Table Of ContentComputational Techniques
for Voltage Stability
Assessment and Control
POWER ELECTRONICS AND POWER SYSTEMS
Series Editors
M. A. Pai and Alex Stankovic
Other books in the series:
REAL-TIME STABILITY IN POWER SYSTEMS:
Techniques for Early Detection of the Risk of Blackout
Savu C. Savulescu, ISBN 0-387-25626-1
ROBUST CONTROL IN POWER SYSTEMS
Bikash Pal, Balarko Chaudhuri, ISBN 0-387-25949-X
APPLIED MATHEMATICS FOR RESTRUCTURED ELECTRIC POWER SYSTEMS:
Optimization, Control, and Computational Intelligence
Joe H. Chow, Felix F. Wu, James A. Momoh, ISBN 0-387-23470-5
HVDC and FACTS CONTROLLERS: Applications of Static Converters in Power Systems
Vijay K. Sood, ISBN 1-4020-7890-0
POWER QUALITY ENHANCEMENT USING CUSTOM POWER DEVICES
Arindam Ghosh, Gerard Ledwich, ISBN 1-4020-7180-9
COMPUTATIONAL METHODS FOR LARGE SPARSE POWER SYSTEMS ANALYSIS:
An Object Oriented Approach
S. A. Soman, S. A. Khaparde, and Shubha Pandit, ISBN 0-7923-7591-2
OPERATION OF RESTRUCTURED POWER SYSTEMS
Kankar Bhattacharya, Math HJ. Bollen and Jaap E. Daalder, ISBN 0-7923-7397-9
TRANSIENT STABILITY OF POWER SYSTEMS: A Unified Approach to
Assessment and Control
Mania Pavella, Damien Ernst and Daniel Ruiz-Vega, ISBN 0-7923-7963-2
MAINTENANCE SCHEDULING IN RESTRUCTURED POWER SYSTEMS
M. Shahidehpour and M. Marwali, ISBN 0-7923-7872-5
POWER SYSTEM OSCILLATIONS
Graham Rogers, ISBN 0-7923-7712-5
STATE ESTIMATION IN ELECTRIC POWER SYSTEMS: A Generalized Approach
A. Monticelli, ISBN 0-7923-8519-5
COMPUTATIONAL AUCTION MECHANISMS FOR RESTRUCTURED POWER
INDUSTRY OPERATIONS
Gerald B. Sheble, ISBN 0-7923-8475-X
ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS
K.R. Padiyar, ISBN 0-7923-8319-2
POWER SYSTEMS RESTRUCTURING: Engineering and Economics
Marija Ilic, Francisco Galiana, and Lester Fink, ISBN 0-7923-8163-7
CRYOGENIC OPERATION OF SILICON POWER DEVICES
Ranbir Singh and B. Jayant Baliga, ISBN 0-7923-8157-2
VOLTAGE STABILITY OF ELECTRIC POWER SYSTEMS
Thierry Van Cutsem and Costas Voumas, ISBN 0-7923-8139-4
AUTOMATIC LEARNING TECHNIQUES IN POWER SYSTEMS
Louis A. Wehenkel, ISBN 0-7923-8068-1
ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY
M. A. Pai, ISBN 0-7923-9035-0
ELECTROMAGNETIC MODELLING OF POWER ELECTRONIC CONVERTERS
J. A. Ferreira, ISBN 0-7923-9034-2
SPOT PRICING OF ELECTRICITY
F. C. Schweppe, M. C. Caramanis, R. D.Tabors, R. E. Bohn, ISBN 0-89838-260-2
Computational Techniques
for Voltage Stability
Assessment and Control
Venkataramana Ajjarapu
Iowa State University
Ames, Iowa, U.S.A.
Springer
Venkataramana Ajjarapu
Iowa State University
Department of Electrical and
Computer Engineering
1122 Coover Hall
Ames, Iowa 50011
U.S.A.
Computational Techniques for Voltage Stability Assessment and Control
Library of Congress Control Number: 2006926216
ISBN-10: 0-387-26080-3 ISBN-10: 0-387-32935-8 (e-book)
ISBN-13: 9780387260808 ISBN-13: 9780387329352 (e-book)
Printed on acid-free paper.
© 2006 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
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The use in this publication of trade names, trademarks, service marks and similar terms,
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Printed in the United States of America.
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Contents
Contents V
Preface XI
1 Introduction 1
1.1 What is voltage stability? 1
1.2 Voltage Collapse Incidents 4
1.3 Two Bus Example 5
1.3.1 Derivation for critical voltage and critical power 7
1.3.2 Q-V curves 11
1.3.3 Discussion on PV and QV Curves 12
1.3.4 Maximum power and power flow Jacobian 15
References 16
2 Numerical Bifurcation Techniques 19
2.1 Various Types of Bifurcation 19
2.2 Bifurcation of Dynamical Systems 22
2.2.1 Center manifold 24
2.3 Detection of Bifurcation Points 26
2.3.1 Static bifurcations 26
2.3.2 Homotopy Method 27
2.3.3 Continuation methods 30
2.3.4 Curve Tracing 32
2.3.5 Direct method in computing the Saddle node bifurcation
point: a one step continuation 38
2.4 Hopf Bifurcation 40
2.4.1 Existence of Hopf bifurcation point 40
2.4.1.1 Direct methods 41
2.4.1.2 Indirect methods 42
2.5 Complex Bifurcation 42
References 45
3 Continuation Power Flow 49
VI
3.1 Introduction 49
3.2 Locally Parameterized Continuation 49
3.3 Formulation of Power Flow Equations 50
3.4 The Predictor-corrector Process 51
3.4.1 Selecting the continuation parameter 53
3.4.2 Identifying the critical point 53
3.5 Examples 54
3.6 Simultaneous Equilibria Tracing in Power Systems 75
3.6.1 Total solution at an equilibrium 76
3.6.2 Traditional approach 76
3.7 Power Flow Methodology and Assumptions 77
3.7.1 Nonlinearity in power flow 78
3.7.2 Slack bus assumption 79
3.7.3 PVbus assumption 80
3.8 Total Power System Equilibria Solutions 81
3.8.1 Formulation of power system DAE model 82
3.8.1.1 Synchronous generators 82
3.8.1.2 Excitation Control system 83
3.8.1.3 Prime mover and speed governor 84
3.6.1.4 Nonlinear load model 85
3.8.1.5 LTC model 86
3.8.1.6 Other models 86
3.8.1.7 Network power equations 89
3.8.1.8 Power system DAE model 90
3.8.2 Bifurcation modeling of power system dynamics 90
3.8.2.1 Saddle-node bifurcation 91
3.8.2.2 Hopf bifurcation 91
3.8.3 Manifold models in power systems 92
3.8.3.1 Manifold 92
3.8.3.2 Natural parameterization 93
3.8.3.3 Local parameterization 93
3.8.4 Equilibrium manifold Tracing of power systems 95
3.8.5 Initialization for power system equilibrium tracing 96
3.8.6 Continuation method with local parametrization 98
3.8.7 Linerization of power system DAE 99
3.8.8 Detection of Saddle Node Bifurcation with System Total
Jacobian 100
3.8.8.1 Detection of saddle-node bifurcation 101
3.8.9 Limits implementation 104
3.8.9.1 Governor limits 104
3.8.9.2 AVR limits 104
3.9 Numerical examples for EQTP 108
VII
References 115
Sensitivity Analysis for Voltage Stability 117
4.1 Introduction 117
4.2 Given State Based Indices 117
4.3 Large Deviation Based Indices 120
4.4 Stability Studies via Sensitivity Analysis 120
4.4.1 Identification of critical elements 121
4.4.2 Eigenvalue sensitivity 121
4.4.3 Modal analysis 122
4.4.4 Sensitivity analysis via CPF 123
4.4.5 Tangent vector, right eigenvector, and right singular vector
of J 124
4.4.6 Voltage stability index from the tangent vector 125
4.4.7 Sensitivity analysis from the tangent vector 127
4.4.8 Bus sensitivities 127
4.4.9 Branch sensitivities 129
4.4.10 Generator sensitivities 131
4.4.11 Qualitative vs. quantitative sensitivities 133
4.5 Margin Sensitivity 133
4.5.1 Transfer margin estimation 136
4.5.2 Multi-parameter margin sensitivity 138
4.5.3 Sensitivity formulas 139
4.6 Test System Studies 142
4.6.1 Two bus example: 142
4.6.2 The New England system 147
4.6.2.1 Exciter parameters 147
4.6.2.2 Network parameters 149
4.6.2.3 Load (scenario) parameters 151
4.6.2.4 Multiple-parameter variations 153
References 154
Voltage Stability Margin Boundary Tracing 157
5.1 Introduction 157
5.2 Natural Parameterization for Margin Boundary Tracing 158
5.2.1 Load parameter space 158
5.2.1 Control parameter space 159
5.3 Formulation of Margin Boundary Tracing 160
5.3.1 Margin boundary manifold of power system 160
5.3.2 Characterization of margin boundary 160
5.3.2.1 Characterization of saddle node bifurcation related
margin boundary tracing 160
VIII
5.3.3 Margin boundary tracing 161
5.3.3.1 Augmentation for bifurcation characterization 161
5.3.3.2 Augmentation for local parameterization 162
5.3.4 Basic Steps Involved in the Margin Boundary Tracing 164
5.3.5 Practical implementation 164
5.3.5.1 Implementation of reduced method 165
5.4 Examples 167
5.4.1 Series compensation between bus 6 and bus 31 176
5.4.2 Shunt Compensation 176
5.4.3 Multiple contingencies 177
5.4.4 Boundary tracing with respect to generation control
parameters 178
5.4.4.1 Load margin vs adjustment of Ka of AVR system 178
5.4.4.2 Load margin versus adjustment of Vref of AVR systeml 79
5.4.5 Control combination 180
5.4.6 Advantages of margin boundary tracing 181
5.5 Formulation of Voltage Stability Limited ATC 181
5.6 Scenario Parameters 184
5.7 Scenario According to Simultaneous Multi-area Transactions. 185
5.7.1 Determination of ^^. 187
5.7.2 Determination of K^. 190
5.8 Numerical Example 191
5.8.1 Description of the simulation system 191
5.8.2 Emergency transmission load relief 201
5.8.2.1 Single transaction case 201
5.8.2.2 Simultaneous transaction case 202
5.8.3 Reactive power Support 202
5.8.3.1 Single transaction case 202
5.8.3.2 Simultaneous transaction case 203
5.8.4 Control combination 204
5.9 Conclusion 205
References 206
Time Domain Simulation 207
6.1 Introduction 207
6.2 Explicit and Implicit Methods 208
6.2.1 Explicit method 208
6.2.2 Implicit method 209
6.2.3 Stiffness and Numerical Stability 209
6.3 Decoupled Time Domain Simulation 212
IX
6.4 Numerical Examples 217
6.4.1 Two bus system 217
6.4.2 New England 39-bus system 223
6.5 Quasi-Steady-State Simulation (QSS) 226
6.5.1 Problem Formulation 227
6.5.2 Steps involved in QSS Method 228
6.5.3 Implementation of the Continuation Method in QSS 230
6.5.4 Consideration of Load Change with respect to Time 230
6.5.5 Numerical Results 232
6.5.5.1 2-bus system 232
6.5.5.2 CQSS Simulation for New England 39-bus system.... 233
References 236
Appendix 239
A. Data of 2-bus test system 239
Al. One line diagram 239
A2. The IEEE format: Base case power flow data of the
2-bus system 239
A3. The dynamic data of the 2-bus system 240
B. Data of New England test system 241
Bl. One line diagram 241
B2. The IEEE format: Base case power flow data of the
New England system 242
B3. The Dynamic Data of the New England System 245
Index 249
Preface
This book is intended to present bifurcation and continuation based com
putational techniques for voltage stability assessment and control.
Chapters 1 and 2 provide background material for this book. Chapter 2 re
views various aspects of bifurcation phenomena and includes numerical
techniques that can detect the bifurcation points. Chapter 3 discusses the
application of continuation methods to power system voltage stability and
provides extensive coverage on continuation power flow. Chapter 4 pre
sents general sensitivity techniques available in the literature that includes
margin sensitivity. Chapter 5 introduces voltage stability margin boundary
tracing. This chapter also discusses application of continuation power flow
for ATC. Chapter 6 finally presents time domain techniques that can cap
ture short as well as long term time scales involved in voltage stability.
Decoupled time domain simulation is introduced in this chapter. Basic
steps involved in various methods in each chapter are first demonstrated
through a two bus example for better understanding of these techniques.
I am grateful to Prof Pai, the series editor, who encouraged me and helped
me to write this book.
I would like to acknowledge the help from my previous and current gradu
ate students who helped me directly or indirectly in many ways to organize
this book. In general would like to thank Srinivasu Battula, Qin Wang,
Zheng Zhou, Gang Shen, Cheng Luo and Ashutosh Tiwari. In particular , I
would like to acknowledge contributions of Colin Christy (for chapter 3) ,
Byongjun Lee (for chapter2) , Bo Long (for chapters 2,3 and 4) , Yuan
Zhou (for chapters 3 and 5), Geng Wang (for chapter 5) and Dan Yang (for
chapter 6).
I would also like to acknowledge IEEE and Sadhana for some of the fig
ures and material I borrowed from my papers in these journals.
Finally I thank my wife Uma for her continued support and encourage
ment.
