ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILI1Y

ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILI1Y

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE POWER ELECTRONICS AND POWER SYSTEMS Consulting Editor Thomas A. Lipo University of Wisconsin - Madison Other books in the series: SPOT PRICING OF ELECTRICITY Fred C. Schweppe ISBN 0-89838-260-2 RELIABILITY ASSESSMENT OF LARGE ELECTRIC POWER SYSTEMS Roy Billinton and Ronald N. Allan ISBN 0-89838-266-1 MODERN POWER SYSTEMS CONTROL AND OPERATION Ati'S. Debs ISBN 0-89838-265-3 ELECTROMAGNETIC MODELLING OF POWER ELECTRONIC CONVERTERS J. A. Ferreira ISBN 0-7923-9034-2 ENERGY FUNCTION ANAL YS/S FOR POWER SYSTEM STABILITY M. A. Pai ISBN 0-7923-9035-0

ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILIlY by M.A. Pai University of Illinois at Champaign/ Urbana... " KLUWER ACADEMIC PUBLISHERS Boston/Dordrecht/London

Distributors for North America: K1uwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 USA Distributors for all other countries: K1uwer Academic Publishers Group Distribution Centre Post Office Box 322 3300 AH Dordrecht, THE NETHERLANDS Library of Congress Cataloging-In-Publication Data Pai, M. A., 1931- Energy function analysis for power system stability / by M.A. Pai. p. cm. - (The Kluwer international series in engineering and computer science. Power electronics and power systems) Bibliography: p. Includes index. ISBN-13: 978-1-4612-8903-6 e-isbn-13: 978-1-4613-1635-0 DOl: 10.1007/978-1-4613-1635-0 I. Electric power systems-mathematical models. 1. Title. n. Series: Kluwer international series in engineering and computer science. Power electronics & power systems. TKlOO5.P329 1989 621.31-dc20 89-15387 elp Copyright 1989 by Kluwer Academic Publishers Softcover reprint of the hardcover 1 st edition 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061

Dedicated to Jawaharlal Nehru

Vll CONIENfS CHAFfERl Preface POWER SYSTEM STABILIlY IN SINGLE MACHINE SYSTEM ix 1 1.1 1.2 1.3 1.4 1.5 1.6 Introduction Statement of the Stability Problem Mathematical Formulation of the Problem Modeling Issues Motivation Through Single Machine Infinite Bus System Chapter Outline 1 4 6 9 10 18 CHAFfER 2 ENERGY FUNCTIONS FOR CLASSICAL MODELS 21 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Introduction Internal Node Representation Energy Functions for Internal Node Models Individual Machine and other Energy Functions Structure Preserving Energy Functions Alternative Form of the Structure Preserving Energy Function Positive Definiteness of the Energy Integral Tsolas-Araposthasis-Varaiya Model 21 22 26 31 32 43 44 45 CHAFfER 3 REDUCED ORDER ENERGY FUNCTIONS 49 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 INTRODUCTION Individual Machine and Group Energy Function Simplilled Form of the Individual Machine Energy Function Cutset Energy Function Example of Cutset Energy Function Extended Equal Area Criterion (EEAC) The Quasi Unstable Equilibrium Point (QUEP) Method Decomposition-Aggregation Method Time Scale Energies 49 50 54 58 63 70 72 73 73 CHAFfER 4 ENERGY FUNCTIONS WITH DETAILED MODELS OF SYNCHRONOUS MACHINES AND ITS CONTROL 87 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Introduction Single Machine System With Flux Decay Model Multi-Machine Systems With Flux Decay Model (Method of Parameter Variations) Lyapunov Functions for Multi-Machine Systems With Flux Decay Model Multi-Machine Systems With Flux Decay Models and A VR Energy Functions With Detailed Models Lyapunov Function for Multi-Machine Systems With Flux Decay and Nonlinear Voltage Dependent Loads 87 88 92 101 112 113 133

viii CHAPTERS REGION OF STABILIlY IN POWER SYSTEMS 145 5.1 5.2 53 5.4 55 5.6 Introduction Characterization of the Stability Boundary Region of Stability Method of Hyperplanes and Hypersurfaces Potential Energy Boundary Surface (PEBS) Method Hybrid Method Using the Gradient System 145 146 151 159 172 184 CHAPTER 6 PRACTICAL APPLICATIONS OF THE ENERGY FUNCTION METHOD 189 6.1 6.2 63 6.4 6.5 6.6 Introduction The Controlling u.e.p. Method Modifications to the Controlling u.e.p. Method Potential Energy Boundary Surface (PEBS) Method Mode of Instability (MOl) Method Dynamic Security Assessment 189 190 194 201 201 206 CHAPTER 7 APPENDIX A REFERENCES INDEX FUTURE RESEARCH ISSUES 10 Machine 39 Bus System Data 219 223 229 239

PREFACE This research monograph is in some sense a sequel to the author's earlier one (Power System Stability, North Holland, New York 1981) which devoted considerable attention to Lyapunov stability theory, construction of Lyapunov functions and vector Lyapunov functions as applied to power systems. This field of research has rapidly grown since 1981 and the more general concept of energy funct ion has found wide spread application in power systems. There have been advances in five distinct areas (i) Developing energy functions for structure preserving models which can incorporate non-linear load models (ii) Energy functions to include detailed model of the generating unit i.e., the synchronous machine and the excitation system (iii) Reduced order energy functions for large scale power systems, the simplest being the single machine infinite bus system (iv) Characterization of the stability boundary of the post-fault stable equilibrium point (v) Applications for large power networks as a tool for dynamic security assessment. It was therefore felt appropriate to capture the essential features of these advances and put them in a somewhat cohesive framework. The chapters in the book rough ly fo llow this sequence. It is interesting to note how different research groups come to the same conclusion via different reasonings. For example the concept of critical and non-critical groups of machines may be looked at from the vulnerable cutset point of view, lowest normalized potential energy margin, slow coherency concept or simply based on acceleration, kinetic energy etc. of the machine at t = 0+ or t = t. Unfortunately the tercr. minology is not standard in the literature and hence there has been some difficulty in explaining in great detail some of the more recent research work. However the book will provide a basis for the researcher, theoretical or prac-

tically oriented one to explore the topic in greater detail. The last chapter lists some of the topics which merit further investigation. I would like to thank all the researchers whose work is explained in the book. I know I migh t have left out some other import ant work also, to whom I apologize. This is the dilema an author faces in writing a book where the literature is extensive and the terminology is not standard. Useful discussions with Professor P. W. Sauer at University of Illinois, Professor K. R. Padiyar at I.I.Sc., Bangalore, India, Professor V. Vittal at Iowa State University, and Professor H. D. Chiang at Cornell University are gratefuly acknowledged. I would like to thank my wife Nandini for the moral and logistic support during the course of the last two years when the book was being written. I would like to thank Kelly C. Voyles of our publications office for her excellent typing of the manuscript and Fred Daab for the fine job of doing the drawings. I would like to acknowledge the support of the National Science Foundation for supporting my research in this area through its grant ECS 84-14677 and ECS 87-19055. The book is dedicated to Mr. Jawaharlal Nehru, the first Prime Minister of India and whose birth Centenary (1889-1989) is being celebrated this year. ~is vision about Science and Technology being the cornerstone for India's development has made many of us knowingly or unknowingly grateful to him. M. A. Pai

ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILI1Y