Optimal Control Applications in Electric Power Systems

204 Pages·1987·6.197 MB·English

by  G. S. Christensen, M. E. El-Hawary, S. A. Soliman (auth.)

Optimal Control
Applications in
Electric Power Systems
MATHEMATICAL CONCEPTS AND METHODS
IN SCIENCE AND ENGINEERING
Series Editor: Angelo Miele
Mechanical Engineering and Mathematical Sciences
Rice University
Recent volumes in this series:
22 APPLICATIONS OF FUNCTIONAL ANALYSIS IN ENGINEERING
• J. L. Nowinski
23 APPLIED PROBABILITY. Frank A. Haight
24 THE CALCULUS OF VARIATIONS AND OPTIMAL CONTROL: An
Introduction • George Leitmann
2S CONTROL, IDENTIFICATION, AND INPUT OPTIMIZATION
• Robert Kalaba and Karl Spingarn
26 PROBLEMS AND METHODS OF OPTIMAL STRUCTURAL DESIGN
• N. V. Banichuk
27 REAL AND FUNCTIONAL ANALYSIS, Second Edition
Part A: Real Analysis. A. Mukherjea and K. Pothoven
28 REAL AND FUNCTIONAL ANALYSIS, Second Edition
Part B: Functional Analysis. A. Mukherjea and K. Pothoven
29 AN INTRODUCTION TO PROBABILITY THEORY WITH STATISTICAL
APPLICATIONS • Michael A. Golberg
30 MULTIPLE-CRITERIA DECISION MAKING: Concepts, Techniques,
and Extensions. Po-Lung Yu
31 NUMERICAL DERIVATIVES AND NONLINEAR ANALYSIS
• Harriet Kagiwada, Robert Kalaba, Nima Rasakhoo, and Karl Spingarn
32 PRINCIPLES OF ENGINEERING MECHANICS
Volume 1: Kinematics-The Geometry of Motion • Millard F. Beatty, Jr.
33 PRINCIPLES OF ENGINEERING MECHANICS
Volume 2: Dynamics-The Analysis of Motion. Millard F. Beatty, Jr.
34 STRUCTURAL OPTIMIZATION
Volume 1: Optimality Criteria. Edited by M. Save and W. Prager
35 OPTIMAL CONTROL APPLICATIONS IN ELECTRIC POWER SYSTEMS
G. S. Christensen, M. E. El-Hawary, and S. A. Soliman
A Continuation Order Plan in available for this series. A continuation order will bring delivery of
each new volume immediately upon publication. Volumes are billed only upon actual shipment.
For further information please contact the publisher.
Optimal Control
Applications in
Electric Power Systems
G. S. Christensen
University of Alberta
Edmonton, Alberta, Canada
M. E. El-Hawary
Technical University of Nova Scotia
Halifax, Nova Scotia, Canada
and
S. A. Soliman
Ain Shams University
Cairo, Egypt
Springer Science+Business Media, LLC
Library of Congress Cataloging in Publication Data
Christensen, G. S. (Gustav S.)
Optimal control applications in electric power systems.
(Mathematical concepts and methods in science and engineering; 35)
Includes bibliographies and index.
1. Electric power systems-Control. I. El-Hawary, M. E. II. Soliman, S. A. III.
Title. IV. Series.
TK1005.C47 1987 621.319 87-7712
ISBN 978-1-4899-2087-4
ISBN 978-1-4899-2087-4 ISBN 978-1-4899-2085-0 (eBook)
DOI 10.1007/978-1-4899-2085-0
© 1987 Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1987
Softcover reprint of the hardcover 1st edition 1987
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
Preface
Significant advances in the field of optimal control have been made over
the past few decades. These advances have been well documented in
numerous fine publications, and have motivated a number of innovations
in electric power system engineering, but they have not yet been collected
in book form. Our purpose in writing this book is to provide a description
of some of the applications of optimal control techniques to practical power
system problems.
The book is designed for advanced undergraduate courses in electric
power systems, as well as graduate courses in electrical engineering, applied
mathematics, and industrial engineering. It is also intended as a self-study
aid for practicing personnel involved in the planning and operation of
electric power systems for utilities, manufacturers, and consulting and
government regulatory agencies.
The book consists of seven chapters. It begins with an introductory
chapter that briefly reviews the history of optimal control and its power
system applications and also provides an outline of the text. The second
chapter is entitled "Some Optimal Control Techniques"; its intent is to
introduce fundamental concepts of optimal control theory that are relevant
to the applications treated in the following chapters. Emphasis is given to
clear, methodical development rather than rigorous formal proofs. Topics
discussed include variational calculus, Pontryagin's maximum principle,
and geometric methods employing functional analysis. A number of solved
examples are included to illustrate the techniques.
"Dynamic Optimal Load Flow" is the title of Chapter 3. This chapter
deals with the problem of economic scheduling of electric power systems
including thermal and hydrothermal generation resources. The network
model utilized is the exact load flow formulation. The case of hydroplants
with essentially constant head is addressed. Computational approaches are
detailed and sample test cases are provided.
v
vi Preface
"Economic Coordination of Hydrothermal-Nuclear Systems" is
treated in Chapter 4. In this chapter the details of optimal economic
operation strategies for systems including nuclear capability are discussed.
For the sake of clarity in presentation, the electric network is represented
by an active power balance model.
The problem of optimally controlling the tie-line flow in an interconnec
ted system so as to maintain acceptable frequency variations in response
to load variations is detailed in Chapter 5, "Optimal Tie-Line Control."
Computational examples are included in this chapter.
"Optimal Control of Turboalternators" is the title of Chapter 6. Open
loop and feedback optimal control strategies for turbo alternator stabilization
are discussed in Chapter 6. Torque and voltage variables are taken as the
control inputs in both cases. Cbmputational examples are provided to
compare the performance of the different methods of approach.
Chapter 7 summarizes the developments discussed in this monograph.
Directions for further improvements in model complexity versus worth are
highlighted.
G. S. Christensen
Edmonton, Alberta, Canada
M. E. El-Hawary
Halifax, Nova Scotia, Canada
S. A. Soliman
Cairo, Egypt
Acknowledgments
During the planning and writing of this book, we have incurred indebtedness
to many people. We wish to acknowledge continuing encouragement
received from Dr. C. R. James, Chairman of the Department of Electrical
Engineering at the University of Alberta. We are grateful to our many
friends and colleagues and, in particular, to Dr. D. H. Kelly of the University
of Alberta, Dr. M. Abdelhalim of the University of Alexandria, Egypt, Mr.
Y. Chen, Chinese Scholar, Dr. R. S. Rao, and Dr. A. Shamaly. We wish to
express our thanks to Ms. Barbara J. Peck, Director, Canterbury Executive
Services, for a neatly done job of typing the many drafts of the manuscript.
This work would not have been possible without the patience and under
standing of our wives, Dr. Penelope Christensen, Dr. Ferial El-Hawary,
and Laila Soliman.
G.S.C.
M.E.E.
S.A.S.
vii
Contents
1. Introduction
1.1. The Role of Optimization and Computational Mathematics 1
1.2. Outline of the Book ................................... 2
2. Some Optimal Control Techniques
2.1. Introduction.......................................... 5
2.2. Calculus of Variations ................................. 5
2.2.1. Unconstrained Optimization ...................... 5
2.2.2. Constrained Optimization. . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3. Pontryagin's Maximum Principle. . . . . . . . . . . . . . . . . . . . . . . . 10
2.4. The Functional Analytic Optimization Technique. . . . .. . .. . 14
2.4.1. Norms ......................................... 15
2.4.2. Inner Product (Dot Product) . . . ... . . . . . .. . . .. . .. . . 15
2.4.3. Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.4. A Minimum Norm Theorem.............. ........ 18
References ........................................... 20
3. Dynamic Optimal Load Flow
3.1. Introduction.......................................... 21
3.2. The All-Thermal OPF .. . .. .. . .. . . . . .. . . .. .. . .. . . .. . . . . . 22
3.3. The Hydrothermal OPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4. A Successive Approximation Solution. . . . . . . . . . . . . . . . . . . . 27
3.5. A Newton-Iterative Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1. The Jacobian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5.2. Initial Guess Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6. Computational Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1. Successive Approximation Algorithm.. . .. . . .. . . .. . . 39
3.6.2. Newton's Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
x Contents
3.7. Conclusions.......................................... 52
References ........................................... 53
4. Economic Coordination of Hydrothermal-Nuclear Systems
4.1. Introduction.......................................... 55
4.2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3. The Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4. The Optimal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.5. A Feasible Multilevel Approach. . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6. Conclusions and Comments ............................ 66
Appendix 4.A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Appendix 4.B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
References ........................................... 68
S. Optimal Tie-Line Control
5.1. Introduction.......................................... 69
5.2. Load Frequency Control of Interconnected Power Systems 70
5.2.1. Mathematical Model. . . . . . . . . . . . . . . . . .. . . .. . .. . . . 71
5.2.2. Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.3. Optimal Solution ................................ 74
5.2.4. Solution Technique .............................. 77
5.2.5. Practical Example ............................... 78
5.3. Optimum Load Frequency Control of
Multiarea Interconnected Power Systems. . . . . . . . . . . . . . . . . 82
5.3.1. Problem Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3.2. Optimal Solution ................................ 86
5.3.3. A Sudden Loss or Gain of Load at One Area ....... 88
5.3.4. A Loss of a Tie-Line between Two Areas. . . . . . . . . . . 91
5.3.5. A Sudden Loss of Generation at One Area. . . . . . . . . . 94
5.4. Optimal Load Frequency Control with Governor Backlash 99
5.4.1. Mathematical Model. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
5.4.2. Problem Formulation. .. . . . . . . . . . . . .. . . . . .. . .. . . .. 101
5.4.3. Solution Technique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.4.4. Practical Example ............................... 106
5.5. Conclusions .......................................... 108
References ........................................... 108
6. Optimal Control of Turboalternators
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 111