See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/41231911 Power System Dynamics. Stability and Control Article · January 2012 Source: OAI CITATIONS READS 1,020 53,821 3 authors, including: Jan Machowski J.W. Bialek Warsaw University of Technology Durham University 90 PUBLICATIONS 2,497 CITATIONS 102 PUBLICATIONS 4,999 CITATIONS SEE PROFILE All content following this page was uploaded by Jan Machowski on 30 May 2014. The user has requested enhancement of the downloaded file. SEE PROFILE P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come POWER SYSTEM DYNAMICS P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come POWER SYSTEM DYNAMICS Stability and Control Second Edition Jan Machowski Warsaw University of Technology, Poland Janusz W. Bialek Durham University, UK James R. Bumby Durham University, UK P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Reprinted with corrections September 2012. This edition first published 2008 C 2008 John Wiley & Sons, Ltd. Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. 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If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Machowski, Jan. Power system dynamics: stability and control / Jan Machowski, Janusz W. Bialek, James R. Bumby. – 2nd ed. p. cm. Rev. ed. of: Power system dynamics and stability / Jan Machowski, Janusz W. Bialek, James R. Bumby. 1997. Includes bibliographical references and index. ISBN 978-0-470-72558-0 (cloth) 1. Electric power system stability. 2. Electric power systems–Control. I. Bialek, Janusz W. II. Bumby, J. R. (James Richard) III. Title. TK1010.M33 2008 621.319 1–dc22 2008032220 A catalogue record for this book is available from the British Library. ISBN 978-0-470-72558-0 Typeset in 9/11pt Times New Roman by Aptara Inc., New Delhi, India. Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents About the Authors xiii Preface xv Acknowledgements xix List of Symbols xxi PART I INTRODUCTION TO POWER SYSTEMS 1 Introduction 1.1 Stability and Control of a Dynamic System 1.2 Classification of Power System Dynamics 1.3 Two Pairs of Important Quantities: Reactive Power/Voltage and Real Power/Frequency 1.4 Stability of a Power System 1.5 Security of a Power System 1.6 Brief Historical Overview 3 3 5 7 9 9 12 2 Power System Components 2.1 Introduction 2.1.1 Reliability of Supply 2.1.2 Supplying Electrical Energy of Good Quality 2.1.3 Economic Generation and Transmission 2.1.4 Environmental Issues 2.2 Structure of the Electrical Power System 2.2.1 Generation 2.2.2 Transmission 2.2.3 Distribution 2.2.4 Demand 2.3 Generating Units 2.3.1 Synchronous Generators 2.3.2 Exciters and Automatic Voltage Regulators 2.3.3 Turbines and their Governing Systems 2.4 Substations 2.5 Transmission and Distribution Network 2.5.1 Overhead Lines and Underground Cables 2.5.2 Transformers 2.5.3 Shunt and Series Elements 2.5.4 FACTS Devices 15 15 15 16 16 16 16 18 18 19 19 19 20 21 25 35 35 35 36 41 43 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents vi 2.6 Protection 2.6.1 Protection of Transmission Lines 2.6.2 Protection of Transformers 2.6.3 Protection of Busbars 2.6.4 Protection of Generating Units 2.7 Wide Area Measurement Systems 2.7.1 WAMS and WAMPAC Based on GPS Signal 2.7.2 Phasors 2.7.3 Phasor Measurement Unit 2.7.4 Structures of WAMS and WAMPAC 54 54 56 57 57 58 58 59 61 62 3 The Power System in the Steady State 3.1 Transmission Lines 3.1.1 Line Equations and the π -Equivalent Circuit 3.1.2 Performance of the Transmission Line 3.1.3 Underground Cables 3.2 Transformers 3.2.1 Equivalent Circuit 3.2.2 Off-Nominal Transformation Ratio 3.3 Synchronous Generators 3.3.1 Round-Rotor Machines 3.3.2 Salient-Pole Machines 3.3.3 Synchronous Generator as a Power Source 3.3.4 Reactive Power Capability Curve of a Round-Rotor Generator 3.3.5 Voltage–Reactive Power Capability Characteristic V(Q) 3.3.6 Including the Equivalent Network Impedance 3.4 Power System Loads 3.4.1 Lighting and Heating 3.4.2 Induction Motors 3.4.3 Static Characteristics of the Load 3.4.4 Load Models 3.5 Network Equations 3.6 Power Flows in Transmission Networks 3.6.1 Control of Power Flows 3.6.2 Calculation of Power Flows 65 65 66 67 72 72 72 74 76 76 83 89 91 95 100 104 105 106 110 111 113 118 118 122 PART II INTRODUCTION TO POWER SYSTEM DYNAMICS 4 Electromagnetic Phenomena 4.1 Fundamentals 4.2 Three-Phase Short Circuit on a Synchronous Generator 4.2.1 Three-Phase Short Circuit with the Generator on No Load and Winding Resistance Neglected 4.2.2 Including the Effect of Winding Resistance 4.2.3 Armature Flux Paths and the Equivalent Reactances 4.2.4 Generator Electromotive Forces and Equivalent Circuits 4.2.5 Short-Circuit Currents with the Generator Initially on No Load 4.2.6 Short-Circuit Currents in the Loaded Generator 4.2.7 Subtransient Torque 127 127 129 129 133 134 140 146 149 150 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents vii 4.3 Phase-to-Phase Short Circuit 4.3.1 Short-Circuit Current and Flux with Winding Resistance Neglected 4.3.2 Influence of the Subtransient Saliency 4.3.3 Positive- and Negative-Sequence Reactances 4.3.4 Influence of Winding Resistance 4.3.5 Subtransient Torque 4.4 Synchronization 4.4.1 Currents and Torques 4.5 Short-Circuit in a Network and its Clearing 152 153 156 159 160 162 163 164 166 5 Electromechanical Dynamics – Small Disturbances 5.1 Swing Equation 5.2 Damping Power 5.2.1 Damping Power at Large Speed Deviations 5.3 Equilibrium Points 5.4 Steady-State Stability of Unregulated System 5.4.1 Pull-Out Power 5.4.2 Transient Power–Angle Characteristics 5.4.3 Rotor Swings and Equal Area Criterion 5.4.4 Effect of Damper Windings 5.4.5 Effect of Rotor Flux Linkage Variation 5.4.6 Analysis of Rotor Swings Around the Equilibrium Point 5.4.7 Mechanical Analogues of the Generator–Infinite Busbar System 5.5 Steady-State Stability of the Regulated System 5.5.1 Steady-State Power–Angle Characteristic of Regulated Generator 5.5.2 Transient Power–Angle Characteristic of the Regulated Generator 5.5.3 Effect of Rotor Flux Linkage Variation 5.5.4 Effect of AVR Action on the Damper Windings 5.5.5 Compensating the Negative Damping Components 169 169 172 175 176 177 177 179 184 186 187 191 195 196 196 200 202 205 206 6 Electromechanical Dynamics – Large Disturbances 6.1 Transient Stability 6.1.1 Fault Cleared Without a Change in the Equivalent Network Impedance 6.1.2 Short-Circuit Cleared with/without Auto-Reclosing 6.1.3 Power Swings 6.1.4 Effect of Flux Decrement 6.1.5 Effect of the AVR 6.2 Swings in Multi-Machine Systems 6.3 Direct Method for Stability Assessment 6.3.1 Mathematical Background 6.3.2 Energy-Type Lyapunov Function 6.3.3 Transient Stability Area 6.3.4 Equal Area Criterion 6.3.5 Lyapunov Direct Method for a Multi-Machine System 6.4 Synchronization 6.5 Asynchronous Operation and Resynchronization 6.5.1 Transition to Asynchronous Operation 6.5.2 Asynchronous Operation 6.5.3 Possibility of Resynchronization 207 207 207 212 215 215 216 220 222 223 225 227 228 230 237 239 240 241 242 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come viii Contents 6.6 Out-of-Step Protection Systems 6.6.1 Impedance Loci During Power Swings 6.6.2 Power Swing Blocking 6.6.3 Pole-Slip Protection of Synchronous Generator 6.6.4 Out-of-Step Tripping in a Network 6.6.5 Example of a Blackout 6.7 Torsional Oscillations in the Drive Shaft 6.7.1 The Torsional Natural Frequencies of the Turbine–Generator Rotor 6.7.2 Effect of System Faults 6.7.3 Subsynchronous Resonance 244 245 248 249 251 253 253 253 259 261 7 Wind Power 7.1 Wind Turbines 7.1.1 Generator Systems 7.2 Induction Machine Equivalent Circuit 7.3 Induction Generator Coupled to the Grid 7.4 Induction Generators with Slightly Increased Speed Range via External Rotor Resistance 7.5 Induction Generators with Significantly Increased Speed Range: DFIGs 7.5.1 Operation with the Injected Voltage in Phase with the Rotor Current 7.5.2 Operation with the Injected Voltage out of Phase with the Rotor Current 7.5.3 The DFIG as a Synchronous Generator 7.5.4 Control Strategy for a DFIG 7.6 Fully Rated Converter Systems: Wide Speed Control 7.6.1 Machine-Side Inverter 7.6.2 Grid-Side Inverter 7.7 Peak Power Tracking of Variable Speed Wind Turbines 7.8 Connections of Wind Farms 7.9 Fault Behaviour of Induction Generators 7.9.1 Fixed-Speed Induction Generators 7.9.2 Variable-Speed Induction Generators 7.10 Influence of Wind Generators on Power System Stability 265 265 269 274 277 280 282 284 286 287 289 290 291 292 293 294 294 294 296 296 8 Voltage Stability 8.1 Network Feasibility 8.1.1 Ideally Stiff Load 8.1.2 Influence of the Load Characteristics 8.2 Stability Criteria 8.2.1 The dQ/dV Criterion 8.2.2 The dE/dV Criterion 8.2.3 The dQG /dQL Criterion 8.3 Critical Load Demand and Voltage Collapse 8.3.1 Effects of Increasing Demand 8.3.2 Effect of Network Outages 8.3.3 Influence of the Shape of the Load Characteristics 8.3.4 Influence of the Voltage Control 8.4 Static Analysis 8.4.1 Voltage Stability and Load Flow 8.4.2 Voltage Stability Indices 299 299 300 303 305 305 308 309 310 311 314 315 317 318 318 320 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents ix 8.5 Dynamic Analysis 8.5.1 The Dynamics of Voltage Collapse 8.5.2 Examples of Power System Blackouts 8.5.3 Computer Simulation of Voltage Collapse 8.6 Prevention of Voltage Collapse 8.7 Self-Excitation of a Generator Operating on a Capacitive Load 8.7.1 Parametric Resonance in RLC Circuits 8.7.2 Self-Excitation of a Generator with Open-Circuited Field Winding 8.7.3 Self-Excitation of a Generator with Closed Field Winding 8.7.4 Practical Possibility of Self-Excitation 321 321 323 326 327 329 329 330 332 334 9 Frequency Stability and Control 9.1 Automatic Generation Control 9.1.1 Generation Characteristic 9.1.2 Primary Control 9.1.3 Secondary Control 9.1.4 Tertiary Control 9.1.5 AGC as a Multi-Level Control 9.1.6 Defence Plan Against Frequency Instability 9.1.7 Quality Assessment of Frequency Control 9.2 Stage I – Rotor Swings in the Generators 9.3 Stage II – Frequency Drop 9.4 Stage III – Primary Control 9.4.1 The Importance of the Spinning Reserve 9.4.2 Frequency Collapse 9.4.3 Underfrequency Load Shedding 9.5 Stage IV – Secondary Control 9.5.1 Islanded Systems 9.5.2 Interconnected Systems and Tie-Line Oscillations 9.6 FACTS Devices in Tie-Lines 9.6.1 Incremental Model of a Multi-Machine System 9.6.2 State-Variable Control Based on Lyapunov Method 9.6.3 Example of Simulation Results 9.6.4 Coordination Between AGC and Series FACTS Devices in Tie-Lines 335 336 336 339 341 345 346 347 349 350 353 354 356 358 360 360 361 364 370 371 375 378 379 10 Stability Enhancement 10.1 Power System Stabilizers 10.1.1 PSS Applied to the Excitation System 10.1.2 PSS Applied to the Turbine Governor 10.2 Fast Valving 10.3 Braking Resistors 10.4 Generator Tripping 10.4.1 Preventive Tripping 10.4.2 Restitutive Tripping 10.5 Shunt FACTS Devices 10.5.1 Power–Angle Characteristic 10.5.2 State-Variable Control 10.5.3 Control Based on Local Measurements 10.5.4 Examples of Controllable Shunt Elements 10.5.5 Generalization to Multi-Machine Systems 10.5.6 Example of Simulation Results 383 383 384 387 387 391 392 393 394 395 395 397 400 404 406 414 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents x 10.6 Series Compensators 10.6.1 State-Variable Control 10.6.2 Interpretation Using the Equal Area Criterion 10.6.3 Control Strategy Based on the Squared Current 10.6.4 Control Based on Other Local Measurements 10.6.5 Simulation Results 10.7 Unified Power Flow Controller 10.7.1 Power–Angle Characteristic 10.7.2 State-Variable Control 10.7.3 Control Based on Local Measurements 10.7.4 Examples of Simulation Results PART III 416 417 419 420 421 423 423 424 426 428 429 ADVANCED TOPICS IN POWER SYSTEM DYNAMICS 11 Advanced Power System Modelling 11.1 Synchronous Generator 11.1.1 Assumptions 11.1.2 The Flux Linkage Equations in the Stator Reference Frame 11.1.3 The Flux Linkage Equations in the Rotor Reference Frame 11.1.4 Voltage Equations 11.1.5 Generator Reactances in Terms of Circuit Quantities 11.1.6 Synchronous Generator Equations 11.1.7 Synchronous Generator Models 11.1.8 Saturation Effects 11.2 Excitation Systems 11.2.1 Transducer and Comparator Model 11.2.2 Exciters and Regulators 11.2.3 Power System Stabilizer (PSS) 11.3 Turbines and Turbine Governors 11.3.1 Steam Turbines 11.3.2 Hydraulic Turbines 11.3.3 Wind Turbines 11.4 Dynamic Load Models 11.5 FACTS Devices 11.5.1 Shunt FACTS Devices 11.5.2 Series FACTS Devices 433 433 434 434 436 440 443 446 453 458 462 462 463 470 470 471 476 481 485 488 488 488 12 Steady-State Stability of Multi-Machine System 12.1 Mathematical Background 12.1.1 Eigenvalues and Eigenvectors 12.1.2 Diagonalization of a Square Real Matrix 12.1.3 Solution of Matrix Differential Equations 12.1.4 Modal and Sensitivity Analysis 12.1.5 Modal Form of the State Equation with Inputs 12.1.6 Nonlinear System 12.2 Steady-State Stability of Unregulated System 12.2.1 State-Space Equation 12.2.2 Simplified Steady-State Stability Conditions 12.2.3 Including the Voltage Characteristics of the Loads 12.2.4 Transfer Capability of the Network 491 491 491 496 500 509 512 513 514 515 517 521 522 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Contents xi 12.3 Steady-State Stability of the Regulated System 12.3.1 Generator and Network 12.3.2 Including Excitation System Model and Voltage Control 12.3.3 Linear State Equation of the System 12.3.4 Examples 523 523 525 528 528 13 Power System Dynamic Simulation 13.1 Numerical Integration Methods 13.2 The Partitioned Solution 13.2.1 Partial Matrix Inversion 13.2.2 Matrix Factorization 13.2.3 Newton’s Method 13.2.4 Ways of Avoiding Iterations and Multiple Network Solutions 13.3 The Simultaneous Solution Methods 13.4 Comparison Between the Methods 535 536 541 543 547 548 551 553 554 14 Power System Model Reduction – Equivalents 14.1 Types of Equivalents 14.2 Network Transformation 14.2.1 Elimination of Nodes 14.2.2 Aggregation of Nodes Using Dimo’s Method 14.2.3 Aggregation of Nodes Using Zhukov’s Method 14.2.4 Coherency 14.3 Aggregation of Generating Units 14.4 Equivalent Model of External Subsystem 14.5 Coherency Recognition 14.6 Properties of Coherency-Based Equivalents 14.6.1 Electrical Interpretation of Zhukov’s Aggregation 14.6.2 Incremental Equivalent Model 14.6.3 Modal Interpretation of Exact Coherency 14.6.4 Eigenvalues and Eigenvectors of the Equivalent Model 14.6.5 Equilibrium Points of the Equivalent Model 557 557 559 559 562 563 565 567 568 569 573 573 575 579 582 589 Appendix 593 References 613 Index 623 P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come About the Authors Professor Jan Machowski received his MSc and PhD degrees in Electrical Engineering from Warsaw University of Technology in 1974 and 1979, respectively. After obtaining field experience in the Dispatching Centre and several power plants, he joined the Electrical Faculty of Warsaw University of Technology where presently he is employed as a Professor and Director of the Power Engineering Institute. His areas of interest are electrical power systems, power system protection and control. In 1989–93 Professor Machowski was a Visiting Professor at Kaiserslautern University in Germany where he carried out two research projects on power swing blocking algorithms for distance protection and optimal control of FACTS devices. Professor Machowski is the co-author of three books published in Polish: Power System Stability (WNT, 1989), Short Circuits in Power Systems (WNT, 2002) and Power System Control and Stability (WPW, 2007). He is also a co-author of Power System Dynamics and Stability published by John Wiley & Sons, Ltd (1997). Professor Machowski is the author and co-author of 42 papers published in English in international fora. He has carried out many projects on electrical power systems, power system stability and power system protection commissioned by the Polish Power Grid Company, Electric Power Research Institute in the United States, Electroinstitut Milan Vidmar in Slovenia and Ministry of Science and Higher Education of Poland. Professor Janusz Bialek received his MEng and PhD degrees in Electrical Engineering from Warsaw University of Technology in 1977 and 1981, respectively. From 1981 to 1989 he was a lecturer with Warsaw University of Technology. Currently he holds the Chair of Electrical Power and Control at Durham University having previously (2003– 2008) held Bert Whittington Chair of Electrical Engineering at the University of Edinburg. Janusz is Fellow of Institute of Electrical and Electronics Engineers (IEEE) and Honorary Professor of Heriot-Watt University, UK. His research deals with achieving stable, secure, sustainable and economic supply of electricity while meeting the challenges of reducing CO2 emissions. His particular expertise is in technical and economic integration of renewable generation in the power system, in preventing electricity blackouts and in analysis of power system dynamics. He has published 2 books and about 130 research papers. He has been a consultant to the UK government, Scottish Government, European Commission, Elexon, Polish Power Grid Company, Scottish Power and Enron. He has been the Principal Investigator of a number of major research grants funded by the Engineering and Physical Sciences Research Council (EPSRC) and Electrical Power Research Institute (EPRI) in USA. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 xiv 13:39 Printer Name: Yet to Come About the Authors Dr Jim Bumby received his BSc and PhD degrees in Engineering from Durham University, United Kingdom, in 1970 and 1974, respectively. From 1973 to 1978 he worked for the International Research and Development Company, Newcastle-upon-Tyne, on superconducting machines, hybrid vehicles and sea-wave energy. Since 1978 he has worked in the School of Engineering at Durham University where he is currently Reader in Electrical Engineering. He has worked in the area of electrical machines and systems for over 30 years, first in industry and then in academia. Dr Bumby is the author or co-author of over 100 technical papers and two books in the general area of electrical machines/power systems and control. He has also written numerous technical reports for industrial clients. These papers and books have led to the award of a number of national and international prizes including the Institute of Measurement and Control prize for the best transactions paper in 1988 for work on hybrid electric vehicles and the IEE Power Division Premium in 1997 for work on direct drive permanent magnet generators for wind turbine applications. His current research interests are in novel generator technologies and their associated control for new and renewable energy systems. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Preface In 1997 the authors of this book, J. Machowski, J.W. Bialek and J.R. Bumby, published a book entitled Power System Dynamics and Stability. That book was well received by readers who told us that it was used regularly as a standard reference text both in academia and in industry. Some 10 years after publication of that book we started work on a second edition. However, we quickly realized that the developments in the power systems industry over the intervening years required a large amount of new material. Consequently the book has been expanded by about a third and the word Control in the new title, Power System Dynamics: Stability and Control, reflects the fact that a large part of the new material concerns power system control: flexible AC transmission systems (FACTS), wide area measurement systems (WAMS), frequency control, voltage control, etc. The new title also reflects a slight shift in focus from solely describing power system dynamics to the means of dealing with them. For example, we believe that the new WAMS technology is likely to revolutionize power system control. One of the main obstacles to a wider embrace of WAMS by power system operators is an acknowledged lack of algorithms which could be utilized to control a system in real time. This book tries to fill this gap by developing a number of algorithms for WAMS-based real-time control. The second reason for adding so much new material is the unprecedented change that has been sweeping the power systems industry since the 1990s. In particular the rapid growth of renewable generation, driven by global warming concerns, is changing the fundamental characteristics of the system. Currently wind power is the dominant renewable energy source and wind generators usually use induction, rather than synchronous, machines. As a significant penetration of such generation will change the system dynamics, the new material in Chapter 7 is devoted entirely to wind generation. The third factor to be taken into account is the fallout from a number of highly publicized blackouts that happened in the early years of the new millennium. Of particular concern were the autumn 2003 blackouts in the United States/Canada, Italy, Sweden/Denmark and the United Kingdom, the 2004 blackout in Athens and the European disturbance on 4 November 2006. These blackouts have exposed a number of critical issues, especially those regarding power system behaviour at depressed voltages. Consequently, the book has been extended to cover these phenomena together with an illustration of some of the blackouts. It is important to emphasize that the new book is based on the same philosophy as the previous one. We try to answer some of the concerns about the education of power system engineers. With the widespread access to powerful computers running evermore sophisticated simulation packages, there is a tendency to treat simulation as a substitute for understanding. This tendency is especially dangerous for students and young researchers who think that simulation is a panacea for everything and always provides a true answer. What they do not realize is that, without a physical understanding of the underlying principles, they cannot be confident in understanding, or validating, the simulation results. It is by no means bad practice to treat the initial results of any computer software with a healthy pinch of scepticism. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 xvi 13:39 Printer Name: Yet to Come Preface Power system dynamics are not easy to understand. There are a number of good textbooks which deal with this topic and some of these are reviewed in Chapter 1. As the synchronous machine plays a decisive role in determining the dynamic response of the system, many of these books start with a detailed mathematical treatment of the synchronous generator in order to introduce Park’s equations and produce a mathematical model of the generator. However, it is our experience that to begin a topic with such a detailed mathematical treatment can put many students off further study because they often find it difficult to see any practical relevance for the mathematics. This can be a major obstacle for those readers who are more practically inclined and who want to understand what is happening in the system without having to refer continuously to a complicated mathematical model of the generator. Our approach is different. We first try to give a qualitative explanation of the underlying physical phenomena of power system dynamics using a simple model of the generator, coupled with the basic physical laws of electrical engineering. Having provided the student with a physical understanding of power system dynamics, we then introduce the full mathematical model of the generator, followed by more advanced topics such as system reduction, dynamic simulation and eigenvalue analysis. In this way we hope that the material is made more accessible to the reader who wishes to understand the system operation without first tackling Park’s equations. All our considerations are richly illustrated by diagrams. We strongly believe in the old adage that an illustration is worth a thousand words. In fact, our book contains over 400 diagrams. The book is conveniently divided into three major parts. The first part (Chapters 1–3) reviews the background for studying power system dynamics. The second part (Chapters 4–10) attempts to explain the basic phenomena underlying power system dynamics using the classical model of the generator–infinite busbar system. The third part (Chapters 11–14) tackles some of the more advanced topics suitable for the modelling and dynamic simulation of large-scale power systems. Examining the chapters and the new material added in more detail, Chapter 1 classifies power system dynamics and provides a brief historical overview. The new material expands on the definitions of power system stability and security assessment and introduces some important concepts used in later chapters. Chapter 2 contains a brief description of the major power system components, including modern FACTS devices. The main additions here provide a more comprehensive treatment of FACTS devices and a whole new section on WAMS. Chapter 3 introduces steady-state models and their use in analysing the performance of the power system. The new material covers enhanced treatment of the generator as the reactive power source introducing voltage–reactive power capability characteristics. We believe that this is a novel treatment of those concepts since we have not seen it anywhere else. The importance of understanding how the generator and its controls behave under depressed voltages has been emphasized by the wide area blackouts mentioned above. The chapter also includes a new section on controlling power flows in the network. Chapter 4 analyses the dynamics following a disturbance and introduces models suitable for analysing the dynamic performance of the synchronous generator. Chapter 5 explains the power system dynamics following a small disturbance (steady-state stability) while Chapter 6 examines the system dynamics following a large disturbance (transient stability). There are new sections on using the Lyapunov direct method to analyse the stability of a multi-machine power system and on out-of-step relaying. Chapter 7 is all new and covers the fundamentals of wind power generation. Chapter 8 has been greatly expanded and provides an explanation of voltage stability together with some of the methods used for stability assessment. The new material includes examples of power system blackouts, methods of preventing voltage collapse and a large new section on self-excitation of the generator. Chapter 9 contains a largely enhanced treatment of frequency stability and control including defence plans against frequency instability and quality assessment of frequency control. There is a large new section which covers a novel treatment of interaction between automatic generation control (AGC) and FACTS devices installed in tie-lines that control the flow of power between systems in an interconnected network. Chapter 10 provides an overview of the main methods of stability enhancement, both conventional and using FACTS devices. The new material P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 Preface 13:39 Printer Name: Yet to Come xvii includes the use of braking resistors and a novel generalization of earlier derived stabilization algorithms to a multi-machine power system. Chapter 11 introduces advanced models of the different power system elements. The new material includes models of the wind turbine and generator and models of FACTS devices. Chapter 12 contains a largely expanded treatment of the steady-state stability of multi-machine power systems using eigenvalue analysis. We have added a comprehensive explanation of the meaning of eigenvalues and eigenvectors including a fuller treatment of the mathematical background. As the subject matter is highly mathematical and may be difficult to understand, we have added a large number of numerical examples. Chapter 13 contains a description of numerical methods used for power system dynamic simulation. Chapter 14 explains how to reduce the size of the simulation problem by using equivalents. The chapter has been significantly expanded by adding novel material on the modal analysis of equivalents and a number of examples. The Appendix covers the per-unit system and new material on the mathematical fundamentals of solving ordinary differential equations. It is important to emphasize that, while most of the book is a teaching textbook written with finalyear undergraduate and postgraduate students in mind, there are also large parts of material which constitute cutting-edge research, some of it never published before. This includes the use of the Lyapunov direct method to derive algorithms for the stabilization of a multi-machine power system (Chapters 6, 9 and 10) and derivation of modal-analysis-based power system dynamic equivalents (Chapter 14). J. Machowski, J.W. Bialek and J.R. Bumby Warsaw, Edinburgh and Durham P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come Acknowledgements We would like to acknowledge the financial support of Supergen FutureNet (www.super gennetworks.org.uk). Supergen is funded by the Research Councils’ Energy Programme, United Kingdom. We would also like to acknowledge the financial support of the Ministry of Science and Higher Education of Poland (grant number 3 T10B 010 29). Both grants have made possible the cooperation between the Polish and British co-authors. Last but not least, we are grateful as ever for the patience shown by our wives and families during the torturous writing of yet another book. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come List of Symbols Notation Italic type denotes scalar physical quantity (e.g. R, L, C) or numerical variable (e.g. x, y). Phasor or complex quantity or numerical variable is underlined (e.g. I, V, S). Italic with arrow on top of a symbol denotes a spatial vector (e.g. F). Italic boldface denotes a matrix or a vector (e.g. A, B, x, y). Unit symbols are written using roman type (e.g. Hz, A, kV). Standard mathematical functions are written using roman type (e.g. e, sin, cos, arctan). Numbers are written using roman type (e.g. 5, 6). Mathematical operators are written using roman type (e.g. s, Laplace operator; T, matrix transposition; j, angular shift by 90◦ ; a, angular shift by 120◦ ). Differentials and partial differentials are written using roman type (e.g. d f/dx, ∂ f/∂ x). Symbols describing objects are written using roman type (e.g. TRAFO, LINE). Subscripts relating to objects are written using roman type (e.g. I TRAFO , I LINE ). Subscripts relating to physical quantities or numerical variables are written using italic type (e.g. Ai j , xk ). Subscripts A, B, C refer to the three-phase axes of a generator. Subscripts d, q refer to the direct- and quadrature-axis components. Lower case symbols normally denote instantaneous values (e.g. v, i ). Upper case symbols normally denote rms or peak values (e.g. V, I). Symbols a and a2 Bµ Bsh D Ek Ep ef eq ed eq ed operators shifting the angle by 120◦ and 240◦ , respectively. magnetizing susceptance of a transformer. susceptance of a shunt element. damping coefficient. kinetic energy of the rotor relative to the synchronous speed. potential energy of the rotor with respect to the equilibrium point. field voltage referred to the fictitious q-axis armature coil. steady-state emf induced in the fictitious q-axis armature coil proportional to the field winding self-flux linkages. transient emf induced in the fictitious d-axis armature coil proportional to the flux linkages of the q-axis coil representing the solid steel rotor body (round-rotor generators only). transient emf induced in the fictitious q-axis armature coil proportional to the field winding flux linkages. subtransient emf induced in the fictitious d-axis armature coil proportional to the total q-axis rotor flux linkages (q-axis damper winding and q-axis solid steel rotor body). P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 xxii eq E Ef Efm Ed Eq E Ed Eq E Ed Eq Er Er m EG f fn F Fa Fa AC Fa DC Fad , Faq Ff G Fe G sh Hii , Hi j iA, iB, iC i A DC , i B DC , i C DC i A AC , i B AC , i C AC id, iq iD, iQ if i ABC i fDQ i 0dq I Id , Iq I S, I R I R, I E 13:39 Printer Name: Yet to Come List of Symbols subtransient emf induced in the fictitious q-axis armature coil proportional to the total d-axis rotor flux linkages (d-axis damper winding and field winding). steady-state internal emf. excitation emf proportional to the excitation voltage Vf . peak value of the excitation emf. d-axis component of the steady-state internal emf proportional to the rotor selflinkages due to currents induced in the q-axis solid steel rotor body (round-rotor generators only). q-axis component of the steady-state internal emf proportional to the field winding self-flux linkages (i.e. proportional to the field current itself). transient internal emf proportional to the flux linkages of the field winding and solid steel rotor body (includes armature reaction). d-axis component of the transient internal emf proportional to flux linkages in the q-axis solid steel rotor body (round-rotor generators only). q-axis component of the transient internal emf proportional to the field winding flux linkages. subtransient internal emf proportional to the total rotor flux linkages (includes armature reaction). d-axis component of the subtransient internal emf proportional to the total flux linkages in the q-axis damper winding and q-axis solid steel rotor body. q-axis component of the subtransient internal emf proportional to the total flux linkages in the d-axis damper winding and the field winding. resultant air-gap emf. amplitude of the resultant air-gap emf. vector of the generator emfs. mains frequency. rated frequency. magnetomotive force (mmf) due to the field winding. armature reaction mmf. AC armature reaction mmf (rotating). DC armature reaction mmf (stationary). d- and q-axis components of the armature reaction mmf. resultant mmf. core loss conductance of a transformer. conductance of a shunt element. self- and mutual synchronizing power. instantaneous currents in phases A, B and C. DC component of the current in phases A, B, C. AC component of the current in phases A, B, C. currents flowing in the fictitious d- and q-axis armature coils. instantaneous d- and q-axis damper winding current. instantaneous field current of a generator. vector of instantaneous phase currents. vector of instantaneous currents in the field winding and the d- and q-axis damper windings. vector of armature currents in the rotor reference frame. armature current. d- and q-axis component of the armature current. currents at the sending and receiving end of a transmission line. vector of complex current injections to the retained and eliminated nodes. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 List of Symbols I G, I L I L J j kPV , kQV kPf , kQf K Eq K Eq K E Ki KL KT l LAA , LBB , LCC , Lff , LDD , LQQ Ld , Lq Ld , Lq , Ld , Lq LS Lxy LS LR LS LSR , LRS M Mf , MD , MQ N p Pacc PD Pe PEq cr PEq (δ), PE (δ ), PEq (δ ) Pg PL Pm Pn PR PrI , PrII , PrIII , PrIV 13:39 Printer Name: Yet to Come xxiii vector of complex generator and load currents. vector of load corrective complex currents. moment of inertia. operator shifting the angle by 90◦ . voltage sensitivities of the load (the slopes of the real and reactive power demand characteristics as a function of voltage). frequency sensitivities of the load (the slopes of the real and reactive power demand characteristics as a function of frequency). steady-state synchronizing power coefficient (the slope of the steady-state power angle curve PEq (δ)). transient synchronizing power coefficient (the slope of the transient power angle curve PEq (δ )). transient synchronizing power coefficient (the slope of the transient power angle curve PE (δ )). reciprocal of droop for the i th generating unit. frequency sensitivity coefficient of the system real power demand. reciprocal of droop for the total system generation characteristic. length of a transmission line. self-inductances of the windings of the phase windings A, B, C, the field winding, and the d-and the q-axis damper winding. inductances of the fictitious d- and q-axis armature windings. d- and q-axis transient and subtransient inductances. minimum value of the self-inductance of a phase winding. where x, y ∈ {A, B, C, D, Q, f} and x = y, are the mutual inductances between the windings denoted by the indices as described above. amplitude of the variable part of the self-inductance of a phase winding. submatrix of the rotor self- and mutual inductances. submatrix of the stator self- and mutual inductances. submatrices of the stator-to-rotor and rotor-to-stator mutual inductances. coefficient of inertia. amplitude of the mutual inductance between a phase winding and, respectively, the field winding and the d- and the q-axis damper winding. generally, number of any objects. number of poles. accelerating power. damping power. electromagnetic air-gap power. critical (pull-out) air-gap power developed by a generator. air-gap power curves assuming Eq = constant, E = constant and Eq = constant. in induction machine, real power supplied from the grid (motoring mode), or supplied to the grid (generating mode). real power absorbed by a load or total system load. mechanical power supplied by a prime mover to a generator; also mechanical power supplied by a motor to a load (induction machine in motoring mode). real power demand at rated voltage. real power at the receiving end of a transmission line. contribution of the generating units remaining in operation to covering the real power imbalance during the first, second, third and fourth stages of load frequency control. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 xxiv PsI , PsII , PsIII , PsIV Ps PS PSIL PsEq (δ) PT Ptie PVg (δ) PVg cr QL QG Qn QR QS R r RA , RB , RC , RD , RQ , Rf RABC RfDQ s s scr Sn SSHC t Td , Td , Tdo Tdo Tq , Tq , Tqo Tqo Ta T vA , vB , vC , vf vd , vq vw vABC vfDQ V Vcr Vd , Vq Vf Vg 13:39 Printer Name: Yet to Come List of Symbols contribution of the system to covering the real power imbalance during the first, second, third and fourth stages of load frequency control. stator power of induction machine or power supplied by the system. real power at the sending end of a transmission line or real power supplied by a source to a load or real power supplied to an infinite busbar. surge impedance (natural) load. curve of real power supplied to an infinite busbar assuming Eq = constant. total power generated in a system. net tie-line interchange power. air-gap power curve assuming Vg = constant. critical value of PVg (δ). reactive power absorbed by a load. reactive power generated by a source (the sum of QL and the reactive power loss in the network). reactive power demand at rated voltage. reactive power at the receiving end of a transmission line. reactive power at the sending end of a transmission line or reactive power supplied by a source to a load. resistance of the armature winding of a generator. total resistance between (and including) a generator and an infinite busbar. resistances of the phase windings A, B, C, the d- and q-axis damper winding, and the field winding. diagonal matrix of phase winding resistances. diagonal matrix of resistances of the field winding and the d- and q-axis damper windings. Laplace operator. slip of induction motor. critical slip of induction motor. rated apparent power. short-circuit power. time. short-circuit d-axis transient and subtransient time constants. open-circuit d-axis transient and subtransient time constants. short-circuit q-axis transient and subtransient time constants. open-circuit q-axis transient and subtransient time constants. armature winding time constant. transformation matrix between network (a, b) and generator (d, q) coordinates. instantaneous voltages across phases A, B, C and the field winding. voltages across the fictitious d- and q-axis armature coils. wind speed. vector of instantaneous voltages across phases A, B, C. vector of instantaneous voltages across the field winding and the d- and q-axis damper windings. Lyapunov function. critical value of the voltage. direct- and quadrature-axis component of the generator terminal voltage. voltage applied to the field winding. voltage at the generator terminals. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 List of Symbols Vs Vsd , Vsq VS, VR Vsh V i = Vi δi VR, VE W W W, U Xa XC XD Xd , Xd , Xd xd , xd , xd xd PRE , xd F , xd POST Xf Xl Xq , Xq , Xq xq , xq , xq XSHC YT Y YGG , YLL , YLG , YLG Yi j = G i j + jBi j YRR , YEE , YRE , YER Zc Zs = Rs + jXs ZT = RT + jXT β γ γ0 δ δg δ̂s δ δ fr ω ε ζ ϑ λR λi = αi + j ρ ρT i 13:39 Printer Name: Yet to Come xxv infinite busbar voltage. direct- and quadrature-axis component of the infinite busbar voltage. voltage at the sending and receiving end of a transmission line. local voltage at the point of installation of a shunt element. complex voltage at node i . vector of complex voltages at the retained and eliminated nodes. work. Park’s modified transformation matrix. modal matrices of right and left eigenvectors. armature reaction reactance (round-rotor generator). reactance of a series compensator. reactance corresponding to the flux path around the damper winding. d-axis synchronous, transient and subtransient reactance. total d-axis synchronous, transient and subtransient reactance between (and including) a generator and an infinite busbar. prefault, fault and postfault value of xd . reactance corresponding to the flux path around the field winding. armature leakage reactance of a generator. q-axis synchronous, transient and subtransient reactance. total q-axis synchronous, transient and subtransient reactance between (and including) a generator and an infinite busbar. short-circuit reactance of a system as seen from a node. admittance of a transformer. admittance matrix. admittance submatrices where subscript G corresponds to fictitious generator nodes and subscript L corresponds to all the other nodes (including generator terminal nodes). element of the admittance matrix. complex admittance submatrices where subscript E refers to eliminated nodes and subscript R to retained nodes. characteristic impedance of a transmission line. internal impedance of an infinite busbar. series impedance of a transformer. phase constant of a transmission line. instantaneous position of the generator d-axis relative to phase A; propagation constant of a transmission line. position of the generator d-axis at the instant of fault. power (or rotor) angle with respect to an infinite busbar. power (or rotor) angle with respect to the voltage at the generator terminals. stable equilibrium value of the rotor angle. transient power (or rotor) angle between E and Vs . angle between the resultant and field mmfs. rotor speed deviation equal to (ω − ωs ). rotor acceleration. damping ratio. transformation ratio. frequency bias factor. eigenvalue. static droop of the turbine–governor characteristic. droop of the total system generation characteristic. P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come List of Symbols xxvi τe τm τω τ 2ω τ d, τ q τ R, τ r ϕg a ad , aq a AC a DC f A , B , C AA , BB , B a AC r a DC r a r D , Q d , q f fa fA , fB , fC ABC fDQ electromagnetic torque. mechanical torque. fundamental-frequency subtransient electromagnetic torque. double-frequency subtransient electromagnetic torque. direct- and quadrature-axis component of the electromagnetic torque. subtransient electromagnetic torque due to stator and rotor resistances. power factor angle at the generator terminals. armature reaction flux. d- and q-axis component of the armature reaction flux. AC armature reaction flux (rotating). DC armature reaction flux (stationary). excitation (field) flux. total flux linkage of phases A, B, C. self-flux linkage of phases A, B, C. rotor flux linkages produced by a AC . rotor flux linkages produced by a DC . rotor flux linkages produced by the total armature reaction flux. total flux linkage of damper windings in axes d and q. total d- and q-axis flux linkages. total flux linkage of the field winding. excitation flux linkage with armature winding. excitation flux linkage with phases A, B and C. vector of phase flux linkages. vector of flux linkages of the field winding and the d- and q-axis damper windings. vector of armature flux linkages in the rotor reference frame. angular velocity of the generator (in electrical radians). synchronous angular velocity in electrical radians (equal to 2π f ). rotor speed of wind turbine (in rad/s) frequency of rotor swings (in rad/s) rotation matrix. reluctance. reluctance along the direct- and quadrature-axis. 0dq ω ωs ωT d , q Abbreviations AC ACE AGC AVR BEES d DC DFIG DFIM DSA emf EMS FACTS HV HAWT alternating current area control error Automatic Generation Control Automatic Voltage Regulator Battery Energy Storage System direct axis of a generator direct current Doubly Fed Induction Generator Double Fed Induction Machine Dynamic Security Assessment electro-motive force Energy Management System Flexible AC Transmission Systems high voltage Horizontal-Axis Wind Turbine P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 13:39 Printer Name: Yet to Come List of Symbols IGTB IGTC LFC mmf MAWS PMU PSS pu q rms rpm rhs SCADA SIL SMES SSSC STATCOM SVC TCBR TCPAR TSO VAWT UPFC WAMS WAMPAC insulated gate bipolar transistor integrated gate-commutated thyristor load frequency control magneto-motive force mean annual wind speed Phasor Measurement Unit power system stabiliser per unit quadrature axis of a generator root-mean-square revolutions per minute right-hand-side Supervisory Control and Data Acquisition surge impedance load superconducting magnetic energy storage Static Synchronous Series Compensator static compensator Static VAR Compensator Thyristor Controlled Braking Resistor Thyristor Controlled Phase Angle Regulator Transmission System Operator Vertical-Axis Wind Turbine unified power flow controller Wide Area Measurement System Wide Area Measurement, Protection and Control xxvii P1: OTE/OTE/SPH P2: OTE fm JWBK257/Machowski August 21, 2012 View publication stats 13:39 Printer Name: Yet to Come