the left side of Fig. 7.6b) is used for monitoring purposes.

7.5 Laboratory Results In the non-linear real-time laboratory experiment, the proposed agent-based LFC scheme was applied to the control area power system described in Fig. 7.4. The performance of the closed-loop system is tested in the presence of load disturbances and time delays. The nominal area load demands PL1 , PL2 and PL3 (Fig. 7.6a) during test scenarios are considered as 0.3 pu, 0.6 pu and 0.6 pu, respectively. For the first scenario, the power system is tested following a step loss of generation 0.05 pu. It is assumed that Gen 4 does not participate in the LFC task, and the participation factors for Gen 1, Gen 2 and Gen 3 are fixed by the DC agent (following the necessary received information from the DAM agent) at 0.4, 0.4 and 0.2, respectively. Also, it is assumed that the total time delay due to data communication and data processing is 6 s. The closed-loop system response, including frequency deviation (Δω ), tie-line power change (ΔPtie ), generation set-points (ui ) and area control error (ACE) are shown in Fig. 7.7. This figure shows that the changes in generator set-point commands are proportional to the value of their assigned participation factors. As a severe test scenario, the power system is examined in the presence of a sequence of larger step load changes. The load change pattern and the system responses are shown in Fig. 7.8. This figure shows that the frequency deviation and tie-line power change of control area are properly maintained within a narrow band using smooth control efforts. In this experiment, the participation factors for Gen 1 to Gen 4 have been already determined by the DC agent, according to the described method in Sect. 7.2.2, as 0.40, 0.25, 0.20 and 0.15, respectively. The obtained results show that the designed controllers can ensure good performance despite load disturbance and indeterminate delays in the communication network. It is shown that the proposed LFC system acts to maintain area frequency and total exchange power closed to the scheduled values, by sending corrective smooth signal to the generators in proportion to their participation in the LFC task. In order to demonstrate the effectiveness of the applied delay-based H∞ control technique on the LFC system performance, it is compared with the addressed conventional (delay-less) optimal H∞ tuning algorithm given in [23]. The real-time laboratory results of the LFC system with a conventional PI-based H∞ controller for the same test scenario are shown in Fig. 7.9. The experiment results (Figs. 7.8 and 7.9) show that the system performance using the proposed H∞ tuning method is significantly better than the conventional (delay-less) H∞ techniques.

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7.6 Remarks • For the sake of real-time laboratory simulation in the present and previous chapters, for simplicity, an infinite bus (bus 5) is used in simulating a power system for the frequency control testing. This assumption is acceptable for demonstrating the performance of proposed frequency controllers. However, it may be unsuitable for evaluating other aspects and services within an AGC system. • In the performed simulations for the proposed frequency controllers in this book, the performance of LFC systems is mainly examined with regards to the proper damping of area frequency deviation and tie-line power changes. Therefore, the slow dynamics and process of conventional AGC systems are not emphasized. Conventional AGC systems must act in accordance with the specified performance criteria of the market and the interconnections in which they operate. The cost of their actions should be considered. The available capacity of a Genco should not be the overall capacity remaining, but rather the capacity accessible within a reasonable window of time (from tens of seconds to a few minutes). In

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a conventional AGC system, to show reliable evidence that the controls properly respond to small fluctuating imbalances that characterize standard system operations, several days of operation data may be needed to be simulated and analysed. • In a deregulated market, in addition to technical aspects, the costs are known as an important issue. A desirable performance implies the provision of acceptable reliability as defined by the market, at minimum cost. In the deregulated electric market systems similar to the one described in this chapter, regulation is often priced separately from energy. Some markets pay for the right to regulate within a band; some pay for unit maneuvering directly, and some pay only for energy. However, if an AGC market imposes excessive uncompensated maneuvering on Gencos, they will simply raise their energy price to make up the losses. As such, regardless of market rules, or even the existence of a market, AGC must still attempt to minimize production costs and unit maneuvering.

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7.7 Summary A two-agent control scenario is proposed for the LFC problem in power systems under restructured market rules. Using simple algorithms, the agents measure/receive the required signals/data and estimate the total power imbalance and generator participation factors. The dynamic PI controllers are tuned using an LMI-based H∞ methodology, as proposed. The simplicity and flexibility of control structure, as well as retaining the fundamental LFC concepts, can be considered as advantages of the proposed methodology. The proposed method was applied to a control area power system using a laboratory real-time non-linear simulator.

References

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References 1. N. Jaleeli, D. N. Ewart and L. H. Fink, Understanding automatic generation control, IEEE Trans. Power Syst., 7(3), 1106–1112, 1992. 2. Ibraheem, P. Kumar and P. Kothari, Recent philosophies of automatic generation control strategies in power systems, IEEE Trans. Power Syst., 20(1), 346–357, 2005. 3. J. Kumar, N. K. Hoe and G. B. Sheble, AGC simulator for price-based operation Part 1: A model, IEEE Trans. Power Syst., 2(12), 527–532, 1997. 4. V. Donde, M. A. Pai and I. A. Hiskens, Simulation and optimization in an AGC system after deregulation, IEEE Trans. Power Syst., 3(16), 481–489, 2001. 5. H. Bevrani, Y. Mitani and K. Tsuji, Robust AGC: Traditional structure versus restructured scheme, Trans Electr. Eng. Jpn, 124(2), 751–761, 2004. 6. B. Delfino, F. Fornari and S. Massucco, Load–frequency control and inadvertent interchange evaluation in restructured power systems, IEE Proc. Gener. Transm. Distrib., 5(149), 607–614, 2002. 7. S. D. J. McArthur, E. M. Davidson, V. M. Catterson, A. L. Dimeas, N. D. Hatziargyriou, F. Ponci and T. Funabashi, Multi-agent systems for power engineering applications – Part I: Concepts, applications and technical challenges, IEEE Trans. Power Syst., 22(4), 1743–1752, 2007. 8. S. D. J. McArthur, E. M. Davidson, V. M. Catterson, A. L. Dimeas, N. D. Hatziargyriou, F. Ponci and T. Funabashi, Multi-agent systems for power engineering applications – Part II: Technologies, standards and tools for multi-agent systems, IEEE Trans. Power Syst., 22(4), 1753–1759, 2007. 9. NEMMCO, FCAS Constraints, NEMMCO, 2006. [Online]. Available: http://www.nemmco. com.au/ancillary services/160–0272.pdf. 10. P. M. Anderson and M. Mirheydar, A low-order system frequency response model, IEEE Trans. Power Syst., 5(3), 720–729, 1990. 11. P. M. Anderson and A. A. Fouad, Power System Control and Stability. Piscataway, NJ: IEEE, 1994. 12. P. W. Sauer and M. A. Pai, Power System Dynamics and Stability. Champaign, IL: Stipes, 2006. 13. K. R. Padiyar, Power Systems Dynamics: Stability and Control. New York, NY: Wiley, 1999. 14. P. Kundur, Power System Stability and Control. Englewood Cliffs, NJ: McGraw-Hill, 1994. 15. J. S. Thorp, X. Wang, K. M. Hopkinson, et al., Agent technology applied to the protection of power systems, Autonomous Systems and Intelligent Agents in Power System Control and Operation, C. Rehtanz, ed., pp. 113–154, Berlin: Springer, 2003. 16. B. H. Bakken and K. Uhlen: Market based AGC with online bidding of regulating reserves, In Proceedings of IEEE PES Summer Meeting, vol. 2, pp. 848–853, 2001. 17. V. Julian and V. Botti, Developing real-time multi-agent systems, Integrated Comput-Aided Eng., 11(2), 135–149, 2004. 18. Foundation for Intelligent Physical Agents (FIPA), Agent Management Specification, 2002. [Online]. Available: http://www.fipa.org/specs/fipa00023/SC00023J.html. 19. S. Bhowmik, K. Tomosovic and A. Bose, Communication models for third party load frequency control, IEEE Trans. Power Syst., 19(1), 543–548, 2004. 20. X. Yu and K. Tomosovic, Application of linear matrix inequalities for load frequency control with communication delays, IEEE Trans. Power Syst., 19(3), 1508–1515, 2004. 21. A. J. Wood and B. F. Wollenberg, Power Generation Operation and Control. NewYork: Wiley, 1984. 22. H. Bevrani and T. Hiyama, A control strategy for LFC design with communication delays, In Proceedings of the Seventh International Power Engineering Conference (IPEC), Singapore, 2005. 23. H. Bevrani, Y. Mitani and K. Tsuji, Robust decentralized load–frequency control using an iterative linear matrix inequalities algorithm, IEE Proc. Gener. Transm. Distrib., 150(3), 347–354, 2004.

Chapter 8

Application of Structured Singular Values in LFC Design

This chapter presents the application of structured singular value theory (μ synthesis and analysis) in load–frequency control (LFC) design. Two robust control methodologies are proposed in two sections. Section 8.1 describes a systematic approach to the design of sequential decentralized load–frequency controllers in a multi-area power system. System uncertainties, practical constraints on the control action and the desired performance are included in the synthesis procedure. Robust performance is used as a measure of control performance in terms of the structured singular value. A four-control area power system example is used to demonstrate the procedure of synthesis and the advantages of the proposed strategy. Section 8.2 addresses a robust decentralized control approach for LFC design in a multi-area power system. In this approach, the power system is considered as a collection of separate control areas in a deregulated environment. Each control area can buy electric power from available generation companies to supply its load. The control area is responsible for performing its own LFC by buying enough power from pre-specified generation companies that are equipped with robust load–frequency controllers. A three-control area power system example is given to illustrate the proposed control approach. The resulting controllers are shown to minimize the effect of disturbances and achieve acceptable frequency regulation in the presence of uncertainties and load variation.

8.1 Sequential Decentralized LFC Design Simultaneous design for a fixed control structure is used in all reported decentralized LFC scenarios, which is numerically difficult for large-scale power systems, and does not provide some of the advantages that are usually the reason for using decentralized control in the first place, such as the ability to bring the system into service by closing one loop at a time and the guarantee of stability and performance in the case of failures. In addition, some proposed methods might not work properly and do not guarantee performance when the operating points vary. H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 8, c Springer Science+Business Media LLC 2009

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In this section, based on structured singular value theory (μ ), a systematic approach to sequential decentralized LFC design in a multi-area power system is described. Because of the advantages it provides, the sequential control design is the most common design procedure in real applications of decentralized synthesis methods. Sequential design involves closing and tuning of one loop at a time. This method is less conservative than independent decentralized design because, at each design step one utilizes the information about the controller specified in the previous step [1], and it is more practical in comparison with common decentralized methods. After introducing the μ -based sequential control framework and pairing inputs and outputs, a single input single output (SISO) controller is designed for each loop (control area). In the LFC design for each control area, the structured singular value [2] is used as a synthesis tool and as a measure of performance robustness. This work shows that μ synthesis can be successfully used for the sequential design of multi-area power system load–frequency controllers that guarantee robust stability and robust performance for a wide range of operating conditions.

8.1.1 Model Description Recalling the simplified LFC model shown in Fig. 8.1 (for control area i), the statespace realization of area i (from m-control area power system) is given as follows: x˙i = Ai xi + Bi ui + Fi wi yi = Ci xi .

Fig. 8.1 Block diagram of control area-1

(8.1)

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Here, the state vector xi , control input ui , disturbance input wi and measured output yi are defined as follows: (8.2) xiT = Δ fi ΔPm i ΔPg i ΔPtie−i , ui = ΔPC i , yi = βi Δ fi + ΔPtie−i , (8.3) wi = ΔPd i .

(8.4)

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8.1.2 Synthesis Procedure 8.1.2.1 Methodology The main goal in each control area is to maintain the area frequency and tie-line power interchanges close to specified values in the presence of model uncertainties and disturbances. To achieve our objectives and to meet the μ synthesis requirements, the control area model can be modified as shown in Fig. 8.2. In comparison with Fig. 8.1, the inter-area connections are removed, and it is considered by ΔPtie−i that it is properly weighted by inter-area connecting coefficients, and is obtained from an integrator block. This figure shows the synthesis strategy for area i. It is noteworthy that for each control area, there are several uncertainties because of parameter variations, model linearization and unmodelled dynamics which are due to the approximation of the rest of the power system. Usually, the uncertainties in the power system can be modelled as multiplicative and/or additive uncertainties [3]. However, to keep the complexity of the controllers reasonably low, dependent on the given control area, it is better to focus on the most important

Fig. 8.2 Proposed strategy for LFC synthesis in area-i

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uncertainty. Sensitivity analysis of frequency stability due to parameter variation is a well-known method for this purpose. In Fig. 8.2, ΔU i models the structured uncertainty set in the form of a multiplicative type and WU i includes the associated weighting function. According to performance requirements and practical constraints on control actions, two fictitious uncertainties WP1i and WP2i are added to the control area model. The WP1i on the control input sets a limit on the allowed control signal to penalize fast change and large overshoot in the control action. The weight WP2i at the output sets the performance goal, for example, tracking/regulation error on the output deviation frequency. Furthermore, it is worth noting that in order to reject disturbances and to assure a good tracking property, WP1i and WP2i must be selected so that the singular value of sensitivity transfer function from ui to yi in the related area can be reduced at low frequencies [4]. ΔU i , Δp1i and Δp2i are the uncertainty blocks associated with WU i , WP1i and WP2i , respectively. The synthesis starts with setting the desired level of stability and performance for the first loop (control area) with a set of (ui , yi ) and chosen uncertainties to achieve robust performance. In order to maintain adequate performance in the face of tieline power variation and load disturbances, the appropriate weighting functions must be used. The inclusion of uncertainties adequately allows for maximum flexibility in designing the closed-loop characteristics, and the demands placed on the controller will increase. We can redraw Fig. 8.2 as shown in Fig. 8.3. g1i and g2i are transfer functions from the control input (ui ) and input disturbance (ΔPd i ) to the control output, respectively. Figure 8.4 shows M −Δ configuration for area i. Gi−1 includes the nominal model of area i, associated weighting functions and scaling factors. As previously mentioned, the blocks Δp1i and Δp2i are the fictitious uncertainties added to assure robust performance, while the block ΔUi models the important multiplicative uncertainty associated with the area model. Now, in step i, the synthesis problem is reduced to design the robust controller Ki . Based on the μ -synthesis, the robust performance holds for a given M–Δ configuration if and only if

Fig. 8.3 Synthesis framework for area-i

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inf sup μ [Mi (jω )] < 1,

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Fig. 8.5 Framework for μ -synthesis

When the controller synthesis has been completed, another robust controller is designed for the second control area with its set of variables and this procedure continues until all the areas are taken into account. During the design of each controller, the effects of previously designed controllers are being considered. The overall framework of the proposed control strategy is given in Fig. 8.5. It is noteworthy that the block G0 is assumed to contain the nominal open-loop model, the appropriate weighting functions and scaling factors according to Δ1 . The block Gm−1 includes G0 and all decentralized controllers K1 , K2 , . . ., Km−1 designed in previous iterations 1, 2, . . ., (m − 1) and related uncertainty blocks. Consider the nominal open-loop state-space representation of the power system as x˙ = Ax + Bu + Fw y = Cx,

(8.12)

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In each step, a μ controller is designed for one set of input and output variables. When this synthesis has been successfully completed, the next μ controller is designed for another set of input–output variables and so on. In every step, the effects of previously designed controllers are taken into account. Therefore, by adding one new loop at a time, the closed-loop system remains stable at each step.

8.1.3 Synthesis Steps To summarize, the proposed method consists of the following steps: Step 1. Identify the order of loop synthesis. The important problem with sequential design is that the final control performance achieved may depend on the order in which the controllers in the individual loops are synthesized. In order to overcome this problem, the fast loops must be closed first, because the loop gain and the phase in the bandwidth region of the fast loops are relatively insensitive to the tuning of the lower loops. In other words, for cases in which the bandwidths of the loops are quite different, the outer loops should be tuned such that the fast loops are contained in the inner loops. This causes a lower number of iterations during the re-tuning procedure to obtain the best possible performance [5]. Obtaining an estimation of the interactions on each control area behaviour to determine the effects of undesigned loops is the other important issue in the sequential synthesis procedure. Methods for determining the performance relative gain array (PRGA) and closed-loop disturbance gain (CLDG) which are given in [6] are useful for this purpose. Step 2. Identify the uncertainty blocks and associated weighting functions according to the first control area input–output set, dependent on the dynamic model, practical limits and performance requirements. There is no obligation to consider the uncertainty within only a few parameters. In order to consider a more complete model, the inclusion of additional uncertainties is possible and causes less conservatism in the synthesis. However, the complexity of computations and the order of obtained controllers will increase. Step 3. Isolate the uncertainties from the nominal area model, generate the Δp1i , Δp2i , ΔU i blocks and perform the M–Δ feedback configuration (formulate the desired stability and performance). Step 4. Start the D–K iteration using the μ synthesis toolbox [7] to obtain the optimal controller, which provides desirable robust performance such that max μ [M(jω )] < 1, ω ∈R

(8.14)

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Step 5. Reduce the order of the resulting controllers by utilizing the standard model reduction techniques, and then, apply μ analysis to the closed-loop system with reduced controller to check whether or not the upper bound of μ remains less than one. It is worth noting that the controller found by this procedure is usually of a high order. In order to decrease the complexity of computation, appropriate model reduction techniques might be applied both to the open-loop system model and to the H∞ controller model within each D–K iteration. Step 6. Continue this procedure by applying the above steps to other loops (control area input–output sets) according to the specified loop closing order in Step 1. Step 7. Retune the controllers which have been obtained to achieve the best performance and check if the overall power system satisfies the robust performance condition using μ analysis. If the objective is the achievement of the best possible performance, the controller that was designed first must be removed and then re-designed. However, this must now be done with controllers that have been synthesized in successive steps, because the first synthesis was according to the more conservative state. The proposed strategy guarantees robust performance for multi-area power systems when the design of load–frequency controllers is followed according to the above sequential steps. The advantage of the procedure by closing one loop for a special control area at a time ensures that this control area gets robust performance, while at the same time, the multi-area power system maintains its stability at each step. Similarly, during startup, the system will at least be stable if the loops are brought into service in the same order as they have been designed [6, 8].

8.1.4 Application Example 1 The proposed control approach is applied to a four-control area power system example shown in Fig. 8.6. The nominal parameter values are given in Table 8.1, [9–11]. The nominal state-space model for this system as a multi-input multi-output (MIMO) system can be constructed as (8.12). The matrices A ∈ R16×16 , B ∈ R16×4 and F ∈ R16×4 are given in Appendix B. The state variables and input/output vectors are considered as follows xT = Δ f1 ΔPm1 ΔPg1 ΔPtie−1 Δ f2 ΔPm2 ΔPg2 ΔPtie−2 (8.15) Δ f3 ΔPm3 ΔPg3 ΔPtie−3 Δ f4 ΔPm4 ΔPg4 ΔPtie−4 , uT = [u1 u2 u3 u4 ] , T

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The nominal open-loop MIMO system is stable. Simulation results show that the open-loop system performance is affected by changes in the equivalent inertia constants Mi and synchronizing coefficient Ti j , and these are more significant than changes of other parameters within a reasonable range. Eigenvalue analysis shows that the considerable change in these parameters leads to an unstable condition for the power system. Therefore, to demonstrate the capability of the proposed strategy for the problem at hand, from the viewpoint of uncertainty, our focus is concentrated on the variations of the Mi and Ti j parameters of all control areas, which are the most important parameters from a control viewpoint. Hence, for a given power system, LFC objectives have been set to area frequency regulation and assuring that robust stability and performance in the presence of specified uncertainties and load disturbances are as follows: 1. Holding stability and robust performance for the overall power system and each control area in the presence of 40% uncertainty for Mi and Ti j , which are assumed the sources of uncertainty associated with the given power system model. 2. Minimizing the effectiveness of step load disturbances (ΔPd i ) on output signals.

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3. Maintaining acceptable overshoot and settling time on the frequency deviation signal in each control area. 4. Set a reasonable limit on the control action signal in the change speed and amplitude view point. Next, the proposed strategy is separately applied to each control area of the given power system. Because of similarities and for brevity, the first controller synthesis is described in detail, whereas only the final results are shown for the other control areas. As the bandwidths of the four loops are similar, the order of closing the loops is not important in regard to the problem at hand. Therefore, the synthesis procedure is started with control area 1.

8.1.4.1 Uncertainty Weight Selection As mentioned, the specified uncertainty in each control area can be considered as a multiplicative uncertainty (WU i ) associated with the nominal model. Correspondˆ ing to an uncertain parameter, as described in Sect. 3.5.3, let the G(s) denotes the transfer function from the control input ui to the control output yi at operating points other than the nominal point. Following a practice common in robust control, this transfer function can be represented as ˆ = G0 (s)[1 + Δu (s)Wu (s)], G(s)

(8.19)

where Δu (s) shows the uncertainty block corresponding to the uncertain parameter, Wu (s) is the associated weighting function and G0 (s) is the nominal transfer function model. Then, the multiplicative uncertainty block can be expressed as ˆ − G0 (s)]G0 (s)−1 ; G0 (s) = 0. |Δu (s)Wu (s)| = [G(s) (8.20) Wu (s) is a fixed weighting function containing all the information available about the frequency distribution of the uncertainty, where Δu (s) is a stable transfer function representing the model uncertainty. Furthermore, without a loss of generality (by absorbing any scaling factor into Wu (s) where necessary), it can be assumed that Δu (s)∞ = sup |Δu (s)| ≤ 1. ω

(8.21)

Thus, Wu (s) is such that its respective magnitude Bode plot covers the Bode plot of all possible plants. Using (8.20), some sample uncertainties corresponding to the different values of Mi and Ti j are obtained and shown in Fig. 8.7. It can be seen that the frequency responses of both sets of parametric uncertainties are close to each other. Hence, to keep the complexity of the obtained control at a low level, the uncertainties due to parametric variations can be modelled by using a single, normbonded multiplicative uncertainty to cover all possible plants and this is obtained as follows 0.15(s2 + 0.004) . (8.22) WU1 (s) = 2 s + 0.1s + 18

151

102

102

100

100

10−2

10−2

10

Magnitude

Magnitude

8.1 Sequential Decentralized LFC Design

−4

10−6

10

10−4 10−6

−8

10−8

−10

10−10

10

10−12 0

1

2

3 4 5 6 7 8 Frequency (rad/sec)

9

10

Wu1

10−12 0

1

2

3 4 5 6 7 8 Frequency (rad/sec)

9

10

Fig. 8.7 Uncertainty plot due to change of (a) Hi ; (b) dotted (Ti j ) and solid (WU1 (s))

The frequency response of WU1 (s) is also shown in Fig. 8.7b. This figure clearly shows that attempting to cover the uncertainties at all frequencies and finding a tighter fit using higher order transfer functions will result in a high-order controller. The weight (8.22) used in our design provides a conservative design at low and high frequencies, but it gives a good trade off between robustness and controller complexity.

8.1.4.2 Performance Weight Selection As discussed in Sect. 8.1.2, in order to guarantee robust performance, adding a fictitious uncertainty block associated with the control area error minimization and control effort is required along with the corresponding performance weights WP11 and WP21 . Based on the following discussion, a suitable set of performance weighting functions that offer a good compromise among all the conflicting time-domain specifications for control area 1 is as follows WP11 (s) =

0.5s s + 0.75 ,WP21 (s) = . 0.01s + 1 150s + 1

(8.23)

The selection of WP11 and WP21 entails a trade off among the different performance requirements. The weight on the control input WP11 is chosen close to a differentiator to penalize fast change and large overshoot in the control input. The weights on output error WP21 are chosen close to an integrator at low frequencies in order to get disturbance rejection, good tracking and zero steady-state error. Additionally, as pointed out in Sect. 8.1.4.1, the order of the selected weights should be kept low in order to keep the controller complexity low. Finally, it is well known that to reject disturbances and to track command signal properties, it is necessary for the singular value of sensitivity function to be reduced at low frequencies, and WP11 and WP21 must be selected to satisfy this condition [12]. Our next task is to isolate the uncertainties from the nominal plant model and redraw the system in the standard M–Δ configuration (Fig. 8.8).

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Fig. 8.8 Standard M–Δ block

By using the uncertainty description and already developed performance weights, an uncertainty structure Δ, with a scalar block (corresponding to the uncertainty) and a 2 × 2 block (corresponding to the performance) is obtained. Having setup our robust synthesis problem in terms of the structured singular value theory, the μ -analysis and synthesis toolbox [7] is used to achieve a solution. The controller K1 (s) is found at the end of three D–K iterations, yielding the value of about 0.893 on the upper bound on μ , thus guaranteeing robust performance. Since the resulting controller has a high order (21st), it is reduced to a fourth order with no performance degradation (μ < 0.998), using the standard Hankel Norm approximation. The Bode plots of the full-order and the reduced-order controllers are shown in Fig. 8.9. The transfer function of the reduced order controller is given as K1 (s) = N1 (s)/D1 (s) with N1 (s) = 6.3905s3 + 0.10604s2 + 44.3998s + 37.994 D1 (s) = s4 + 18.9617s3 + 182.1594s2 + 739.3578s + 0.7393.

(8.24)

Using the same procedure and setting similar objectives, as already discussed, achieves a set of suitable weighting functions for the remaining loop synthesis (Appendix B). The order of the other obtained controllers without model reduction was 29 (K2 ), 37 (K3 ) and 45 (K4 ). These controllers can be approximated by lower order controllers which are given in (Table B.1 in Appendix B).

8.1.5 Simulation Results The proposed control scenario was applied to the four-control area power system as shown in Fig. 8.6. To test the system performance, a step load disturbance of

8.1 Sequential Decentralized LFC Design

153

Fig. 8.9 Bode plots comparison of full-order (original) and reduced-order controller K1 (s)

ΔPd i = 0.01 pu is applied to each control area, using the nominal plant parameters and those with uncertainty parameters by different percentage uncertainties. Since the system parameters for the given four control areas are identical and the ΔPtie between the two neighboring areas k and j is caused by Δ fk − Δ f j , the system performance can be mainly tested by applying the disturbance ΔPd i in the presence of the parameters uncertainties and observing the time response of Δ fi in each control area. Some selected time response simulation results are given in Figs. 8.10 and 8.11. Figure 8.10 shows the frequency deviation and the control action signal in control areas 1 and 2, following the simultaneous step load disturbances of ΔPd1 = 0.01 pu and ΔPd2 = 0.01 pu. Figure 8.11 shows the simultaneous power system response for a step load disturbance in each area and a 40% decrease in uncertain parameters. These simulation results demonstrate the effectiveness of the proposed strategy to provide robust frequency regulation in multi-area power systems. Because of our tight design objectives which include the consideration of several simultaneous uncertainties and input disturbances, the order of the resulting robust controllers is relatively high. However, the proposed method performs well from the aspect of disturbance rejection and frequency error minimization in the presence of model uncertainties.

154

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8 Application of Structured Singular Values in LFC Design

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Frequency deviation (Hz)

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0

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Δf1

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−0.04 0

Δf4

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8

10 12 Time (sec)

14

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Fig. 8.11 Frequency deviation in the presence of 40% changes in Hi and Ti j , and a step load disturbance of 0.01 pu in all areas

8.2 A Decentralized LFC Design This section addresses a decentralized design of robust load–frequency controllers based on the structured singular value theory for multi-area power systems. The power system is considered as a collection of control areas interconnected through

8.2 A Decentralized LFC Design

155

high voltage transmission lines or tie-lines. Each control area has its own load– frequency controller and is responsible for tracking its own load and honouring tieline power exchange contracts with its neighbors. The proposed strategy is applied to a three-control area example. The results obtained show that the controllers which have been designed guarantee robust stability and robust performance for a wide range of operating conditions.

8.2.1 Synthesis Methodology The general scheme of the proposed control system for a given area is shown in Fig. 8.12. βi and λ Pi are properly setup coefficients of the secondary regulator. The robust controller acts to maintain area frequency and total exchange power close to the scheduled value by sending a corrective signal to the assigned Gencos. This signal, which is weighted by the ACE participation factor αi j , is used to modify the set points of generators. Consider the state-space model (8.12) and analogously to the traditional area control error, let the output system variable be defined as follows y = Cx + Ew,

(8.25)

where C = C1 C2 · · · CN CN+1 ,Ci = βi 0 0 ,CN+1 = [1]1×N ; and

E = 1 θ , wT = [ΔPL d] .

i = 1, 2, . . . , N (8.26)

θ is a zero vector with the same size as the disturbance vector (d) and the ΔPL is the contracted load change. To achieve the LFC objectives in accordance with the structured singular value theory requirements, a control strategy applicable for each control area is proposed as shown in Fig. 8.13. ΔU models the structured uncertainty set in the form of multiplicative type and WU includes the associated weighting function(s).

Fig. 8.12 General scheme for the proposed control system

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8 Application of Structured Singular Values in LFC Design

Three fictitious uncertainties WP1 , WP2 and WP3 are added to the power system model. The WP1 on the control input sets a limit on the control signal according to the existing practical constraint. This is necessary to guarantee the feasibility of the proposed controller. At the output, the weights WP2 and WP3 set the performance goal, for example, tracking/regulation of the output area control signal. ΔP is a diagonal matrix that includes the uncertainty blocks Δp1 , Δp2 and Δp3 associated with WP1 , WP2 and WP3 , respectively. Figure 8.13 can be redrawn as a standard M–Δ configuration, which is shown in Fig. 8.14. G includes the nominal model of the given control area, associated weighting functions and scaling factors. The block labelled M, consists of G and controller K. Based on the μ synthesis, robust stability and performance will be satisfied for a given M − Δ configuration, if and only if

Fig. 8.13 The synthesis framework

Fig. 8.14 M–Δ configuration

8.2 A Decentralized LFC Design

157

inf sup μ [M(jω )] < 1. K ω ∈R

(8.27)

The well-known upper bound for μ can be determined by using (8.10) or (8.11). To summarize, the proposed method for each control area consists of the following steps: Step 1. Identify the uncertainty blocks and associated weighting functions for the given control area, according to the dynamic model, the practical limits and required performance requirements as shown in Fig. 8.13. Step 2. Isolate the uncertainties from the nominal control area model, generate Δp1 , Δp2 , Δp3 and ΔU blocks and perform M–Δ feedback configuration (formulate the robust stability and performance). Step 3. Start the D–K iteration by using μ synthesis toolbox [7], in order to obtain the optimal controller. Step 4. Reduce the order of the resulting controller by utilizing the standard model reduction techniques and apply μ -analysis to the closed-loop system with the reduced controller to check whether or not the upper bound of μ remains less than one. The proposed strategy guarantees robust performance and robust stability for the closed-loop system.

8.2.2 Application Example 2 A sample power system with three control areas is shown in Fig. 8.15. Each control area has some Gencos and each Genco is considered as a generator unit (Gunit). Here, it is assumed that one generator unit with enough capacity is responsible to regulate the area–load frequency. A control area may have a contract with a Genco in the other control area. For example, control area 3 buys power from G11 in control area 1 to supply its load. The power system data is given in Appendix B (Table B.2). Next, the synthesis procedure in control area 1 is described in detail, and just the final results are presented for other two areas. Assuming the output as shown in (8.25), the state-space model of control area 1 can be obtained in the form of (8.12).

8.2.2.1 Design Objectives Control area 1 delivers enough power from G11 and firm power from other Gencos to supply its load and support the LFC task. In case of a load disturbance, G11 must adjust its output to track the load changes and maintain the energy balance. Simulation results show that the open-loop system performance is affected by individual changes of H1 and H3 (inertia constants), which are more significant than changes to the other parameters of the control areas within a reasonable range. Eigenvalue analysis shows that a considerable change in these parameters leads the

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Fig. 8.15 Three-control area power system

power system to an unstable condition. In this case, therefore, from the aspect of uncertainty, our focus is concentrated on the variations of H1 and H3 parameters, these are the sources of uncertainty associated with the control area model and important parameters from the aspect of control issue. The objectives are considered as follows for the control area 1: 1. Hold robust stability and robust performance in the presence of 75% uncertainty for H1 and H3 (This variation range leads the control area to an unstable condition) 2. Hold robust stability and desired reference tracking for a 10% demand load change in control area (0 ≤ ΔPL (%) ≤ 10) 3. Minimize the impacts of step disturbance from outside areas (d) through the L12 and L13 4. Maintain acceptable frequency amplitude changes and zero steady-state error 5. Set a reasonable limit on the control action signal with regard to changes in speed and amplitude

8.2.2.2 Uncertainty Weight Selection The related uncertainty weighting function in each control area is easily determined using the method described in the Sect. 8.1.4.1. Some sample uncertainties which correspond to different values of H1 and H3 are shown in Fig. 8.16. This figure shows that the frequency response of parametric uncertainties is close to each other. Hence to keep the complexity of the obtained controller at a low level, and to cover

8.2 A Decentralized LFC Design

159

Fig. 8.16 Uncertainty plot due to change of H1 (dotted) and H3 (solid)

all possible plants, the uncertainties due to H1 and H3 variation can be modelled by using a single norm-bonded multiplicative uncertainty as follows. (The frequency response of WU (s) is also shown in Fig. 8.16) WU (s) =

−10(s + 0.04) . s + 15

(8.28)

8.2.2.3 Performance Weight Selection The performance weight selection in a μ -based LFC synthesis is explained in the Sect. 8.1.4.2. Here, the weight on the control input WP1 is chosen to penalize fast change and large overshoot in the control input. The weights on the input disturbance from other areas (WP3 ) and output error (WP2 ) are chosen to get disturbance rejection, good tracking and zero steady-state error. In order to reject disturbances and track the command signal property, it is required that the singular value of sensitivity function be reduced at low frequencies. WP2 and WP3 must be selected such that this condition is satisfied. For the problem at hand, a suitable set of performance weighting functions that offers a good compromise among all the conflicting timedomain specifications is WP1 (s) =

0.1s 0.005s + 1 0.9s + 0.9 , WP2 (s) = , WP3 (s) = . 0.01s + 1 35.7s + 0.04 100s + 1

(8.29)

The next task is to isolate the uncertainties from the nominal plant model and redraw the system in the standard M–Δ configuration. Using the uncertainty description and

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8 Application of Structured Singular Values in LFC Design

Fig. 8.17 Bode plots comparison of the original and reduced-order controllers

performance weights that have been developed, an uncertainty structure Δ with a scalar block (corresponding to the uncertainty) and a 3 × 3 block (corresponding to the performance) is obtained. The controller K1 (s) is found at the end of the third D–K iteration, yielding the value of 0.994 on the upper bound on μ , thus guaranteeing robust performance. The resulting controller has a high order (29th). Using the standard Hankel Norm approximation, it is reduced to a seventh order with no performance degradation. The Bode plots of the full-order controller and the reduced-order controller are shown in Fig. 8.17. The transfer function of the reduced order controller is given as K1 (s) = N1 (s)/D1 (s) with N1 (s) = 226.28s6 + 23,024.16s5 + 20,719s4 + 153,700s3 + 245,730s2 + 162,930s + 844 D1 (s) = s7 + 3,240s6 + 70,777s5 + 710,490s4 + 362,130s3

(8.30)

+ 3,853, 000s2 + 24,901s + 21. Using the same procedure and setting similar objectives, as already discussed, gives us the desired robust load–frequency controllers for control areas 2 and 3. The associated polynomials with K2 (s) and K3 (s) are given in Appendix B.

8.3 Summary

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0.04 0.02 0

25

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20

25

Fig. 8.18 Area 1 response, following a 10% load increase; (a) frequency deviation and (b) power change

8.2.3 Simulation Results The proposed load–frequency controllers are applied to a three-control area power system described in Fig. 8.15. Figure 8.18a shows the frequency deviation in control area 1, following a 10% increase in the area–load. Δ f11 , Δ f12 , Δ f13 and Δ f14 display the frequency deviation at Gencos G11 , G12 , G13 and G14 , respectively. At steady state, the frequency in each control area reaches its nominal value. Figure 8.18b shows the changes in power which come to control area 1 from its Gencos. It is seen that the power is initially coming from all Gencos to respond to the load increase and will result in a frequency drop that is sensed by the speed governors of all machines. After a few seconds, however, and at a steady state, the additional power comes only from G11 and the other Gencos do not contribute to the LFC task. Figure 8.19 demonstrates the disturbance rejection property of the closed-loop system. This figure shows the frequency deviation at generation units in control area 1, following a step disturbance of 0.1 pu on area interconnection lines L12 and L13 at t = 17 s. The power system is already started with a 10% load increase in each area. Figure 8.20 shows the frequency deviation in control area 2 and 3, following a 10% load increase in each control area. As another simulation case, the random demand load signal shown in Fig. 8.21a, which represents the expected area demand load fluctuations, is applied to control area 1. The frequency deviations are shown in Fig. 8.21b, c. Power changes and control signals are given in Fig. 8.21d, e. These figures show that the controller tracks the load fluctuations effectively.

8.3 Summary A systematic method for sequential μ -based LFC design in a multi-area power system is proposed. At each design step, the information about the controllers which have been designed in the previous steps is taken into account. Therefore,

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8 Application of Structured Singular Values in LFC Design

0 −0.02

Δf (Hz)

−0.04 −0.06 −0.08

Δf11 Δf12

−0.1

Δf13 Δf14

−0.12

5

10

15

20 25 Time (sec)

30

35

40

Fig. 8.19 Frequency deviation in control area 1, following a 0.01-pu step disturbance on interconnection lines at t = 17 s and 10% load increase at t = 0 s

b

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−0.08

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−0.16 20

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Fig. 8.20 Frequency deviation at Gencos in (a) control area 2, (b) control area 3, following a 10% load increase in each area

the method is less conservative than independent LFC design and more practical than proposed simultaneous decentralized load–frequency controller designs. In Sect. 8.2, a robust decentralized LFC synthesis is addressed by using structured singular value theory. The proposed methods are examined on multi-area power system examples. It was shown that the controllers that have been designed will guarantee robust stability and robust performance under a wide range of parameter variation and area–load conditions.

References

163

a ΔPL1 (pu)

0.2 0 −0.2 0

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Fig. 8.21 System response to random demand; (a) Demand load, (b) Δ f11 , Δ f12 , (c)Δ f13 , Δ f14 , (d) power change at G11 and (e) control effort

References 1. M. Chiu and Y. Arkun, A methodology for sequential design of robust decentralized control systems, Automatica, 28, 997–1001, 1992. 2. J. C. Doyle, Analysis of feedback systems with structured uncertainties, IEE Proc., Pt. D, 129, 242–250, 1982. 3. M. Djukanovic, M. Khammash and V. Vittal, Structured singular value theory based stability robustness of power systems, in Proceedings of IEEE Conference on Decision and Control, pp. 2702–2707, 1997.

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4. H. Bevrani, Robust load frequency controller in a deregulated environment: A μ -synthesis approach, in Proceedings of IEEE International Conference on Control applications, pp. 616–621, 1999. 5. S. Skogestad and I. Postlethwaite, Multivariable Feedback Control, Wiley, New York, NY, pp. 397–448, 2000. 6. M. Hovd and S. Skogestad, Sequential design of decentralised controllers, Automatica, 30, 1610–1607, 1994. 7. G. J. Balas, J. C. Doyle, K. Glover, A. Packard and R. Smith, μ -Analysis and Synthesis Toolbox for Use with MATLAB, The MathWorks, Natick, MA, 1995. 8. M. Djukanovic, M. Khammash and V. Vittal, Sequential synthesis of structured singular value based decentralized controllers in power systems, IEEE Trans. Power Syst., 14, 635–641, 1999. 9. T. Hiyama, Design of decentralised load–frequency regulators for interconnected power systems, IEE Proc. Pt. C, 129, 17–23, 1982. 10. T. C. Yang, H. Cimen and Q. M. Zhu, Decentralised load frequency controller design based on structured singular values, IEE Proc. Gener. Transm. Distrib., 145(1), 7–14, 1998. 11. T. C. Yang, Z. T. Ding and H. Yu, Decentralized power system load frequency control beyond the limit of diagonal dominance, Electr. Power Energy Syst., 24, 173–184, 2002. 12. H. Bevrani, Y. Mitani and K. Tsuji, Robust load frequency regulation in a new distributed generation environment, in Proceedings of 2003 IEEE-PES General Meeting (CD Record), Toronto, Canada, 2003.

Chapter 9

Frequency Control in Emergency Conditions

Following a large generation loss disturbance, a power system’s frequency may drop quickly if the remaining generation no longer matches the load demand. Significant loss of generating the plant without adequate system response can produce extreme frequency excursions outside the working range of plant. As mentioned in Chaps. 1 and 2, large frequency deviations can degrade load performance, overload transmission lines and even lead to system collapse. Depending on the size of the frequency deviation experienced, emergency control and protection schemes may be required to maintain power system frequency. One method of characterizing frequency deviations experienced by a power system is in terms of the frequency deviation ranges and related control actions shown in Table 2.3. The frequency variation ranges (Δ f1 , Δ f2 , Δ f3 and Δ f4 ), shown in the table, are identified in terms of different power system operating conditions. The small frequency deviations (Δ f1 and Δ f2 ) can be attenuated by the governor natural autonomous response (primary control) and LFC system. For larger frequency deviations and in a more complex condition (such as Δ f3 and Δ f4 frequency deviation events) the emergency control and protection schemes such as under-frequency load shedding must be used to restore the system frequency. Many frequency response models and control scenarios have attempted to address such contingency conditions using emergency control strategies over the years. A centralized emergency control approach is used in many of the proposed emergency control schemes.

9.1 Frequency Response Model In response to a large disturbance, a rapid change in generation–load would be required to initially arrest the decline in system frequency, and to restore the frequency to the nominal level. For a large change in system frequency, such as might arise from a multiple contingency events, the combined response of the primary and supplementary controls may be insufficient and unreliable. Additional contingency H. Bevrani, Robust Power System Frequency Control, Power Electronics and Power Systems, DOI 10.1007/978-0-387-84878-5 9, c Springer Science+Business Media LLC 2009

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9 Frequency Control in Emergency Conditions

ancillary services or emergency control plans may be required to avoid market failure and to stop further frequency decline, to the point where generating units are beyond their reliable operating limits.

9.1.1 Modelling As described in Chap. 2, for the sake of dynamic frequency analysis in the presence of sudden load changes, it is usual to model the multi-machine dynamic behaviour by an equivalent single machine. Using the concept of an equivalent single machine [1,2], we can simplify the control area block diagram (Fig. 2.5) as shown in Fig. 9.1. Here, ΔPL , Rsys , and Msys (s) are the system load change, drooping characteristic and governor–turbine dynamic model, respectively. The system frequency deviation Δ f , equivalent inertia H and equivalent load damping coefficient D are defined as follows [3–5] & N

Δ f = Δ fsys = ∑ (Hi Δ fi ) i=1

N

N

N

i=1

i=1

i=1

∑ Hi , H = Hsys = ∑ Hi , D = Dsys = ∑ Di .

(9.1)

The supplementary control (LFC) dynamic is slower than emergency control dynamics, and the LFC response is not usually taken into account in emergency control schemes and their analysis (ΔPC = 0). According to Fig. 9.1, one can write Δ f (s) =

1 [ΔPm (s) − ΔPL (s)] 2Hs + D

(9.2)

or taking the inverse Laplace transform ΔPm (t) − ΔPL (t) = ΔPD (t) = 2H

dΔ f (t) + DΔ f (t). dt

(9.3)

ΔPD (t) shows the load–generation imbalance proportional to the total load change. The magnitude of total load–generation imbalance immediately after the occurrence of disturbance at t = 0+ s can be expressed as follows

Fig. 9.1 Simplified frequency response model

9.1 Frequency Response Model

167

ΔPD = 2H

dΔ f . dt

(9.4)

The “Δ f ” is the frequency of the equivalent system. To express the result into a form suitable for sampled data, (9.4) can be represented in the following difference equation: 2H [Δ f1 − Δ f0 ], (9.5) ΔPD (TS ) = TS where TS is the sampling period. Δ f1 and Δ f0 are the system equivalent frequencies at t0 and t1 (the boundary samples within the assumed interval). Equation (9.4) shows that the frequency gradient in a power system is proportional to the magnitude of total load–generation imbalance. The factor of proportionality is the system inertia. In fact, the inertia constant is loosely defined by the mass of all the synchronous rotating generators and motors connected to the system. For a specific load decrease, if H is high, then the frequency will fall slowly and if H is low, then the frequency will fall faster. Similarly, it is easy to find a relationship between frequency gradient, system load damping (D), and total load change (ΔPL ). With no speed governing at t = 0+ s (ΔPm = 0), (9.2) is reduced to Δ f (s) =

−ΔPL (s) . 2Hs + D

(9.6)

For a step change in load by ΔPL , Laplace transform of the load change is ΔPL s

(9.7)

−(ΔPL /D) . s[1 + (2H/D)s]

(9.8)

ΔPL (s) = and, from (9.6) Δ f (s) = Taking the inverse Laplace transform:

t 2HΔPL − (2H/D) ΔPL Δ f (t) = − + e . 2 D D Hence, the frequency gradient at t = 0+ s is proportional to ΔPL /D. ΔPL dΔ f (t) ∼ . =− dt t=0+ D

(9.9)

(9.10)

The results given in (9.4) and (9.10) are helpful as a first step in establishing the basis for emergency control plans such as load-shedding schemes.

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9 Frequency Control in Emergency Conditions

9.1.2 Considering of Emergency Control/Protection Dynamics As mentioned, in the case of a large generation loss disturbance, the scheduled power reserve may not be enough to restore the system frequency and the power system operators may follow an emergency control plan such as under-frequency load shedding (UFLS). The UFLS strategy is designed so as to rapidly balance the demand of electricity with the supply and to avoid a rapidly cascading power system failure. Allowing normal frequency variations within expanded limits will require the coordination of primary control and scheduled reserves with generator load set points; for example under-frequency generation trip (UFGT), over-frequency generation trips (OFGT), or over-frequency generator shedding (OFGS) and other frequency-controlled protection devices. The conventional frequency response model shown in Fig. 9.1 gives the free response of the primary control system following a contingency. In the case of contingency analysis, the emergency protection and control dynamics must be adequately modelled in the frequency response model. Since they influence the power generation–load balance, the mentioned emergency control dynamics can be directly included to the system frequency response model. This is made by adding an emergency control/protection block to the block diagram of Fig. 9.1 as shown in Fig. 9.2. The ΔPUFLS (s), PUFGT (s) and ΔPOFGT (s) represent the dynamics effects of the UFLS, UFGT and OFGT actions, respectively. In order to cover the variety of generation unit types in a power system, different governor–turbine models Mi (s) are considered.

Fig. 9.2 The frequency response model considering the emergency control/protection dynamics

9.1 Frequency Response Model

169

Fig. 9.3 L-step UFLS scheme

The emergency control schemes and protection devices dynamics are usually represented using incremented/decremented step behaviour. Thus, in Fig. 9.2, for simplicity, the related blocks can be represented as a sum of incremental (decremental) step functions. For instance, as shown in Fig. 9.3, for a fixed UFLS scheme [6], the function of ΔPUFLS in the time domain could be considered as a sum of the incremental step functions of ΔPj u(t − t j ). Therefore, for L load-shedding steps: ΔPUFLS (t) =

L

∑ ΔPj u(t − t j ),

(9.11)

j=0

where ΔPj and t j denote the incremental amount of load shed and time instant of the jth load-shedding step, respectively. Similarly, to formulate the ΔPOFGT , ΔPUFGT and other emergency control schemes, appropriate step functions can be used. Therefore, using the Laplace transformation, it is possible to represent ΔPEC (s) in the following summarized form N ΔPl −tl s e , (9.12) ΔPEC (s) = ∑ l=0 s where ΔPl is the size of equivalent step load/power changes due to a generation–load event or a load-shedding scheme at tl .

9.1.3 Simulation Example A two interconnected control areas is considered as study system to support the addressed analytic discussion. The study system simulates an actual LFC scheme [7, 8]. The configuration of the study system is shown in Fig. 9.4. Area I and Area II are interconnected through a 500 kV tie-line. Area I consists of four sub-areas A, B, C and D. The sub-areas A, B, C and D have eight, five, seven and three thermal units, respectively. Sub-area A and B are responsible to maintain the frequency close

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9 Frequency Control in Emergency Conditions

Fig. 9.4 Two-control area power system

to specified value among the Area I. Here, we have mainly focused on dynamic characteristics and frequency response of Area I. Five and three LFC units are there in the sub-areas A and B, respectively. The participation factors for the participated LFC units in these sub-areas are given in Table 9.1. The LFC system and power system parameters are considered similar to the practical system, which is described in [7, 8]. The power generation from the nuclear units (in the sub-areas D and C), from the hydro-units (in the sub-area D) and also from the other utility units are not included

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171

Table 9.1 LFC participation factors Sub-area

A

PL, A (Mw)

Generation unit Participation factor

PL, B (Mw) Δ Ptie (Mw) Δ PS, A (pu)

Gen 2 0.0676

Gen 3 0.1689

Gen 4 0.1689

Gen 5 0.1689

Gen 9 0.0998

Gen 10 0.6603

Gen 11 0.2399

6400 6200 0

Δ PS, B (pu)

Gen 1 0.4257

B

100

200

300

400

500

600

700

800

100

200

300

400

500

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700

800

100

200

300

400

500

600

700

800

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400 Time (sec)

500

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2600 2400 0 200 0 −200 0 0.02 0.01 0 −0.01 0 0.02 0.01 0 −0.01 0

Fig. 9.5 Tie-line power fluctuation and supplementary control actions following a ramp and a random load changes in sub-areas A and B

in the simulation model because they are not in use for the frequency regulation purpose. In the non-linear simulations, the generation rate constraints, dead bonds and time delays are considered. It is assumed that the maximum LFC reserved power in Area I to track the area power imbalance is fixed at 1,000 MW. For the first scenario, to evaluate the efficiency of the LFC system during the non-linear simulations, a random load change in the sub-area A and a ramp load change of 200 MW for 100 s in sub-area B are considered. The frequency response and the LFC system performance are shown in Figs. 9.5 and 9.6. Figure 9.5 shows the applied load change patterns in sub-areas A and B, tie-line power change between Areas I and II, and LFC control signals in sub-areas A and B. The relatively large fluctuation of the tie-line power (nearly 200 MW) is mainly caused by the random load change in the sub-area A.

Δ fEX (Hz)

Δ fD (Hz)

Δ fC (Hz)

Δ fB (Hz)

Δ fA (Hz)

172

9 Frequency Control in Emergency Conditions 0.05 0 −0.05 −0.1 0 0.05 0 −0.05 −0.1 0 0.05 0 −0.05 −0.1 0 0.05 0 −0.05 −0.1 0 0.05 0 −0.05 −0.1 0

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Fig. 9.6 Frequency deviations following a ramp and a random load changes in sub-areas A and B; with supplementary control (solid), and only primary control (dotted)

Frequency deviations in different locations of system are shown in Fig. 9.6. To clarify the impact of the LFC control action, the system response with and without supplementary control is examined. The dotted line in Fig. 9.6 shows the pure governors response following the applied disturbances. In steady state, the frequency deviation (Δ fss ) reaches the value given by (7.2). Using the supplementary control loop, since the frequency deviation remained within permitted range, and the available LFC power reserve could match the power demand, the system recovered within tens of seconds to few minutes. As second test scenario, the system frequency response is tested following a step loss of generation 500 MW in sub-area D (Fig. 9.7a). The frequency deviation and the corresponding frequency gradient for all sub-areas (in Area I) and the external area are shown in Fig. 9.7b, c. The highest frequency rate change occurs in sub-area D. Recalling (9.4), this behaviour is easily understandable. The rate of frequency change is proportional to the power imbalance, and it also depends on the area system inertia. From Fig. 9.7, it can be concluded that the disturbance location affects the frequency behaviour of power systems and consequently the design and the selection of a suitable emergency control plan. As mentioned, the supplementary (LFC) control is responsible for maintaining of system frequency by using the available instantaneous reserve to raise the frequency

9.1 Frequency Response Model

173

a PL (Mw)

5500

D

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4500 48

D f (Hz)

b

49

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51 52 Time (sec)

53

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0.02 0

−0.02 −0.04

48

dD f/dt (Hz/s)

c

Initial rate of frequency change

0.2

0

−0.2 48

Fig. 9.7 System response following a step load change in sub-area D at 50 s; (a) load disturbance, (b) frequency deviation, (c) frequency gradient; sub-area A (dotted), sub-area B (line-dotted), subarea C (dashed), sub-area D (bold solid), Area II (solid)

back to the nominal level. In case of a large enough disturbance, or an insufficient reserved power, the frequency (in steady state) may stay in an off-normal condition. As a sever test scenario, consider the system frequency response following a large load disturbance of 3,000 MW in sub-area A (Area I). Here, the total area load demand is higher than the available LFC reserve. The normal and LFC controls are not able to maintain the frequency at the nominal value and the frequency may pass the specified threshold frequency ft . Figure 9.8 shows the simulation results for this case. In this case, the system is in an emergency condition and a suitable load disconnection (load shedding) procedure is needed to recover the system frequency. Considering the applied disturbance (Fig. 9.8a) and the overall area load disturbance magnitude (9.4), assume that a three-staged load-shedding plan is investigated and started at 70 s, as shown in Fig. 9.9. The loads 0.15 pu, 0.1 pu and 0.05 pu are disconnected at 70, 80, and 85 s, respectively (here, MW base is 1,000 MW) ΔPUFLS (t) = 0.15u(t − 70) + 0.1u(t − 80) + 0.05u(t − 85).

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9 Frequency Control in Emergency Conditions

PLA (Mw)

a 10000 8000 6000 45

50

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0 −0.2 −0.4 −0.6 −0.8 45

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0.5 0 −0.5 −1 45

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75

80

85

90

Δ f (Hz)

b

dΔ f/dt (Hz/s)

c

Δ Ptie (Mw)

d 0 −2000 −4000 45

Fig. 9.8 System response for insufficient frequency regulation following a large load disturbance; (a) load disturbance, (b) frequency deviation in all sub-areas of Area I, (c) rate of frequency deviation in all sub-areas of Area I and (d) tie-line power change

9.2 Under-Frequency Load Shedding 9.2.1 Why Load Shedding? Load shedding is an emergency control action designed to ensure system stability by curtailing system load to match generation supply. The emergency load shedding would only be used if the frequency falls below a specified frequency threshold. Typically, load-shedding protects against excessive frequency or voltage decline by attempting to balance real and reactive power supply and demand in the system. Most common load-shedding schemes are the UFLS schemes, which involve shedding pre-determined amounts of load if the frequency drops below specified frequency thresholds. Under-voltage load-shedding (UVLS) schemes, in a similar manner, are used to protect against excessive voltage decline. It is noteworthy that some limitations affect the generator prime mover abilities to recover the system frequency following a significant disturbance, so that the available real generation reserve for frequency regulation can be less than the remaining total generation. In addition to the existing limit on the ability of a boiler to pick up a significant amount of load, the speed governors of thermal generating units have a

9.2 Under-Frequency Load Shedding

a

175

Δ PUFLS (pu)

0.4 0.2

Δ f (Hz)

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0 45

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dΔf/dt (Hz/s)

c

0.5 0 −0.5 45

Δ Ptie (Mw)

d 2000 0 −2000 −4000 45

Fig. 9.9 (a) Load-shedding plan, (b) frequency deviation in all sub-areas of Area I, (c) rate of frequency deviation in all sub-areas of Area I and (d) tie-line power change

time delay of 3–5 s. Furthermore, the generation can be increased only to the limits of available spinning reserve within each affected control area [1]. For the above-mentioned reason, in many situations after severe system disturbances, the UFLS schemes are employed to prevent extended operation of tripping of generating units by under-frequency protective relays (Fig. 2.14). UFLS schemes reduce the connected load to a level that can be safely supplied by real available generation. A UFLS scheme is usually composed of several stages. Each stage is characterized by frequency threshold, amount of load and delay before tripping. The objective of an effective load-shedding scheme is to curtail a minimum amount of load, and provide a quick, smooth and safe transition of the system from an emergency situation to a normal equilibrium state [9, 10].

9.2.2 A Brief Literature Review on UFLS There are various types of UFLS schemes discussed in literature and applied by the electric utilities around the world. A classification divides the existing schemes into static and dynamic (or fixed and adaptive) UFLS types. Static load shedding curtails

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9 Frequency Control in Emergency Conditions

the constant block of load at each stage, while dynamic load shedding curtails a dynamic amount of load by taking into account the magnitude of disturbance and frequency characteristics of the system at each stage. A static UFLS usually disconnects more or less load than is required [11]. Although the dynamic load-shedding schemes are more flexible and have several advantages, most real-world UFLS plans are of static type [9]. Several dynamic or adaptive UFLS are proposed over many years [6,12–21]. Recently, the rate of frequency change has been used as an additional control variable to improve UFLS by many researchers [12–14, 18–28]. Some of these methodologies are given for setting the under-frequency relays, based on the initial rate of frequency change at the relays. Various modified UFLS schemes have been promoted in support of improved protection, including (1) adaptive UFLS schemes that utilized both local frequency and frequency rate information, (2) dynamic UFLS schemes that dynamically adjust the size of load shed stages and (3) optimized UFLS schemes, amongst others. However, the co-ordination between emergency UFLS schemes and other aspects of the power system operation are not clearly considered in the mentioned reports. Recent cascade failure events have highlighted the importance of the complicated interactions between various aspects of a power system [29–33]. These recent events have helped to identify hidden failure and line overloading as two important propagation mechanisms in cascade failure. In particular, overloaded lines can contribute to cascade failure through a variety of mechanisms including (1) increased risk of flashover faults [32]; (2) decreased synchronizing power causing transient instability or the unstable growth of small-signal power oscillations and (3) heavy reactive power flows inducing transient voltage instability [30, 32, 34]. In recent years, numerous avenues for reducing cascade failure risks have been identified, including (1) general minimization of fault risks [34], (2) the exploitation of flexible ac transmission systems and HVDC links and (3) improved, more coordinated emergency controls [29, 30, 32, 34]. In general terms, these suggestions are attempts to improve the co-ordination of power system design and operation to decrease cascade failure risks caused by line overloading and large reactive power transfers. One key example is improved consideration of cascade failure issues in the automatic or manual decisions undertaken during emergency situations. The load characteristics and demand side management (e.g., distributed interruptible loads control) have been considered to provide an effective load-shedding design [35, 36]. Recently, some improvements have been added to the conventional optimal load flow to minimize the load curtailments necessary to restore the equilibrium of operating point during load shedding [37]. A renewed investigation of the load shedding for frequency protection is necessary because decentralized load shedding can actually induce temporarily overloaded power lines and/or increase voltage support requirements [28, 32, 38]. Wide-area, or centralized, load-shedding approaches appear to be one obvious candidate framework for developing load-shedding schemes that offer better coordination with other cascade failure considerations [29, 38–41]. Numerous widearea load-shedding studies have demonstrated the role of disturbance size and

9.3 UFLS in Multi-Area Power Systems

177

location, load-shedding size and location and shed delay time in the effectiveness of load shed actions [20, 30, 32, 38, 39]. However, when suitable, local approaches are still desirable due to reliability and cost issues [30].

9.3 UFLS in Multi-Area Power Systems 9.3.1 On Targeted Load Shedding Emergency control design approaches have typically been based on the assumption that contingency events are rare and have independent probabilities. That is, the possibility of simultaneous contingency events can safely be ignored. However, experiences throughout the world [29] and examinations of historic power system contingency data [33] demonstrate that contingency probabilities are not independent, and the possibility of multiple contingencies cannot be safely ignored. Cascade failure is one important multiple contingency failure mode that has been emphasized by recent system events [29–34]. Unfortunately, from a cascade failure perspective, standard UFLS schemes tend to share load-shedding responsibilities throughout the system. This sharing behaviour arises as a natural consequence of a power system’s tendency to distribute power adjustments though-out the system according to the machine inertias (although the initial impact of any disturbance tends to be distributed according to synchronizing power coefficients) [21]. This load-sharing behaviour is undesirable from the perspective that overloaded lines have been identified as an important source of the observed cascade failure behaviour [30, 31]. In comparison, recently proposed wide-area load-shedding schemes have demonstrated that the optimal action often rapidly sheds load near the source of power imbalance, and hence minimizes the impact on inter-area power flows [32, 38, 39]. This suggests that there are two basic paradigms for load shedding: a shared loadshedding paradigm and a targeted load-shedding paradigm. The first paradigm appears in the well-known UFLS schemes, and the second paradigm in some recently proposed wide-area load-shedding approaches. Using simulations for a multi-area power system (as shown in Sect. 9.4), it is easy to illustrate the difference between these two paradigms, following generation loss in one area. Sharing load-shedding responsibilities (such as induced by UFLS) is not necessarily an undesirable feature and can be justified on a number of grounds. For example, shared load-shedding schemes tend to improve the security of the interconnected regions by allowing generation reserve to be shared. Further, UFLS approaches can be indirectly used to preferentially shed the least important load in the system. However, sharing load shedding can have a significant impact on interregion power flows and, in certain situations, might increase the risk of cascade failure.

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9 Frequency Control in Emergency Conditions

a

Bris Syd Melb Adel

0

DF/DT Variation 0.15

B S M A

0.1

−0.02

0.05

−0.04

df/dt

Frequency (Hz)

b

Frequency Variations

−0.06

0 −0.05 −0.1

−0.08

−0.15

−0.1 41.98

42

42.02

42.04

Time (minute)

42.06

42.08

−0.2

41.97 41.98 41.99 42 42.0142.02 42.03 42.04 42.05

Time (minute)

Fig. 9.10 Regional frequency response following a major protected event (the August 13 event in the main cities Brisbane, Sydney, Melbourne and Adelaide of the affected regions; (a) frequency deviation and (b) frequency gradient

Although both shared and targeted load-shedding schemes may be able to stabilize overall system frequency, the shared load-shedding response leads to a situation requiring more power transmission requirements. In some situations, this increased power flow might cause line overloading and increase the risk of cascade failure. Recent serious real-world power system events demonstrate this fact, clearly. In the Australian network, the National Electricity Market Management Company (NEMMCO) coordinates the National Electricity Market (NEM) and states that the policy is to share the load-shedding requirements. Figure 9.10 shows the regional power system frequency and its rate deviations in four region centres following a significant incident on Friday, 13 August 2004, in Australia. An equipment failure in New South Wales (NSW) led to the loss of six major electricity generating units in that region, resulting in some customers in NSW, Queensland, Victoria and South Australia losing supply. For this event, approximately 1,500 MW of customer load was automatically shed from the system and power was progressively restored within 2.5 h of the incident occurring [42]. Of particular significance, it is noteworthy that load shedding in Queensland and the resulting increased transfer to NSW almost caused line overload and line trip events. A better load-shedding strategy, such as selected load shedding in NSW could have significantly reduced the risk of reaching transfer limits, the tripping of more generators and further cascade events. The initial frequency gradient strongly suggests that NSW had the fastest initial acceleration, and a biased shedding approach for NSW could be used to significantly increased load shedding in that state. Analysis of this event show that targeted (area-based) load shedding is desirable and feasible and, in this situation, would have limited the peak stresses on interconnections. In weakly inter-connected power systems, due to the delay in the propagation of frequency changes throughout the power system, there is some tendency for the localization of power adjustments following large events. The size and delays of load-shedding actions through the system depends on both the electrical distance and the inertias of the regional generation involved [21].

9.3 UFLS in Multi-Area Power Systems

179

9.3.2 A Centralized UFLS Scheme Consider the following emergency control cost function for an ideal centralized load-shedding problem (ignoring power losses) in a power system with N interconnected areas: N

F(uls ) = ∑ Ci ΔPLi (uls )

(9.13)

i=1

subject to the existing constraint on inter-area power flows and load–generation imbalance (stability). That is, ignoring losses, total system generation must equal total system load, and the generation–load imbalance in either area must not exceed the inter-region power flow limits. Here, Ci is a per unit impact factor, and ΔPLi (uls ) is the load change in the area i following load-shedding decision uls . The new load level in area i, can be shown as follows PLi (uls ) = PLi − ΔPLi (uls ).

(9.14)

This cost function penalizes load-shedding decisions in weighted proportion to the amount of load shed, whilst the constraint ensures system stability and no overloading of inter-area power lines. One can use different representations of load-shedding costs [6], but the additional features in these representations are unimportant in the context of this section. Here, a protected event is defined as a large unplanned generation loss event (or equivalent) for which the system is expected to remain stable, perhaps following the application of an emergency control action. For the purposes of this paper, we will also divide protected events into minor-protected and major-protected events according the risk of inducing overloaded lines. Following a protected event, an optimal centralized load-shedding design prob lem is to determine the load-shedding amounts ΔPL1 , . . . , ΔPLN that minimize the cost function (9.13). For example, in a power system with two areas, the regionally based emergency control problem is to determine optimal load shed amount ΔPL1 (uls ) and ΔPL2 (uls ) that minimizes customer impact in the sense of achieving min C1 ΔPL1 (uls ) +C2 ΔPL2 (uls ) (9.15) uls

subject to the following constraint on power flow and load–generation imbalance PG1F + PG2F = PL1 (uls ) + PL2 (uls ) 1F PG − PL1 (uls ) ≤ ΔP12 ,

(9.16)

where PG1F and PG2F denote the post-event generation levels in areas 1 and 2, respectively. ΔP12 represents the power flow constraint between areas 1 and 2. The total post-event generation PGF and the total change in system generation ΔPGF can be considered as follows:

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9 Frequency Control in Emergency Conditions

PGF = PG1F + PG2F ΔPGF = PG1 + PG2 − PG1F + PG2F .

(9.17)

This constrained optimization problem has only one degree of freedom, due to the power balance equation ΔPL1 (uls ) = ΔPGF − ΔPL2 (uls ). Further, the linear nature of the cost ensures that if an optimal solution to the constrained problem exists, then a solution can be found at a constraint boundary. Rearrangement of constraints and some algebra gives the following optimal solution. If load losses in area 2 cause larger customer impact, that is C2 > C1 , then an optimal emergency control action, u∗ls , is given in terms of the optimal load levels as ! max PG1F − ΔP12 , 0 if PGF − PL2 < max PG1F − ΔP12 , 0 1 ∗ , (9.18) PL (uls ) = otherwise PGF − PL2 ΔPL2 (u∗ls ) = ΔPGF − ΔPL1 (u∗ls )

(9.19)

and it also follows that ΔPL1 u∗ls = PL1 − PL1 u∗ls and ΔPL2 u∗ls = PL2 − PL2 u∗ls . Alternatively, if load losses in area 1 cause large customer impact, that is if C1 > C2 , then an optimal emergency control action, u∗ls , is given in terms of the optimal load levels as ! max PG2F − ΔP12 , 0 if PGF − PL1 < max PG2F − ΔP12 , 0 2 ∗ , (9.20) PL (uls ) = otherwise PGF − PL1 ΔPL1 (u∗ls ) = ΔPGF − ΔPL2 (u∗ls ) .

(9.21)

Of primary interest, it is noteworthy that this emergency control load-shedding rule exhibits two distinct regions of behaviour. When operating inside the power flow constraints (i.e., PG1 − PL1 ≤ ΔP12 ), then it is optimal to shed the cheapest load. If the power flow constraint is reached (i.e., PG1 − PL1 = ΔP12 ) then the ability to share load shedding has been reached and the remaining load must be shed in the more expensive region.

9.3.3 Targeted Load Shedding Using Frequency Rate Change The above centralized load-shedding solution suggests that load-shedding schemes that protect inter-area power lines should exhibit three distinct regimes of behaviour. The first regime of desired behaviour is a no load-shedding response to N − 1contingencies events (an N − 1 contingency event is defined as an unplanned generation loss event, or equivalent, for which the system is expected to remain stable without the application of an emergency control). The second regime is a shared load-shedding behaviour in response to minor protected events. Finally, the third regime is a targeted load-shedding behaviour in response to major protected events (so that changes to inter-area power flows are minimized).

9.3 UFLS in Multi-Area Power Systems

181

Typical implementations involve decentralized load-shedding control approaches where local shedding decisions, based on local information, are independently made throughout the system, rather than centralized control decisions based on overall system information. In decentralized approaches, the size and location of disturbances is not directly known. However, it is shown that disturbance size is related to the average frequency rate experienced in the system [20, 21]. Moreover, local frequency change is related to the electrical distance from the disturbance, and inter-area power requirements are minimized by shedding load near to the source of generation–load imbalance [21, 32, 38]. Together, these results suggest that local frequency rate information might be useful in targeting load shedding to the disturbance location and minimizing inter-region power flows. This idea is used to propose the following adaptive load-shedding scheme for regional protection. Let fi denote the local frequency in area i at time t, and assume that the power system allows K fixed blocks of load shedding in each area. The proposed load-shedding algorithm is to shed load block j in area i, if at time t fi ≤ fthr( j) ,

for j = 1, . . . , K

(9.22)

( ' 0 ˙ (9.23) fthr( j) = min fthr( j) + f off , f LS , 0 = f0 0 ˙ where fthr thr(1) , . . . , f thr(K) is the vector of frequency thresholds; f off is the threshold bias and fLS is the earliest load-shedding frequency. 0 is a vector of UFLS thresholds used to define the load-shedding beHere fthr haviour in response to minor protected events; fLS prevents unnecessary load shedding in response to minor frequency adjustments and f˙off is an offset used to bias load shedding towards the location of generation–load imbalance, if a major protected event is experienced. The threshold bias f˙off can be defined as follows ) ˙ ˙ ˙ ˙foff = α f0i if f0i ≥ fthr , (9.24) 0 otherwise and

where f˙0i is the initial post-contingency frequency rate experienced in power system area i, the gain α describes the bias rate towards the location of generation–load imbalance during major protected events, and f˙thr is a major event threshold used to discriminate between minor and major protected events. As shown in the block diagram of Fig. 9.11, the proposed scheme involves the use of frequency rate information to modify frequency thresholds, and a proportional change to threshold values driven by the size of the disturbance. The desire to minimize inter-region power flows whilst ensuring stability (rather than an exclusive focus on minimal frequency deviation) motivates the use of threshold adjustments based on initial frequency rates f˙0i (rather than the frequency rates f˙i ). Acceptable f˙0i estimation techniques may be system dependent (for example, may depend on the measurements available), but a reasonable f˙0i estimate is the maximum frequency rate within a short time window surrounding a significant frequency excursion event.

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9 Frequency Control in Emergency Conditions

Fig. 9.11 The proposed logic for targeted load-shedding scheme

9.3.3.1 Load-Shedding Settings 0 The three key parameters of the proposed scheme are (1) the UFLS thresholds fthr( j) for j = 1, . . ., K, (2) the major event threshold f˙thr and (3) bias gain α . To simplify this process, it is important to recognize that parameter tuning can be conducted 0 in three stages. The first stage would be the selection of fthr( j) by evaluating performance against the N − 1 contingencies and minor protected events (in much the same way as existing UFLS settings can be designed). The second stage would be to determine a suitable f˙thr by examining the initial frequency rate experienced by the system in response to a selection of minor and major protected events. In the third and final stage, a suitable α gain could be determined by variation until the scheme provides suitable protection against major protected events. 0 A structured design path is based on optimization techniques to determine fthr 0 and α . For example, suitable UFLS settings fthr( j) could be determined using an 0 and f˙ have been selected, a suitable optimization approach such as [6]. Once fthr thr gain α can be determined using a modification of the optimization approach used to 0 obtain fthr( j) settings. It is noteworthy that the f˙thr threshold choice delineates load shedding between “shared” and “targeted” behaviours. Hence, this threshold indirectly determines the amount of power importation allowed between connected regions. A natural

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robustness to errors in f˙thr threshold detection is an attractive feature of the proposed scheme. In the event of a f˙thr threshold failure, the system’s worst behaviour is either the standard UFLS response or a targeted load-shedding outcome, and either outcome is often reasonable. In comparison, consider the poor system protection provided by fully centralized wide-area load-shedding scheme during communication failure.

9.3.4 Simulation Example The proposed load-shedding scheme is demonstrated through simulation studies on a three area power system shown in Fig. 9.12. Each multi-generator area is connected to other two areas. The power system parameters are given in Appendix B (Table B.3). It is assumed that our nominal system frequency is 50 Hz and that a frequency decline below fLS = 49.75 Hz is required before any load shedding can be triggered. The presented three region model allows the examination of regional and line overload aspects of load shedding properly. It is assumed that generator units are of steam type represented by the classical 2nd order model (Fig. 2.4a). In each area, 0 = [49.75, 49.5, 49.25] Hz and the size our basic load-shedding thresholds were fthr of load shed blocks was fixed at 0.2 pu. The acceleration threshold was f˙thr = −1 and the bias gain was α = 0.1. As a test scenario, consider the system frequency response following a 0.5-pu load step disturbance (generation loss) in area 1. In this case, the system frequency passes the first threshold frequency (49.75 Hz). The total load demand is much

Fig. 9.12 Block diagram of three-area power system

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higher than the regional power reserve, and, therefore, the primary and LFC controls are unable to maintain the frequency at the nominal value. In this scenario, the system is in an emergency condition and load shedding is required to help maintain system frequency. The first load-shedding event is triggered at 2.12 s and is quickly followed by a second required load-shed event (note that load-shedding actions are simulated to occur immediately after passing the relevant frequency thresholds). The system response (frequency deviation, frequency rate change and load shedding in each area) for the proposed load-shedding scheme is shown in Fig. 9.13. Tie-line power changes in each region, following the specified major protected event, are shown in Fig. 9.14. In order to illustrate the difference between the proposed (targeted) loadshedding scheme and conventional (shared) load-shedding schemes, the simulation was repeated and these simulation results are shown in Figs. 9.15 and 9.16. The results show that both shared and targeted load-shedding schemes are able to stabilize the interconnected power system and stop frequency decline. However, the comparison of tie-line power flows in Figs. 9.14 and 9.16 show that to compensate the yield frequency deviation, the shared load-shedding response leads to a

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situation with larger inter-region power flows. As mentioned, in certain situations these larger inter-region power flows might cause line overloading, and increase the risk of cascade failure.

9.4 Remarks • The unpredictable nature of power system contingency events means that it is impossible to optimize the regional load-shed properties by a centralized emergency control scheme for all contingency incidents. Thus, for each control area, the first objective is to ensure that the area will remain secure for the loss of its interconnections with adjoining areas. The performance of regional emergency control schemes also depends on the amounts of frequency control ancillary service enabled in each region at the time of the incident. • The results show that the power system areas as authorized control areas can share the frequency-based emergency control plans such as under-frequency load shedding, before and after separation. This does not mean that there is no need for centralized supervision. In a multi-area market, an independent supervisor organization is still needed to organize the interchange powers and stabilize the overall market.

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• Here, it is assumed that the advanced computing techniques and fast hardware facilities are available to measure the rate of area frequency changes in the appropriate time (e.g., less than 0.5 s) to prevent spurious operation. The initial assessment, based upon studies on the Australia power system, suggest that load would have to be shed within less than 0.5 s to be effective in the preventive loss of interconnections on the first swing. • The frequency gradient provides a good measurement index to describe the regional behaviour, and it does not include all necessary knowledge of the actual event. Therefore, to decrease the potential risks of unintended adverse consequences (e.g., over-shedding of load leading to excessive over frequency, or unnecessarily shedding of load following a minor event) other information and parameters may be needed. • The local and inter-modal oscillations during large disturbances can cause d f /dt relays to measure a quantity at a location that is different to the actual underlying system d f /dt [28]. Using Δ f /Δt setting, which is derived over an appropriate time interval, gives values closer to the real rate of system frequency changes which are not influenced by other oscillations.

9.5 Summary This chapter introduced a generalized frequency response model suitable for the analysis of a power system in the presence of significant disturbances and emergency conditions. The effects of emergency control/protection dynamics are properly considered. Under-frequency load-shedding strategies are reviewed and decentralized area-based load-shedding design is emphasized. Finally, the potential benefits of target load shedding compared to more conventional shared load-shedding approaches are examined using simulation of a three-control area power system.

References 1. P. Kundur, Power System Stability and Control. Englewood Cliffs, NJ: McGraw-Hill, 1994. 2. P. M. Anderson and M. Mirheydar, A low-order system frequency response model, IEEE Transactions on Power Systems, 5(3), 720–729, 1990. 3. P. M. Anderson and A. A. Fouad, Power System Control and Stability, Piscataway, NJ: IEEE, 1994. 4. P. W. Sauer and M. A. Pai, Power System Dynamics and Stability. Champaign, IL: Stipes, 2006. 5. K. R. Padiyar, Power Systems Dynamics: Stability and Control. New York, NY: Wiley, 1999. 6. D. L. H. Aik, A general-order system frequency response model incorporating load shedding: analytic modeling and applications, IEEE Transactions on Power Systems, 21(2), 709–717, 2006. 7. T. Hiyama, S. Koga and Y. Yoshimuta, Fuzzy logic based multi-functional load frequency control, In Proceedings of IEEE PES Winter Meeting, vol. 2, pp. 921–926, 2000.

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8. T. Hiyama and G. Okabe, Coordinated load frequency control between LFC unit and small sized high power energy capacitor system, In Proceedings of International Conference on Power System Technology, pp. 1229–1233, 2004. 9. A. A. Mohd Zin, H. Mohd Hafiz and W. K. Wong, Static and dynamic under-frequency load shedding: A comparison, International Conference on Power System Technology POWERCON 2004, Singapore, pp. 941–945, 2004. 10. C. Concordia, L. H. Fink and G. Poullikkas, Load shedding and on an isolated system, IEEE Transactions on Power Systems, 10(3), 1467–1472, 1995. 11. G. S. Grewal, J. W. Konowalec and M. Hakim, Optimization of a load shedding scheme, IEEE Industry Application Magazine, 4(4), 25–30, 1998. 12. J. G. Thompson and B. Fox, Adaptive load shedding for isolated power systems, IEE Proceedings-Generation, Transmission, and Distribution, 141(5), 492–496, 1994. 13. P. M. Anderson and M. Mirheydar, An adaptive method for setting under-frequency load shedding relays, IEEE Transactions on Power Systems, 7(2), 647–655, 1992. 14. H. Bevrani, G. Ledwich and J. J. Ford, On the use of df/dt in power system emergency control, in Proceedings of IEEE Power Systems Conferences and Exposition (CD Record), Seattle, Washington, USA, 2009. 15. J. Jung, C-C Liu, S. L. Tanimoto and V. Vittal, Adaptation in load shedding under vulnerable operating conditions, IEEE Transactions on Power Systems, 17(4), 1199–1205, 2002. 16. D. Prasetijo, W. R. Lachs and D. Sutanto, A new load shedding scheme for limiting underfrequency, IEEE Transactions on Power Systems, 9(3), 1371–1378, 1994. 17. P. M. Anderson and M. Mirheydar, An adaptive method for setting underfrequency load shedding relays, IEEE Transactions on Power Systems, 7(2), 720–729, 1992. 18. V. N. Chuvyvhin, N. S. Gurov, S. S. Venkata and R. E. Brown, An adaptive approach to load shedding and spinning reserve control during underfrequency conditions, IEEE Transactions on Power Systems, 11(4), 1805–1810, 1996. 19. S. J. Huang and C. C. Huang, An adaptive load shedding method with time-based design for isolated power systems, Electric Power Energy Systems, 22(1), 51–58, 2000. 20. V. V. Terzija, Adaptive underfrequency load shedding based on the magnitude of the disturbance estimation, IEEE Transactions on Power Systems, 21(3), 1260–1266, 2006. 21. H. You and V. Vittal, Self-healing in power systems: an approach using islanding and rate of frequency decline-based load shedding, IEEE Transactions on Power Systems, 18(1), 174–181, 2003. 22. P. M. Anderson, Power System Protection. New York, NY: IEEE/Wiley, 1999. 23. H. You, V. Vittal and Z. Yang, Self-healing in power systems: an approach using islanding and rate of frequency decline-based load shedding, IEEE Transactions on Power Systems, 18(1), 174–181, 2003. 24. L. J. Shih, W. J. Lee, J. C. Gu and Y. H. Moon, Application of df/dt in power system protection and its implementation in microcontroller based intelligent load shedding relay, In Proceedings of Industrial and Commercial Power System Technical Conference, pp. 11–17, 1991. 25. W. J. Lee and J. C. Gu, A microprocessor based intelligent load shedding relay, IEEE Transactions on Power Delivery, 4, 2018–2024, 1989. 26. C. J. Durkin, J. E. R. Eberle and P. Zarakas, An underfrequency relay with frequency decay rate compensation, IEEE Transactions on Power Apparatus and Systems, 88(6), 812–820, 1969. 27. B. C. Widrevitz and R. E. Armington, A digital rate-of-change underfrequency protective relay for power systems, IEEE Transactions on Power Apparatus and Systems, 96(5), 1707–1714, 1977. 28. D. E. Clarke, Tasmanian Experience with the Use of df/dt Triggering of UFLSs, Final Report, Transend Networks PTY LTD, No. D08/22185, 2008. 29. G. Andersson, et al. Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic performance, IEEE Transactions on Power Systems, 20(4), 1922–1928, 2005. 30. L. Wehenkel, Emergency control and its strategies, Proceedings of 13th Power System Computation Conference, pp. 35–48, Trondheim, Norway, 1999.

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Chapter 10

Renewable Energy Options and Frequency Regulation

As the use of renewable energy resources (RESs) increases worldwide, there is rising interest in the impacts on power system operation and control. This chapter presents an overview of the key issues and new challenges on frequency regulation, concerning the integration of renewable energy units into the power systems. The impact of power fluctuation produced by variable wind and solar renewable sources on system frequency performance via a simulation study is analysed. An updated LFC model is introduced, and the need for the revising of frequency performance standards is emphasized. Finally, a brief survey on the recent studies on the frequency regulation in the presence of RESs and associated issues is presented.

10.1 RESs and New Challenges The increasing need for electrical energy in the twenty-first century, as well as limited fossil fuel reserves and the increasing concerns with environmental issues for the reduction of carbon dioxide (CO2 ) and other greenhouse gasses [1], call for fast development in the area of RESs. Renewable energy is derived from natural sources such as the sun, wind, hydro-power, biomass, geothermal, oceans, and fuel cells that replenished themselves over a relatively short period of time. The power system architecture of the future incorporating RESs will look very different from that of today. The RESs revolution has already commenced in many countries, as evidenced by the growth of RESs in response to the climate change challenge and the need to enhance fuel diversity. Renewable energy currently provides 14% of the world’s energy supply [2]. The European Union has set as a target 12% of electricity supplied by renewable generation by 2010. According to a recent directive of the European parliament [3], this is translated to an electricity production of 22.1% from RESs. It is predicted to connect the major volumes of renewable generation sources so that 20% of the overall electricity consumption is supplied from renewable sources by 2020 [4].

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Limiting green house gas emissions, avoidance of the construction of new transmission circuits and large generating units, diversification of energy sources to enhance energy security, quality and reliability, and support for competition policy are some important drivers in environmental, commercial, and national/regulatory aspects behind the growth of RESs [5]. Currently, wind is the fastest growing and most widely utilized renewable energy technology in power systems. This renewable energy has experienced accelerated expansion in recent years and its global production is predicted to grow to 300 GW in 2015 [6]. It has been predicted that wind power global penetration will reach 8% by 2020, about 400 GW installed worldwide [4]. According to the European Wind Energy Association (EWEA), European wind power capacity is expected to be 180 GW in 2020 [7]. Since the primary energy source (wind) cannot be stored and is uncontrollable, the controllability and availability of wind power significantly differs from conventional power generation. In most power systems, wind power producers are exempted for some or all ancillary services including frequency regulation and voltage support. The RESs affect the dynamic behaviour of the power system in a way that might be different from conventional generators. Conventional power plants mainly use synchronous generators that are able to continue operation during significant transient faults. In order to protect the unit converters, the variable speed wind turbines are disconnected from the network during a fault. If a large amount of wind generation is tripped because of a fault, the negative effect of that fault on power system control and operation, including frequency control issue, could be magnified [8]. As the use of RESs increases worldwide, there is rising interest in the impacts on power system operation and control. High renewable energy penetration in power systems may increase uncertainties during alter/abnormal electricity industry operation and introduces several technical implications and opens important questions, as to whether the traditional power system control approaches to operation in the new environment are still adequate. Introducing a significant number of RESs into power systems, adds new societal, economical, environmental and technical challenges associated with RESs. Research on the integration of RESs has already received increasing attention. However, the impact analysis techniques and the appropriate modelling and control synthesis are in the early stages of development. Continued work is needed to identify the key distributed RESs and grid characteristics that determine the technical/economical impact dynamics and to design effective compensation methods. The proposed studies found that the renewable integration impacts are non-zero and become more significant at higher size of penetrations. Some studies represent a range of estimates based on different system characteristics, penetration levels and study methods. However, a common thread of all methods was the focus on RESs effects on the interconnected power system, rather than in an isolated one. The technical issues associated with renewable energy compatibility relate to the ability of renewable energy equipment to function effectively as part of the electricity industry as it exists today. There may also be technical means at the system

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level to reduce the variability of the aggregated output from RES units. The RES units must meet technical requirements with respect to voltage, frequency, ability to rapidly isolate faulty parts from the rest to the network and have a reasonable ability to withstand abnormal system operating conditions. However, the novel nature of some RES technologies, such as wind turbines and photovoltaic systems, leads to uncertainties in their technical performance, particularly during abnormal power system operating conditions when power system security may be at risk. It also leads to challenges in developing mathematical models that can adequately predict power system behaviour with high renewable energy penetration. High RESs penetration increases the risk of tie-line overloading. A large renewable energy source such as a wind farm that is located away from major load centres and existing conventional generation units may require network augmentation, and possibly additional interconnections to avoid flow constraints. A sudden reduction in a large RESs power production, not properly forecasted, may also lead to overload problems in interconnection lines, which will be required in the future development of ACE and performance monitoring tools to identify, in advance, the expected behaviour of the system regarding such incidents. As mentioned, among all RESs, the progress in wind power development in recent years is impressive. Considerable developments have been recently made on the technological front, and in the above respect, the development of micro-turbines and novel energy storage technologies is potentially the most challenging. However, there are still unresolved issues for wind energy integration, particularly in the area of forecasting and in the general enhancement of frequency regulation. The variable and non-storable nature of key renewable energy forms, such as wind and solar energy, leads to a need for the accurate forecasting of resource availability and consequent electricity production [2]. The important impacts of a large penetration of variable generation in power system operation and control can be summarized in the following directions [7]: regional overloading of transmission lines in normal operation as well as in emergency conditions, reduction of available tie-line capacities due to large load flows, frequency performance, grid congestions, increasing need for balance power and reserve capacity, increasing power system losses, increasing reactive power compensation, and impact on system security and economic issues. This chapter covers the issues concerning the integration of new renewable power generation (particularly from wind power) in power systems with the frequency regulation problem. The recent investigation studies indicate that relatively large-scale wind generation will have an impact on power system frequency regulation as well as other operation issues and costs; however, these impacts in many countries are relatively low at penetration rates that are expected over the next several years [9]. Increasing wind power generation in the future leads to a higher frequency deviation. With increasing wind power the frequency deviation rate following a disturbance will be increased [10, 11].

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10.2 Impact on Frequency Regulation: A Simulation Study The power outputs of some RESs such as solar and wind power generation systems are dependant on weather conditions, seasons and geographical locations. Wind turbines and solar units are all asynchronous and the generated power is not completely controllable. The total power output characteristics of a power system with a large number of wind and solar systems may be different from that of the conventional system. This section provides a simulation study on the impacts of solar and wind power units on the power system frequency. For this purpose, as shown in Fig. 10.1, an isolated small power system including a 7,500 kW diesel generation unit, a 200 kW photovoltaic (PV) unit, and a 1,000 kW wind turbine unit is considered as a study system to simulate the impact of existing RESs (PV and wind turbine units) on the system frequency performance. The modelling of the study system using three-phase instantaneous values has been performed in the Simulink environment of Matlab software. It is assumed that the diesel generator is responsible for regulating system frequency using a simple PI controller. The system parameters are assumed the same as used in [12]. For the sake of simulation, random variations of solar isolation and wind velocity have been taken into account. Dynamics of the windmill including the pitch angle control of the blade is also considered. The variation of produced powers by wind turbine and PV sources (due to change in wind velocity and solar isolation, together with the variation of load (simulated by a variable load) performs the source of frequency variation in the study system. The maximum amounts of fixed and variable load are assumed about 3,750 kW (PL1 ) and 800 kW (PL2 ), respectively. The total load in the system is equal to PL , where PL = PL1 + PL2 .

(10.1)

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VW (m/s), the blade pitch angle of windmill Beta (deg), the output power of wind turbine PWT (kW), the output power of PV unit PPV (kW), the power consumption of load PL (kW), the produced power by diesel generator PG (kW), the terminal voltage at load VL (pu) and the system frequency f (Hz). The start-up, rated and shutdown wind velocities for the windmill are specified as 5 m/s, 10 m/s and 24 m/s, respectively. Furthermore, the pitch angle control for the wind blade is activated only beyond the rated wind velocity, and the pitch angle is fixed to 10◦ at the lower wind velocity below the rated one. Since the power generation from the wind turbine and solar sources is generally treated as negative load, the variation in overall system load is then a combination of actual load variation ΔPL and RES generation fluctuation. For power systems with small amounts of RESs, the additional variation from RESs is small. However, for a large RESs penetration, the conventional LFC reserve may be insufficient to maintain frequency within the bounds for service quality. Figure 10.3a shows the system response without connecting RESs (PV and wind turbine units). The system response following connection of the low capacity PV unit (only) is shown in Fig. 10.3b; and finally, the frequency variation for the three cases (i.e., without RESs, with PV unit and with both PV and wind turbine units) are compared in Fig. 10.4. When wind power is a part of the power system, additional imbalance is created when the actual wind output deviates from its forecast. Fast movements in wind power output are combined with fast movements in load and other resources. Scheduling conventional generator units to follow load (based on the forecasts) may also be affected by wind power output. Errors in load forecasts are generally uncorrelated with errors in wind forecasts. As shown in the simulation results, in the presence of RESs, the frequency regulation performance is significantly decreased. The diesel generator unit, equipped with the conventional PI controller, is unable to provide a desirable frequency regulation following fluctuation caused by system load–generation imbalance. Since the system inertia determines the sensitivity of overall system frequency, it plays an important role in this consideration. A large interconnected power system generally has sizeable system inertia, and frequency deviation in the presence of wind and solar power variations is small. In other words, larger electricity industry may be more capable of absorbing variations in electricity output from RESs. However, the combination of RES systems to system inertia of a small isolated power system must be considered, and the LFC designs need to consider altering their frequency control strategies to avoid long rates of change of the system frequency. In these cases, using compensation devices such as energy capacitor systems (ECS) [12] and batteries [13, 14], and robust/intelligent tuning techniques such as those described in the previous chapters, can be useful to improve the system frequency control performance. A small-sized ECS with a multi-agent-based coordinator for a similar power system example is suggested in [12]. A controllable battery system in order to suppress the fluctuation of the total power output of distributed generation and area frequency control is introduced in [14]. In this study, battery output is controlled by the LFC signal and it is shown that

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if battery and LFC participant generators are controlled cooperatively, installation of battery with a sufficient capacity makes it possible to decrease the LFC capacity of conventional generators units. Installing batteries and dump loads can absorb the fluctuating solar and wind powers. However, these methods have the disadvantages of high cost and low efficiency. Regarding the nature of RESs output power variation, power systems with large shares of the power producers with fast dynamics such as hydro-units in their LFC systems may be able to respond in a more effective way to response this type of frequency regulation demand.

10.3 Considering RESs Effect in LFC Model The increase in the share of RESs production in power system network is increasingly requiring an analysis of the system dynamic behaviour of some incidents that may occur through an effective modelling. A proper dynamic modelling of the RES units, for dynamic behaviour studies, is a key issue to gaining an adequate idea of the impact in the network resulting from the presence of these generation units following some disturbances. In such dynamic analysis for frequency regulation, only low frequency phenomena are of interest. Therefore, when comparing different wind integration studies, it is important to adopt a clear definition of the timescales involved. Frequency regulation impacts are defined to be those impacts that occur on the basis of a few seconds to minutes.

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10.3.1 LFC Model with RESs When renewable power plants are introduced into the power system, an additional source of variation is added to the already variable nature of the system. To analyse the additional variation caused by RES units, the total effect is important, and every change in RES power output does not need to be matched one for one by a change in another generating unit moving in the opposite direction. Instantaneous fluctuations in load and RES power output might amplify each other, be completely unrelated to each other, or they may cancel each other out [15]. Due to the unpredictable amount of RES power available at any instant, such as solar and wind units, these powers cannot be regarded as a main power reserve for frequency regulation purposes. Recent studies show that the operational impacts of individual fast fluctuations are largely absorbed by the large mechanical and thermal time constants as well as control dead bands of conventional thermal units [16]. However, the slow RES power fluctuation dynamics and total average power variation negatively contribute to the power imbalance and frequency deviation, which should be taken into account in the LFC control scheme. This power fluctuation must be included in the conventional LFC structure. A simplified LFC model in the presence of RES is shown in Fig. 10.5. Here, the filtered total effect of power fluctuation ΔPRES is considered. Also, for a large RESs penetration, the resulting ACE signal must reflect the total RESs power generation changes which is usually smoothed compared to variations from the individual RES units ACE = β Δ f + ∑ (PCon,act − PCon,sched ) + ∑ (PRES,act − PRES,estim ),

Fig. 10.5 Simplified LFC model with considering RES power fluctuation

(10.2)

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10 Renewable Energy Options and Frequency Regulation

where PCon,act , PCon,sched , PRES,act and PRES,estim are actual conventional power, scheduled conventional power, actual RESs power and estimated RESs power, respectively. Because of range of use and specific dynamic characteristics such as a considerable amount of kinetic energy, most recent attempts have focused on the consideration of wind power fluctuation in the conventional LFC structure [17–21]. Similar to conventional plants, the inertia constant for a wind turbine is given by [22] HW =

2 J ωW , 2S

(10.3)

where J is the inertia of the wind rotor, ωW is the rotational speed of wind turbine and S is the nominal apparent power. The inertia constant for wind power is time dependant. The typical inertia constant for the wind turbines is about 2–6 s [23]. Therefore, wind turbines have a significant amount of kinetic energy stored in the rotating mass of their blades. In case of variable-speed wind turbines, since the rotational speed is decoupled from the grid frequency by power converter, this energy does not contribute to the inertia of the grid. Thus, the inertia contribution of wind turbines is much less than of conventional generator units. Depending on the type of generator units, typical inertia constants for the grid power generators are in the range of 2–9 s [24].

10.3.2 Required Supplementary LFC Reserve In steady-state operation, assuming that the total RES production level can be defined as PRES and total consumption level in PL , the amount of power to be produced by the conventional units (PG ) is PG = PL + PLosses − PRES .

(10.4)

This means that the steady-state impacts are largely dependent on the final dispatch solution to be adopted. However, the variable renewable electricity output may or may not be available during peak demand and abnormal periods. It might be that intermittent resources cannot contribute to the overall system frequency regulation and reliability. For power systems with small amounts of RESs, the additional variation from RESs is small. However, for a large RESs penetration, the conventional LFC reserve may be insufficient to maintain frequency within the bounds for service quality. To allow for increased penetration of RESs, a change in LFC reserve policy may be required. In this direction, in addition to deregulation policies, the amount and location of RES units, renewable generation technology and the size and the characteristics of the electricity system must be considered as important technical aspects. Recently, several studies have been conducted on the required LFC reserve estimation in the presence of various RES units. A mathematical model to evaluate the

10.3 Considering RESs Effect in LFC Model

201

impact of small PV power generating stations on economic and performance factors for a large-scale power system is developed in [25]. Based on the results of [25], an electrical power system containing a 10% contribution from PV stations would require a 2.5% increase in LFC capacity over a conventional system. Using multipoint observation data, the required LFC capacity for the output fluctuation of PV systems is estimated in [26]. It was found that wind power, combined with the varying load, does not impose major extra variations on the system until a substantial penetration is reached [27]. Large geographical spreading of wind power will reduce variability, increase predictability and decrease the occasions with near zero or peak output. It is investigated in [27] that the power fluctuation from geographically dispersed wind farms will be uncorrelated with each other, hence smoothing the sum power and not imposing any significant requirement for additional frequency regulation reserve, and required extra balancing is small. It is estimated in [28] that for a 10 GW installed wind capacity (in UK power system), 126–192 MW additional continues conventional LFC power is required. According to a study for Denmark and Germany [29], the supplementary control must provide 6.6 MW/min of additional capacity per 1,000 MW of installed wind power to keep the nominal frequency. Usually, the demand on supplementary control is specified by the wind power production forecast error. That is, the difference between the forecast and the actual power productions. The fluctuation of the aggregated wind power output in a short term (e.g., tens of seconds) for a larger number of wind turbines are much smoothed. It is investigated that the wind turbines aggregation has positive effects on the regulation requirement. Relative regulation requirement decreases whenever larger aggregations are considered. Based on a record [30], a 202-MW wind plant would have required 18.2 MW of regulation during a particularly volatile week if it had to compensate for its variability independently, but would require only 9.4 MWh when integrated into the control area (48% reduction). The fast, random fluctuations associated with regulation are typically uncorrelated. Consequently, the total frequency regulation requirement is not the sum of the regulation requirements of the individual loads and uncontrolled generators, but is instead the sum of the correlated components.

10.3.3 RESs and Frequency Performance Standards Power system frequency control is an issue that may evolve into new guidelines. The increasing share of renewable energy, which is difficult to predict accurately, may have an adverse impact on frequency quality. The existing frequency operating standards need to change to allow for the introduction of renewable power generation, and allow for modern distributed generator technologies. It is investigated that the slow component of renewable power fluctuation negatively affects the performance standards such as policy P1 of UCTE (Union for the

202

10 Renewable Energy Options and Frequency Regulation

Coordination of Transmission of Electricity) performance standard [63], or the control performance standards CPS1 and CPS2 introduced by NERC (North American Electric Reliability Council) [16]. The standards redesign must be done in both normal and abnormal conditions, and should take account of operational experience on the initial frequency control schemes and again used measurement signals including tie-line power, frequency and rate of frequency change settings. The new set of frequency performance standards are under development in many countries [32, 33]. The new standards introduce the update high and low trigger, abnormal, and relay limits applied on the interconnection frequency excursions. The revised standards may bring an element of a more centralized frequency control through a better coordination among control areas, delegating more authority to the control areas performing frequency monitoring functions, and perhaps creating distributed or inter-area control centres to decentralized frequency control through the creation of corresponding ancillary service markets [34, 35]. It is shown that the rate of frequency change (d f /dt) following a disturbance is proportional to the power imbalance, and it also depends on the equivalent system inertia [36, 37]. Recalling (10.3), since large wind farms can considerably increase the overall system inertia, the d f /dt will be significantly changed. From an operational point of view, a larger variable renewable power in the power system causes a smaller frequency rate change following a sudden loss of generation or load disturbance. The initial frequency rate change for the given simulation example (Sect. 10.2), following 0.01 pu step load disturbance is shown in Fig. 10.6. A similar test is repeated in the presence of a larger wind power penetration (1,200 kW) and a lower frequency rate change is achieved (shown as dotted line in Fig. 10.6). This issue is important for those networks that use the protective d f /dt relays to re-evaluate their tuning strategies. For high wind penetration, not only frequency relay settings need revision, but also current and voltage relays need to be coordinated [38, 39]. Protection schemes for distribution and transmission networks are one of the main problems posed by RESs in power systems. Change of operational conditions and dynamic characteristics influence the requirements to protection parameters. The performance standards revision has already commenced in many countries [7, 33, 34, 40]. In Australia, the Australian Electricity Market Commission (AMEC) is proposing revised technical rules for generator connection, including wind generators. As well as meeting technical standards, generators are required to provide information on energy production via the system operator’s SCADA system [40]. National Electricity Market Management Company (NEMMCO) sets out functional requirement for an Australian Wind Energy Forecasting System (AWEFS) for wind farms in market regions. In the USA, NERC is working to revise the conventional control performance standards [33]. The existing market rules and priority rules for the transport of RES electricity is also under re-examination by UCTE in Europe [7].

10.4 A Survey on Recent Studies

a

203

1

Δ f (Hz)

0.5 0 -0.5 -1

b

0

10

20

30

40

60

70

0.15 0.1

Δ f (Hz)

50

Initial rate of frequency change

0.05 0 -0.05 -0.1 22

23

24

25

26

27

28

Time (sec)

Fig. 10.6 (a) Frequency deviation following 0.01 pu step load disturbance at 25 sec, in the presence of 1000 kW (solid) and 1200 kW (dotted) wind power; (b) A zoomed view around 25 sec

10.4 A Survey on Recent Studies This section presents a brief critical literature review and an up to date bibliography for the proposed studies on the frequency regulation in the presence of RESs and associated issues. A considerable part of attempts has focused on wind power generation units. An automatic generation control system for a wind farm with variable speed turbines is addressed in [41]. The proposed integrated control system includes two control levels (supervisory system and machine control system). A distributed control system for frequency control in an isolated wind system is given in [28, 42] describes the impact of wind generation on the operation and development of the UK electricity systems. Impacts of wind power components and variations on power system frequency control are described in [16, 17]. The capability of providing a short-term active power support of a wind farm to improve the primary frequency control performance is discussed in [43]. A method of quantifying wind penetration based on the amount of fluctuating power that can be filtered by wind turbine generation and thermal plants is addressed in [44]. A small power system including three thermal units (equipped with LFC system) and a wind farm is considered as a test example. Using the Bode diagram of system transfer function between frequency deviation and real power fluctuation signals, the permitted power fluctuation for 1% frequency deviation is approximated.

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Recently, several works are reported on considering the effect of wind power fluctuation in a generalized LFC structure [17–21]. Although any model that involves the complete interactions of wind power with conventional power system operation requires a number of simplifying assumptions, most proposed models do not account the uncertainty of wind generation in a frequency regulation timescale. The technology to filter out the power fluctuations (in result frequency deviation) by wind turbine generators for the increasing amount of wind power penetration is growing. The new generation of variable-speed, large wind turbine generators with high moments of inertia from their long turbine blades can filter power fluctuations in the wind farms. A method is presented in [22] to let variable-speed wind turbines emulate inertia and support primary frequency control. Integrating energy storage systems (ESSs) or ECSs into the wind energy system to diminish the wind power impact on power system frequency has been addressed in several reported works [12, 18, 19, 45–47]. In [18], an ESS-based wind power filtering algorithm is proposed. It is shown that power systems are more sensitive to the power fluctuations in the medium frequency region (between 0.01 and 1 Hz), in which the majority of wind power fluctuations are located and below. In [45–47], different ESSs by means of an electric double-layer capacitor (EDLC) and superconducting magnetic energy storage (SMES) and energy saving are proposed for wind power levelling. Some preliminary studies showed that the kinetic energy stored in the rotating mass of a wind turbine can be used to support primary frequency control for a short period of time [22]. To support primary frequency control for a longer period, some techniques such as using a combination of wind and fuel cell energies are suggested [48, 49]. The amount of installed fuel cell energy capacity needed to compensate frequency deviation is discussed in [48]. In [20, 50], the application of rotary frequency converter is studied as a buffer to connect the wind farm and the power system grid. This converter usually includes a synchronous motor and a doubly fed adjustable speed machine (DFM). DFMs are currently in use [51] and have control capability of input power during pumping operation, and can contribute to grid frequency control. If the amount of frequency deviation is within the allowable slip range of the DFM, it can be absorbed in the rotary frequency converter as mechanical energy, and its influence on the power system can be cancelled. The impacts of wind power on tie-line power flow in the form of low frequency oscillations due to insufficient system damping are studied in [52, 53]. A control scheme based on controllable distributed generators is addressed in [54] to attenuate the mentioned tie-line flow deviation. Some recent studies analyse the impacts of RESs on power market operation [2, 9, 40, 55–57]. Some of those are reviewed in [9]. A study is conducted in [55] to help determine how wind generation might interact in the competitive wholesale market for regulation services and a real-time balancing market. This study recognized that wind integration does not require that each deviation in wind power

10.4 A Survey on Recent Studies

205

output be matched by a corresponding and opposite deviation in other resources, and the frequency performance requirement must apply to the aggregated system, not to each individual generator. A year of actual wind speed data and hourly load data for a region is used to determine the optimal sizes and locations of local power plants in [56]. This analysis has focused on the impact of wind plants on hourly system imbalance, and physical requirements that wind would impose on the electrical supply. An electrolyser system with a fuzzy PI control is used in [58] to solve power quality issues resulting from micro-grid frequency fluctuations. As mentioned, some variable speed turbine technologies use power electronic converters to create an electrical decoupling between the machine and the grid. Such decoupling leads to an even lower participation of wind turbines to the system stored kinetic energy. Using the kinetic ESS (blade and machine inertia) to participate in primary frequency control is addressed in [22]. To ensure a regular primary reserve even when the wind generator works under rated power, without any wind speed measurement, a fuzzy logic supervisor is proposed in [59]. This supervisor is used to simultaneously control the generator torque and the pitch angle to keep a primary reserve. Using modal techniques, the dynamic influence of wind power on the primary frequency control is studied in [60]. This study shows that the wind turbines excite the power system in the electromechanical modes. An increase in the wind power leads to an increase in the frequency because the load on synchronous machines is reduced and the speed drop characteristics of the speed governors lead to an operational frequency slightly above the rated. Similarly, a reduction in the wind power leads to a decrease in the frequency. While the amount of generation to participate in the LFC task to compensate the additional variation will grow, the rising RES market share will reduce the amount of generation that actually capable of providing frequency support. To overcome the above problem, several approaches have been proposed. A demand-based frequency control idea is presented in [61] to provide frequency control support where conventional LFC reserve is not enough or unavailable. Several works on solar (PV) energy, batteries, and energy capacitor units are being performed in Japan [12, 13, 25, 26]. The Japan has set an ambitious target of 4.5 GW of electricity to be generated by PV systems by the year 2010 [25]. The influence of PV system on power system frequency control is discussed in [26]. Using Redox Flow (RF) batteries for supplementary control and maintenance of power quality in the presence of distributed power resources is suggested in [13]. It is shown that the LFC capacity of RF battery systems is ten times that of fossil power systems, due to quick response characteristics. There are few reports on the role of RESs and distributed generation in emergency conditions. The impact of distributed utilities on transmission stability is addressed in [62], and an optimal load shedding strategy for power systems with distributed sources is introduced in [31].

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10.5 Summary The key issues and new challenges on frequency regulation, concerning the integration of renewable energy units into the power systems, are discussed. The impact of power fluctuation produced by variable wind and solar renewable sources on system frequency performance is analysed, and a simulation study is performed. An updated LFC model is introduced, and the need for the revising of frequency performance standards is emphasized. Finally, a brief survey on the recent studies on the frequency regulation in the presence of RESs and associated issues is presented.

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Appendix A

Table A.1 Applied data for the given simulation in Chap. 4 Parameters

Genco

MVAbase (1,000 MW)

1

Rate (MW) Bi (pu/Hz) Di (pu MW/Hz) Ri (Hz/pu) 2Hi / f0 (pu s) Tt i (s) Tg i (s) αi Ramp rate (MW/min)

1,000 800 1,000 1,100 900 1,200 850 1,000 1,020 0.3483 0.3473 0.3180 0.3827 0.3890 0.4140 0.3692 0.3493 0.3550 0.015 0.014 0.015 0.016 0.014 0.014 0.015 0.016 0.015 3.00 0.1677 0.4 0.08 0.4 8

2

3.00 0.120 0.36 0.06 0.4 8

3

3.30 0.200 0.42 0.07 0.2 4

4

2.7273 0.2017 0.44 0.06 0.6 12

5

2.6667 0.150 0.32 0.06 0 0

6

2.50 0.196 0.40 0.08 0.4 8

7

2.8235 0.1247 0.30 0.07 0 0

8

3.00 0.1667 0.40 0.07 0.5 10

9

2.9412 0.187 0.41 0.08 0.5 10

209

210

Appendix A

Table A.2 Generating unit parameters for the real-time simulation in Chap. 5 Parameters

Gen 1

Gen 2

MVA R (Hz/pu) T1 (s) T2 (s) T3 (s) T4 (s) T5 (s) β (pu/Hz) D (pu/Hz) 2H (s) TH (s) TI (s) TL (s) KH (pu) KI (pu) KL (pu) M1 (pu/min) M2 (pu/min) M3 (pu/min) N1 (pu/min) N2 (pu/min) N3 (pu/min)

100 3.00 0.08 0.10 0.10 0.40 10.0 0.3483 0.0150 8.05 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Gen 3

60 3.00 0.06 0.10 0.10 0.36 10.0 0.3473 0.0150 7.00 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

100 3.30 0.07 0.10 0.10 0.42 10.0 0.3180 0.0150 8.05 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Table A.3 Power system parameters for the real-time simulation in Chap. 6 Parameter MVA R (Hz/pu) T1 (s) T2 (s) T3 (s) T4 (s) T5 (s) β (pu/Hz) D (pu/Hz) 2H (s) TH (s) TI (s) TL (s) KH (pu) KI (pu) KL (pu) M1 (pu/min) M2 (pu/min) M3 (pu/min) N1 (pu/min) N2 (pu/min) N3 (pu/min)

Gen 1 1, 000 3.00 0.08 0.10 0.10 0.40 10.0 0.3483 0.0150 8.05 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Gen 2 600 3.00 0.06 0.10 0.10 0.36 10.0 0.3473 0.0150 7.00 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Gen 3 1, 000 3.30 0.07 0.10 0.10 0.42 10.0 0.3180 0.0150 8.05 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Gen 4 900 3.30 0.07 0.10 0.10 0.3 10.0 0.3827 0.0150 6.00 0.05 0.08 0.58 0.31 0.24 0.45 0.50 0.050 2.00 −0.50 −0.20 −0.50

Appendix B

• State-space model matrices for Example 1 in Sect. 8.1.4 ⎡ ⎤ 1 1 Di 0 − − ⎢ 2Hi 2Hi 2Hi ⎥ ⎡ ⎤ ⎢ ⎥ 1 1 A11 A12 A13 A14 ⎢ ⎥ 0 − 0 ⎢ ⎥ ⎢ A21 A22 A23 A24 ⎥ ⎢ ⎥ Tt i Tt i ⎢ ⎥ A=⎣ = , A ⎢ ⎥, ii 1 A31 A32 A33 A34 ⎦ ⎢− 1 0 − 0 ⎥ ⎢ ⎥ Tg i A41 A42 A43 A44 ⎢ Ri Tg i ⎥ ⎣ ⎦ 0 0 2π ∑ Ti j 0 j

⎡

⎤

0 000 ⎢ 0 0 0 0⎥ ⎥ Ai j (i = j) = ⎢ ⎣ 0 0 0 0 ⎦, −Ti j 0 0 0 ⎤

⎡

−1 000 0 000 ⎢ Tg1 ⎢ −1 ⎢ 000 ⎢0 0 0 0 0 0 Tg2 ⎢ B=⎢ ⎢0 0 0 0 0 0 0 0 0 0 ⎢ ⎢ ⎣ 00 0 000 0 000 00

⎡ −1 ⎢ 2H1 ⎢ ⎢ 0 ⎢ F =⎢ ⎢ ⎢ 0 ⎢ ⎣ 0

000 0 000 −1 000 2H2 000 0 000

000

000 0 000

0 000 0 0

⎥ ⎥ ⎥ 0 0 0 0 0 0⎥ ⎥ ⎥, −1 0 0 0 0 0⎥ ⎥ Tg3 ⎥ −1 ⎦ 0 000 0 Tg4 ⎤ 0 000 0 000 ⎥ ⎥ 0 0 0 0 0 0 0 0⎥ ⎥ ⎥. ⎥ −1 0 0 0 0 0 0 0⎥ ⎥ 2H3 ⎦ −1 0 000 000 2H4 211

212

Appendix B

• Low-order controllers for control areas 2, 3 and 4 of power system Example 1 in Sect. 8.1.4 K2 (s) =

N2 (s) N3 (s) N4 (s) , K3 (s) = , K4 (s) = , D2 (s) D3 (s) D4 (s)

where N2 (s) = 140.756s5 + 164, 530.87s4 + 194, 365.253s3 + 98, 449.36s2 + 546, 138.32s + 723, 970.37 D2 (s) = s6 + 387.75s5 + 35, 235.403s4 + 67, 819.44s3 + 2, 742, 801.2s2 + 626, 558.42s + 126, 075.23, N3 (s) = 526.29s5 + 1, 287.18s4 − 1, 416.26s3 + 6, 371.23s2 + 12, 698.7s + 633.53 D3 (s) = s6 + 7, 229.77s5 + 6, 809.8s4 + 93, 877.3s3 + 101, 675.4s2 + 4632.21s + 23.39, N4 (s) = 560.94s6 + 8, 329.72s5 + 4, 783.48s4 + 1, 246.86s3 + 19, 675.43s2 + 2, 638.25s + 93.49 D4 (s) = s7 + 18, 945.33s6 + 12, 511.83s5 + 76, 432.43s4 + 836, 228.94s3 + 42, 388.23s2 + 1, 612.47s + 532. • Low-order controllers for control areas 2 and 3 of power system Example 2 in Sect. 8.2.2 K2 (s) =

N2 (s) N3 (s) , K3 (s) = , D2 (s) D3 (s)

where N2 (s) = 145s5 + 1, 445, 267s4 + 178, 943, 657s3 + 96, 405, 249s2 + 274, 613, 248s + 323, 019, 700

Table B.1 Weighting functions for control area loops 2, 3 and 4 of power system Example 1 in Sect. 8.1.4 Area-2

Area-3

Area-4

WU2 (s) =

0.1s2 + 0.001 s2 + 0.2s + 21

WU3 (s) =

0.5s2 + 0.005 s2 + 0.05s + 10

WU4 (s) =

0.11s2 + 0.004 s2 + 0.11s + 15

WP12 (s) =

0.005s 10−5 s + 4.5

WP13 (s) =

0.01s 10−4 s + 1

WP14 (s) =

0.009s 10−6 s + 15

WP22 (s) =

s + 0.1 93(s + 0.001)

WP23 (s) =

s + 1.1 100(s + 0.1)

WP24 (s) =

s + 0.22 83(s + 0.02)

Appendix B

213

Table B.2 Applied data for simulation of power system Example 2 in Sect. 8.2.2 Quantity

G11

G12

Rating (MW) Hi (s) Di (pu MW/Hz) Ri (%) 2Hi / f0 Tt i Tg i Kt i , Kg i Ti Ti j (MW/rad)

1,600 600 5 4 0.02 0.01 4 5.2 0.167 0.134 0.5 0.5 0.2 0.1 1 1 0.2 0.1 T12 = 60

G13

G14

G21

800 800 4 5 0.01 0.015 5.2 5 0.134 0.167 0.5 0.5 0.15 0.1 1 1 0.1 0.2 T13 = 60

G22

600 1,200 4 5 0.01 0.02 5.2 4 0.134 0.167 0.5 0.5 0.1 0.2 1 1 0.1 0.2 T23 = 100

G23

G24

G31

G32

G33

800 4 0.01 5.2 0.134 0.5 0.15 1 0.1

1,000 5 0.015 5 0.167 0.5 0.1 1 0.2

1,400 5 0.02 4 0.167 0.5 0.2 1 0.2

600 4 0.01 5.2 0.134 0.5 0.1 1 0.1

600 4 0.01 5.2 0.134 0.5 0.1 1 0.1

Table B.3 Applied data for performed simulation in Chap. 9 (Sect. 9.3.4) Areas

Area-1

Generator unit Rating (MW) Hi (s) Di (pu MW/Hz) Ri (%) Tt i Tg i Ki Ti j (pu/Hz)

G11 G12 1,200 600 6.0 4.0 0.05 0.08 3.0 3.0 0.40 0.36 0.30 0.20 1.0 1.0 T12 = 0.2 T13 = 0.25

G13 800 5.0 0.05 3.2 0.42 0.07 1.0

Area-2 G14 800 5.0 0.04 2.7 0.45 0.10 1.0

G21 G22 600 1,200 5.0 5.0 0.05 0.08 2.7 2.6 0.44 0.32 0.30 0.20 1.0 1.0 T21 = 0.2 T23 = 0.12

Area-3 G23 800 4.0 0.05 2.5 0.40 0.15 1.0

G31 G32 1,400 600 6.0 5.0 0.07 0.05 2.8 3.0 0.30 0.40 0.15 0.15 1.0 1.0 T31 = 0.25 T32 = 0.12

D2 (s) = s6 + 288s5 + 20, 235s4 + 767, 219s3 + 17, 402, 801s2 + 226, 558, 154s + 226, 075, N3 (s) = 226.3s5 + 22, 873s4 − 1, 616s3 + 137, 110s2 + 126, 934s + 533 D3 (s) = s6 + 3, 239.8s5 + 68, 092s4 + 638, 727s3 + 3, 016, 725s2 + 16, 332.2s + 13.3.

G33 600 5.0 0.04 3.0 0.41 0.20 1.0

Index

A Adaptive load shedding, 181 Analog power system simulator (APSS), 92–94, 104, 113, 123, 132, 133 Ancillary service, 9, 30, 45, 56, 123, 125, 166, 186, 192, 202 Area control error (ACE), 8, 22, 24, 28, 39, 40, 42, 44, 45, 48, 49, 56, 57, 66, 67, 75, 77, 80, 88, 89, 94, 96, 97, 99, 105–107, 114, 117–120, 124, 126, 130, 131, 133, 135, 136, 138, 155, 193, 199 Area interface, 39, 40, 67, 88, 107, 131 Automatic generation control, 5, 8, 15, 203 Automatic voltage regulator (AVR), 5, 7, 8 B Bias factor, 23 Bilateral contracts, 47–49, 81 Bilinear matrix inequalities (BMI), 69 C Cascade failure, 176–178, 186 Centralized control, 181 Classical load-frequency control model, 39, 60 Communication delays, 56, 57, 85–88, 92, 96, 97, 99, 103, 105–107, 110, 116, 117, 121, 130 Constraints, 9–11, 29, 39, 55–60, 64, 67, 71, 73, 77, 86, 89, 92, 96, 103, 104, 107, 109, 112, 113, 117, 141, 144, 156, 171, 179, 180, 193 Continuous maximum rating (CMR), 55 Control area concept, 20 Control framework, 63, 66–69, 77–79, 81, 89, 106, 123, 126, 131, 142 Control performance, 28, 39, 56, 57, 60, 141, 147, 196, 202, 203

Control performance standards (CPS), 28, 202 Conventional control, 202 Conventional frequency response model, 168 Conventional LFC, 47, 123, 196, 199–201, 205 D D–K iteration, 145, 147, 148, 152, 157, 160 Damping coefficient, 17, 39, 124, 166 Damping control, 9, 88 Data acquisition and monitoring (DAM), 123, 126, 129, 133, 135 Dead band, 29, 39, 55–58, 60, 199 Decentralized control, 141, 146 Decentralized load frequency control (LFC), 10, 11, 63, 141–162 Decentralized load shedding, 176, 181 Decentralized property, 10, 85 Decision and control (DC), 123, 125–127, 130, 133, 135 Deregulated environment, 44–46, 56, 81, 85, 86, 123, 126, 141 Deregulation, 26, 30, 44, 47, 63, 123, 126, 200 Disco participation matrix, 47 Distributed generation, 2, 196, 205 Doubly-fed adjustable speed machine (DFM), 204 Droop characteristic, 17, 39, 41, 42, 124, 166 Dynamic load shedding, 176 Dynamic performance, 39–60 Dynamical model, 25, 29, 30, 39, 43–44, 47, 55, 60, 81, 89, 96, 107, 131, 147, 157, 166 E Electric double-layer capacitor (EDLC), 204 Emergency conditions, 165–187, 193, 205

215

216 Emergency control, 6, 27, 165–169, 172, 174, 176, 177, 179, 180, 186, 187 Emergency control/protection schemes, 6 Emergency operating, 27 Energy capacitor system (ECS), 196, 204 Excitation control, 5, 94, 113, 132 Experimental study, 92–96, 113–117 F Fictitious output, 67, 77, 79, 89, 107, 113, 117 Fictitious weights, 58 Frequency decline rate, 165, 166, 174, 183, 184 Frequency operating standards, 26–28, 30, 201 Frequency performance standards, 28, 29, 191, 201–203, 206 Frequency response model, 16–20, 124, 165–174, 187 Frequency stability, 1, 7, 11, 144 Frequency threshold, 27, 174, 175, 181, 184 G Generation loss, 26, 165, 168, 177, 179, 180, 183 Generation participation matrix (GPM), 47, 49–52, 54 Generation rate, 11, 29, 39, 55–56, 60, 113, 171 Generation rate constraint (GRC), 11, 29, 55, 56, 58, 113, 171 Genetic algorithm (GA), 29, 63, 79 H H∞ control, 63–65, 69, 71, 73, 75–80, 82, 85–87, 103–106, 112, 113, 121, 135 H∞ performance, 73, 89, 96, 105–107, 112, 121, 131 H2 performance, 105–107, 109 Hankel norm approximation, 152, 160 High voltage direct current (HVDC), 2, 5, 6, 8, 30, 176 Hydraulic amplifier, 16 I Independent contract administrator (ICA), 45 Independent power producers (IPPs), 30, 45 Independent system operator (ISO), 45 Inertia constant, 17, 20, 39, 125, 149, 157, 167, 200 Instability phenomena, 2–5 Interconnected power system, 9, 16, 20–25, 28, 42, 85, 87, 99, 184, 192, 196 Inter-modal oscillations, 187

Index Isolated power system, 20, 194, 196 Iterative linear matrix inequalities (ILMI), 63, 73–81, 106, 108–110 K Kalman filter, 64 Kharitonove’s theorem, 29, 63 L Laplace transform, 17, 21, 43, 166, 167, 169 LFC model, 47, 49, 50, 55, 58, 90, 106, 123, 142, 191, 198, 199, 206 LFC reserve, 171, 173, 196, 200–201, 205 Load disturbance, 16, 19, 25, 41, 48, 75, 78, 79, 82, 88, 96, 98, 106, 117, 121, 124, 126, 135, 144, 149, 152–154, 157, 173, 174, 202, 203 Load-generation imbalance, 125, 166–167, 179, 196 Load relief, 125 Local oscillation, 114 Lyapunov inequalities, 64 M M-Δ configuration, 144–145, 147, 151, 156, 157, 159 Mixed H2/H∞ control, 10, 103–106, 113 Model linearization, 143 Modelling, 1, 2, 10, 16–20, 29, 30, 47–50, 57, 59, 103, 108, 166–167, 192, 194, 198 Multi-agent control system, 123 Multi-area control system, 15, 30 Multi-area frequency control, 10, 15, 30 Multi-area power system, 10, 20, 22, 24, 29, 57, 63, 81, 87, 88, 99, 121, 124, 141–142, 148, 153, 154, 161, 162, 177–186 Multi-input multi-output (MIMO), 1, 148, 149 Multi-objective control, 103–121 Multiple delays, 86, 99, 104 Multiplicative uncertainty (W Ui ), 59, 108, 116, 144, 145, 150, 159 N N-1 contingency, 180 Non linear simulations, 56, 96, 98, 99, 117, 138, 171 Normal operating, 6, 27 North American Electric Reliability Council (NERC), 28, 202 O Operating conditions, 2, 6, 10, 27, 58, 59, 63, 88, 111, 121, 123, 142, 155, 165, 193 Operating states, 6–7

Index Optimization, 1, 29, 58, 64, 70, 71, 99, 103, 108, 109, 121, 180, 182 Over-frequency generation trips (OFGT), 168, 169 Over-frequency generator shedding (OFGS), 168 Overloading, 176, 178, 179, 186, 193 P Participation factors, 8, 25–26, 39, 45, 47, 54, 88, 94, 97, 114, 117, 121, 123, 126–128, 130, 135, 138, 155, 170–171 Photovoltaic (PV), 30, 193–198, 201, 205 Physical constraints, 10, 29, 39, 55–60, 73, 107 Power fluctuation, 171, 191, 199–201, 203, 204, 206 Power imbalance, 8, 123, 125–126, 130, 138, 171–172, 177, 199, 202 Power system analysis, 10, 17, 20, 28, 40, 42, 47, 92, 93, 124, 187, 192, 198 control, 1–11, 59, 87, 92, 107, 112, 192 control and operation, 192 dynamics, 8, 17, 56, 93 frequency control, 8–11, 28–30, 39, 44, 55, 60, 85, 201, 203, 205 operation, 8, 9, 26, 30, 39, 47, 58, 60, 85, 123, 176, 191–193, 204 restructuring, 39, 60 stability, 1–4, 6–9, 11, 29, 112, 113, 141, 142, 148, 149, 179, 205 Power system stabilizer (PSS), 3, 5, 8 Primary control, 16, 26, 27, 39, 40, 42, 124, 165, 168, 172 Primary control loop, 16, 42 Proportional integral (PI) based LFC, 63, 66, 67, 69, 85, 86, 89, 91, 99, 103, 106, 107, 111, 112, 121 Protected event, 178–182, 184, 185 PSS. See Power system stabilizer R Reactive power, 5, 29, 174, 176, 193 Real power, 4, 8, 15, 27, 29, 30, 143, 203 Real-time simulations, 86, 96, 98, 99, 104, 117, 121, 136, 138, 210 Reliability, 8–10, 26, 28, 30, 123, 126, 137, 177, 192, 200, 202 Remote terminal units (RTUs), 57, 88, 105 Renewable energy resources (RESs), 2, 10, 30, 191–206 RESs penetration, 193, 196, 199, 200

217 Robust control, 2, 9, 10, 23, 29, 43, 59, 63, 73, 77, 108, 130, 141, 144, 146, 147, 150, 153, 155 Robust control techniques, 10, 29 Robust frequency control, 11 Robust LFC, 10, 11, 87 Robust performance, 10, 29, 73, 75, 78, 79, 82, 108, 112, 141, 142, 144, 147–149, 151, 152, 155, 157, 158, 160 Robust performance indices, 75, 78, 112 Robust PI control, 73–75, 78, 98, 114, 121 Robustness, 9, 10, 75, 96, 106, 107, 112, 116, 142, 145, 151, 183 Rotating mass, 26, 40, 42, 59, 167, 200, 204 Rotor angle instability, 3 RTUs. See Remote terminal units S SCADA. See Supervisory control and data acquisition system Schur complement, 66, 87, 91 Sensitivity, 144, 145, 151, 159, 196 Shared load shedding, 177, 178, 180, 184–187 Single input single output (SISO), 142 SMES. See Superconductivity magnetic energy storage SOF stabilization, 64, 65, 108 Solar power, 191, 194, 196, 198, 199, 206 Speed governor, 16, 17, 24, 55, 56, 124, 161, 174, 205 Speed governor dead band, 29, 55, 56 Stability, 1–4, 6–11, 29, 93, 96, 104, 106, 109, 112, 113, 115, 141, 142, 144, 147–149, 155–158, 162, 163, 174, 179, 181, 205 Stability margin, 112 State feedback, 29, 63, 64, 86, 87, 91 State-space dynamic model, 43–44, 96, 109, 130, 148, 155, 157 Static load shedding, 175 Static output feedback (SOF), 63–67, 69–71, 81, 86, 89, 90, 92, 96, 99, 103, 104, 106–110, 112, 121, 130–132 Strictly positive real (SPR), 92, 103, 112 Structured singular value theory, 141, 142, 152, 154, 155, 162, 163 Superconductivity magnetic energy storage (SMES), 30, 204 Supervisory control and data acquisition system (SCADA), 2, 202 Supplementary control, 2, 6, 16, 19, 22–27, 29, 39, 40, 42, 43, 57, 124, 128, 165, 166, 171, 172, 191, 201, 205

218 Supplementary control loop, 16, 22, 43, 57, 172 Synthesis procedure, 10, 11, 58, 59, 85, 112, 113, 141, 143, 147, 150, 157 T Targeted load shedding, 177–178, 180–184 Tie-line bias, 15, 30 power, 10, 20–22, 24, 28, 39, 45, 48, 51, 53–55, 67, 77, 79, 88, 89, 97, 103, 117, 124, 130, 135, 136, 143, 171, 174, 175, 184–186, 202, 204 Time delays, 11, 39, 56–58, 60, 85–99, 103, 107, 108, 113, 115, 117, 121, 130, 133, 135, 171 Timescales, 7–8, 11, 198, 204 Transmission system operator (TSO), 45 U Uncertainties, 2, 7, 10, 11, 29, 39, 58–60, 85, 103, 106–108, 113, 116, 121, 141, 143,

Index 144, 147, 149–151, 153, 156–159, 163, 192, 193 Under-frequency generation trip (UFGT), 168, 169 Under-frequency load shedding (UFLS), 168, 169, 173–183 Under-frequency/over-frequency generation trips, 168 Under-voltage load shedding (UVLS), 6, 8, 174 Union for Coordination of Transmission of Electricity (UCTE), 46, 201, 202 Unmodelled dynamics, 59, 112–113, 143 V Vertically integrated utility (VIU), 30, 45 Voltage instability, 3, 176 Voltage stability, 2, 7–9 W Wide-area measurement (WAM), 85, 99 Wind penetration, 202, 203 Wind power, 192–194, 196, 200–205

Power Electronics and Power Systems Series Editors:

M.A. Pai University of Illinois at Urbana-Champaign Urbana, Illinois

Alex Stankovic Northeastern University Boston, Massachusetts

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