Nonlinear Control Systems and Power System Dynamics (The International Series on Asian Studies in Computer and Information Science)

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS The Kluwer International Series on ASIAN STUDIES IN COMPUTER AND...

Author: Qiang Lu | Yuanzhang Sun | Shengwei Mei

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NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS

The Kluwer International Series on

ASIAN STUDIES IN COMPUTER AND INFORMATION SCIENCE Series Editor

Kai-Yuan Cai Being University ofAeronautics andAstronautics, Being, CHINA

Editorial Advisory Board war

vii

Han-Fu Chen, Institute of System Science, Chinese Academy of Sciences Jun-Liang Chen, Beijing University of Post and Telecommunication Lin Huang, Peking University Wei Li, Beijing University of Aeronautics and Astronautics Hui-Min Lin, Institute of Software Technology, Chinese Academy of Sciences Zhi-Yong Liu, Institute of Computing Technology, Chinese Academy of Sciences Ru-Qian Lu, Institute of Mathematics, Chinese Academy of Sciences Shi-Tuan Shen, Beijing University of Aeronautics and Astronautics Qing-Yun Shi, Peking University You-Xian Sun, Zhejiang University Lian-Hua Xiao, National Natural Science Foundation of China Xiao-Hu You, Southeast University Bo Zhang, Tsinghua University Da-Zhong Zheng, Tsinghua University Bing-Kun Zhou, Tsinghua University Xing-Ming Zhou, Changsha University of Technology >>.

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Also in the Series: COMMON WAVEFORM ANALYSIS: A New and Practical Generalization of FourierAnalysis, by Yuchuan Wei and Qishan Zhang; ISBN: 0-7923-7905-5

DOMAIN MODELING-BASED SOFTWARE ENGINEERING: A Formal Approach, by Ruqian Lu and Zhi Jin; ISBN: 0-7923-7889-X AUTOMATED BIOMETRICS: Technologies and Systems, by David D. Zhang, ISBN: 0-7923-7856-3 FUZZY LOGIC AND SOFT COMPUTING, by Guoqing Chen, Mingsheng Ying, KaiYuan Cai; ISBN: 0-7923-8650-7 INTELLIGENT BUILDING SYSTEMS, by Albert Ting-pat So, Wai Lok Chan; ISBN: 0-7923-8491-1 Calf

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by

Qiang Lu

Yuanzhang Sun Shengwei Mei

Tsinghua University, Beijing, China

0 KLUWER ACADEMIC PUBLISHERS

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Distributors for North, Central and South America:

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Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 USA Telephone (781) 871-6600 Fax (781) 681-9045 \:l

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IT Electronic Services Library of Congress Cataloging-in-Publication Data

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Lu, Qiang, 1936Nonlinear control systems and power system dynamics / Qiang Lu, Yuangzhang Sun, Shengwei Mei. p. cm.-- (Kluwer international series on Asian studies in computer and information science ; 10) Includes bibliographical references and index. ISBN 0-7923-7312-X (alk. paper) 1. Automatic control. 2. Nonlinear control theory. 3. Electric power system stability. I. Sun, Yuanzhang, 1954- II. Mei, Shengwei, 1964- III. Title. IV. Series.

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vii

TJ213.L72 2001 629.89-dc2l

2001016020

Copyright © 2001 by Kluwer Academic Publishers

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

Printed on acid free paper. Printed in the United States of America The Publisher offers discounts on this book for course use and bulk purchases. For further information, send email to <[email protected]>.

To Our Alma Mater Tsinghua University

SERIES EDITOR'S ACKNOWLEDGMENTS

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I am pleased to acknowledge the assistance to the editorial work by Beijing University of Aeronautics and Astronautics and the National Natural Science Foundation of China

Kai-Yuan Cai Series Editor Department ofAutomatic Control Beijing University of Aeronautics and Astronautics Beijing 100083 China

Contents

Preface ...............................................................................................mod

Chapter 1

Introduction .........................................................................................1 --'

1.1 Overview ................................................................................................ 1 1.2 Outline of the Development of Control Theory ..................................... 3 1.3 Linear and Nonlinear Control Systems ................................................ 12 1.0

.-' ...

1.4 Modeling Method of Approximate Linearization ................................ 16 1.5 Stable and Unstable Equilibrium Points .............................................. 19

1.6 References ........................................................................................... 22

Chapter 2 Basic Concepts of Nonlinear Control Theory ................................25 k/'1

2.1 Introduction .......................................................................................... 25 2.2 Coordinate Transformation of Nonlinear Systems ............................... 26 2.2.1 General Concepts of Coordinate Transformation ..................... 26 2.2.2 Coordinate Transformation of Linear Systems ......................... 28 2.2.3 Nonlinear Coordinate Transformation and Diffeomorphism .... 29 2.2.4 Mapping .................................................................................... 30 2.2.5 Local Diffeomorphism .............................................................. 30

N¢;

2.2.6 Coordinate Transformation of Nonlinear Control Systems....... 32

2.3 Affine Nonlinear Control Systems ....................................................... 33 2.4 Vector Fields ........................................................................................ 34 2.5 Derived Mapping of Vector Fields ....................................................... 36 2.6 Lie Derivative and Lie Bracket ............................................................ 38

2.6.1 Lie Derivative ........................................................................... 38

viii


2.6.2 Lie Bracket................................................................................ 41 2.7 Involutivity of Vector Field Sets .......................................................... 45

2.8 Relative Degree of a Control System ................................................... 47 2.9 Linearized Normal Form ...................................................................... 50 2.10 Summary ............................................................................................ 56 2.11 References ......................................................................................... 58

Chapter 3 Design Principles of Single-Input Single-Output Nonlinear Control Systems ..............................................................59 3.1 Introduction .......................................................................................... 59

3.2 Design Principles of Exact Linearization via Feedback....................... 60 'p.

3.2.1 Linearizing Design Principle as Relative Degree r Equals n for an nth-order System .............................................. 61 3.2.2 General Linearization Design Principle .................................... 70 3.2.3 Conditions for Exact Linearization ........................................... 72 3.2.4 Algorithm of Exact Linearization ............................................. 80 3.3 Zero Dynamics Design Principle ......................................................... 90 3.3.1 First Type of Zero Dynamic Design Method ............................ 91 3.3.2 Second Type of Zero Dynamic Design Method ........................ 97 3.3.3 Discussion of Some Problems................................................. 101 3.4 Zero Dynamics Design Method for Linear Systems .......................... 104 3.5 Design of Disturbance Decoupling .................................................... 109 I'D

3.6 References ......................................................................................... 120

Chapter 4 Design Principles of Multi-Input Multi-Output Nonlinear Control Systems ............................................................121 4.1 Introduction........................................................................................ 121 4.2 Relative Degrees and Linearization Normal Forms ........................... 122 4.2.1 Relative Degree ........................................................................ 122 4.2.2 Linearization Normal Form ...................................................... 125

4.3 Zero Dynamics Design Principle ....................................................... 136 4.4 Design Principles of Exact Linearization via State Feedback............ 147

4.4.1 Conditions for Exact Linearization via State Feedback.......... 148

Contents

ix

4.4.2 Algorithm of Exact Linearization via State Feedback ............ 151 4.5 References ......................................................................................... 164

Chapter 5 Basic Mathematical Descriptions for Electric Power Systems ...................................................................165 5.1 Introduction ........................................................................................ 165 5.2 Rotor Dynamics and Swing Equation ................................................ 166 5.3 Output Power Equations for a Synchronous Generator ..................... 170 5.4 Output Power Equations for Synchronous Generators in a Multi-Machine System ................................................................ 179 5.4.1 Output Power Equations for a Generator in a One-machine Infinite-bus System ....................................... 179 5.4.2 Practical Output Power Equations for Synchronous Generators in a Multi-machine System .................................. 181 5.5 Electromagnetic Dynamic Equation for Field Winding ..................... 185 5.6 Mathematical Description of a Steam Valving Control System......... 186 5.7 Mathematical Description of a DC Transmission System ................. 191 5.7.1 Dynamic Equations of a DC Transmission Line ..................... 191

5.7.2 Mathematical Model of a DC Control System ........................ 196 5.8 References ......................................................................................... 198

Chapter 6 Nonlinear Excitation Control of Large Synchronous Generators .......................................................................................199 6.1 Introduction ........................................................................................ 199 6.2 Development of Excitation Control ................................................... 200 6.3 Nonlinear Excitation Control Design for Single-Machine Systems .................................................................... 208 6.3.1 Exact Linearization Design Approach .................................... 209

6.3.2 Discussions on the Implementation of Nonlinear Excitation Control ................................................. 217 6.3.3 Effects of Nonlinear Excitation Control .................................. 219 6.4 Nonlinear Excitation Control Design for Multi-Machine Systems ..................................................................... 223

x


6.4.1 Dynamic Equations of Multi-Machine Systems ..................... 223 6.4.2 Exact Linearization Design Method for Excitation Control ............................................................ 225 6.4.3 Practical Nonlinear Excitation Control Law ........................... 234

6.4.4 Discussion on the Nonlinear Excitation Control Law ............. 235 6.4.5 Effects of the Nonlinear Excitation Control ............................ 237 6.5 References ......................................................................................... 244

(!1

Chapter 7 Nonlinear Steam Valving Control .................................................245

7.1 Introduction ........................................................................................ 245 7.2 Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System .................................................... 246 7.2.1 Mathematical Model ............................................................... 246 7.2.2 Exact Linearization Method .................................................... 249 7.2.3 Physical Simulation Results of Nonlinear Valving Control in a One-Machine Infinite-Bus System ..................... 256

7.2.4 Digital Simulation Results of Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System ....... 259

7.3 Nonlinear Steam Valving Control in a Multi-Machine System ......... 261 7.3.1 Mathematical Model ............................................................... 261 7.3.2 Exact Linearization Method .................................................... 263 .-..

7.3.3 Effects of Nonlinear Steam Valving Control in a Multi-Machine System ........................................................... 271

7.4 Discussion on Some Issues ................................................................ 273

7.5 References ......................................................................................... 276

Chapter 8 Nonlinear Control of HVDC Systems ...........................................277 8.1 Introduction ........................................................................................ 277 8.2 Characteristics and Conventional Control of Converter Stations ...... 278 8.2.1 Voltage-Current Characteristics on Rectifier Side .................. 278 8.2.2 Voltage-Current Characteristics on Inverter Side .................... 279

8.2.3 Conventional Control with Constant DC Current at Rectifier and Constant Extinction Angle at Inverter .............. 280

Contents

xi

8.2.4 Conventional Control with Constant DC Current at Rectifier and Constant DC Voltage at Inverter....................... 282 8.2.5 Power Modulation in DC Transmission Systems.................... 284 8.3 Nonlinear Control of Converter Stations ........................................... 285 8.3.1 Nonlinear Control with Constant Current and Constant Extinction Angle .............................................. 285 8.3.2 Nonlinear Control with Constant Current at Rectifier and Constant DC Voltage at Inverter....................... 297 8.4 Nonlinear Control of DC Systems and Stability of AC/DC Systems ............................................................................ 302 8.4.1 Modeling for Nonlinear Stabilizing Control Design of AC/DC Systems ................................................................. 302 8.4.2 Nonlinear Control Design for Stabilizing

AC/DC Systems ..................................................................... 304

8.4.3 Effects of Nonlinear Control for Stabilizing AC/DC Systems ..................................................................... 306 8.5 References ......................................................................................... 308

Chapter 9 Nonlinear Control of Static Var Systems ......................................309 9.1 Introduction ........................................................................................ 309 9.2 Fundamentals of Reactive Power Compensation ............................... 310 9.2.1 Reactive Power Flow in a Transmission System .................... 310 9.2.2 Two Basic Types of Reactive Power Compensators ............... 312 9.2.3 Effects of the Midpoint Compensator on the Stability Limits ............................................................ 314 9.3 Configuration of Static Reactive Compensators ................................ 319 9.3.1 Thyristor-Controlled Reactor (TCR) ....................................... 319 9.3.2 Thyristor-Switched Capacitor (TSC) ...................................... 325

9.4 Conventional Control Strategies of SVS ........................................... 330 9.5 Nonlinear Controller Design for SVS ................................................ 333 9.5.1 Modeling of SVS Control Systems ......................................... 333 9.5.2 Exact Linearization Design Approach .................................... 335 9.5.3 Effects of the Nonlinear Control of SVS ................................ 339 9.6 References ......................................................................................... 342

xi

i


Chapter 10 Nonlinear Robust Control of Power Systems ...............................343 10.1 Introduction ...................................................................................... 343

10.2 Basic Concepts ................................................................................. 344 10.2.1 L2-space ................................................................................ 344 10.2.2 L2 -gain .................................................................................. 345

10.2.3 Penalty Vector Function ........................................................ 348 10.2.4 Dissipative Systems .............................................................. 348

10.3 Nonlinear Robust Control ................................................................ 350 10.3.1 Description of Nonlinear Robust Control ............................. 350 10.3.2 General Form of the Nonlinear Robust Control Law............ 351 10.3.3 Hamilton-Jacobi-Isaacs Inequality ........................................ 355

10.4 HJI Inequality of Linear Control System - Riccati Inequality....... 358 10.5 Nonlinear Robust Excitation Control (NREC) ................................. 359 10.5.1 Introduction ........................................................................... 359 10.5.2 Regulation Output Linearization ........................................... 361 10.5.3 Analysis of Robustness of the Closed Loop System ............. 363 'v,

10.5.4 Nonlinear Robust Excitation Control .................................... 365

10.5.5 Simulation Results ................................................................ 368 10.5.6 Summary ............................................................................... 369 10.6 References ....................................................................................... 371

Index .................................................................................................373

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List of Figures

Figure 1.1 Figure 1.2 'ti

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Figure 1.3 Figure 2.1 boo

Figure 2.2

Structural diagram of linear optimal control system ..................... 8

R-L--C Circuit ............................................................................. 12 A one-machine infinite-bus system ............................................ 14 a, b, c coordinates and d, q, 0 coordinates of a synchronous generator ........................................................ 27 Mappings between X coordinate system and

Z coordinate system ................................................................... 30 Figure 3.1

Diagram illustrating the design principle of exact linearization via feedback .................................................. 68

Figure 3.2

Relations among the coordinate transformations of

Figure 3.3

Structural diagram of a system with

V'1

the spaces X, W and Z ............................................................... 85 outputs decoupled from disturbances .........................................112 Relationship between different reference axes

Figure 5.2

used to measure the motion of a generator's rotor .................... 167 Coordinate axes a, b and c fixed on the stator and

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Figure 5,1

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coordinate axes d, q and 0 fixed on the rotor ............................ 171

Figure 5.3 Figure 5.4

A one-machine, infinite-bus power system ............................... 180

Figure 5.5

Electric potential vector diagram of

Figure 5.6

Positive directions of the current and voltage of

Figure 5.7

a field winding .......................................................................... 185 Physical configuration of the steam valving control system

A 6-generator power system and its equivalent circuit ............. 182 an n-generator power system .................................................... 183

L."

for a large generator set with reheater ....................................... 187

Figure 5.8

Transfer function block diagram of the control system

for a steam turbine with reheater .............................................. 188

xiv


Figure 5.9

Transfer function block diagram of a steam valving control system with reheater ..................................................... 189

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Figure 5.10 Transfer function block diagram of a steam valving control system with reheater ........................................ 190 Figure 5.11 Transfer function diagram for a steam valving control system. 190 Figure 5.12 Basic configuration of a DC transmission system ..................... 191 Figure 5.13 Equivalent circuit of a DC transmission line ............................. 192 Figure 5.14 Converting loop of the rectifier side ......................................... 193 Figure 5.15 Converting process of a rectifier ............................................... 193 Figure 5.16 Equivalent circuit for a rectifier ................................................ 193 Figure 5.17 Converting loop of an inverter .................................................. 194 Figure 5.18 Converting process of an inverter ............................................. 194 Figure 5.19 Equivalent circuit for an inverter when Vd, is expressed by the inverter firing angle /3.................................... 194

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Figure 5.20 Equivalent circuit for an inverter when Vd; is expressed by the extinction angle y........................................... 195 Figure 5. 21 Schematic diagram of a rectifier's a -regulator ....................... 196 Figure 5.22 Schematic diagram of a inverter's (3-regulator .......................... 197 Figure 6.1 The structure of generator self-shunt excitation ........................ 200 Figure 6.2 The block diagram of transfer function of single variable excitation control .............................................. 201 Figure 6.3 A single-input single-output closed-loop system ...................... 202 Figure 6.4 The excitation regulator transfer function dividing the amplification into static and dynamic amplifications ................ 204 Figure 6.5 The transfer function block diagram of PSS ............................. 205 Figure 6.6 Schematic diagram of a generator LOEC (analogous) in a one-machine, infinite-bus system ...................................... 207 Figure 6.7 The schematic diagram of microcomputer nonlinear excitation controller ......................... 219 Figure 6.8 The one-machine, infinite-bus system diagram and its parameters ...................................................................... 219 Figure 6.9 The generator power-angle curve under nonlinear excitation control....................................................... 221 Figure 6 10 The structure diagram of the 6-machine system ....................... 238 C1.

.,.

List of Figures

xv

Figure 6.12 Figure 6.13 Figure 6.14 Figure 6.15 Figure 6.16 Figure 6.17 Figure 7.1

The system's dynamic response curves with PSS ..................... 241


A one-machine, infinite-bus system ......................................... 256

Figure 7.4

Physical simulation results for improving the transient stability by

Figure 7.5

using steam valving nonlinear control under permanent faults. 258 Computer simulation results of

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The comparison of generator small disturbance response curves under various excitation controls strategies ................... 239 iU.

Figure 6.11

.-.

The system's dynamic response curves with LOEC ................ 241 The system's dynamic response curves with NOEC................. 242.

The system's dynamic response curve with PSS ...................... 242 The system's dynamic response curves with LOEC ................ 243 The system's dynamic response curves with NOEC................. 243 The structure diagram of transfer function for nonlinear control of steam valves .............................................. 255 Physical simulation results for improving transient stability by

using nonlinear steam valving control under temporary faults. 257

nonlinear steam valving control ................................................ 259

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Figure 7.6 Figure 7.7 Figure 7.8 Figure 7.9 Figure 7.10

Computer simulation results with permanent fault .................... 260

Dynamic responses under three-phase fault .............................. 272 Dynamic response curves under three-phase fault ................... 273 Dynamic responses under three-phase fault .............................. 274 Physical testing results of re-synchronizing by using the nonlinear steam valve control .................................... 275

Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4

Vd,-Id, characteristic of rectifier with various firing angles....... 278 Vd, Id; characteristic of rectifier with various firing angles ....... 279

Vd,. Id, characteristic of inverter with various extinction angles 280

Converter controller characteristic with constant

current and constant extinction angle ...................................... 281

Figure 8.5

Converter controllers with constant current and constant extinction angle .................................................... 281

Figure 8.6

Converter controller characteristic with constant current and constant voltage ....................................... 283


Inverter controllers with constant DC voltage ........................... 283

The transfer function block diagram of the power modulator in a converter ........................................... 284

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS




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Figure 8.11

A 6-machine AC/DC power system .......................................... 294 System dynamic response under conventional controllers with constant current rectifier control and constant extinction angle inverter control ................................ 295 System dynamic response under nonlinear controllers with constant current at rectifier and constant extinction angle at inverter .......................................... 296 System dynamic response under conventional controllers with constant current at rectifier and constant voltage at inverter..... 300 System dynamic response under nonlinear controllers with constant current at rectifier and constant voltage angle at inverter .............................................. 301 An AC/DC power system .......................................................... 302 ,..

xvi

System dynamic response under nonlinear controllers for AC/DC system stability ....................................................... 307

Figure 9.1

Reactor and capacitor treated as load ........................................ 310

Figure 9.2 Figure 9.3 Figure 9.4 Figure 9.5

Reactor and capacitor modeled as power supply ........................311


The equivalent diagram of transmission system ....................... 314

Figure 9.8 Figure 9.9 Figure 9.10 Figure 9.11 Figure 9.12

Single phase diagram of TCR and its waveforms ..................... 320

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A node with three branches ....................................................... 312 Simple circuit and its vector diagram ........................................ 312 '-n

The circuit with the capacitive compensator ............................. 313 Effects of reactive power compensation on the L/?

transmitted power of symmetrical lossless line ........................ 319 Current and voltage waveforms at different gating angle ......... 321 The control characteristic of BTCR ............................................. 322 The configuration of TCR-FC type compensator ..................... 323

Voltage/current characteristics of Thyristor-Controlled

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if allow us to summarize in one sentence, is an integration of modeling o

theory in state space and mathematical method with linear algebra. The modeling theory and method in state space can be summarized as {x, (t), x2 (t), , x,, (t)}

or a

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follows. Firstly, we define a set of variables

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vector X(t) = [x, (t) x2 (t) ... X" (t)]T as the system's state variable set or state vector which can uniquely determine the dynamic behavior of the system by

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these minimum number of variables. A space R" formed with all of the state variable axes is called the state space. One state of the system can be represented by a certain point in the state space, and the dynamic process o

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starting from an initial state X,, will be characterized as a trajectory in the space starting from X.. If the dimension of the state vector of the dynamic system is n, we call it an n`h-order system. It is well known that, an n'h-order linear dynamic system can be modeled as an n`h-order constant-coefficient ordinary differential equation. But if we model a dynamic system as an n'order one, what appear in the model are only the input and output of the system. In order to investigate multi-variable systems, modern control theory requires using n first-order differential equations to describe an nth-order linear dynamic system. The standard form of the mathematical model is 0

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0

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(t) = a, x, (t) + a12 X2 (t) + ... + al ^ xn (t) + hi l u l (t) + ... + b, u, (t) +

dX1

dt

+

+

dX2(t) = a21x1 (t) + a22X2 (t) +... + a2n X. (t) + b21u1 (t) +... + b2ru,(1) dt

(1.5)

dt

The output equations are +

+

II

y1(t)= CII X1 (t) + C12 X2 (t) +... + C1n X (1)

+

+

+

II

ym(t) = CmIXI (t)+

Cm2X2(t)+... +

where, x; (t), i = 1, 2, , n, represent the state variables; y, (t), l =1, 2, , m, the output variables; uk (t), k =1, 2, , r, the control variables; a,,, b,,, and

Introduction

7

c,,, j =1, 2, , n, I =1, 2, , m, the coefficients. If all of the coefficients are

constant, the system is linear and time-invariant. Otherwise, the system is linear time-varying. Eq.(1.5) can be rewritten in the following matrix form X(t) = AX(t) + BU(t) Y(t) = CX(t)

(1.6)

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where, X(t) is an n-dimensional state vector; Y(t) an m-dimensional output vector; U(t) a r-dimensional control vector; A, B, and C are coefficient matrixes. The standard form given in Eq.(1.5) or (1.6) is the state space equation of a linear dynamic system, or state equation for short. With the initial state X(to) = X0, the solution of the state equation is represented by a trajectory starting from point Xo in the state space, which is the state locus. It is a description of the dynamic process of the system, which is commonly called the transient process in the literature of power systems. N..

'-,

...

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.-.

Since all the linear dynamic systems can be modeled as the state equation in matrix form as shown in Eq.(1.6), almost all the methods in matrix algebra (linear algebra) can be used to investigate and analyze the various problems of linear dynamic systems, such as the problems of c°)

controllability, observability, dynamic performance, stability, identification

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of parameters and compensation techniques, etc. So, as mentioned previously, state space description method, combined with linear algebra fro

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theory, formed the complete multi-variable linear control theoretical system, and the control theory has stepped onto its second stage of development. It is worth noting that there exists a matured and widely-used branch in .+,

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this theoretical system, i.e. the linear optimal control [16]. Problems of optimal control can be defined as the following. To a given linear system or nonlinear system linearized at one of its equilibrium points as follows X(t) = AX(t) + BU(t) Y(t) = CX(t)

(1 7)

4-+^

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For the LOC (Linear Optimal Control) design of the system, a quadratic form performance index is usually chosen as

(1.8) DC'

J = to (X' (t)QX(t)+ UT(t)RU(t))dt

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where, Q is a positive definite or positive semi-definite weighting matrix and R a positive definite weighting matrix, and the superscript T denotes the transpose of the vector. The goal is to find the control law U'(X(t)) such that the performance index J given in formula (1.8) achieves its extremum. Whether the extremum is maximum or minimum depends on the physical meanings of the problem itself and the index. The control U'(X(t)) that

8


makes the index J achieve its extremum is called the optimal control. As we know, the quadratic index J shown in expression (1.8) is a functional of state vector X(t) and control vector U(t). So from the viewpoint of mathematics, to solve the optimal control problem is to find the conditional extremum of functional J[X(t),U(t)] subject to the constraint of state equations. This is a --4

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typical problem of variation with constraints and the Euler-Lagrange equation is the foundation for solving that problem. In 1957, R. Bellman also proposed the dynamic programming approach to the linear optimal control

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problem [38]. Pontryagin (A. C. H o H T p x r u H) generally solved this problem by discovering the so-called Pontryagin's maximum principle in 1958. No matter which method was used, the same final result was reached that the solution to this linear optimal problem was found to be a certain linear combination of state variables X(t), i.e.

U' =-K'X(t)

(1.9)

where, K' is the optimal gain matrix which is expressed as K' = R-'BTP*

(1.10)

This is a constant matrix as the system under consideration is linear and time-invariant and P' is the positive definite solution of the Riccati matrix equation ATP+PA-PBR-'BTP+Q = 0

(1.11)

Therefore, the linear optimal control system can be graphically represented by the structural block diagram given in Fig. 1.1. The linear optimal control problem with quadratic performance index is called the LQR problem, that is the linear-quadratic-Riccatian problem. The multi-variable linear optimal

To solve: PA + ATP - PBR-'BTP + Q = 0 Figure 1.1

Structural diagram of linear optimal control system

9

Introduction +-+

control theory made such a rapid progress that it could be said that all the

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theoretical problems were solved during the period from the beginning of the 1960s to the mid of the 1970s. In summary, the modem control theory as mentioned above has some significant features as follows: (1) It models the system under consideration

as a set of basic linear differential equations in time domain; (2) The mathematical tools used are mainly the theory of linear time-invariant ordinary first order differential equations and linear algebra, rather than

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Laplace transformation and algebra of polynomials used by the classical control theory; (3) The modeling theory and mathematical methods make it possible to handle the linear multi-input multi-output system ; (4) A systematical theory and design methods of optimal control have been established which enables the control laws so acquired to guarantee the performance index functional achieving its extremum; (5) To systems whose

parameters possibly vary in large ranges, optimal control design method combined with parameter identification could achieve the so-called adaptive 0

800Ò

control systems. These are the main characteristics and development backgrounds of the second stage of control theory, i.e. the linear multiin'

variable control theory. The third stage of the development is nonlinear control theory [3, 17, 28, 46]. As we know, most engineering control systems are nonlinear in nature. Admittedly many systems can be linearized at one of their equilibrium points, still satisfying the basic engineering requirements. These systems can certainly be analyzed and synthesized using theories and methods of linear ;-.

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control system. But some system models such as the models of power

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systems are not fitting and proper when established by adapting the approach

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of approximate linearization as their stability and dynamic responses are analyzed under large disturbance [24]. Other systems such as robot arm systems, automatic pilot systems, and some chemical processes control systems could not get the ideal control effects when the approximately -40)

linearized mathematical models are used in design. In one word, the development of engineering and science calls for a new systematical nonlinear control theory urgently. coo

During the past decade, differential geometry has manifested to be an effective means of analyzing and designing nonlinear control systems, and formed a new scientific field [5, 7-9, 11, 14, 15, 19, 21-23, 27, 29, 30, 41 ..cd

-45, 49, 51-53]. In that field, a branch-exact linearization via feedback has been rapidly developed and has seen success in some challenging boo

engineering applications. Its essential points are stated as:

An affine nonlinear) system is described by the following state equations and output equations ') The concept of an affine nonlinear system will be discussed in Chapter 2.


10

dxl (t) dt

=f(xl,...,x

)+g

+...+

dx2 (t)

dt

n

(xl,...,x 11

n

+g (x l ,...,x n )u 2

)u l

21

9.1(x1,..., xn)um

(xl,...,xn

--(xl,...,x)+g12

+...+

)u +g22(x l 1

,...,x n )u2

9'2(x1,...,xn)um

=Jn(xl,...,xn)+g1n(xl,...,xn)ul

dt

+g2n(xl,...,xn)u2

+...+gmn(xl,...,xn)um

y1 (t) = hl (xl , ... , xn ) y2(t)=h2(xl,...,xn)

ym(t)=h.(xl,...,xn)

where, u1 , u2 ,

and

are the state variables; y, , , ym the output variables; , um the control variables; f,, i =1, 2, 3, , n;; h j =1, 2, 3, , m; x1,

, x,,

the scalar functions of the state variables. Rewrite the above equation into the compact form as X(t) = f (X(t)) +

g 1(X)u

(1.12)

Y(t) = h(X(t))

where X = [x, x2

xn IT is a state vector

y = [y, Y2

Y. IT is an output vector

f(X)=[fi(X) J2 (X) ... fnMIT gi(X)=[gri(X) 9!2(X) ... g;,,(X)]T h(X) = [h1 (X) h2 (X) ... hm (X )]T

The target is to find a nonlinear feedback U = a(X) + 13(X)V

(1.13)

such that the nonlinear system (1.12) can be transformed into a completely controllable linear system via a proper coordinate transformation Z(t) = cb(X(t)), that is

Z(t) = AZ(t) + By Y(t) = CZ(t)

(1.14)

where, a(X) is an m -dimensional vector function, R(X) an m x m matrix

Introduction

11

function, V an m-dimensional vector, Z an n -dimensional vector, A an n x n state coefficient matrix, B an n x m control coefficient matrix, Y(t) an m -dimensional output vector, C an m x n output coefficient matrix. It is obvious that this linearization approach is entirely different from

that obtained by discarding high-order terms of the Taylor expansion of nonlinear function f(X) at a given equilibrium point, that is, to replace the increment of a nonlinear function f (X) at a given equilibrium point Xa

Af(X)=.f(X)-f(X0) with the total differential of the function at the same point df(X) = af(xo)Ax, +...+ of(Xo)Ax,, CX1

ax',

.°u

--J

fro

;-H

As a result, the approach which will be emphatically discussed is called not approximate but exact linearization. R. W. Brockett was a representative founder of exact linearization [44]. B. Jakubczyk, et al. gave the sufficient and necessary conditions of local exact linearization of nonlinear multi-variable control system [6], which were predigested by R. Su and L. R. Hunt [43]. A. Isidori and T. J. Tam and D. Cheng gave the even briefer sufficient and necessary conditions of the global exact linearization of affine nonlinear systems [8], which hinted the corresponding algorithm. In the application area, G. Meyer used the theory of exact linearization in the helicopter automatic flight control [12, 13]; T. J. Tarn successfully used it in the control systems of the robot arms [5, 47]; The authors of this book cultivated its new applications in nonlinear control of power systems [33, 36, 37, 56]. In recent years a new academic branch, the nonlinear robust control or say nonlinear disturbance attenuation control, is developing [3, 4, 34, 35, 48, 50]. Drawing on the concept of H. control of linear systems, some authors named that branch as nonlinear H. control. The statement of this problem is as follows. Consider an affine nonlinear system modeled by equations of the form ,...

X = f(X)+g,(X)W +g2(X)U Z = h(X) + K(X)U

(1.15)

The Eq.(1.15) describes a dynamic system with state X E R", control input

U e R' and disturbance input W e R' ; and Z E R' denotes the penalty vector.

The statement of the nonlinear robust control problem is that to find the control strategy U' for the system in Eq.(1.15) such that under the action of control input U' the Lz-gain G2 of the closed loop system from disturbance

12


input W to penalty output Z is less than or equal to a prescribed positive number y, which can be expressed as follows

VT>0

foDIZ(t)IIZdt _ y2foIIW(t)II2dt

(1.16)

In fact, formula (1.16) can be considered as a performance index for the control system (1.15). In this sense the nonlinear robust control is a kind of nonlinear optimal control. As the approach to linear robust control problem

.Go

is deduced to getting the solution of an algebraic Riccati equation, the

.gon

o.0

CDi

nonlinear robust control one is deduced to solving the so-called HamiltonJacobi-Isaacs inequality, a partial differential one. However, up to date there is no general algorithm for obtaining its solution. In Chapter 10 of this book, we will see how the problem is solved for the nonlinear robust control of a multi-machine system, and how this control law works in practice.

The foregoing paragraphs have presented an overview of the three stages of development of control theory. We hope that this brief introduction will be helpful to readers in understanding the main developing trends of control theory from a high position and in generally mastering the contents of this book.

1.3

LINEAR AND NONLINEAR CONTROL SYSTEMS ."a

'a'

Control systems, according to their dynamic characteristics, or mathematical models, can be classified into two categories: linear and nonlinear. >a.

It is of course correct if we define linear control systems as those that can be modeled by linear differential equations, and nonlinear systems as those that can be modeled by nonlinear differential equations. This definition, te".

.`3

however, does not bring more information about the problem. In order to deepen the understanding of this classification issue, let us first take a look at

the R - L - C circuit (Fig. 1.2). Take the outer voltage source V(t) as the

input, the voltage imposed on the capacitor R

VV (t)

as the output, the

L

i(t)

V'(t) -C

V(t)

T Figure 1.2

R-L-C Circuit

Introduction

13

differential equation of this system is LCd2V(t)+RC dt

dt2

t)+V (t)=V(t)

(1.17)

`

where R, L and C denote the resistance, inductance and capacitance, respectively. If we set x,(t)=V,(t) X2 (t) =

dxi(t) di

dt

then, Eq.(1.17) can be rewritten as z, (t) = x2(t)

HIV

-IV

iR

x2(t)=-rI X,(t)- x2(t)+ I V(t)

(1.18)

or as

r

Lx2 (t)]

0

1 LC

0 I

1

- RL

[x2 (t)] +

V(t)

(1.19)

LC

The previous equation can be written as its compact form (1.20)

i(t) = AX(t) + BV(t)

This is the general form of state equations of linear control systems. In Eq.(1.20), if A and B are both constant matrices, the system modeled by this equation is a time-invariant linear system, which is also called the constant linear system. Suppose that we want to consider the aging effects of the components,

that is to consider the effects of parameter variations on the circuit as time vii

goes by, then, R, L and C should be considered as functions of time t, which are represented as R(t), L(t), and C(t) respectively. In this case, A and B in Eq.(1.19) are no longer constant matrices, instead they are function matrices of time t, and the R-L-C circuit under discussion should be modeled as j ((t) = A(t)X(t) +B(t)V(t)

(1.21)

This equation gives the general form of state equations of time-varying linear systems.

From the discussion above, we know that linear systems have two varieties, namely, time-varying and time-invariant systems. Now we turn to discuss the mathematical models of nonlinear systems.

To make the discussion more illustrative, we still take the R-L-C `C3

circuit as an example. Now consider a special capacitor whose capacitance C is an inverse function of the voltage V. That is

14


C=.f(VV)=k, 1 fr

=k 1

(1.22)

x,

where k, is a constant. Substitute Eq.(1.22) into Eq.(1.18), and let a, _ -1/(k,L) , a2 = -R/L, a, =1/(k,L) , which are obviously constants, we have

x,(t)=x2(t)

(1.23)

x2 (t) = a, x, (t) + a2 x2 (t) + a, x, (t)V (t)

Apparently, the state equation of this circuit can not be written in the form of X = AX+BU, instead, it can only be written as

x,(t)1=[

I x2(t)J

X2 (t)

1 +[

2

La,x, (t)+a2x2(t)J

ax,(t).1

(1.24)

V(t)

In the previous equation, let f(X(t))

g(X(t)) = [a3

= [aixi () +ta2x2 (t)]

(t)]

u(t) = V (t)

Eq.(1.24) can be expressed as

X(t) = f(X(t)) + g(X(t))u(t)

(1.25)

,02

.'3

This is the general form of state equations of nonlinear systems, which are often encountered in engineering. Let us take another example of a simple power system to illustrate this nonlinear model. Suppose a simple power system where a large steam-turbine generator .=.

set connects to an infinite bus, as shown in Fig. 1.3. According to the 5

N governor F*X 6 1

2

infinite system

4

Figure 1.3 A one-machine infinite-bus system

I- generator; 2- transformer; 3- transmission line; 4- steam turbine; 5- steam valve; 6- regulated valve; 7- governor

Newton's second law, the motion equations of the machine's rotor can be written as

Introduction

15

HddtZt)

M,(t)-Me(t)

d8(t) =w(t)

(1.26)

(1.27)

dt

where, 8(t) is the generator's rotor angle; Mm the mechanical input torque; M, the electric output torque; H the moment of inertia of the shaft of the machine; co the speed. From the electric machinery, we know that the electric torque can be expressed as Me (t) = XVS sin S(t)

(1.28)

(CD

chi

where E9 is the transient potential of q-axis of generator; V. the voltage of the infinite bus, which is a constant; XL the sum of the transient inductance of the generator, the inductance of the transformer and the inductance of the transmission lines. To be simple, assume the transient potential Eq is constant. So the ..N

system's control input variable is the mechanic torque of the steam turbine M.. Now, Egs.(1.2,7) and (1.26) can be written as S(t) = w(t) tiw(t) = -a, sin S(t) + a2M, (t)

(1.29)

where _ a'

E9VS

HX'

aZ

H

Let x, (t) = S(t)

xz (t) = w(t)

u(t) = M. (t)

then Eq.(1.29) can be rewritten as x,(t) _ z2 (t)]

x2(t)

0 u(t) - [- a, sin x, (t)] + [a2l

(1.30)

Therefore, the state equation of the power transmission system can be written in the affine nonlinear equation form X=f(X)+g(X)u

(1.31)

The only difference is that g(X) in Eq. (1.30) is a constant vector. It

should be noted that Eq.(1.31) is just the general form of

mathematical models of nonlinear systems of one particular class. This form

is featured with its nonlinearity relative to the state vector X(t) and its

16


linearity relative to the control vector u(t). The most general form of state equations of nonlinear control systems is dx,(t) =J1(x1(t),...

xn(t),u1(t),...

dt dxdt2

=

f2(x1(t),...,x

um(t))

(t),u1 (t), ... U. (t))

4(x1(t),...

(1.32)

xn(1),u1(t),...,um(t))

dt Y1 (t) = h,(Xi(t)r ...>xn(t))

(1.33) y. (t) =

L3.

where Eq.(1.32) is the state equations,,and Eq.(1.33) the output equations, in which x, (t), , x,,(t) are the state variables, u, (t), , um (t) the control variables, y, (t), , ym (t) the output variables, f, , f2 ..... f could be considered as the generalized velocity functions, and h, h2, ., hm the output functions. Eqs.(1.32) and (1.33) can be written in the matrix form as X(t) = f(X(t), U(t)) Y(t) = h(X(t))

(1.34)

where X(t) is an n -dimensional state vector, U(t) an m -dimensional control variable, f an n -dimensional vector function of X.

MODELING METHOD OF APPROXIMATE LINEARIZATION E"1

1.4

.+'

Strictly speaking, almost all the practical dynamic systems are nonlinear.

In the design of a control system, the traditional method is to linearize the

original system at one of its equilibrium points, say X, to get its approximately linearized mathematical model, which is a transfer function or '+'

a linear state equation set; then the system can be analyzed and designed cad

according to the methods of linear control theory. Generally speaking, a nonlinear function is feasible to be linearized if it is monodromic and differentiable in the neighborhood of the point Xr. For example, we suppose there is a nonlinear function of two variables 100

y_,

x, and x2 Y=f(x1,x2) Rewrite it using Taylor expansion

Introduction

17

Y =! (xl, x2) = J (xa +&l,xe2 +A2) of of =f(xel,xe2)+ Al + axe A2 ax, IX=x,

+1

+...

(7f Axl +&2)(2)

2! ax,

7x2

In the previous equation,

x=x,

is the selected equilibrium point, and

(al Ox + of Ox2)(2) = a2 f Ox; + aZ f Ox2 +2 7x2

ax,

(1.35)

x=x

7x2

ax,

"f

OxOx2

ax ax2

4.+

Those first-order and high-order partial derivatives as given above are evaluated at the equilibrium point If the second-order and highorder derivative terms are ignored, we have

Ox,+O

Y=f(xel,xe2)+L

2 X.

X,

from which we immediately get the linear expression of Ay in the form as f(xl,x2)-f(Xel,xe2)

AY =

Ox, +

of

0x2 =k, Ox, +k2i\x2

(1.36)

2 x=x,

1 x=x

where k, =

2f__1

ox,

= const

k2 =

of = const 7z2

x=x,

x=x,

This is the principle and method of approximate linearization. From formula 1'?

(1.36) we know that the right side of the expression Ay is the total of function y = at point Xe . So the

differential

essence of approximate linearization is to replace the nonlinear function's increment Of (X) near the equilibrium point Xe for the state equation of the system by the total differential df(X) near the same point. For example, in a one-machine, infinite-bus power system modeled by Eq.(1.30), we can write the increment of nonlinear function Me = a, sins at 8o as a(al sin 8)

?C'

AM,,

L'.

08 = a, cos50A8 = ks08

(1.37)

Thus the system modeled by Eq.(1.30) can be expressed by the corresponding deviation form as follows


18

AS(t) = AC(t)

Am(t) = k,A8+a,AM,,,

or expressed with the form of X = AX + BU as

]m IAwi _ [ka

(1.38)

0] [Am] + [a2

In this way, we get the approximately linearized state equation of the onemachine power system. The approximate linearization has its serious limitations which can not ?.'

be ignored. As the online operating point of the system is far from the selected equilibrium point, the state equations obtained by approximately linearized method will lose its qualification for describing the original nonlinear system. The inevitable result is that the controllers designed

C15

..,

according to the mathematical model obtained by this means can not exert its expected effects on the practical systems when they are operating at a point

far from the equilibrium state used in design. They even create negative vii

...,

actions in some situations. Let us recall the power system just discussed as an example. From Eq.(1.37) we know that in matrix A the element a2,(= k8) equals a, cos80 . If we choose 80=60' as the equilibrium for forming the

model, now suppose the power angle swings to

130°

tossed by large

'c'

disturbance, then the ratio of the designed value of a21(a, cos60°) to its postdisturbance value (a, cos130°) is about 230%. It would not surprise that the

a.,

G:.

Ll.

well-designed controllers according to the mathematical model obtained through approximate linearization can only be able to improve the systems' stability under small disturbance, but reveals their inability under large CAD

disturbance. Why, however, the method of approximate linearization is still widely used in engineering design even today in spite of so many limitations? There

19-

?'_

are several contributing factors: firstly, indeed some of nonlinear control systems usually will not slip very far away from its specified equilibrium point used in design, and its mathematical model obtained via approximate linearization satisfies the engineering requirements; secondly, to some systems, although it is obvious that the state point of them could run very far away from the selected equilibrium point, the linearized model is easy to handle for design, while the nonlinear control theory and method have not been commonly mastered by the technicians and engineers since it only appears in recent years and the relevant literature about it is written in coo

ors

.>7

abstruse mathematical forms.

It is time for researchers, engineers and technicians to master and use nonlinear control approach.

Introduction

1.5

19

STABLE AND UNSTABLE EQUILIBRIUM POINTS The most general description of nonlinear control systems is (1.39)

X(t) = f(X(t),U(t))

where, X E R" is the state vector, U E R' the control vector. If state feedback is used, the control variables u, , u 2 , , um are functions of the state variables X, expressed as U = U(X). Substituting this vector function into

Eq.(1.39), we obtain the general mathematical description of nonlinear control systems in the form as X(t) = f(X(t))

(1.40)

In the above equation, let

.-.

f(X)=0 The zeros of i(X) are precisely the solutions of the above algebraic equation. Each solution Xe determines one equilibrium state of the nonlinear system. Since each certain state corresponds to a certain point in the state space, the equilibrium state X. is also called an equilibrium point.

From Eq.(1.40), we know that the physical meaning of f(X) is the generalized velocity set. Because Xe is one of the solutions of the equation

f(X) = 0 , that implies, the "velocity" of every state variable x; of the system at point Xe along its own axis is zero, namely, =

dt = dt

l(xel,xe2,...,x")0

f2(xelIX.21

=4

`-'

dx,{ dt 72

(Xel,xe2,...,xen)

=0

,.,

This means that absolutely without any disturbances, the state point of the system will stay at the point X. for ever. This is why the Xe is called an equilibrium point of the system. A hyperbolic linear system has only one equilibrium point, i.e. the coordinate origin. Since the state equation is i(t) = AX and its equilibrium equation is AX = 0 , therefore X,=O, because hyperbolic means A has only non-zero real part eigenvalues and hence A is non-singular. As a result, if a 'C3

a°.

20


linear system is stable at Xe= 0, then the system is stable. According to the Lyapunov's stability principle, the necessary and sufficient condition for a CAD

linear system to be asymptotically stable is that the real part of every eigenvalue of matrix A is negative, i.e.

R{2;}(Z) dZt = 5V(Z) f(Z)

where f(Z) = f(X8 + Z)

(1.42)

Introduction

21

satisfies

for Z=0 for Z#0

V(Z)=0 V(Z) 0

for Z=0 Z

for

0

then point Xe(Z = 0) is an unstable equilibrium point. `.
-1) where the function c(X) is singlevalued and smooth, and the existing inverse function is single-valued, smooth as well, which can be denoted as z,

[:']=-'(ZZ2)=[Z

2

+ z,

Thus we know that the above expression Z = cD(X) is a local coordinate transformation or a local diffeomorphism defined on a certain domain. How to test whether a nonlinear mapping d>(X) is a local

diffeomorphism at X°? The answer is that (D(X) is a local diffeomorphism in an appropriate neighborhood-of X° including X° if the vector function c(X) has a nonsingular Jacobian matrix at X°. That is to say, it is a valid coordinate transformation or mapping. Let us present the property of the above nonlinear function by a proposition.

Proposition 2.1 Suppose c1(X) is a smooth function defined on a certain subset U of space R". If the Jacobian matrix at X= X° a0p,

ar,

ax2

awe

a

ax

F ap,

ax

awl

...

ate,

ax awl

&2

X=X°

L ax,

ax2

axn

X=X°

is nonsingular, c1(X) is then a local diffeomorphism in an open subset including X°. Now let us illustrate it with an example. Example 2.1 Consider the following function

U°

32


[z2]-(D(x"x2)-[sinx221

which is defined for all points in space R2 . Its Jacobian matrix _ rl

ax

o

cosx2

is nonsingular at X° =[0 0]T and has rank 2. Hence this function cp(X) is a

local diffeomorphism in a subset including x° = [0 O]''. What can be confirmed further is that this open subset including X° is

U°_{(x,,x2)IIx2I ... , xnv)

How to define the derivative of this scalar function 2(X) along the vector field f(X) ? The answer is that the derivative is defined as the scalar product (point product or inner product) between VA(X) and f(X), where V2(X) is the gradient of the function 2(X) VA(X) = I

That is

ap (x)

as (X)

ax

axe

...

a 2(X)

ax

Basic Concepts of Nonlinear Control Theory

39

or

< aA(X),f(X) >

ax

< V2(X),f(X) >

(2.41)

This formula defines a new scalar function which is called the Lie derivative of 2(X) along f(X) and denoted by L f2(X) . Now we give the precise definition of Lie derivative.

Given a differentiable scalar function 2(X) of and a vector field f(X)=[f, the new scalar

Definition 2.2

X=[x, ... function, denoted by L12(X) j s obtained by the following operation

Lf2(X)= a2(X)f(X)-Ya2,(X) f; (X) ax

ax;

(2.42)

and called the Lie derivative of function A(X) along the vector field f(X).

Example 2.4 the vector fields

Find the Lie derivative of the function h(X) = x3 along e'!

0

and

g(x) = e.'

f (X) =

,

..y

0

x+ x, -x2 y..,

According to the definition of Lie derivative we first obtain the gradient vector of the scalar function h(X) = x, as Oh(X) = a(x3) = [0

0

ax

ax

1]

The desired Lie derivatives are er2

Lgh(X)

aaX)g(X)=[0

0

1]I ex,

I=0

Lfh(X)_[0

r-,

0

0

1]

0

x, +x2 =x,-x2 El XI -x2

From Definition 2.2, we know that Lie derivative is a scalar function, so it is possible to repeat use of this operation to obtain the Lie derivative L f2(X) along another vector field g(x), i.e.

LgL f

(X ) =

a(L,2(x)) ax

g(x)

(2.43)


40

Certainly, we can also derive the k`' order Lie derivative of 2(X) along f(X) recursively as

L f(L fi(x)) = L fi(x) 2

=

a(L fA (x)) ax f(x)

(2.44) L

f (x)

The k'"-order Lie derivative of 2(X) along f(X), L2(X) is still a scalar function and thus can be used to get Lie derivative along another vector field g(X), i.e. k

LgLf2(X) =

a(Lk2(X))

ax

(2.45)

g(X)

Let us use an example to show the method of obtaining high order Lie derivative.

Example 2.5

Given a scalar function

h(X)=x3 and vector fields 0

f(X) = x, +x2 x, -x2

ex,

g(X) = ex'

and

e X3

find high order Lie derivative of h(X) along f(X) and g(X)

.

According to Definition 2.2, we obtain

Lfh(x)= a(h(X)) f(x)=[o ax

0

1]

x, I x1

= x, - x2

From formula (2.44) we know that LZfh(X)=

a(Lfh(x)) f(X) = a(x, - x2)f(x) ax ax 0

=[I -1 0] x, + x2 Lxi - x2 J

-(x, +X2 )

With the above second-order Lie derivative of h(X) along f(X), we can further obtain


a(

LJh(X)

ax

41

x2) X + X 2 X, -X ' 0

=[-I

0] x, +x2

-2x2

x, - x2

=-2x2(x, +x2) According to formula (2.45) we obtain 2

L9Lfh(X) =

a(- x, - x2 )

ax

= -er' -

2.6.2

g(X) = [-1

- 2x2

2x2exi

Lie Bracket

Assume two vector fields of the same dimension are given as

fi(x...... x,,)l .f2(xl,...,x»)

g(X) =

and

I

III

f(X) =

frr(xl,...,x,r)

mss'

fit

How to define the derivative of the vector field g(X) along the other vector field f(X) ? What should be clarified first is that the derivative of one vector field along another is still a vector field rather than a scalar function. And it is defined as follows. Let [f(X),g(X)] denote the derivative of g(X) along f(X), then ''b

ag,

L'9 2

ag2

x

cps

If, g1(X) = ax,

,r

ax

of

of

cps

ag,

rf,

fi

a

f2

al2

,r

...

- ax

fig, l g2

ax,r

(2.46)

...

Oxag'

g In

...

ax,,

af, a,

afn axn

...

gn

which can be abbreviated as Vet

If, g1(X) = ax

f ax

(2.47) g

where ag/aX and of/aX are the corresponding Jacobian matrixes.

42


Expression (2.47) shows a new vector field, namely Lie bracket, for its accustomed denotation [f, g], which can also be denoted by adfg . The precise definition of Lie bracket is given below.

Definition 2.3 Suppose two vector fields f(X) = [f f2 - f ]T and g ]T. The following operation denoted by [f(X), g(X)] or g(X) _ [g, g2 adfg

agf-a ax

[f,g]=adfg=

g

obtains a new vector field which defines the Lie bracket of g(X) along f(X). `C3

E-'

The following two examples are presented to explain the computing

method.

Example 2.6

...

Given vector fields 0

I X3(1+X2)

+X1 )- x3

f(X)=

g(X)= 1+x2

and

x, 1 x2(1

compute `".

[f(X), g(X)] = ad fg(X) C"'

According to the definition of Lie bracket, we first figure out the ,ti

Jacobian matrixes of f(X) and g(X) I

f(X) ax

f

J s= ag(X) ax

O2

0

0

x2

1+X,

0

0

0

1

0

0

0

-1

0

0

From Definition 2.3

[f(X), g(X)] = ad fg(X) = 0

=0 0

a gm f(X) - af(X) g(X) ax ax

0

0

x3(1+x2)

0

x3

1+x2

0

1

0

x,

1

0

0

1+x2

x2

1+x,

0

-x3

-1 X2(1+x1 )

0 0

X,

-x2(1+x1 )

0

-

0

(1+x,)(1+x2)

43

Basic Concepts of Nonlinear Control Theory 0 X1

-(1+x,)(1+2x2)

Since the Lie bracket of g(X) along f(X) is a new vector field, it can be used to calculate the Lie bracket along f(X) once more. Similarly we can find k"` order Lie bracket inductively as ad f g(X) = [f,[f,g]](X) (2.48)

ad jg(X) _ If, ad j'g](X) Suppose two vector fields f(X) and g(X) are as in Example 2.6, calculate ad 2g . From Eq. (2.48) we know that nPO

Example 2.7

ad fg(X) = [f(X),ad,-g(X)] y!'

f(X) - af(X) ax (ad fg(X )) 0^Q

ax

AIM

= a(ad fg(X ))

From Example 2.6, we have 0

ad f9 =

x,

-(1+x,)(1+2x2) Therefore 0

0

1

0

0

L- (1 + 2x2)

- 2(1 + x,)

0

7C)

adfg=

0

1 + x211

0

x3

1

0

0

x2

1 + x,

0

0

x3(1 + x2)

-2x,(l+x,)-x3(1+x2)(1+2X2) [x1x3 -(1+x1)(1+x2)(1+2x2)1 0

x1(1+x 1) - x1x3 + (1 + x1)(1 + x2)(1 + 2x2) x3 (1 + x2 )

-x3(1+x2)(1+2x2)-3x1(1+x,)

44


The following gives the main operational rules of Lie bracket. Rule 1 Lie bracket is skew symmetric, i.e. (2.49)

If, g] = -[g, f ] This can be shown from the definition clearly.

Rule 2

If c, and c2 are two real numbers, then

[f,c,g, +c2g2](X)=c,[f,g1](X)+c2If,g2](X) Rule 3

(2.50)

If p(X) is another vector field, then

If,[g,P]I+[P,If,g]I+[g,[P,f]]=0

(2.51)

,+y

Formulae (2.50) and (2.51) can be deduced from the definition directly. Rule 4 If f and g are vector fields, A(X) is a scalar function, then the Lie derivative of ,,(X) along vector field If (X), g(X)] is L1J.g]A(X) = LJLs'%(X) - LgL f..%(X)

(2.52)

O.=

We give the proof for Rule 4 below while leaving the proof for the other three rules to the reader as an exercise.

Proof for rule 4: From the definition of Lie derivative we have

ax ax

V'1

a.. (X) ag _ of f g) ox (aX ox - a2(X) ag f _ a,,(X) of

LtJ,gl't(X) =

(2.53)

ax ax g

According to the definition we know L9X

a f=X (ax g).f

...

a(L9A)

V'1

e(,

LJLgA(X)

(2.54)

f) - g

(2.55)

and

a(L 1 A) U.&

g=

aax ax (aa,

V'1

L g Lf A(X) =

The above two expressions can be further evolved as

a (aa,) + aA ag )f ax ax ax ax gTax(

L JL9A(x) = (gT

(2.56)

ax )f+axaxf

L 9 L fA(X) = f T

From these two formulae we get

a (aa. )g + aA of g ax ax

ax ax

(2.57)

can

45


L fLg2(X) - LgL fZ(X) =

aa. ag f - aa, of

ax ax

g

ax ax

(2.58)

Comparing formula (2.53) with formula (2.58) we have L(f g] A(X) = La,,gA(X) = LfL9A.(X)- L 9 L f %(X) Thus Rule 4 above has been proved.

These rules about Lie derivative and Lie bracket yield a kind of algebraic operation, i.e. Lie algebra. According to Rule 1,

If M, g(X)] # [g(X), f(X)] Thus we know that Lie algebra does not satisfy commutative law and is a type of non-commutative algebra.

2.7

INVOLUTIVITY OF VECTOR FIELD SETS

c0)

Topics in this section will be: first, what is involutivity; second, how to comprehend its geometrical meaning. Let k n-dimensional vector fields be of the following forms 9k1(x11...1x11)

g11

gk(X)=

g1(X)= to form the matrix g11 g21 ... gkl `.Y

G = g12 922 ... 9k2

= [g1(X) g2(X) ... gk(X)]

(2.59)

gL, g2n ... gb,

If the matrix has rank k at X = X° and the augmented matrix [g1

g2

...

gk

1909,1 1

(2.60)

has the same rank k at X = X° to the arbitrary number pair i and j where I < i, j :- k , then the vector field set (2.61)

is called an involutive one or we say it has the property of involutivity. Sp {g 1, g 21 , g k } , the spanned space of the above vector field set is also called the distribution in geometry. Hence the distribution satisfying the above conditions is called an involutive distribution.

46


The definition of involutivity has been clarified so far, but how to understand it geometrically? Since the matrices in expressions (2.59) and (2.60) have an equal rank value, the new vector field obtained by Lie bracket operation of two arbitrary vector fields from the original k vector fields (i.e. evaluating the derivative of one vector field along the other) is not linearly independent but linearly dependent with the previous k vector fields. In other words, the new vector field [g,, gJ ] is still in the original spanned space of .-J

the k vector fields and does not form a new direction. This is just the .r..'

geometrical meaning of involutivity. S3.

It is very important to confirm the involutivity of vector field sets in nonlinear control systems. A simple example is used as follows to explain how to check the involutivity of a vector field set.

Example 2.8 Let the vector fields f(X) and g(X) be given as in Example 2.6. We need to check the involutivity of vector field set L."

{g(X),ad fg(X)} near X° =0. From Example 2.6 we know that 0

0

g(X)= I+x2

ad jg(X)=

x,

-(I+x,)(1+2x2)

-x3 At X = 0 , the matrix

0

0 .-.

[g(X) ad rg(X )] =

1

0

0

-1

has rank 2 (i.e. r=2).

To check the involutivity of {g(X), ad fg(X)} at X = 0 , we first calculate the Lie bracket of vector field ad fg along g at X = 0 [g, ad fg]=

0

0

0

0

I

0

0

1+x2

0

-x3

-(1+2x2) -2(1+x,) 0

0

0

0

-0

1

0

X,

0

0

At X = 0 , the augmented matrix

-1 -(1+x, )(1+2x2)

47


[g

ad fg

[g, ad fg] ] =

0

0

0

1

0 -1

-3

0

0

has the rank that still equals 2. Therefore the set {g, ad fg} is involutive.

2.8

RELATIVE DEGREE OF A CONTROL SYSTEM

Another important concept, relative degree of a control system denoted by r, is introduced in this section. Suppose a single-input single-output nonlinear control system

X = f(X)+g(X)u

(2.62)

y=h(X) where X E R", U E R, y r= R, f(X) and g(X) are vector fields. If

(i) The Lie derivative of the function Lfh(X) along g equals zero in a

neighborhood 52 of X = X ° , i.e. LgLfh(X)=0,

k(X)g(X)=0 D2 + 42> (X)g(X) + k22> (X)ad jg(X) = 0

`g(X)=0

D;

(3.59)

(X)ad""'g(X)=0

Formula (3.59) implies that the vector field D, is a linear combination of vector fields g(X), adfg(X), adfg(X), , ad'f'g(X), i=1, G1.

.L"

Step 3. Let us introduce a symbol (D`, (X°) , which represents the integral curve of the vector field f(X)_[f(X) f2(X) f,,(X)]T with the 0

initial value V, which is also the solution to the set of differential equations dx, (t)

.fi(xi,x2,...,x")

dx2 (t)

=J2(x1,x...... x

n)

dtt) =

with the initial value X° = [x,° x2

x°]T

.

The task of Step 3 is to calculate the mapping X = F(W) from the state

space R" of the new coordinates W to the state space R" of the former

n vector fields D D2, , A,,

...

coordinates X. This mapping can be expressed by the integral curve of the i.e.

by the solution

a).6,,

corresponding differential equation, i =1, 2, , n, namely the mapping

of the

82


F(w, w2

(3.60)

ocI

If we still feel a little puzzled about the meaning of formula (3.60), the following discussion will make it clear. Formula (3.60) can be calculated by the following steps: first calculate the integral curve (DD- (X0) of the vector field D with the initial value X0

and the independent variable w . That is to compute the solution of the differential equation x,(0)

d

x2 x1

=D

X(0) = x2(0)

x

-X0

(3.61)

x (0)

The solution (DD. (X0) of the differential equation (3.61) is the vector function of the independent variable w , namely x,(wn)

(D D.(XOX\wn)

x2(wn)

Equation (3.60) tells us that the next step is to calculate the integral curve of the vector field with the independent variable and the initial

condition cp°; (X0) which is to find out the solution to the differential equation x,

d

X2

(3.62)

dwn-, X1,

Solving the differential equation (3.62) we can get the integral curve (DD._, (X 0) ° De (X0), which is the vector function of the independent variable

i.e.

(X

q)

(D

D.

We use the above as the initial value and compute (DD:-;

(X0) ° D;=; (Xo) ° D; (X0) = X(wi-2,

w)

83

Design Principles of Single-Input Single-Output Nonlinear Control Systems

Similarly, we can obtain

vector

field DI , which is the solution to the differential equation x' xl(w2,... wn)

d

x2

Dl

( X 0 ) . . _..,OD, (X°)

X(0) = (D

dw,

x2(w2,...,wn)

(3.63)

=

Di

`XJW2,...I WO]

Solving equation (3.63) we obtain the mapping (3.60)

defined by

x1

X= x2 =F(W)= F2(wl,w2, .,wn)

(3.64)

xn

From the above we can calculate the inverse mapping of F wl(x1,x2, ',xn) w2(x1,x2, ,xn)

Wl

W=

IV2

=F-'(X)=

(3.65)

Lwn(xl,x2, ,xn)J

LwnJ

Step 4. Calculate the derived mapping F.-' (f) of the vector field f(X) of the system (3.1) under the mapping F-' in Eq. (3.65). From the definition of the vector field derived mapping in Section 2.5 we know F.-'(f) = J F-' f(X)IX=F(W)

(3.66)

In the above formula, JF_, represents the Jacobian matrix of F-'(X), i. e. awl (X) axl

JF =

'1(X) ax2

awl (X) axn

awl(X) aw2(X) ... 3w2(X) axl vc2 n 11

(3.67)

,,..

UWn(X) n(X) axl

ax2

5w,(X) axn

Let .fl'0) (W)

f(0)(W)= fz°)(W) = F. ' (f) = J FA f(x)l x=F(W) fn(o)

(W)

(3.68)

84


In order to compute the coordinate transformation and state feedback for exactly linearizing the nt'-order affine nonlinear system, we also need to define R R2, , one by one. First we define the transformation R, as follows f2(0) (W)

z1(')

z(° =(°' f(W) 2

(3.69)

z(1) = f(o) (W) n

1

Z(1) = Wn n ..o

In the above formula f(') (W), i = 2, , n, has been given in Eq. (3.68). Subsequently we find out f(1)(W)

f(') (W)=(R)).f(0)(W)=JR f(0)(W)

(3.70)

The Jacobian matrix in (3.70) is

(3.71)

JR =

awl

L awl

SWn J

Then we continue to define the transformation Ri as ZI(i) = f{ (i-1) (W)

f

Z(2j)

-1) (W)

Z(i ) = f "U-1) (W)

(3.72)

-1

Zi) = z (i1)(= n

From the above we can calculate the f(i)(W) f (i' (W)

f(i)(W) = f(i )(W)

=j .(W)f(i (W)

(3.73)

fn(i) (W)

JR, in formula (3.73) denotes the Jacobian matrix of the mapping Ri which

has the same form as J. in Eq. (3.71). Following the computing procedure shown by (3.72) we can calculate each transformation we need until the (n -1)1h transformation

Design Principles of Single-Input Single-Output Nonlinear Control Systems z("-1)

Z2,-1)

85

= f,("-" (W) { 3(11-2) (W) =J

f

(»-1)

z("-'

Z"

11

(3.74)

(W) )

is actually the eventual coordinate transformation we need. From (3.74) we know that the transformation is the change of coordinates from W space to Z("-') space. What we need is to calculate the mapping from X space to Z("-') space. Note that calculating the coordinate transformation from X space to Z("-° space is equivalent to calculating the composite transformation T, defined by T = R,,-, F-1 (refer to Fig. 3.2). The

Figure 3.2

Relations among the coordinate transformations of the spaces X, W and Z

transformation T can be calculated in terms of the following formulae

f2(-2) (W)=TA(X)

zin-1)

W=F" (X)

f3(n-2)(W) W=F"i(X) c..

z(2-1)=

=T2 (X)

(3.75) Z(. -I)

n

1

=

f,(1-2)

(W)IW=F-I(X) = Tn-I(X)

N

z("-1) = w n W."-, (X) = Tn (X)

According to the coordinate transformation T, we could transform the vector fields f(X) and g(X) of the original nonlinear system (3.1) to f(X) and g(X) which represent the transformed vector fields, i.e. i(X) = JT(X)f(X) = [fi(X) f2(X) g(X) = JT(X)f(X) =181(X)

0

...

.._

Of

fn-1(X) f (X )]T

(3.76)

(3.77)

86


where g(x) m o, and aT,(X)

aT, (X) aT, (X)

8x2

axt

azn

aT2(X) aT2(X)

JT(X) =

ax!

aT2(X)

ax2

axn

aTn(X) aTT(X) ax1

Step 5.

ax2

(3.78)

aT°(X) ...

aXn

The final coordinate transformation Z=F(X) could be

immediately obtained by setting >C.

Z, = w (X) Z2 = f,(( )

(3.79)

Zn-1 = 3 (X)

Z. =f2(X) The inverse transformation is (3.80)

X = F-' (Z)

If we define U

f (X)

1

v)IX-F i(Z)

- ( 81(X) + 81(X) then the original nonlinear system (3.1) is transformed into a Brunovsky normal form ii = Z2 Z2 = Z3

(3.81) Zn

in = V

and the state feedback law of the original nonlinear system is achieved as follows __ _

1,(X) 1 v 81(X) + g1(X) 2C0

U(X)

(3.82)

Formula (3.82) gives the control law which is sought for. Above is the exact linearization algorithm proposed by Ref. [1]. Now

+-'

we can understand that the essence of this algorithm is to transform the problem of solving partial differential equations into the problem of of a series of ordinary obtaining the solution

87


cep

0

differential equations, and then to carry on the transformation step by step using the solution. It is worthy of noting that before taking the steps of exact linearization transformation, we should check whether the discussed affine nonlinear system satisfies the exact linearization conditions of Theorem 3.2.

After obtaining the affirmative answer, we can carry on the deduction of exact linearization. Let us explain the whole process of this algorithm with an example. Example 3.3

Consider the system

x,+3x; +Sxl

x [x2 ]

1

+[1+X11 ] u

l +X2

0

-x, +x2

(3.83)

x2 (0) = 0

X1 (0) = 0

vii

First we should check whether the system satisfies the conditions for exact linearization. To save space, we suppose that the system satisfies the

conditions of Theorem 3.2. Now let us carry on the deduction of exact linearization via state feedback.

Step 1. Calculate the Lie bracket of g(X) along f(X). Since n = 2, we only need calculate adfg(X) (1+x; )-2x,

ad rg( X )

0

x,+3xl +Sx5 0 0 [

[(1 +0x1 2 )

[

1+x ,

-x +x2

1 2x2

l

J

J

-1

=

1

l_+_X;

A..

-(1+x; )-22x,(x, +2x, /3+x; 15) +(l+x, )-'(1+2x, +x, ). where The two vector field sets composed are D,=(g) and D2 = {g, ad rg} 1

D, _{ 1 +x ;

-1

1

}

D 2 ={l+x,

0

l+x

0

r..

t71

I}

1

z

Step 2. Select the simplest n = 2 linearly independent vector fields ED, and DZ E D2. We determine the functions k;',(X), k;2)(X) and

k(2) (X) based on formula (3.59).

1

L

0

[01

88

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS I

D2 +k;2'(X) l+x2 +k(22) (X) 0

In order to make D, and D2 as simple as possible, we choose

k;' (X)=-(1+x k,2)(X)=-(1+x,)2 k22)(X)=-(1+x;) Then we have

Step 3.

D2 =[0 If

Of

D, =[1

The next task is to compute the mapping using formula (3.60) .(DD, (X,)

X0 =0

So we need to calculate the integral curve (DW, (Xo) .Solving the differential

equations dx,

chv2

dx2

with [°1

[x2 (0)

I

=

[01

dw2

we have

x2]7 =[0 w2]f

[x1

Then we calculate cb o 5' (Xo), i.e. solve the differential equations z

r, l dw,

[o]

dx2

dw,

with the initial condition X(0) = [0

w2 ]

T

Thus we obtain [x21

F(wl+x'z)-[F2(w'],w2)][x'2]

Obviously, the inverse mapping is w2 J

[F (x, xz) =

W1 (X I X2) X. J [w2 (xI , x2 )J = [x2


Step 4.

89

Calculate the derived mapping F'(f), thereby obtain

0)(W). First we need to compute the Jacobian matrix JF_, of F-' ,(X)

aw1(X) axz _

ax,

JF'

1

ax,

0

01

aw2(X) aw2(X) axe

Thus, from formula (3.68) we have

1f f(0)(W)=

ro)(W) -JF`f(X)IX=F(W)

0]

0

1

W1+3W1+SW

W1+W,3 +SW1 3

1

N_"'

[1

l+w

l+w

_WI +W2

-w,+w2

Since in this example n = 2, it is necessary to compute the R,. From (3.69) we have R, transformation Z;') = A°' (W) = w2 - w, = W2 Zco 2

Accordingly, the transformation T = R,F-' is >

Z,

( z - W, ) = W2 I

W-F_'(X)

(1)

Z2 = W21 W=F-'(X)

2 = X2 -X,

_ - x2

w++

From transformation T we can get the final transformation f = T.(f) = JT(X)f(X) x , +2X 1

2x2

0

+x S

'

1+x

1

- x, + x2

X1+?x; +-x;

t+x

5

2

1

+2x2(-X, +x2)

-x, +x2 = T.(9) = JT(X)f(X)

=r-1

2x21 1

1

-1

l+x; = l+x 0

0

90


Step 5.

Define z,

=.f2(X) =-x, +x2

z2 =

and take the state feedback as u

f (X)

i (3.84)

(X) + S1 (X) v

=-(x,+3x1 +5x,)+2x2(-x,+x2)(1+x, )-(1+x, )v 2

Thus the system of the example has been exactly transformed into a linear and controllable system in the new coordinates Z Il =z2 Z2 =V'

In order to strengthen the understanding of the problem, we may substitute the control law u(X) (3.84) into the system (3.83), then we can get a closed loop system whose state vector is X X = f(X)+g(X)u(X)

(3.85)

After that, using the inverse coordinate transformation given by the Step 5 of this example X, = z,2 - z2 x2 = z,

we transform the system (3.85) from X space into Z space and have the Brunovsky normal form, i.e. a, = z2

z2=v

3.3

ZERO DYNAMICS DESIGN PRINCIPLE In the foregoing section, we discussed the necessary and sufficient

conditions for the exactly transforming a SISO affine nonlinear system into a

completely controllable linear system (Brunovsky normal form), and introduced how to calculate an appropriate coordinate transformation and the corresponding nonlinear state feedback law, which will be able to greatly improve stability and dynamic performance of the nonlinear system even when the system state changes in a wide range. Actually the state feedback calculated by the above method compensates or "counteracts" the original system's nonlinear characteristics and transforms it into a controllable linear

c^0


91

any

system with good dynamic performance. In dealing with this problem, getting the proper nonlinear state feedback is essential for obtaining the system's exact linearization, while getting the appropriate coordinate transformation is only the external appearance of the way in which the Q..

linearization problem is solved. In the algorithm provided in above section, seeking for coordinate transformation is closely connected with getting the state feedback. The control law obtained by the algorithm can completely and exactly linearize the original nonlinear system. It is clear that the control law designed by this method is quite complicated, which can be seen from the given examples. In this section we shall discuss and explore another design method, which could be called zero dynamics design method. This method does not need to exactly linearize all system's state equations, but just a part of them. In fact dynamic behavior of a system could be classified as external dynamics and internal dynamics. From the perspective of application, we are mainly concerned with the system's external dynamics, which should have stability and good performance. But as to the internal dynamics, what we need is only stability. According to this idea the designed control law may sometimes be more practical and simple. This is the basic idea of the zero dynamics design method. Next we will discuss this design principle and method. There are two types of the zero dynamics design method, which will be discussed one by one. G1.

pry

o

3.3.1

First Type of Zero Dynamic Design Method

Consider a nonlinear system X = f(X)+g(X)u y(t) = h(X(t))

(3.86)

a,.

The relative degree r from the output equation h(X) to the system is less than the system's order n. Under this condition, from formula (2.91) we know that the original system (3.86) can be transformed into the following normal form

.-.

III

2-1 = Zr Zr = Lrfh((D-' (Z))+LgLrj'h((D-' (Z))u

Zr+i = Lfco, i(I '(Z))

(3.87)

92


(3.87)

z = L f(p (10-' (Z)) where Lrfh((D-'(Z)) denotes Lrfh(X)Ix=(D-,(z), and so on. z,

...

h(X) Lfh(X) 0000,

...

O00

Zr

(D(X) _

(3.88)

L.i' (X)

zr+l

(Pr+i(X)

L z. J

The selected functions (pr+ , (p (X) satisfy (3.89)

LgcPr+1 (X) = Lg(Pr+2 (X) _ ... = Lg(p (X) = 0

And the Jacobian matrix

n(X)

Jm

ax

x=x°

is nonsingular. For the sake of convenience, we introduce here the following notations:

denotes the first r state variables in Eq. (3.87),

n the remnant n-r state

variables, ' = [z1 Z2 ... Zr ]!

n=[zr+l zr+2 ...

(3.90) z,,]T

(3.91)

Then Eq. (3.87) can be written as Z, =z2

(3.92)

Zr- = Zr

z, =a(t, n)+b(b, n)u n)

n

where L fh(X)j,=0-,(4'

n) = Lf h((I)

' (r , n))

b(t, Ti) = Lg Lr!' h(X)I x=a -,(;, n) = Lg Lf' h((D '

(3.93)

n))

and

qr+l(r'f n)

Lf(Pr+,(b'(`', n)) (3.94)

n) q,,

n)

Ti))


93

Generally speaking, we can always choose the output equation h(X(t))

which is equal to zero at the equilibrium point X°, that is h(X)=O. Therefore the output y(t)=h(X) is the actually dynamic deviation of the practical output (dynamic response) from the output at an equilibrium point.

If we use the control means to impose on that dynamic deviation of the output of the system keeps zero at any time, i.e.

y(t)=h(X(t))=0

0- 0, then the purpose to keep the generator's terminal voltage constant under the influence of disturbances will be fulfilled. Here is another example. To the generators in a power system, we hope that the generators can not only keep their terminal voltages constant, but also hope keep their rotor speeds invariant. Then we may choose the following output equation y(t) = h(X(t)) = (V, (t) - V,EF )2 + (w(t) - w°)2

where w° is the rated rotor speed of a generator. For the time being, we will not discuss whether or not the requirement

o=.

of , y(t) = h(X(t)) = 0 for t >- 0, can be actually fulfilled in practice but clearly it is reasonable and logical that the condition which makes the external dynamics (output) of a system zero at any time is regarded as an 0

objective function, i.e. a performance index of a control system in controller design. As we have mentioned above, the condition of the system output

p.,

w`
1, then Lgh,(X)=Lg2h,(X)=0 So Eq. (4.4) can be written as (4.5)

Wl = L fh(X) = 4/2

We can derive similar results for rp; until (4.6)

-1 = rPr,

As stated by the definition of relative degree, are at least not all equal to zeros. So we have

Lg,

h, (X) and Lg2 Lrj ' h, (X)

01=Lrrh,(X)+Lg Lrf'h,(X)u, +Lg2Lrj'h,(X)u2 Similar to the above we can get

(4.7)

126


(4.8)

V/r,

VV, = L)h2(X)+Lg,Lrf'h2(X)u, +Lg, rl h2(X)u2

combining Eqs. (4.5) through (4.8), we know that when the relative degrees satisfy r = r, + r2 = n, under the coordinate transformation shown in Eq. (4.3), the system (4.2) can be transformed into the following normal form 01 =P2 Or,-1 = lP,

(p,. = Lrfh,(X)+Lg Lj'h',(X)u, +Lg2 Lrf'h,(X)u2

(4.9a)

W2

Vri

yrr= = Lrjh2(X)+Lg Lrj'h2(X)u,

and the output Y, (t) _ -P 1(X (t ))

Y2 (t) = V/1 (X(0)

(4.9b)

The above Eq.(4.9) is called the first type normal form of MIMO affine nonlinear system. It corresponds to the condition that the sum of system

+-N

relative degrees r = r, + r2 + + rm = n . Let us discuss the condition that the sum of system relative degrees r,, +'A

i =1, 2, , m is less than the system's order n . If for a system as shown in Eq. (4.2), its relative degrees satisfy r = r, + r2 < n, then after choosing a coordinate transformation as shown in Eq.(4.3), the last (n - r) coordinates

could always be found as i =171(X) ......

(4.10)

17n-r = 17 -r (X)

such that the Jacobian matrix of the vector function O(X)=[,P1(X), ..., p, (X); V, (X), ..., V , ,(X); 171(X),

(4.11)

is nonsingular at X = X°. Thus, we have chosen a set of qualified local coordinates mapping. Under this condition, the following (n - r) equations should be added to the system described in new coordinates besides the equation as shown in Eq. (4.9), i.e.

Design Principles of Multi-Input Multi-Output Nonlinear Control Systems

127

r), = Lj77i(X)+Lg r7i(X)u, (4.12)

Lj'ln_r(X)+Lg,rln-r(X)u, +Lg,77n-r(X)u2

Therefore, the system after transformation is 01 =qP2 (Pry

gyp, = Lrjh1(X)+Lg Lrf'h,(X)u, 'VI=W2

(4.13a)

Wr,= Wr, =L'jh2(X)+Lg,L'j-'h2(X)u,+ Lg,Lrr'h2(X)u2 r/, = Ljrl,(X)+Lg rl,(X)u, +Lg 171(X)u2

n-r = L1rln-r(X)+

Lg,rl'-r(X)u2

together with output equations Y1 = Vi Y2 =W1

(4.13b)

.-.

What is expressed by Eq. (4.13) can be called the second type normal form. Let us come to the third type normal form. It can be verified that, if the vector field set {91(X)192(X), (X),.. ., g.(X)} shown in (4.1) is involutive, then n-r (X) can surely be found such that coordinates mappings 71, (X), rl, (X),

Lgrl,(X)=0 L" 77, (X) = 0

(4.14)

Lgarl;(X)=0

Thus, (4.13) can be transformed into '1 =P2 Or,-] = V"

rpr = Lr}h,(X)+Lg Lr}'h,(X)u, +Lg,L' 'h,(X)u2 =W2 Wr,-1 -Wr,

yrr2 =L)h2(X)+LgL'}- h2(X)u,+Lg Lj'h2(X)u2 Tll = L1211(X)

n-r = Liri-r (X )

(4.15a)

128


with output equations yI = VI

(4.15b)

YZ = V1

In Eq. (4.15),

r/, (X), 171 (X),

,

(X)

are the solutions of the partial

differential equation set (4.14). What is expressed by Eq. (4.15) can be in'

called the third type normal form. It should be pointed out that the third type normal form in Eq. (4.15) is

simpler than the second one in Eq. (4.13) in the last (n - r) equations. However, this simplicity is at the cost of solving the set of partial differential equations in Eq. (4.14). Based on all the above, the following propositions could be formed.

+O-'

Proposition 4.1 Suppose that there exists a system as shown in Eq. (4.2), with the sum of relative degrees r = r, + r2 = n. Choose a coordinate transformation Z = cD(X) as follows z1

=97'(X)= h (X)

Z2 = P2 (X) = L fh, (X) Zr

-'Pr,(X)=L}-'h,(X)

Z,+I =y/1(X)=h2(X) Z".2 = V2(X) If h2(X)

(4.16)

z, =wr,(X)=Ef-'h2(X)

-e.

a.+

then the system can be transformed into the first type normal form in the new coordinates Z 0I =IP2

rpr = Z. (Z) + b11(Z)u1 + b12 (Z)u2

(4.17)

WI =1V2 1Vr,-I = yfrz

,r, = a2 (Z) + b2, (Z)u1 + b22 (Z)u2

where

a(4.18)

Z=IV, ...

-Pr

WI ... V, IT

A(Z)=[,(Z)]=[LihjX)]

lei

a2

)

Ljh2(X)

x=m-I(Z)

Design Principles of Multi-Input Multi-Output Nonlinear Control Systems g(Z) = (bI1(Z)

b12(Z)

__

C/1

Proposition 4.2

Lg L! h1(X) Lg,L[ ' h2(X)

Lg Lrr Ihl(X)

(4.19)

Lg L'!I "2(X)

Suppose there exists a system as shown in Eq. (4.2) 0-0

rt.

k1(Z) b22(Z) ]

129

with the sum of relative degrees

r = r, -i-r2 < n .

Apart from the chosen

coordinate transformation as shown in Eq. (4.16), the last (n-r) coordinates can always be found as Z,+l = 771 = 771(X)

(4.20)

='in-, (X)

Zn

such that the Jacobian matrix of the vector related to the coordinate mapping (D(X) = [p, (X) ... P, (X) Il/, (X) ... V/r, (X) 771(X) ... 77n-r (X )iT

is nonsingular at X°, thereby the original system can be transformed into the second type normal form, which is formed by replenishing the form shown by Eq. (4.17) with the following (n - r) equations (4.21 a)

n= 4(Z)+P,(Z)u1 +P2(Z)u2

which may also be written as (4.21b)

7

n=9(Z)+[PI(Z)+P2(Z)] [UJ where "_ Inl

_

77n-rlT

772

91 (Z)

Lf771(X)

(4.22)

9(Z) 9n-r(Z) Pu (Z)

Lf77n-,(X) x=m-'(z) Lg.171(X) (4.23)

Lp 1(n-r)(T) T21 (Z)

Lgl7]nr(X ) x=m-'(Z) Lg,>11(X )

(4.24)

i'2 (Z) P2(n-r) (Z)

Proposition 4.3

Lg,77n-r(X) x=o-1(z)

Suppose there exists a system as shown in Eq. (4.2)

with the sum of relative degrees

r = r, + r2 < n .

If its vector field set

{g1(X),g2(X)} is involutive, the last (n-r)coordinates 171(X), can be chosen to satisfy

772(X),...,?7.-,(X)

130

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Lg 77; (X) = 0

(4.25)

Lg77;(X)=0 rip

such that the original system can be transformed into the third normal form,

which is formed by replenishing the form shown by Eq. (4.17) with the following (n - r) equations rt= q(Z)

(4.26)

In addition, for the above three propositions, their output equations are the same as Y1 =(P,

(4.27)

Y2 = 4/1

The problem discussed in this section so far is, for an affine nonlinear

system with m inputs and m outputs, if the sum of its relative degrees

...

in'

phi

r = r, + r2 + r,, is equal to or less than system's order n, how to transform the system into the three types of normal forms. What if a system has no relation degree, i.e. if the conditions described in the relative degree definition for MIMO systems (Definition 4.1) are not satisfied, can such a system still be transformed into a type of normal form? The answer is yes. As we can see in the following, the normal form for such a system is called

the fourth type of normal form. As before, we first discuss a system with two inputs and two outputs as shown below X(t) = f(X(t)) + g, (X)u, + g2 (X)u2

Y,(t)=h,(X(t)) y2(t)=h2(X(t)) To begin with, choose the first two coordinates as z, =rp,(X)=h,(X) z2 =P2(X)=h2(X) and we will get dynamic equations i, = Lfh,(X)+Lg h,(X)u, +Lglh,(X)u2 i2 = Ljh2(X)+Lg h2(X)u, +Lg=h2(X)u2

(4.28)

(4.29)

(4.30)

which can also be written as +

r ml [Z2]-[Lf h2(X)]+[Lgh2(X) Lgh2(X)j u2J

Now assume that for the matrix in the above formula

(4.31)


B(X)-1Lg, h2 (X)

131

(4.32)

Lg,h2(X)I

.Do

its two row vectors do not equal zero at X° but are linearly dependent. That is to say, for all the X in a neighborhood of X°, matrix B(X) is singular. From the definition of relative degree for MIMO systems (Definition 4.1) we know that, the system in Eq. (4.28) has no relative degree. Therefore, it can not be transformed into the above three types of normal forms. Next, we will discuss how to handle this case.

Since the two row vectors of matrix B(X) are linearly dependent, a scalar function r(x) defined in the neighborhood of X° can be found such that [r(X)

Lg h, (X)

Lg,h, (X) = [0

Lg,h2(X)

Lg,h2(X))

1]

0]

(4.33)

Now let us define a scalar function w(X) w(X) = [r(X) 1) Lfh2(X)I=r(X)Lfhl(X)+Lfh2(X)

(4.34)

k

Suppose that the matrix B, (X) = L

Lg

(X)

L" w(X)

is nonsingular in the neighborhood of X° 1).

i2 can be rewritten by adding and subtracting same terms to and from the right side of the second equation of Eq. (4.30), namely i2 = L f h2 (X) + Lg h2 (X )ul + Lgz h2 (X)u2 + r(X)(L fh, (X)

+Lg h1(X)u, +Lgsh1(X)u2)-r(X)(Lfh,(X)+Lg h1(X)u, +Lg2h1(X)u2)

Manipulating above formula while considering the relations implied in Eqs. (4.33) and (4.34), we have i2 = w(X)-r(X)(Lfhl(X)+Lg hl(X)u, +Lg,h,(X)u2)

(4.35)

We choose the third coordinate transformation as

z3 =P3(X)=w(X)

(4.36)

correspondingly, the dynamic equation will be obtained as i3 =L fw(X) + Lg w(X)u, + Lgsw(X)u2

(4.37)

Our next step is to choose the rest (n - 3) coordinates quite arbitrarily

') If this matrix is still singular, the corresponding processing method will be discussed at the end of this section.

132

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS z4 = -P4 (X)

(4.38)

Z. = P (X)

such that the Jacobian matrix of the vector function CD(X) _ [(Pl (X)

P2 (X)

... Pn (X)]T

co3 (X)

is nonsingular at X°, such that the chosen mapping cD(X) is an eligible coordinate transformation. If a vector n is used to represent the (n - 3) coordinates from z4 through z, , i.e. n=[z4

z5

... Z

(4.39)

then (n - 3) dynamic equations can obviously be written as A=q(X)+p1(X)u1+p2(X)u2

(4.40)

where LfV4(X)

91(X)

(4.41)

q(X)

Lg,tP4(X)

Pu(X)

...

Pl (X) 14,

P,(,,-3) (X)

P21(X)

=

(4.42)

(X)

Lg,

9P4(X)

...

P, (X) _

(4 . 43)

From the first equation in equation set (4.30), and Eqs. (4.35), (4.37) and (4.40), we will immediately get the dynamic system described by the form in

the new coordinates Z as =(Lf4(X)+Lg,hl(X)ul +Lg,hl(X)U2)I,--,,(z)

Zl

tit

z2 =(Z3 -r(X)Lfh1(X)+Lg,h1(X)u1 +Lg,hi(X)u2)IX-m-'(z) Z3=(Lfw(X)+Lg,x(X)ul+Lg2%(X)u2)Ix=1, (Z)

(4.44)

n=(q(X)+P1(X)ul +P2(X)u2)I-'(z)

together with Yl (t) = Zl = hl (X (t)) Y2 (t) = z2 = h2 (X (t))

4.45)

Now we can summarize the above into the following proposition.

Proposition 4.4

Suppose there exists a system as shown in Eq. (4.2).


133

If it has no relative degree, namely the row vectors of matrix B(X) - [Lg'hz (X)

Lg, hz (

)]

are linearly dependent, then we can get the fourth normal form in the new coordinates Z = (D(X)

i,

a,

+b12(Z)u2

i2 = Z3 - r (Z)(a1 (Z) + b11 (Z)u1 + b12 (Z)u2) i3 = a3 (Z) + b31 (Z)u, + b32 (Z)u2

(4.46a)

it = q(Z) + p, (Z)u, + Pz (Z)uz

with output equations Yl = z1

(4.46b)

Yz = Z2

where A(Z) = a1(Z) _ a1() Ca3 (Z)J - La3 (ix=-(z)

g(Z) =

__

L f17, ()

Lf

(4.47a) X=m-, (z)

_ b11(X) b12(X)1

b11 (Z)

b12 (Z)

631 (Z)

b3z(Z)] - Lb31(X)

_ [L" h, (X)

x)

b,2(X)J,=.-,(z) (4.47b)

Lg3 h1 (X)

Lg=w(X) ]

Lg w(X)

x-m (z)

L fcp4(X)

(4.48)

4(Z) _ ILf c (X)

x=m-'(z)

N1 ILg,q'4(X) ooh

(Z)

(4.49)

Lg,tV"N10-1g) L92 V34 (X)

(4.50)

Pz(Z 1Lg2t,11 (X) x=o-,(z)

and for w(X) = r(X)L fh,(X)+L fh2(X)

r(X) satisfies the condition in Eq. (4.33). The coordinate mapping is

134


Z, =,p1(X)=h,(X) z2 = TP2 (X) = h2 (X) z3 = qP3 (X) = w(X)

A= n(X) Nom,

where n= [z4 ... z ]T . The fourth type normal form as shown above in Eq. (4.46) is obtained with the assumption that for all the X in a neighborhood of X°, the matrix [B, (X)

Lgh1(X)

Lg2hI(X)1

Lg w(X)

L,, w(X)

11.

is nonsingular. If the row vectors of the above matrix are still linearly dependent, a scalar function r, (X) of X can surely be found such that [rl (X)

11

L,,4(X) L"h, (X)1= lL Lg w(X)

Lg,w(X) JJ

[0

0]

(4.51)

Define a function p(X) as p(X) = r, (X)L f h, (X) + If w(X)

(4.52)

Suppose that matrix Lgh,(X) Lg,h,(X)

.tr

B2(X)=

Lg,u(X)

L,,/j(X)

is nonsingular at X°. Now let us make the following choice for coordinate transformation z, =Ta1(X)=hl(X) z2 = (p2 (X) = h2(X) z3 = qq3 (X) = w(X)

(4.53a)

z4 =gP4(X)=u(X)

Besides, choose z5 = 95(X)

......

(4.53b)

z^

such that the Jacobian matrix of the vector function D(X) = NP1 (X)

92 (X)

...

P. (X)] T

is nonsingular at X°. Taking steps similar to those for Eq. (4.46), we can have a normal form i1 = a1 (Z) + b1 I (Z)u1 + b12 (Z)u2

_

22 = Z3 - r(Z)(al (Z) + bll (Z)u1 + b12 (Z)u2 )

(4.54)

CA)


135

i3 = Z4 - r, (Z)G', (Z) + b (Z)u, + b12 (Z)u2 )

...

i4 = a4 (Z) + b41(Z)u, + b42 (Z)u2 = 9(Z) + ' P , (Z)u, + P2 (Z)u2

with Y, = Z,

(4.55)

Y2 = Z2

where Z = [Z,

Z2

Z4

Z3

a4 (Z) = a4 (X)

11]T

IX=a-, (z) = L f u(X) IX=,-' (z)

(4.56)

(4.57)

b41(Z) = b4, (X)

Lg,N(X) IX=m-'(z)

(4.58)

b42 (Z) = b42 (X) Ix=m-'(z)

Lg, p(X) IX=d '(Z)

(4.59)

Lfco5(X)

(4.60)

9(Z) L ooh

X=m-' (Z)

Lg,'Ps(X) (4.61)

Lg, Pn(X)

X=m-,(2)

101

Lg, q 5 (X) (4.62)

Pz (Z) _ X=a-' (z)

,u(X) = r, (X)L fh,(X) + L fw(X)

and the scalar function

(4.63)

satisfies Eq. (4.51). The coordinate

r, (X)

transformation is Z, =h1(X) Z2 =h2(X) Z3 = w(X)

Z4=u(X) Tl= n(X)

where TL=[ZS

...

In summary this section presents four types of normal forms for MIMO

136

nonlinear systems

C3.

affine


obtained in

terms

of certain coordinate

transformations. These four types of normal forms will be used in zero dynamics design approaches, which is the subject of the next section.

4.3

ZERO DYNAMICS DESIGN PRINCIPLE

For MIMO systems, the fundamental notion of zero dynamics design is to search control laws u, (X), u2 (X), , u,,, (X) , in order to make the control system satisfy such a performance index that its output responses y, (t), y2 (t), , y, (t) keep equaling zero at any time t >_ 0 . Under this ..,

condition, the stability of the zero dynamics equation set (called zero .-r

'r3

dynamics for short) of the system should be tested. What this zero dynamics equation set describes is in fact the internal dynamic characteristics of the

Mao

system. Obviously, if the zero dynamics is stable, the "external" and

-'O

"internal" parts of the closed loop system are both stable, and so is the whole system. Consequently, the control system could be regarded as optimal as

coo

ro.

-CS

0.-

viewed from output responses. Generally outputs of a control system are chosen as deviations of state variables from their references values (being constants or dynamic trajectories), in other words, the tracking errors. As a result, if a controller thus designed can keep the outputs equaling zero at any time, then, at least, the external dynamics of the control system is optimal.

In the foregoing section, we have obtained four different types of transformation normal forms aiming at different conditions of system relative degrees. In the following, zero dynamics design method will be discussed for these different normal forms. (`JD

Let us begin with a system that has the third type of normal form as shown in Proposition 4.3. The starting point of zero dynamics design is to make the outputs satisfy y,(t)= y2(t)=0 for any t>_0 namely, h,(X(t)) = h2(X(t)) = 0

for any t >_ 0.

such that L"h,(X(t)) = 0 h,(X(t)) = Lrh2(X(t)) _ ... = L''h2(X(t)) = 0 h,(X(t)) = L jh,(X(t))

4.64)

From Eq. (4.3) we know Eq. (4.64) means that for any time t >_ 0 tP, MO) = tP2 MO) = VI MO) = V2 (X(t)) =

q', MO) = 0 V", MO) = 0

4.65)


137

hold. Furthermore Eq. (4.65) implies that 0

dam'

dt dyr,

(4.66) 0

dt

With Eq. (4.66), we can obtain from the two equations about Eq. (4.17) that a, (Z) + b (Z)ul + b1z (Z)u2 = 0

gyp, and Wrl in

(4.67)

a2 (Z) + b21(Z)u, + b22 (Z)u2 = 0

Cal (Z) ] + bl1(Z) a, (Z)] b2l (Z)

°Ì

which can be written in a matrix form [u,

bl2 (Z) bzz

(Z)]

=0

Luz

I=

(4.68)

From Eq. (4.68) we can figure out the control vector U ui 1 _ _ bi' (Z)

uz]

f al (Z)

b_12 (Z)

bzl(Z) bzz(Z)

(4.69)

La2(Z)]

III

which can also be written as U

(4.70)

(Z)A(Z)

Thus, for the third type of normal form we have calculated out the control vector U(Z) corresponding to such a performance index of "zero outputs". In fact, what we really need is the state feedback law U(X). Therefore, we

need to substitute Z = cD(X) for Z in the expression of U given in Eq. _>~

(4.69) or Eq. (4.70). Thus, we obtain bll (X)

U = ul Cut ] -

[b21 (X)

bl2 (X) b22

al (X)11

'

(X)]

(4.71)

[a2 (X)J

i.e.

r-7

U = [u11= -B-1(X)A(X)

(4.72)

z

where, from Eqs. (4.18) and (4.19) we know A(X) = L}b,(X) L;h2(X)]

mss"

B(X) = Lg Lr l(X)

LgLj h2(X) LgA

(4.73)

1h2(X) (X)

(4.74)

138


l-,

7C'

In Eq. (4.72) and the state feedback expressions given in Eqs. (4.73) and (4.74), since the task of control law is to force the system outputs y, (t) and y2(t) into zero, which means Yz(r'-i)

Yz (t) = Yz (t) _ ... =

`-'

Y,(t)=Y,(t)_...=

(t) = 0 (IQ

y.'

as the result of control, X(t) should satisfy the following constraints Ys'

h,(X(t)) = LfA(X(t)) _

= Lrj'h,(X(t)) = 0 hz(X(t))=Lfhz(X(t))=...=Ly1hz(X(t))=0

Namely, if the control strategy in Eq. (4.73) is adopted, the system state vector X(t) will be confined within a subset 91 as shown below

i=1,2) .h?.

S2={XER" f p'+

The last step of the design is to examine the stability of the zero dynamics equations. Substituting Eq. (4.65) into Eq. (4.26), we can immediately get the zero dynamics as follows (4.75)

or be concisely written as `-.

(4.76)

n= q(0, rl)

The stability of differential equations in Eq. (4.76) should then be examined. Described above is the design method for systems with the third type of normal form. Now let us come to the problem that a system has the second type of normal form.

Suppose a system has the second type of normal form as shown in .14

Proposition 4.2. Under this condition, in a way similar to the above we have state feedback as shown in Eq. (4.71) or Eq. (4.72). To search the system zero dynamics equations we still adopt the following symbols

Z=[ , i]T

(4.77)

where = [(P,

L171

... fir, wi ...

yir= ]T

... tl"-rIT

(4.78) (4.79)

Thus, Eq. (4.70) can be rewritten as

U=ru ' L

z

=_'

'(

,

n)A( ,

n)

(4 . 80)

JI

According to the zero dynamics design principle, it should be set that r=o, i.e.


U(0, n) = -B-'(0, n) A(0, n)

139

(4.81)

r,'

Substituting Eq. (4.81) into Eq. (4.21) while setting =o in Eq. (4.21), we can obtain the zero dynamics equation set as follows n= 4(0, n) - p(0, n)B-' (0, n)A(0, n)

(4.82)

where P11(O, n)

P(O, n) _ [PI (O, TO

P21(O, n)

(4.83)

P2 (O, n)l = I0.

hl(n-r)(0, n)

P2(,,-r)(0, n)

In the following let us discuss another probably more important condition as viewed from practical applications, i.e. the condition that the system has the fourth type of normal form as shown in Proposition 4.4. According to the main idea of design principle of zero dynamics, for Eq. (4.46b) we set that Y1(t) = z1 (t) = 0

Y2 (t) = z2 (t) = 0

for any t ? 0

(4.84)

for any t > 0

(4.85)

Thus i1(t) = i2 (t) = 0

Substituting Eqs. (4.84) and (4.85) into the second equation of Eq. (4.46a), we know that under this condition it holds that for any t >- 0

z3 (t) = 0

(4.86)

Thereby M t) = 0

Substituting the above obtained it = 0 and i, = 0 into Eq. (4.46a), we have al (Z) + b,1(Z)ul + bl2 (Z)u2 = 0 a, (Z) + b31(Z)u1 + b32 (Z)u2 = 0

(4.87)

which may be rewritten as Cal(Z) + 11(Z)

a3(Z)

b31 (Z)

b12(Z)

rui =0

b32(Z)

Lugs

(4.88)

or

A(Z) + B(Z)U = 0

(4.89)

where the meaning of A(Z), B(Z) and U will be clear by comparing with Eq. (4.88). From Eq. (4.89), we can figure out the control vector U(Z)

140

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEMDYNAMICS b,2(Z)

U=-B-1(Z)A(Z)=_ Cb3,

biz (Z)]

(Z)

'ra,(Z)

(4.90)

L°3 (Z)]

Obviously, what we need is the control law U(X). Thus, we need to replace Z in Eq. (4.90) by function of X, namely, substituting Z=(D(X) (IQ

C)'

r..

for Z into Eq. (4.90). Referring to (4.46) and (4.47), we get the state feedback U=

Lg,h,(X) LA(X)

Lg w(X)

L1b,(X)

Lg,w(X)] [L fw(X)]

(4.91a)

.°o

By adopting the control strategy in the above formula X(t) will be confined within the following subset

S2= {XER"Ih,(X)=h2(X)=w(X)=0}

(4.91b)

where it is known from Eq. (4.34) that w(X) = r(X)L fh,(X)+ L fh2(X)

(4.92)

The r(X) in the above formula satisfies the relation in Eq. (4.33). We have, so far, reached our main goal, that is, having acquired the state feedback law as shown in (4.91) which will make the outputs y1(t) and y2 (t) of the system keep the tendency of zero outputs for all t > 0.Our next task is to check whether the zero dynamics of the system is asymptotically stable. To this end, following steps we are familiar with, we shall divide the coordinate Z into two parts, i.e. Z = [z,

z2

(4.93)

n]T

z3

From Eqs. (4.84) and (4.86) we can see that

Z=[0 0

0

(4.94)

n]T

Substituting Eq. (4.94) into Eq. (4.90) and the last equation set of Eq. (4.46a), we have n = 4(0, 0, 0, n) - p(0, 0, 0, n)B-' (0, 0, 0, n)A(0, 0, 0, n)

(4.95)

where 9'(0,

0, n)

LfcP4(X)

(4.96)

_

4(0, 0, 0, n) _ 4n-3(0, 0, 0, n)

Lf lpn(X)

X-0-' (0, 0.0.',)

Design Principles of Multi-Input Multi-Output Nonlinear Control Systems (0, 0, 0, n)

141

...

p21 (0, 0, 0, n)

(0,0,0,rl)_

(0, 0, 0, n)

(X)

L.,

Lg,,pn(X)

`rte

PI(n-3) (0, 0, 0, n)

Lgz cn(X) x=m(0,0, o, n)

b (0, 0, 0, n)

K. (0, 0, 0, n)

63 (0, 0, 0, tl)

b32 (0, 0, 0, n)J

L91

-[L A(0, 0, 0, n) = a,

[

h,

(X) w(X)

L

(0, 0, 0, n)

3 (0, 0, 0,

(4.98)

zw(X)]

---

9(0, 0, 0, n) =

(4.97)

Ls,-P4(X)

_ L fh, (X)

n)]

L fw(X)

(4.99) x=4'

(o, o, a, n)

-t4

What we just described is the zero dynamics design method for a system that can be transformed into the fourth type of normal form as O..'

C-'

described in Proposition 4.4. If a system has a normal form as shown in Eq. (4.54), in order to obtain its state feedback and zero dynamics equations, the .fl

design methods and procedures described previously in this section can almost be entirely followed. Thus, without giving any unnecessary details, we can obtain

U=

B i(X)A(X)

'[[u2J (4.100)

Ib4l(X)

b42 (X),

...

-

a4(X)J

Substituting Eqs. (4.57) through (4.59) into the above formula we can have the concise expression of U(X)

uzi

_

Lg h,(X)

-[Lg,p(X)

Lg h,(X)l

Lfh,(X)l

(4.101)

>'C

u _-[u,

Lg,,u(X)J [Lfu(X)J

The state feedback expression given above will confine X(t) within the following subset

(={XaRnIh1(X)=h2(X)=w(X)=p(X)=0} V'1

where, p(x) has been given in Eq. (4.52), r, (X) satisfies the relation given by Eq. (4.51), and w(X) has been given in Eq. (4.34). Substituting the relations of z, =z2 =z3 =z4 =0 and U =-B-'(0,0,0,0, n)A(0,0,0,0, n) into the last equation set of Eq. (4.54), we will obtain the zero dynamics equation set of the system as n= q(0, 0, 0, 0, n) - p(0, 0, 0, 0, r0-B-'(0, 0, 0, 0, rt)A(0, 0, 0, 0, n)

(4.102)

142


where it (0, 0, 0, 0, Ti)

L fcos (X )

(4.103)

y (o, 0, 0, 0, n) _

L1(X)

(0, 0, 0, 0, n)

X=0- (0,0,0,0,,1)

Al (0, 0, 0, 0, n)

p21(0, 0, 0, 0, Ti)

PI(,,-4) (0, 0, 0, 0, n)

T.(-4)(0101010'r')

p(0, 0, 0, 0, n) = Lg,,Ps(X)

Lg,'PJX)

B(0, 0, 0, 0, Ti) =

(4.104)

Lg2T5(X) x=m-'(o,o,o.o,n)

b i (0, 0, 0, 0, n) ' b12 (0, 0, 0, 0, n) b41(0, 0, 0, 0, n) b42 (0, 0, 0, 0, n) Lg2 h, (X)1

Lg ft(X)

Lg=,u(X)Jx=m-'(o,0,0,0,'0

(4.105)

...

Lg h, (X)

-

A(0, 0, 0, 0, Ti) =

a(OOnan n) [ Q4(0. 0,0,0,)] I

Lh(X) Lf u(X) 1

(4.106) X=m-1(0.0.0,0.n)

With the above illustration of a nonlinear system with two inputs and 'o,

E:0

°o.

two outputs, the zero dynamics design method is discussed under the conditions that relative degree is less than the order of the system or the system has no relative degree, etc. These design methods tell us how to +y+

search state feedback law U and corresponding zero dynamics equations. Readers can easily extend the above algorithms to systems with m inputs ON''

and m outputs. To enhance the understanding and grasp of the zero dynamics design method, next we expand the algorithm of the zero CAt

dynamics design method to a more general form. Assume that there exists an nd' order nonlinear system in the form of m

...

X(t) = f NO) + Y g1 M OX (t)

(4.107a)

i=1

..,

YI (t) = h, (X(0) Yz (t) = h2 (X(t))

(4.107b)

Ym(t) = hm(X(t))

Before discussing the general method of zero dynamics design, let us first give specific definitions for the symbols to be adopted. (1) U is an m -dimensional vector U = [u1

u2

...

um ]T

(4.108)


143

(2) g; (X) is an n -dimensional vector g(X) = [g,1(X) g,2 (X) ...

(3) g(X) is an n x m

(4.109)

g;,, (X)]T

matrix composed of g; (X) , i=1,2,.-.,m X00

gl l (X) g2 (X) ... gml (X) g12(X) gn(X) ...

g(X)=[g1(X) g2(X) ... g,,(X)]=

^^C

gln(X) g2, (X)

(4.110)

... gmn(X)

(4) h(X) is an m -dimensional function vector h(X) _ [hl (X)

h2 (X)

...

(4.111)

hm (X )]T

(5) Suppose that W(X) is a k -dimensional vector W(X)=[w,(X) w2(X)

wk(X)f

(4.112)

then L fW(X) is defined as a k-dimensional vector

LfW(X)=[Lfw,(X) Lfw2(X)

..

Lfwk(X)]T

(4.113)

...

Lgawl(X) Lg.W2(X)

(4.114)

...

L..Wk(X)

Besides LgW(X) is defined as a k x m matrix Lg3wl(X)

W2(X)

L92W2(X)

Lg,wk(X)

Lgxwk(X)

91

...

...

LgW(X) =

Lg wl(X)

After making clear the meaning of the above symbols, we will introduce the general approach of zero dynamics design for affine nonlinear

systems with m inputs and m outputs given by Eq. (4.107). The general a¢)

approach involves the following main steps: Step 1. Consider the equation L fh(X) + Lgh(X)U = 0

(4.115)

Suppose that in the above equation the rank of the m x m matrix Lgh(X) is

invariant near X° and equal to r and that adjustments have been done to the component arrangement sequence of vector field h(X) _ [h, (X), , hm(X)]T

such that the first r, row vectors of matrix Lgh(X) are linearly independent. From linear algebra we know that, there must exist an (m - r,) x m matrix rl1(X)

...

ro-,(X)

1

...

1

...

1

[R, (X) 1] _

(4.116)

r(.-,), (X) ...

such that

r(m-r

)r (X)

1

(m-r, )xm

144


[R, (X) 1] Lgh(X) = 0

(4.117)

Now HI(X) is used to represent the vector composed of the first r, components of vector field h(X), i.e. H1(X)_[h1(X) ... h,(X)]T

(4.118)

W(X)=[R(X) 1] Lrh(X)

(4.119)

Set

where obviously W, (X) is an (m - r,) -dimensional vector. At the end of this step, set H 2 (X)

=CWH, M T

(X)]

where H2(X) is an m -dimensional vector. Step 2. Consider the equation LfH2(X)+LgH2(X)U=0

4.120)

(4.121)

Suppose that in the above equation the rank of the m x m matrix LgH2 (X) equals r2, which is equal to or greater than r, , namely r2 >- r, (for all the X near X°), and that adjustments have been done to the component arrangement sequence of vector W, (X) such that the first r2 row vectors of m x m matrix LgH2(X) are linearly independent. Similarly, there must exist

an (m-r2)xm matrix [R2(X) 1], where R2(X) is an (m-r2)xr2 matrix, such that

[R2(X) 1]LgH2(X)=0

(4.122)

Let H2I (X) represent the vector composed of the first r2 components of vector H2(X), and set W2(X) as W2(X)=[R2(X) 1] L fH2 (X)

(4.123)

which is an (m - r2 )-dimensional vector.

Finally, set

H3(X)=I

(4.124)

where the dimension of H,(X) is M. Step 3. The following steps are basically the same as the above ones. To take the k`h step for example, consider the equation LJH, (X) + LgHk (X)U = 0

(4.125)


145

Assuming the rank of LgHk (X) is rk , construct a vector (4.126)

Hk+1(X) _ [Wk(X)l

where HkI (X) represents the first rk components of Hk .

If the calculation is continued to the I" step, where the rank of is m, namely, where matrix LgH,(X) is nonsingular, the

LgH,(X)

calculation terminates at this step. Besides, since the matrix LgH, (X) is invertible, we can solve L JH, (X) + LgH, (X)U = 0

for the control vector U as u,

U= uz =-(LgH,(X))-'LfH,(X)

(4.127a)

Lum J

Through the adoption of the above control strategy, X will be confined within the subset below 111={XeR"

(4.127b) ,,,

a.'

04; E-°

Obtaining Eq. (4.127a) is the main aim of our design. Finally, we need to find the system zero dynamics equation set in order to examine its stability. To obtain the zero dynamics equation set, we need the following calculation. (i) For the mapping Z = d>(X) h(X) W, (X)

Z=rnl= b(X)=

W, (X)

(4.128)

(X)

V (X)

where X=

p = (I + 1)m - (r, + r2 +

+ r,),

calculate

its

inverse

mapping

-' (Z) = (D-'(;, q).

(ii) Substitute X = - (e, n) for X into the Eq. (4.127a), and set = 0, which result in U = -(Ls H, (X))-' L f H i (X)

x- _m_, (o, n)

(iii) Substitute Eq. (4.129) into the equation set

(4.129)

146

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS .M.

ii= 4(0, n)+p(0, n)U

(4.130)

we obtain (o.n>

where I Lfco 1(X)

.^-N

il= (9(O, II) -P(0, A)(LgH,(X))-'LfH,(X)) IX_m

(4.131)

(4.132)

Lf-p"(X)

X=m-I(Z)

Lgz,pP+i(X)

Lg,-p"(X)

Ls,-p"(X)

...

Lg.gP+i(X)

..

(4.133) X=m-,(Z) t17

Lg,q'p+,(X)

The subscript p in the above formula is the same as that in Eq. (4.128). Eq. (4.131) stands for the system's zero dynamics. If the equation set in Eq. (4.131) is asymptotically stable at X°, so is the whole control system in the

neighborhood of V. The above algorithm is called the zero dynamics algorithm. If for each step k >-1, matrix LgHk (X) has constant rank for all the X near X° , and NCD

If.

so does the Jacobian matrix of H k (X) , then X° is called a point of

regularity for the system zero dynamics algorithm. Now let us summarize the above algorithm by the following proposition.

Proposition 4.5 For a nonlinear system as shown in Eq. (4.107), suppose X° is a point of regularity for the zero dynamics algorithm. Then the algorithm ends at the step 1:5 n, in the sense that matrix LgH, (X) is nonsingular at the point X°. In such a case the equation L fH,(X) + LgH,(X)U = 0 ti,

can be solved for U = U' (X) U' (X) = -[LgH, (X)]-' L jH, (X)

1.:

where U' (X) is the unique control vector that can force the output responses Y(t) = h(X) to be zero for any time t >- 0. Namely, U' (X) can guarantee confining the state vector X(t) within the following subset

0 Substituting the above expression of U"(X) and the constraint of X into the previous dynamic equation set

147


X = f (X) + g(X)U'

we will have the zero dynamics of the system. If the solution of this equation set is asymptotically stable, then so is the whole system.

So far we have described in detail the zero dynamics design algorithm for MIMO nonlinear systems. To avoid misunderstanding, one thing needs to

a...

be brought up again. Some people may think that for control systems I.,

designed according to the zero dynamics algorithm, their dynamic responses in actual operation would be really kept to zero. Or reversely, if the dynamic responses of a system designed with this method were not always equal to zero in simulation or actual operation, they would doubt the reasonability and validity of zero dynamics design approach itself. In fact, there are some misunderstandings in both two thoughts. The objective of zero dynamics

design method is to make outputs

y, (t), y2 (t),

, ym (t) equal zero at any

time. This objective is in fact a dynamic performance index, which should be correctly understood as to let the outputs always keep their minimum values under the effect of control. Since many actual factors in real control systems have not been considered in design, as the time delay and tiny dead areas in

control signals or some elements of the system, the amplitude-limiting v0,

poi,

characteristics of control variables, the nonlinear characteristics in some of the system parameters themselves (for example, the saturation characteristics

of generator reactance X4) and errors in measuring components of the controller, the control system designed and manufactured with this method can not really make the dynamic outputs strictly equal zero. However, there is one thing we can be sure, that is a control system correctly designed and manufactured according to this method should have the ability to make its dynamic outputs maintain minimum values.

4.4

DESIGN PRINCIPLES OF EXACT LINEARIZATION VIA STATE FEEDBACK

MIMO affine nonlinear systems, which satisfy certain condition can also be transformed into the following Brunovsky normal form by means of coordinates mapping and appropriate state feedback Zl = Z2

..

Z2 = Z3

(4.134)

Zn-m = Zn-m+l Z'?-m+l

= Vl

148


i = vi,, .1:

The approach of exact linearization has a broad application prospect in C)- te,

engineering practice. In this section, the conditions and algorithms of exact linearization will be discussed for MIMO systems.

4.4.1

Conditions for Exact Linearization via State Feedback

First, the actual meaning of "exact linearization via state feedback" should be discussed. Given the nonlinear system (4.135)

X = f (X) + > g; (X)u; i=l

If matrix g(X) is set as g(X)=[g1(X) 92(X) ...

gm (X)1

and control vector U as U=

Iul

u2

... U. lT

we can rewrite Eq. (4.135) as X = f(X) +g(X)U III

(4.136) ^°o

The problem of exact linearization via state feedback is presented as follows.

x^,

For a system as shown in Eq. (4.136), set feedback U = A(X)+B(X)V (where A(X) is an m -dimensional vector field, B(X) an m x m matrix, and V an m -dimensional vector) and coordinates mapping (D: Z = I (X), such that the feedback system X = f(X) + g(X)A(X) + g(X)B(X)V

(4.137)

is converted into a linear and controllable system (4.138)

Z = ADZ+B,V

=°'

,pi

...

The above is the presentation of exact linearization via state feedback. In fact, from the first type of normal form given in Proposition 4.1, we have no difficulty in obtaining a linearized system as shown in Eq. (4.134), as long as in Eq. (4.17) setting that a, (Z) + b (Z)u1 + bu (Z)u2 = VI a2 (Z) + b21 (Z)ul + b22 (Z)u2 = v2

(4.139)


149

we can get the following linearized system it = z2

z2 = Z3

Z4-I = zr zr, = V,

(4.140)

zq+I = Zr,+2 zr,+2 = Zr,+3

z.-, = z z = v2

From Eq. (4.139) we have the corresponding "control" u,

(Z)

_

Lu2]

b21(Z)

r-a,(Z)+v, (Z)] L-a2(Z)+v2]

b12(Z) b22

(4.141)

To obtain the state feedback law, we only need to substitute coordinate X for Z according to Z = (D(X) . Namely, u,

(bi,(X) U-Cut]-Lb21(X)

-a, (X)+v,

b12(X)

L-a2(X)+v2]

b22(X)]

(4.142)

From Eq. (4.18) and Eq. (4.19) we know, Eq. (4.142) can be concretely written as

u2

Lg,Lj Ihz(X)

Lg,L 'h,(X)

-Ljh1(X)+v,

Lg=LI'h2(X)

L-Lrjh2(X)+v2

.a3

U=[u11_ Lg,L71h,(X)

(4.143)

Thus, we have the desired state feedback law. However, from Proposition 4.1 we can see, the conditions for the above linearization are (1)

Lg h,(X)= Lg Lfh,(X)=

= Lg Lfh,(X)=0

Lg,h,(X)=Lg Ljh,(X)=

=L,e)h,(X)0 Lk h2(X)=0

Lgh2(X)=Lg Lg. h2 (X) = Lg. L J h2 (X) =

k, 1V

`n.

of power systems. With these two points in mind, the block diagram of ,'3

CAD

..+

transfer function of a steam valving control system with the reheater can be further simplified as shown in Fig. 5.10. According to the transfer functions shown in Fig. 5.10, the dynamic equations of the control system composed of the steam control valve and high-pressure turbine can be obtained as

PH =

PH=-1PH+1CHPH TH TH

(5.72a)

1 1

(5.72b)

THg

PH +

TH.-

ul

,

PH (0) =PHI

and the dynamic equations of the control system are composed of the medium-pressure regulated valve and medium-pressure and low-pressure turbines (under normal operating conditions, the relay contact r is not closed, so the medium-pressure valve keeps away from control) II The same principle can also apply to the excitation control system design. But obviously the

effect of limits must be considered in the digital or physical simulation test, otherwise the control effect would be exaggerated beyond allowance.

190


PH

1

1

1+T, s

1+THgs

PHO PMO

1

1+TMLs

1+ Ugs

coo

Figure 5.10 Transfer function block diagram of a steam valving control system with reheater

PM/ =

,

PML _ TMg

PM +

1 C.pM

PML +

(5.73a)

TML

TML

TMg

u2

,

PM (0) = PM0

(5.73b)

The total output mechanical power P. supplied by the prime mover should be the summation of the output mechanical power of the highpressure turbine Py and the output mechanical power of the mediumP. = PH + PML

can

III

pressure and low-pressure turbines P,,,,L, i.e. (5.74)

PMLO =CMLP."

Figure 5.11 T11 E =TH +THg ; T,,,:

Transfer function diagram for a steam valving control system +T,; PHO-initial steady value of the mechanical power of

the high-pressure turbine; PMLO-initial steady value of the mechanical power of the mediumand low-pressure turbines; P,0-initial steady value of the total mechanical power

Basic Mathematical Description for Electric Power Systems

191

oho

..,

`-o

In the block diagram of transfer function shown in Fig. 5.10, all the values of the time constants THg, TH, T,,,g and TML are small (about 0.2 seconds). Therefore the mathematical models can be further simplified by replacing Fig. 5.10 by Fig. 5.11 with THE = TH +THg and T,,,g = TM, +TM8

.

From Fig. 5.11, we could write the differential equations for the steam valving control system as follows PH (t) = - 1

PH (t) + CH ul (t)

,

PH (O) = CHPm0

(5.75a)

,

PML(O)=CMLP.

(5.75b)

THE

THE

INn

UPI

PML(t)=- 1 PML(t)+CMLu2(t) THE

THE

5.7

MATHEMATICAL DESCRIPTION OF A DC TRANSMISSION SYSTEM

5.7.1

Dynamic Equations of a DC Transmission Line

oho

A DC transmission system mainly consists of converter transformers, converters (rectifier and inverter) [6], DC transmission lines, smoothing reactors and the AC and DC side filters. Its basic configuration is shown in Fig. 5.12. _ _.smoothing

smoothing wave reac ce ;----

wave DC line

converter transformer

DC filter

DC filter

11 11

-GD AC filter

DC filter

L

AC filter

DC line

power converter

Figure 5.12

DC filter

power converter

Basic configuration of a DC transmission system


1 92

If Vd, and Vdi denote respectively the mean voltages at DC terminals of the rectifier and inverter, the DC transmission system can be represented by "'S

an equivalent circuit shown in Fig. 5.13, where Ld, and Ld, represent respectively the inductance of the smoothing reactors on the rectifier and inverter sides; Ldand Rd respectively half of the inductance and resistance of the DC transmission line; Cd, the total line-to-earth capacitance of the DC transmission line; Id, and Id; respectively the DC currents of the rectifier and inverter sides; VV the voltage on the line capacitance. All the above variables are measured in per unit value.

Figure 5.13 Equivalent circuit of a DC transmission line

According to the DC line equivalent circuit shown in Fig. 5.13, from circuit theory, the dynamic equations of a DC transmission line can be written as LdrE ddr = -RdIdr +Vd, -V. LdiEdi

Cfr

uV

= -Rd I d, - Vdr + V,

(5.76)

= 1dr- Idi

+

where LdrE = Ldr + Ld and Ldt = Ld, + Ld

.

Generally a converter is composed of three-phase controllable bridge-

type circuit, as shown in Fig. 5.14. Fig. 5.15 illustrates the converting (O)

process from valve 1 to valve 3 as shown in Fig. 5.14. Angle a in the figure .`3

V)'

is called the rectifier firing angle (or control angle), and p is the commutation overlap angle.

When the AC side voltages e0(t), e,,(t) and e,(t) are symmetric, the rectifier is equivalent to the circuit shown in Fig. 5.16, with the voltage as V,0 =

3

-

Vor(i) cos a

(5.77)

and the interior equivalent resistance as 3 R, =-X,

(5.78)


193

v,,

e. (t)

e, (t)

wt

Figure 5.14 Converting loop of the rectifier side

Figure 5.15

Converting process of a rectifier

where x, represents the commutation reactance of a rectifier. R,

id,

Figure 5.16 Equivalent circuit for a rectifier p,+

From Fig. 5.16, the DC output voltage of the rectifier can be written as

Vdr=3V-V,(t)cosa-3xrld, It

(5.79)

It

where Va,.(t) is the line voltage on the secondary side of the rectifier '-h

011

...

transformer (close to the rectifier), i.e. V,(t)=If3Er, where E, is the effective value of the phase voltage on the secondary side of the rectifier transformer.

Oar

Similarly, the converting circuit of the inverter side is shown in Fig. 5.17. The converting process from valve 4 to valve 6 as presented in Fig.

5.18 is shown in Fig. 5.17, where 6 is called the advance angle (or

I'0

inverting angle), y the extinction angle.

Similar to the rectifier, the inverter can also be represented by an equivalent circuit shown in Fig. 5.19, with the equivalent voltage as

194


Wt 10

u)t

',t

Figure 5.17 Converting loop of an inverter (At the moment from valve 4 to valve 6)

Vdo =

Figure 5.18 Converting process of an inverter

32 Vim; (t) cos /3 Ir

(5.80)

and the internal equivalent resistance as

Mgt;

R. =

3 x;

(5.81)

where x, denotes the commutation reactance of inverter.

From Fig. 5.19, the DC input voltage Vdi of the inverter can be expressed as Vdi = 3 r2- VQi(t)cos/3+ 3 xildi X n

(5.82)

In the above equation, V, (t) = %F3E,, where E; is the effective value of the Ri

Figure 5.19

Equivalent circuit for an inverter when Vdi is expressed by the inverter firing angle j6

195


phase voltage at the side of converter transformer close to the inverter. If the extinction angle is denoted by y , the inverter can be represented by the equivalent circuit shown in Fig. 5.20 with the voltage V;io =

3,F2-

if

V,(t)cosy

(5.83)

and the internal resistance -R,.

From Fig. 5.20, we can obtain another expression for Vd;, the DC - R.

Idi

= Vdio

Vdi

T Figure 5.20

Equivalent circuit for an inverter when Vd, is

expressed by the extinction angle y

voltage on the inverter side

3,2 (5.84) 3 xildi if Vo,(t)cos7- if According to the condition that the voltages expressed in Eqs. (5.82) and (5.84) are equivalent to each other, the inverter's extinction angle y can be expressed as [IQ

Vdi =

y = cos-' (cos /3 +

2x,

(5.85)

1d,) C17

r17

Substituting Eqs. (5.79) and (5.82) into Eq. (5.76), the dynamic equations for a DC transmission line can be given as -Rd Id, +

dld;

LdiE

dt = -Rd 1d, -

C,dVd =1dr-1d,

3 if

n xr1dr -V Vo; (t) cos/1 -

3

if

x,,, + V,

l/1

LdrE ddtr

(5.86)

196


Mathematical Model of a DC Control System

5.7.2

cry

When the DC system is operating under steady state, the steady DC current can be calculated according to Fig. 13, Eqs. (5.79) and (5.82) to give 3

(Va, (t) cos a - Va; (t) cos Q)

Id=ld,=le;=-

(5.87)

2Rd+3x,+3x; r

,r

The above equation shows that the DC current Id or the power Pd transmitted

over the DC transmission line can be adjusted by regulating the rectifier firing angle a or the inverter firing angle Q . On the rectifier side, in order to regulate the firing angle a, the signal amplifier, phase controller, and firing circuits are the basic elements for the a -regulator as shown in Fig. 5.21.

Figure 5. 21

Schematic diagram of a rectifier's a -regulator

f].

'C7

In Fig. 5.21, Ua represents the control signal of the a -regulator. When the constant DC current control is adopted, we have the following equation ua =1d, - I&, = AI&

(5.88)

.vim

where Id, is the steady value of the DC current. When the proportional control of the deviation of the firing angle a is used, the transfer function of the signal amplifier block can be written as

Ga(s)=

a

--'t

Ua

=

ka

1+Tas

(5.89)

where ka represents the magnification factor of the regulator; Ta the time constant of the regulator; and Aa the deviation of angle a .


197

Expressing Eq. (5.89) in form of differential equation, we have dt

(-a+ao +kaua)

T0

(5.90)

where ao represents the given value of the rectifier firing angle in its nominal operation state.

In a DC transmission system, the configuration of the phase control circuit for the inverter side (see Fig. 5.22) is similar to the rectifier side:

phase control and trigger circuit

Figure 5.22

Schematic diagram of a inverter's (3-regulator

In Fig. 5.22, up is the control signal of the /3 -regulator. When the constant extinction angle control is adopted for the inverter side, we will have up =Y-Yo =L\Y

(5.91)

where yo represents the allowable rated extinction angle in the inverter's nominal operation. Then the transfer function of the signal amplifier block is (5.92) ONE

-k,6 OQ = 1+T,6s ua [rte

G. (s) = =O,

where k. is the gain of the /i -regulator; and T,6 the time constant of the regulator. Rewriting Eq. (5.92) in form of differential equation, we have d,8=T (-,6 +,80 -kflu")

(5.93)

T,6 000

where fl, represents the given firing angle of the inverter in its nominal operation state.

198


5.8 REFERENCES 1.

A. E. Fitzgerald and C. Kingsley, Electric Machinery, Second Edition, McGraw-Hill,

2. 3. 4.

7.

A. R. Bergen, Power Systems Analysis, Prentice-Hall, New Jersey, 1986. B. Adkins, The General Theory of Electrical Machine, Chapman and Hall, 1964. C. Concordia, Synchronous Machines, John Wiley & Sons, 1951. E. F. Church, Steam Turbines, McGraw Hill, 1950. E. Uhlman, Power Transmission by Direct Current, Wiley-Intersci-ence, 1975. G. R. Slemon, Magnetoelectric Devices, John Wiely & Sons,1966.

S.

G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis,

1961.

5.

6.

McGraw-Hill, 1968.

10. 11.

J. Machowski, J. Bialek and J. Bunmby, Power System Dynamic, Chichester, U.K, .SC

9.

Wiley, 1997. V. A. Venikov, Transient Phenomena in'Electric Power Systems, Pergamon Press, New York, 1964. Y. N. Yu, Electric Power System Dynamics, Academic Press, 1983.

Chapter 6

Nonlinear Excitation Control of Large Synchronous Generators

6.1

INTRODUCTION .`t

Improving power system stability is of great significance, since if the

cab

vii

stability is lost, power collapse may occur in a large area and serious damages will be brought to national economy and the residents' comforts.

Not to mention earlier examples, only since 1960's, large-area power ..O

collapses took place in many large power networks around the world and led

to disastrous losses. This inevitably resulted in grave concern by each r+,

ors

country on the stability and security of power systems. Over a long period of time unremitting researches have been carried out on the subject of power

5°.y

system stability. ,.t

Early in the 1950's, a number of scholars emphasized the importance of synchronous generator's magnetic field regulation to improve power system stability [1, 2). Since then great attention has been paid to research in this field, which mainly covers two areas: one is focused on melioration of the "ate

SS.

t°.

main excitation system, while the other is aimed at the improvement of excitation control strategy.

C1.

CAD

cad

In recent times, static excitation technology has seen great advance. Since the static exciter with thyristor-controlled self-shunt (self-excited) excitation has the advantages of simple structure, high reliability and low cost, it is widely adopted by many large power systems around the world. The design principle and approach of large generator nonlinear excitation control to be discussed in this chapter will be mainly aimed at this type of excitation. For the static excitation shown in Fig. 6.1, since an intermediate inertial

200

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS Rectifying transformer

Adjuster

Figure 6. / The structure of generator self-shunt excitation coo

O'+

element of exciter is omitted, it is featured by the rapidity of its regulation. This rapidity of excitation has two-opposite influences to the power systems. On the one hand, if the excitation control technology and strategy are not

correspondingly improved and proportional or voltage deviation PID

.....

(proportional integral differential) control are still used, the wide adoption of ,4?

rapid excitation will exacerbate the damping characteristics of power Ll.

systems even to the extent of negative damping, which will cause further low frequency oscillation in power systems. On the other hand, however, it is the rapid response of this excitation control that provides favorable conditions for the full play and good effect of advanced control strategies. Summing up

all the above, we can see that the wide adoption of rapid excitation makes

the new excitation control research practically more important and meaningful.

6.2

DEVELOPMENT OF EXCITATION CONTROL The adoption of nonlinear excitation control is the inevitable

development trend of excitation control. In order to motivate the approach to

nonlinear excitation system design, let us return for a moment to the development history of excitation control. Roughly speaking, since 1940's, the excitation control has gone through three development stages.

Stage 1: single variable control

The control strategy of this stage is regulating according to the proportion of generator terminal voltage deviation AY, or to the proportional, integral, and differential of AV, (PID regulation). The transfer functions are respectively as follows.


201

Proportional regulation u

_ kP

(6 . l a)

AY,

PID regulation ( 6 . 1b) z VU

In the above two equations, the generator terminal voltage deviation is AV, =VR

-V,(t), where VRer is the reference voltage, and V, (t) the average

of three-phase effective values of real-time generator terminal voltage. Corresponding to the formulae (6.1a) and (6.1b) the block diagrams of transfer function for the closed-loop system are shown in Fig. 6.2(a) and (b).

V

Controlled object

V.f 'AY'

V'(t)

Controlled object

I+k,s (b)

(a)

Figure 6.2

kP +kDs

The block diagram of transfer function of single variable excitation control (a)

Proportional regulation

(b)

PID regulation

For the single variable control as shown in formula (6.1). or Fig. 6.2,

applying the Frequency Response Method or Root Locus Method of ''',

classical control theory, we can determine the proper value range of each gain, k,, kD and k, in the transfer functions. Since this method is assumed to be familiar to the readers, no details are given here. In the following, we will make some conceptual explanation for the PID regulation as shown in '-.+

Fig. 6.2(b). a-.

,.0

CS'

From formula (6.Ib) or Fig. 6.2(b) it can be seen that, the transfer function structure of the regulator is the proportional block kP added by differential block kDs and then in series with the inertial block 1/(1 + k,s) . As the time constant k, is large enough, the first term in the denominator

caw

polynomial can be omitted, and the inertial block can be approximated by an integral block l/k,s. Thus, we call this regulation mode as PID regulation of terminal voltage deviation AV, . In the following paragraphs, we will discuss the physical essentials of this regulation mode. First, as is well known, the block diagram of transfer function of a linear variable feedback regulation system can always be changed into the form in Fig. 6.3. In this figure XR(s), ,Ca

Y(s), and E(s) are respectively the Laplace transformation functions of input variable xR(t), output variable y(t) and regulation error e(t);

202


0,2

kEG(s) is the forward path transfer function, and H(s) is the feedback path transfer function, which can all be represented in the fraction of polynomials

of s. According to classical control theory, for the closed-loop system in Fig. 6.3, as the gain kp increases, the dominant zeros) of the characteristic E( S)

X R (S)

'Ey

tH(s)Y(s)

Y(S)

G(s)

kp

H(

Figure 6.3 A single-input single-output closed-loop system

polynomial of closed-loop system will move rightward in the complex plane. When the gain kp exceeds its critical value kc,; , a pair of characteristic roots of the closed-loop system will appear in the right half complex plane, which means that the system will turn unstable and its dynamic response will be an amplifying oscillation which may be called the spontaneous oscillation.

Hence, the gain

of single variable feedback proportional regulation system must be confined within a certain interval, i.e. k, o0

4-r

requirement of system stability (to avoid the spontaneous oscillation) and 0

that of steady state voltage regulation precision. .O^'

Ò,

If we divide the regulator gain into dynamic amplification kD (whose influence to control objects has no additive time lag) and static amplification ks (whose influence to control objects should pass an additive heavy inertial

block). The block diagram of transfer function of excitation regulator corresponding to this is shown in Fig. 6.4(a), which can be converted to Fig. 6.4(b).

(b)

(a)

Figure 6.4 The excitation regulator transfer function dividing the amplification into static and dynamic amplifications

According to Fig. 6.4(b) and Initial Value Theorem, we know that at the

dynamic process, of the beginning of the moment, namely when t = 0+, if e(t) is a unit step function, i.e. E(s) =1/s, then the control u(0+) at this time is u(0+)

= lim u(t) = lim sU(s) = lim(s 1o' _-» = lim

(kD + kS) + kDTs 1

1+Ts

s

(6.12a)

(kn +ks)+kOTs _ kD

r-»

1 + Ts

,CD

Besides, from the Final Value Theorem it is known that, for the input of unit step function input E(s) =1/s, the control u at steady state is u(ao) = lim u(t) = lim sU(s)

r..

S-,o

+I±

= 1im(s

(kD+k5)+kDTs 1 _ 1 +Ts s)

k° +S k

(6.12b)

'C3

"i7

According to Eq. (6.12), if the regulator gain is divided into two parts as static ks and dynamic k°, during the dynamic process the control effect is equivalent to proportional regulation with gain of kD , while at steady state its effect is equivalent to proportional regulation with gain of kD +ks . The transfer functions shown in Fig. 6.4(b) is consistent with that of the PID regulator shown in Fig. 6.2(b). Thus, we may realize that PID regulation is essentially equivalent to proportional regulation which divides the whole

205


regulator amplification into two parts as dynamic and static ones.

Although to a certain extent the PID regulation as stated above mitigates the contradiction above mentioned, it can hardly improve stability of power systems effectively. This is because, as we know from the above, the effect of the PID regulator in the power system transient is no more than a proportional regulator which can, in some sense, automatically change its gain.

Stage 2: linear multi-variable control To further improve power system dynamic performance and small disturbance stability, the multi-variable feedback excitation control get gradually developed.

The Power System Stabilizer (PSS) In 1969, F. D. deMello and C. Concordia [4) put forward the supplementary excitation control called Power System Stabilizer (PSS). In the 00..

,.f

control strategy, besides the normal part of proportional, integral and .-y

differential of generator terminal voltage deviation A V, , a second-order lead "CJ

compensation element with respect to generator angular speed deviation Aw or the frequency deviation Af is used as the supplementary feedback block. The block diagram of transfer function of this excitation controller is shown in Fig. 6.5. We can see from Fig. 6.5, the PSS, in fact, is a linear control with a supplementary single signal input Acv or Af .

0

VAEF

I Af (Aw)

9

1+T,s

II

l+kDs l+kDs Hp Flp 1 +k, s l+k , s

III k

IV

Controlled Object

C

+5%

Ts

Ts

-5%

Figure 6.5 The transfer function block diagram of PSS V, : generator terminal voltage; Af : frequency deviation; VREF : reference voltage; Aw : angular speed deviation; VV : control voltage

In the voltage deviation PID path of the above diagram, to meet the requirement of generator steady state voltage regulation precision, its gain kR should be adjusted to be within 100-200; the TD in transfer function of this path is generally selected between 0.3 and 1.0, while the T, may be chosen between 4.0 and 6.0. In Fig. 6.5, the PSS path is made up of two

206


'.0

'C3

'LS

first-order lead compensation elements of I and II, an amplification element III, a reset element so called as "wash out " element IV and limiter of ±5%. For the elements I and II, the relationship between the coefficients kD and c/)

k, should always be kept as k >> k,, and the actual values need to be t].

`:1

carefully designed according to calculation for specific power systems. The selection of gain ks in element III also requires calculation, and its value is

c4:-00

a
i3

The dynamic equations for a steam regulating system: First, we consider only the case in which the high-pressure (HP) turbine

is controllable, the intercept valve 6 in Fig. 5.7 does not take part in +,,

controlling. That is, we study the nonlinear steam valving control problem, disregarding the effect of "fast valving". The dynamic equations constituted by high-pressure turbine and control valve (CV) are

boy

dP_(t)

=-T PH(t)+ H

diu1 (t) =

dt

I THg

lUH (t) +

(7.3a)

H'UM(t)

TN

1 THg

PHO (t) +

1 THg

ul

(7.3b)

own

where, PH (t) is the mechanical power generated by the HP turbine in per unit; pH (t) is the high pressure regulated valve opening in per unit; TH and THg are time constants of HP turbine and the oil-servomotor respectively, in

seconds; u, is the control signal from G, (see Fig. 5.8); CH is the power fraction contributed by HP turbine being roughly 0.3. The total output mechanical power supplied by the prime mover is the

summation of the mechanical power generated by the high-pressure (HP), medium-pressure (MP) and low-pressure (LP) turbines. That is, P., = PH + PML = CH P. + PMT

(7.4)

In the case when the medium-pressure valve keeps away from control process, the power PML generated by MP and LP turbines keeps constant as

248

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS i..

PMLO during the dynamic process being under stability investigation.

Combining Eqs. (7.1a), (7.2), (7.3) and (7.4), we can write the state equations describing the steam valving control system of a synchronous machine connected to an infinite bus as PH (t) _ -

1

,

PH (t) +

0.,

TN

1

(7.5a)

P, (t)

PHO +

THg

THg

PH(t)_-

1

THg

H PH(t)+-PH(t) TH

(7.5b)

D

EQVS

[w(t) - wo ] H PH(t)+±0-P'L'H H

w(t)

0

Hx

sin 8(t)

(7.5c) (7.5d)

S(t) = w(t) - w0

s-.

0.-

where, PML° =CMLPmo represents the initial steady state mechanical power generated by MP and LP turbines, Pmo is the mechanical power of the initial steady state.

If the practical equation (5.75) is adopted, then the state equations for steam valving control of a thermo-generator set connected to an infinite bus can be written as

=_ '

PH + CH

THE

h=

C

wo PH

H

(7.6a)

Pm0 + CH u,

THE

THE

+ !0 H CML Pmo

D (w - wo ) H H

E9V'

sins

7 6b

.°.

PH

(7.6c)

8 = co - coo

where THE = TH + THg .

The above equations can be written in the general form of an affine nonlinear state equation

X(t) = f(X(t))+g(X(t))u

(7.7)

where, 0.'

THE

f(X) =

(ti

I PH+CPm0 T,

D(w-wo HOPH +OC H ML PmO - H w-wo

Hx

Vs

s in S

(7 . 8a)

249

Nonlinear Steam Valving Control

C g(x) _ [TH 0 0

11

T

(7.8b)

HE

(7.8c)

X=

[I'H

S]T

0)

(7.8d)

The state equations given by Eq. (7.6) disregard the control of the medium-pressure valve, i.e. disregarding the effect of so called "fast valving".

7.2.2

Exact Linearization Method

-'d

From the Theorem 3.2 in Chapter 3, it can be seen that the conditions for exact linearization of system (7.6) are (i) The matrix

C =[g(X) adg(X) adjg(X)] is non-singular at a neighborhood n of Xo (ii) The vector field set

.

{g(X),ad fg(X)}

is involutive in the neighborhood Q. From the Lie bracket expression (2.47), we can calculate IT

ad fg(X) =

C. T 2.

CH

adg(x) j

[T

E

LOCH

- HT.

(7.9)

0

-(woCH + HT.'

co,CH HTHE

H 2THE

Therefore, the matrix C=[g adfg addg] becomes CH

T. C=

0

0

Cf11

THz E to0CH

0

3T. CH

[)OCH

COODCH

+ woCH HTHE

So the determinant of matrix C will be

T

(7.10)

250


Co#0

det(C) = -

,

for all X

...

The above determinant is not equal to zero for all X, so the condition for exact linearization (i) as depicted above in this section is satisfied. Since the g(X) and ad fg(X) are both constant vector fields, their Lie-

bracket [g, ad fg] is a zero vector. Of course, the zero vector field can be a

number of any vector field set, so the set {g, ad fg} is involutive. That implies that the above condition (ii) has been fulfilled. According to the above analysis we know that the nonlinear equations

for the steam valving control can be exactly linearized in a large range. According to the algorithm given in Section 3.2.4 we can transform the system (7.6) into a controllable linear system, which will involve five main steps:

Step 1.

For system (7.6), the following three vector fields can be

composed: D, ={g(X)}

D2 ={g(X),ad fg(X)}

D3= (g(X), adfg(X), adf2g(X))

where the adfg(X) and ad fg(x) are given by Eqs. (7.9) and (7.10).

t71

Step 2. Since the g(X) fields, we can choose D, =[I 0 Of a D,

,

ad fg(X) and ad 2g(X) are constant vector Of a D2

1

D3 = [0

0 1]T a D3

171

D2 = [0

where j5, , D2 , D3 are linearly independent vector fields. Step 3. Calculate the mapping F(w1,w2,w3)=(IW,

oI

W2

o D. (X0)

(7.11)

where, X = [Pt5Co S]T, X0 = [CH Pmo we S0 ]T . From Eq. (3.61), we know

that 4V, (X0) is the solution of the following equations dPH (t) dw3

do (t))

=0

-

CH PmO

=0

X(0) =

dw3

co,

80

d.5(t) dw3

By solving them, we get PH =CHPm0

co=0)0

8=w3+50

In order to calculate D °Z o (Dw 3 (X0) , we must solve the following differential equations

Nonlinear Steam Valving Control dPH (t)

251

=0

dw2 dW2

CHPmo

.-.

do(t)

=1

X(O) =

co,

w3 +5.

da(t) = 0 dw2

Then we have 8=W3+80

o = w2 + w0

PH =CHPmo

Furthermore we calculate 0D, o cD, o 0w Xo dPH (t)

,

=1

dw, CH Pm0

'ZS

dw(t) dw, 'ZS

d8(t)

X(0) = w2 +wo

=0

W3 +80 0

dw,

Therefore, we obtain mapping F as PH=w,+CH Pmo

(7.12)

CO = w2 + CO'

S=w3+80 The inverse mapping F-' of F is W, = PH -CHPmo (7.13)

w2 = CO - w°

W3 = 8 -C10

Step 4. To calculate the derived mapping F.-' (f) . From (7.13) the Jacobian matrix of F-', JF_, is an identity matrix. Therefore, f (0) (W) = F.-' (f )IX=F(W) = f(X)IX=F(W)

--W, 1

333

THE

.f,(°) (W) .f2(°) (W)

f'(0) (W)

D

w0

H

wt +

0 Pmo -

H

H

w2 W2

Let us compute the transformation R,

690

-E-V,

H x'

sin(w3 +80)

(7.14)

252


Zi1) =f(0)(W) = Lo w, +±0 Pmo

H

H

E°V,

-D w2 - W0 H H x,

sin(w3 +8°)

Z2') = f-3(0 (W) = w2

(7.15)

(1)

Z3 = w3

The Jacobian matrix of R, should be calculated in order to derive the transformation Ri_1 = R2 :

ale

H

s , cos(w3 +'50)

H

H

(7.16)

JR = 0

1

0

0

0

1

Therefore the f(')(W) can be determined as fl(l) (W)

f(l)(W)=JR f(0)(W)=

f2(1) (W)

.f3(')(W)

H

f,(°)(W)-H f(0)(W)-

H

xiV! cos(w3 +s°)f3(°)(W) a,

f2(0) (W) f3(0)

(W)

From the above we get the transformation R2 as follows Z(2) =f2(o)(W)

H

w, +

H

P,Oo -Dw2

H

Co. E9 Vs

H

sin(w3 +80)

(7.17)

Z22) = f3(0) (W) = w2

(2) - Z3 0) = w3

Z3

The following composite transformation T can be obtained. We know from Eq. (3.75) Z,(2)

=f (')(W) w=F- (x)

_coOP +9°C P H H

Z22) (2) Z3

H M1.

f3(1)(W)IW=F-'(X)

ie0

D(o)-(00)- 0)0 E-Vssin8 H

-H

-0O-CJ°

`w3(W)W_F_i(X) =8-80

Based on the transformation T, the following transformation can be


253

made with respect to f(X) and g(X)

f(X)=JT(X)f(X)=[fi(X) f2(X) f3(X)]T HY z PH+H

HT

.-Hfz(X)-HP,

(7.19)

f2(X)

=1

Aw T

1(X) = [g, (X) 0 O]T

=LOWOC TH

0

(7.20)

0

In the Eq. (7.19), f2(X)=th=AM.M; Aw=co-r,; F, =

E9vs

cos8 Aw, so Eq.

X'E

(7.19) can be rewritten as w0 PH HT",

f (X) l f (X) _

- HD Aw - H0 Pe + O)OCH HTjjM

f2(X)

D

Aw

f3 (X)

Aw

Step 5.The final transformation Z = T(X) is obtained as (7.21a)

Z2 = f3(X)=w-wo =Am

(7.21b)

°11

z, =w3(X)=8-80 =A8

z3 = f2(X)=Aw (7.21c) The next task for us is to obtain the control law. From expression (3.82), we know the nonlinear control law has the form of f' (X) + v*

(7.22)

p02

u, (X) _

k, (X)

where, v' is the "optimal control" input of the following linear system ZI = Z2

(7.23)

a2 = z3

Z3 =v

According to the linear optimal control theory stated in Section 3.2.1, if the weighting matrix is chosen as a diagonal one, i.e. Q = diag(50, 10, 0)

R =1.0

then the optimal gains for the system (7.23) are k; = 7.07

k2 = 7.36

k3 = 3.84

254


As a result, the optimal solution for the linear system (7.23) is (7.24)

v' =-7.07z, -7.36z2 -3.84z3 =-7.0708-7.360co-3.84Ao)

Substituting the above expression into formula (7.22), and rewrite 08 as Jo Acodt, we can get the state feedback 7.36HT w°C"

C" 'THE

J° 't o dt-

0

-7.07

7 25 a) (,

ti. t-.

CSI

T

H

If the damping coefficient in expression (7.25a) is ignored, i.e. let D = 0, then we can obtain the control law for the steam valving control as U, = P. - PmO +

Pc

C"

- 7.36 HT y. AO) w0C"

7 25b

i,>

-7.07 HT"£ J0'Ocodt-3.84 "T' Ow r0°C"

cv°C"

Expression (7.25b) presents the nonlinear steam valving control law of the HP turbine with reheater for a generator set connected to an infinite bus. 's'

If we replace, in expression (7.25b), the equivalent time constant of the valving control system of the HP turbine T E with the equivalent time constant of the regulated valve of MP turbine T,,,E, and coincidently replace

the power fraction contributed by the HP turbine C with the one contributed by MP and LP turbines C,,,, then we can obtain the nonlinear valving control law for the MP turbine, or the so-called fast valving control law as

u2-Pm-PmO+T_ CML

CO.CML

- 3.84 - 7.07 HT- J' &codt 0

(7.26) HTML Ad) wOCML

wOCML

In light of the expressions (7.25) and (7.26), we may realize that the nonlinear steam valving control strategies u, (control for the high-pressure Sam"-"

valve) and u2 (control for the medium-pressure valve) of a generator set connected to an infinite bus possess the following properties: 4-+

(1) The nonlinear valve opening control law u, (u2 as well) in expression (7.25b) involves the terms of the speed deviation Oco, the integral of the Om and the derivative of the Aw, which are of negative 7.+

feedbacks. So when a fault occurs in the power system, the increase of the .-O

speed will cause the valve opening decrease to decelerate, which is beneficial to the stability of the power system.

255


.C3

(2) The control law also involves the derivative of active power P,, which is a positive feedback, so the valve opening will be automatically

2.C

reduced when the active power decreases in case a fault occurs.

f17'

(3) The feedback coefficients are only related to the time constants of oil-servomotors, the time constants of HP, MP and LP turbines, the power fractions contributed by HP, MP and LP turbines, and the moment of inertia H. They are independent of parameters of the power network. The change of topology and parameters of power network will be sensed by the controller via the variation of speed Aw, active power AP, and mechanical power (4i

AP., etc., which can be correctly regulated by the control strategies expressed in Eqs. (7.25) and (7.26). This feature of the control law, which the parameter of the network is not explicitly included in expressions, has made the control completely adaptable to the change of the topology and parameters of the power network. According to the nonlinear steam control law given by the Eq. (7.25), we can draw the transfer function block diagram as shown in Fig. 7.1. iP,(t)=P (t) - P (to)

Tur CH U,

7.4HTHE w0CH

w(t)

HTHE 7.1

woCH S

3.SHT,,s w0CH

CH

Figure 7.1

1

1

1+THs

1+THRs

The structure diagram of transfer function for nonlinear control of steam valves

256


7.2.3

Physical Simulation Results of Nonlinear Valving Control in a One-Machine Infinite-Bus System

A one-machine, infinite-bus system under investigation is shown in Fig. 7.2. The parameters have been indicated in the diagram. V,

Vs

xT

Figure 7.2 A one-machine, infinite-bus system xd =0.7989, xd -0.2992, xq x.479, H=11.73s, Tdo=Ss D=1 .0, x7. =0.1134, xL, =xL 2 =0.5682, SB=300KVA

7.2.3.1 The effect on improving transient stability

chi

vii

According to the system shown in Fig. 7.2, a physical dynamic simulation has been performed under the conditions of a temporary three-phase short circuit fault and two-phase short circuit fault respectively (The fault occurred at 0 second, the line was tripped at 0.15 second and re-closed successfully at 0.75 second). Tab. 7.1 lists the test results for transient stability limits under various fault cases by adopting nonlinear steam valving control and conventional control laws. We can see from the table that if the +O+

nonlinear control strategy is adopted, the transient stability limit will +C+

increase by 21% comparing with the conventional control. Table 7.1

Dynamic simulation results under temporary short circuit fault

Fault type

Control scheme

Maximum power

Increased by %

Three-phase temporary fault

Conventional

1.099

0.0

Nonlinear

1.33

21.6

Two-phase temporary fault

Conventional

1.3

0.0

Nonlinear

1.58

21.5

Fig. 7.3 shows the test results of nonlinear controller on improving the transient stability limit under three-phase fault for the system given in Fig. 7.2. As shown in the figure, the system collapses soon after the fault occurs

at the point K when the active power output P., achieves 1.0, if the generator -is equipped with conventional governor. However the system


257

180 .N+

120

vV

60

0

1.04

°z 1.0

0.98

2.0 1.30

0.0 2.0 1.3 ci. 4E

0.0

1

I

0.0

I

1.0

1

2.0'

I

I

1

3.0

4.0

5.0

Time (s)

Figure 7.3 Physical simulation results for improving transient stability by using nonlinear steam valving control under temporary faults Curve 1-with conventional governor Curve 2-with nonlinear controller

((D

C'1

`01

remains stable even when the active power Pro reaches 1.3 under the same fault case as the nonlinear steam valving control is applied to the governor (see curve 2 in Fig. 7.3). Moreover, the dynamic performances, the lasting time of transient process and the number of times of oscillation have also been significantly improved.

7.2.3.2. The effect on improving the transient stability under permanent fault A transmission system shown in Fig. 7.2 is used as an example to test the effect of the nonlinear valving control developed in this chapter for improving the transient stability. The permanent faults (three-phase short circuit and two-phase short circuit) occur at the point K at 0 second, the line will be tripped at 0.15 second and re-closed at 0.75 second (the faults still exist) and tripped again at 0.9 second. The transient stability limits under various fault cases are listed in Tab. 7.2. It can be seen from the table that

258


nonlinear steam valve control can effectively improve the stability limit under three-phase and two-phase permanent faults. Table 7.2

Dynamic simulation result under permanent faults

Fault type

Control scheme

Power limit

Increased by %

Conventional

0.655

0.0

Nonlinear

0.929

41.8

Two-phase Permanent fault

Conventional

1.072

0.0

Nonlinear

1.257

17.25

z

Three-phase

Permanent fault

40)

The physical simulation results for the dynamic responses of the system under three-phase fault are shown in Fig. 7.4. As shown in this figure, when the fault occurs, the electric power' output P, drops immediately. In the meantime, the mechanical power input P. is reduced. And the governor with the nonlinear control function can automatically limit the opening of the 00)

a.+

180 120

47 0

1.04

ci

1.0

0.98

2.0

°'

ti

P,,=0.93

0.93

2.0

Pmu 0.93

0.93

I

4

0

1.0

f

I

2.0 3.0 Time (s)

1

I

4.0

5.0

.N-.

Figure 7.4 Physical simulation results for improving the transient stability by using steam valuing nonlinear control under permanent faults Curve 1-with conventional governor Curve 2-with nonlinear controller

259


valve to reduce the post-fault mechanical power lower than its pre-fault C3.

value Pmo , hence preventing the power system from the loss of the dynamic stability of a post-permanent fault. Moreover the dynamic performances of the system are improved radically. `CS

Digital Simulation Results of Nonlinear Steam Valving Control in a One-Machine Infinite-Bus System

7.2.4

Fig. 7.5 shows the computer simulations carried out to compare the effect between the nonlinear valve control and the linear optimal valve control.

150.0

Angle

100.0

50.0

0,0 i 0.0

I

I

1

1.0

2.0

3.0

1

1

4.0

5.0

4.0

5.0

Time (s)

Mechanical Power (p.u.)

2.0

1:5

1.0

0.5

1.0

2.0

3.0

Time (s) Computer simulation results of nonlinear steam valving control Curve 1-with linear optimal steam valving control Curve 2-with nonlinear steam valving control 0

Figure 7.5

260


The results are shown in Fig. 7.5 with a temporary three-phase fault occurred on point K (the pose-fault switching operation sequence is the same .ti

as assumed in Section 7.2.3). The swing curves of rotor angle and gyp

mechanical power are exhibited in the figure for comparing the effect of the linear optimal steam valving control with the nonlinear control. As shown in the figure, when the linear optimal governor is used, the regulated valve

opening increase too fast at the moment of fault clearing, such that the system loses the stability. When the nonlinear valving control is adopted, the

regulated valve acts properly such that stability is maintained with less

ivy

:='

COD

oscillation and shorter transient process time, which manifests that the latter is far superior to the former. Fig. 7.6 shows the digital simulation results of the dynamic response of the rotor angle S(t) and mechanical power P, (t) under three-phase permanent fault. It can be seen from the results that when linear optimal steam valving control is used, it can not keep the stability of the system. As the nonlinear steam valving control is applied, it can automatically limit the

Angle

150.0

0.01

0.0

Mechanical Power (p.u.)

'"

I

I

2.0 3.0 Time (s)

4.0

L

2.0

I

1.0

1.5

1.0

0.5

1

1.0

I

1

3.0 2.0 Time (s)

1

4.0

1

5.0

Figure 7.6 Computer simulation results with permanent fault Curve 1-with conventional governor Curve 2-with nonlinear governor

261


valve opening so as to maintain the dynamic stability after a permanent fault,

and to ensure the maximum transmission power (the limit of transmission power). Comparing Fig. 7.4 with Fig. 7.6, it can be seen that the physical simulation results coincide with the computer simulation results.

NONLINEAR STEAM VALVING CONTROL IN A MULTI-MACHINE SYSTEM

7.3

In this section, the theory and design method of decentralized control on the steam valving control in a multi-machine power system will be discussed based on the nonlinear control theory of MIMO systems explored in Chapter

4. An example of multi-machine system simulation results will be used to illustrate the effect of the nonlinear control law to improve power system stability and the post-fault dynamic performances.

7.3.1

Mathematical Model

Consider an n-machine system. Based on the rotor motion equations

and the generator power output equations in a multi-machine system described in Sections 5.2 and 5.4, we can obtain the swing equations of the i"' generator S; = w; - w0

wi =

w0

H;

PH +

(7.27a) w0

CM P.Oi

D, H;

H;

- W, (E,; G;; +E,,EvB, Hi

(CO

0

(7.27b)

j=1 jmi

where, 8, is the power angle between the q-axis electrical potential vector Eqi and a reference bus voltage vector Vu,. in the system, in rad; w; is the speed of the i'h generator, in rad/s; P,,, is the mechanical power of HP turbine, in per unit; E,, and E, are the q-axis internal transient electric potential of the 1" and j'h generator, respectively, in per unit; 8;, = 8; - 8j, in rad; PmO; is the initial mechanical power of the i'h generator, in per unit; Hi, C,,, are moment of inertia in second and the power fraction of HP turbine, respectively; G;; and B j are self-conductance of the i'h bus and the mutual-conductance between the i'h and the j' bus respectively. If only the HP controlled valve is considered without consideration of

262


the fast valving control, according to Eq. (5.75) the dynamic equation of

PH, = 1 PH, + CH

'ti

steam valving control system is Pm0i

+L_ THE

THE

C'1

THE,

p (7.28)

PH,

where THE, = THa. +TH, is the equivalent time constant of HP turbine, THg, the

t].

CAD

COD

time constant of oil-servomotor of regulated valve of HP turbine, TH the time constant of HP turbine, uH, the electrical control signal from the controller for the regulated valve.

Combining Eq. (7.27) with Eq.(7.28), we get the state equation of the CAD

steam valving control system for n-machine system in the form as

X=f(X)+g1(X)u1 +.. +gn(X)uH.

(7.29a)

which is an affine nonlinear system, where X =1PH

PH... PH^ ; w1 , w2, ..., w, ; '51,

'521

..., 5]?

(7.29b)

CH

PH, +

THE

!

Pm of

THE,

CH

1

THE

PH. +

m

Pm0n

f(X) =

(7.29c)

...

wn - wo

L

0

CH

THE,

0

0 0 0

0

0

0 0

F01

CH=

THE,

g1(X) _

j

CH. THE,

0 f

g2(X)

0

g n (X) _

0

I

0 0

0

(7.29d)


263

and

= H PN, + -WO

H.

CML, Pmoi

- Hi (wi - wo )

H (E' G;;+E;>EvBjsin(8,-Sj) j=,

jxi

In (7.29c) we assume E9, , , EQ are constants. Eq. (7.29a) can be written in the compact form as

X = f(X)+

(7.30)

g,(X)uH,

Now we have established the mathematical model of a nonlinear steam valving control system in an n-machine system. The principle and design method of exact linearization will be discussed in the next section.

7.3.2

Exact Linearization Method

First of all, let us examine that whether the affine nonlinear system given in Eq. (7.29) satisfies the conditions for exact linearization. As stated in Section 4.4.2, for the system in Eq. (7.29), we choose the N indexes. They are N

3

1=,

;=,

n,=n, n2=n, n3=n, so N=3, and >n;=Y_n;=3n, 1-1

where n is the number of generators in a power system. Based on the 'C3

Theorem 4.1 presented in Chapter 4, the conditions for linearization are (i) The following 3n vector fields

g,(X), ...,

adfg,(X), ...,

adfg,(X), ..., adfgn(X) (7.31)

are linearly independent. (ii) Every one of the following 3n vector fields sets D, = {g, (X)}

D = {g, (X), g 2 (X), ..., g (X)} {D,,, ad fg, (X)}

(7.32)

D2

ad fg,(X),adfg2(X), z

={DZ,,,adfg,(X)} D3n

={D2i,,adZtg,(X),ad

264


is involutive near X0. In order to verify whether the above two conditions are satisfied, the Lie bracket for system (7.29) will be calculated. They are r

tea.

adfgi(X)=LTH,,0,...,0;_

1T

TH.,0,0,...,0;0,...,o

T

adlg" (X) = 0, ..., 0, CH,

Tl

;

0, ..., 0, _ OCH ; 0, ..., 0

(7.33)

H,,THE T

dz (X) = alg,

CH' ;

0, ...

adz g"( X)=

p ...

p

0; _ ( w0CH, z

H,T,E,

T,

+

aOD,CH, z

H, T Z

WOCH, ), 01 ...10

0,...,

H

01

T

f

CH.. 0, ..., 0, _ a OCH + (

COOD"CH

T,3

); 0,...,0,

a'OCH-

From Eqs. (7.29d) and (7.33), the determinant of matrix C is

Cs z det(C) = (-1)"r1

2;

0, for all X

+_1 H. THE,

The above formula implies that the 3n vector fields of (7.31) are Q..

linearly independent. So the condition (i) is satisfied. Moreover, the 3n vector fields expressed in Eqs. (7.29d) and (7.33), adjg..... adfg", ad 2g , ad f2 g. are all constant vector fields, the Lie-bracket of any two of them is a zero vector which can be the member of any set. So the set of the

mot`

3n vector fields is involutive. That implies the condition (ii) is satisfied. It is

concluded that the nonlinear valving control system for a multi-machine power system (7.29) can be exactly linearized to a controllable linear system Brunovsky normal form.

Subsequently, let us find the nonlinear valve control strategy for a multi-machine system according to the algorithm given in Section 4.4.2.

Step 1. For the system given in (7.29), N = 3 index numbers have been chosen, n, = nz = n, = n , where n is the number of machines, so is the

number of the dimension of the control vector U = [uH, uH2

uH ]T . The

Lie-brackets of each g,(X),..,g"(X) along f(X) from the first order through

(N-1)"' order have been calculated in (7.33). The 3n sets of the vector fields D, , , D3,, have been built up. Step 2. such that

We choose 3n linearly independent vector fields,


265

D, _ [1, 0, 0; 0 D2 = [0, 1, 0,

0; 0

0] T E D,

..., 0;

0, , Of E D2

Dan = [0, ..., 0; 0,

0; 0, ...,

, 0; 0,

....

(7.34)

0, 1] T E Dan

Step 3. We calculate the mapping F(w1,w2,' ,w3")=(D°'

.(DD,,

o...oCj)D'"(X0)

where, X0 =[CH, Pm01, ..,CH.PmOn; (00,-400; 810, "',9,01T. Similarly to the Step 3 of Section 7.2.2, the mapping F is PH =wI+CHIPm01

+

PH = W. + CH. P.O. ()l = Wn+1 + COO

III

(7.35) co,, = w2n + 0')0 81 = w2n+l +1510

15n = w3n + 15"0

The inverse mapping of F is

F-'

.

wl = PH - CH, Pmo1 0.@

Wn = PH. - CH,, Pmon

Wn+1 =[O1(00

(7.36) W2. = 0)n - 0)0 W2n+l = 151 - 810

w3n = 8n - 8n0

Step 4. Calculate the derived mapping of f(X) under F-', F.-'. As know from Eq. (7.36), the Jacobian matrix JF_, = 7F-' /oX is an identity matrix, thus F.-'(f) = JF-'f(X)IX=F(W) Set

f(X)IX=F(W)

266


f (X)

...

fl(0) (W)

4(°)(W) fn( (W)

fn (X) fn+1 (X)

(7.37)

f(0)(W)= f"(.0' (W)

(W)

f2n(X) f2n+1 (X)

A. (X)

X=F(W ) 'fl

, f3 (X) have been given by Eq. (7.29c). Then we look for transformations RI and R2 In the Eq. (7.37),

f,, (X),

RI (1)

ZI

ro)

- fn+1 (W)

=(

(cot

H,

PH, + H, C., Pm01 - H'

- a)o )

N]=

-H (E']Gll+E9,E,B,)sin8,)))Ix=F(w) HI j=2

Z(0)

Hn PH.+H C.. P.M

Nn

(art-Co.) (7.38)

n-1

0 (E Hn Zn+)1

E,Bn)sin 8,y))

x=F(w)

)=1

= f2(n+, (W)=(0)l - W0)Ix=F(W)

Z(n1) =Jan°)(W)=(Con -CVO)Ix=p(w) (1)

_

Z2n+l = w2n+1 = (8 - Slo) x=F(w) (1) _ _ Z3,, - Wan - (sn - Sn0) x=F(W)

cps

where s;; = s; - s; . In light of Eq. (7.38), the f())(W) can be calculated

f°'(W) = JR (W)f(0)(W)

f(1)(W) =

f3(W)

(7.39)


0)° HI coo

Hn

267

+i (X) -

HI

0 E'1 Ev BI; cos 81;./zn+l (X) HI j=2

o°

L(0)(X)- D"

Hn

Hn

%=,

f-) (X) A. M W1 -(0)°

COn - O)0

L

JX=F(W)

From f°)(W), we obtain the transformation R2 42)

H,

Zn2)

H-

P, + NI

CO,

HI

(w, -coo)

(E9, G + Ey, Y-Ev BI; sin SI; ))I x=F(w) J=z

= A (n') (W ) (

Hn P". +

Hn

C. P.. Hn (0n -

- jH° (E"GM + EQn

n-I i=1

n

WO)

Ew B,y sin 8,, ))I x=F(w)

(7.40)

Zn+l = f,(.,+) 1(W = (0) 1 -' o)IX-F(W)

Z2(n) = f ,(n') (W) = (a ,, - (AO )IX=F(W ) (2)

Z2n+I = w2n+1 = (SI - 810) x=F(W)

(2)

Z3n -w3n=(Sn-Sn0)x=F(W)

From R2, the composite transformation T can be derived as

(n Tj(X)=ZI(2) =!'r+l(W)W=F-'(X) =

0)0

H,

PH +

H,

'

0

H,

PMLO

D' (CO, - (00 ) HI

(EgiG,,+E',Y_EvB,;sins,;) j=2

(7.41)


268

T (X)

W=F'(X)

Pnno -H

PHq+

n-1

0),

- Hn

j=1

=w, -CVo T,+1(X)=(W)W=F-,(X) f2('+'.

T2n (X) = 3n) (W)

W=F-,(X) = (0n -W0

=8, -8io

T2.+[ (X) =

T3n (X) = wan I W=F-'(X) = 8n -15,10

The Jacobian matrix can be derived from (7.41) as H, 0

o

0

o

...

°L0

D,

p

w° ape,

w° ape,

H, a8,

H, a8n

0 ...

p

p

D.

- w open

Hn

Hn a8,

I

H,

H.

_

Co. aPen

Hn a8n

.............. ........ ........ .............. ................................... ......... ........ ..... .... ................ ...... .......... .......................

0

...

0

00

...

1

1

Jr(X)= Onxn

(7.42)

0.-

............... ......................o.............. .......................................... .............. ......................... ..................... 1

o

...

1

0

0

...

1

Onxn

Onzn

Now, we should take the transformation of f(X) and g(X) , which will be !(X) = Jr (X)f (X)

[.f (X) ... f, (X) f,+(X) ... 7 2,, (X) K

_ [7 i

r ... b

...

0),

[p

wl - (00

(X) ... .fan (X )l ...

r C)n - Cvo l

where w0

H;Tn

PH -

D,

Hi

wi - !0 ( ape, (0), - COO +

0.I

woCH

as,,

H;T,

...+-(w,, -CUO))+ Pei

Hi as,

ape

Ev Bj sin CSj

E"Gjj + Eqi j=1

jxi

PmOi

i=1,2,.-,n

r

(7.43)


269 O)o C"

I

0

0

...

H,T,,E

0

g(X) = JT(X)g(X) =

n lines

(0OCH.

......................................................................

(7.44)

0

0

n lines

.

0 0 ............................................................. 0

0

n lines 0

0

,DC

...

Step 5. The final coordinate transformation is obtained as follows Zl =

(X) = 8l - 8l0 pmt

Zn = wa,, (X) = (5n - (5nO

Zn+l = J 2n+1 (X) = 0)l - 0)0

(7.45) Z2» =f3n(X)=Con -a0 Z2n+1 =L+l (X) =")1

Q"'

Z3n = f2n (X) = d)n

Comparing Eqs. (7.43) and (7.44) with Eqs. (4.169) and (4.170), we know that

AN an(X)j

,V2

f2(X)

a(X) =

J,(X)

(O PH-D'Aw,tr)o

H,T.

PH

wOCHi

(7.46) CH,

H,

Hl 88,

Hn

Hn

as,,

H1TH1

P.

DnA(wn-rHO(----Aal+...+BP-Lmn)+woCH.Pmon

0

...

88,

88n

H,THE

0

H1THE

b(X) =

0

wOCH,

0

H2THi ...

0

0

(OCH.

(7.47)


270

The inverse matrix of b(X) is

H,T,, 0

0

wOCH

0

H2THL,

b-' (X) _

0

(7.48)

wOCH2

HHTnz.

wOCH, J

As known from Eq.(4.173), the control vector U(X) = -b-'a + b-'V' is THE

u H, uH2

CH'

_

(Pm2

oOCH' w0CH2

UH,,

v1

oOCH'

-Pmo2)+TRZ' Pa +T 2D2 Awe + CH,

,

Ato1 +

+

U(X)

H1T

THE D1

(Pm1 -Pm01)+=Pe1 +

H2THE2

v2

a 0CH2

(7.49)

THE, Dn

Awn + (Pmn - Pmon) + T HE" Pan + wOC, N CH. wOCH

HnTHE_ vn

aP"

where, PU = aP`; Aw, + aP" A0)2+---+ Awn; Am = w, - wo . as, as2 as,

v, v,,

, v,,

are the quadratic optimal "control" (an input) of the linear

system (7.50)

i=1,2, ,n v, = -(k; z, +k;,+iZn+i +k2n+iZ2n+i)

i

(7.51)

where, k, , k,,+i and k2n+i are the optimal "control" gains. As discussed in Section 7.2.1, for the linear system given in (7.50), if the weighting matrix is chosen as Q = diag(50, 10, 0)

R =1.0

Then the optimal "control" gains are k, = 7.07

7.36

k2n+i = 3.84

Substituting Eq. (7.51) into Eq. (7.49) and considering Eq. (7.45) and rewriting AS, as Jo Awi dt , we acquire the control law for the HP turbine


271

valve control of the i' turbo-generator set in a multi-machine system as

T'

7.36H;T, w0CH

(7.52a)

CH'

-7.07

H,THE'

o.0CH

J'4co'.dt- H1THS' (3.84- D' )erar p Hi 0o0CH

If we let D; = 0 in the above formula, we have THE,

uH, =(Pmi -Pma)+Pe! -

7.36H;T Ow; H'THE'

r

wOCH

(7.52b)

w0CH'

CH

-7.07 H,THE, J'Ow,dt p

3.840w

r

w0CH

ti.

Following the derivation steps for obtaining the expression (7.52), we can write the control law for the fast valving control of a steam turbine with reheat of i"' generator set in a multi-machine system as

T,

7.36H;T"

UMi =(Pmi - PmOi)+CPei AII.

-7.07

H,TMS' wOCML,

w0

C

Qwi ML

H,T &' J'Ew.dtp r

3.840cbi

OOCML,

Effects of Nonlinear Steam Valving Control in a Multi-Machine System

7.3.3

(CD

In order to compare the effects of different control strategies in a multimachine power system, the conventional governor, linear optimal governor and nonlinear governor are adopted respectively by No.2 generator set in a 6-machine system shown in Fig. 6.10, while other generators are equipped with conventional governors. Table 7.3

Critical clearing time for different control strategies Control strategy

64-

Critical clearing time

Improved by %

0.198

0.00

0.206

4.04

0.220

11.1

Conventional governor

for No.2 generator r".

Linear optimal valving control for No.2 enerator

Nonlinear valving control for No. 2 generator

272


A permanent three-phase fault is assumed to occur on the bus 11 (see Fig. 6.10) for the large disturbance test. The fault occurs at 0 second lasting for a short duration, the 500KV transmission line between bus 11 and bus 12 +.+

.._.10 p..

is tripped and not reclosed again. The critical clearing time for different control strategies are indicated in Tab. 7.3. It is shown in Fig. 7.7 the

dynamic responses of rotor angles and mechanical powers of all the generators in the 6-machine system, when all the generators are equipped with conventional governors. We can see from the figure, the entire system loses stability in very short period of time.

Fig. 7.8 exhibits the dynamic response curves of the system in which

the No.2 generator set is equipped with linear optimal steam valving -CDs

controller and the conventional governors are applied to the other generators. 160.0

Angle

80.0

0.0

-80.0

Power (p.u.)

-160.0

6.0

4.0

2.0

0.0 i

i

I

I

I

I

0.0

1.0

2.0

3.0

4.0

5.0

F"'

Time (s)

Figure 7.7 Dynamic responses under three-phase fault No.2, 3, 4, 5 generators are equipped with conventional governors. Fault occurres

at 0 second, the line between bus 11 and bus 12 is tripped at 0.20 second, SB=100MVA


273

80.0

0.0

-80.0

-160.0

Mechanical Power

6.0

4.0

2.0

0.0 I

I

I

I

I

I

0.0

1.0

2.0

3.0

4.0

5.0

Time (s)

Figure 7.8 Dynamic response curves under three-phase fault No.2 generator is equipped with linear optimal governor; the others are equipped with conventional governors. Fault occurred at 0 second, the line between bus 11 and bus 12 is tripped at 0.21 second, SB 100MVA

r-:

,..

.N.

Although the linear optimal valving control is adopted, the system is still out of synchronism. The effect of nonlinear steam valving control is shown in Fig. 7.9. The nonlinear control is used on the No.2 generator, and the other generator sets are still equipped with conventional governors. The entire system remains stable with the help of the new control strategy. "C7

7.4

DISCUSSION ON SOME ISSUES p.'

Comparing the Eq. (7.26) with Eq. (7.53), we realize that the expressions of control strategies for one-machine infinite-bus system and for 'C3

fir"

274


Angle

80.0

- 160.0

Mechanical Power

6.0

4.0

2.0

f

P.3

P.4

0.0 I

I

I

I

0.0

1.0

2.0

3.0

1

4.0

5.0

Time (s)

`D.

1'+

Figure 7.9 Dynamic responses under three-phase fault No.2 generator is equipped with nonlinear governor; the others are equipped with conventional governors. Fault occurred at 0 second, the line between bus 11 and bus 12 is cut off at 0.21 second, SB I OOMVA

multi-machine system are the same.

Similar to the nonlinear excitation control law, the nonlinear steam valving control law of u, for the i' generator in a multi-machine system relates only to the local state variables and local output variables such as the mechanical power Pm, , the active power P, and the rotor speed w,. The control law is independent of the state or output variables of other generators. So, the requirement of decentralized control has been achieved. The control law is independent of the parameters of power network. In other words, the control variable relates only to local parameters such as inertia constant H;, time constants of oil-servomotor and HP turbine and T,,, , and the fraction power of HP turbine c,,, . It does not involve the parameters of power networks, which makes this type of control strategy automatically

275 "'A

Nonlinear Steam Calving Control

adaptive to the network parameters and change of network configurations.

The results, including both physical simulations and digital one for a one-machine, infinite-bus system and the digital simulation

for a

O's

multimachine system mentioned above, show that the nonlinear steam valving control technology can improve the stability of a power system ISA

-"Al

°p'

remarkably. Moreover, when a permanent fault occurs, the regulated steam

cA-

valve opening can be automatically limited to maintain the post-fault dynamic stability of the power system. ."r

L."

In addition, by applying the nonlinear steam valving control, the generator which loses its stability can be pulled into synchronism r'3

'C7

automatically in certain short time (several seconds). This result is obtained by physical dynamic simulation experiments. The power system for testing is shown in Fig. 7.2, and the physical simulation results are illustrated in Fig. 7.10. From it we can see that as the power system suffers from a severe fault and loses stability, the steam valve closes as fast as possible to reduce the mechanical power P. (see Fig. 7.10), about 3 seconds later the generator is pulled into synchronism and stable operation is resumed.

0.0

cep

0.e'

coo

..4.i

The nonlinear steam valving control technology developed in this

r-,

b tlz

G.-

-0. 92

1.0

-1.0

0

1.0

2.0

3.0

4.0

5.0

Time (s)

Figure 7.10 Physical testing results of re-synchronizing by using the nonlinear steam valve control

276


Chapter can not only improve both dynamic and transient stability of power systems, but also can resynchronize the generators. This kind of functions and effects can not be achieved by other types of linear control strategies.

1.

.--.

E. F. Church, Steam Turbines, McGraw Hill, 1950.

2.

H. T. Akers, J. D. Dickinson and J. W. Skooglund,

4.

"Operation and Protection of Large Steam Turbine Generators under Abnormal Conditions", IEEE PAS, Vol. 87, pp. 1180-1199, April, 1968. IEEE Committee Report, "Dynamic Models for Steam and Hydro Turbines in Power System Studies", IEEE PAS, Vol. 92. pp.1904-1915, Nov./Dec. 1973.

IEEE Working Group Report, "Dynaipic Models for Fossil Fueled Steam Units in

5'0°0 Sao

f+1

3.

>0.

REFERENCES

7.5

V'1

(]w

Power System Studies", IEEE PWRS, Vol. 6, No. 2, pp. 753-761, May. 1991. 5.

IEEE Working Group Report, "Hydraulic Turbine and Turbine Control Models for

C"'

System Dynamic Studies", IEEE PWRS, Vol.7, No. 1, pp. 167-179, Feb. 1992.

M. S. Baldwin and D. P. McFadden, "Power Systems Performance as Affected by Turbine-generator Control Response during Frequency Disturbance", IEEE PAS, '-1

6.

.-.

Vol.100. pp. 2468-2494, May, 1981. l-:

7.

oo)

P. Kundur, D. C. Lee and J. P. Bayne, "Impact of Turbine Generator Overspeed Controls on Unit Performance under System Disturbance Condition", IEEE PAS, Vol.104. pp. 1262-1267, June 1981. T. D. Younkins and L. H. Johnson, "Steam Turbine Overspeed Control and Behavior During System Disturbance", IEEE PWRS, Vol. 6, No. 2, pp. 753-761, 1991. BOO

8.

Chapter 8

Nonlinear Control of HVDC Systems

INTRODUCTION

8.1

.::

A DC Electric Power Transmission System, with its inherent advantages, such as fast regulation, flexible operation, high-power transmission, and the ability to rapidly regulate active power flow, has been widely applied to highpower transmission over long distance, regional power systems inter- connection,

.J.""

?'.

(~D

.-.

and undersea electric power transmission. So far, dozens of DC transmission lines have been put into operation in the world. As of 2000 in China, there were two main DC transmission lines in operation, which are the TianshengqiaoGuanzhou and the Gezhouba-Shanghai ±500-KV HVDC transmission lines. It is expected that more AC/DC power systems will be developed along with the s..

°o.

power system expansion in China. [z.

For many years, many researches have been done to improve the mar,

performance and stability of a AC/DC parallel system by taking advantage of the fast regulation capability of a DC transmission system, which has ..t

r."

become the direction of the research on AC/DC systems [1-9, 11-15]. This chapter starts with a brief review of the control principle of the DC transmission system and some conventional control methods. It then discusses how to apply the nonlinear control approach explained in Chapter 3 and Chapter 4 to DC converter station control [16]. Finally the chapter concludes by showing some relevant digital simulation results.

278


8.2

CHARACTERISTICS AND CONVENTIONAL CONTROL OF CONVERTER STATIONS

8.2.1

Voltage-Current Characteristics on Rectifier Side

According to the mathematical description in the section 5.7 and formula (5.79), the DC voltage on rectifier side (at the sending end) can be written as Vdr =

N

n

Va,(t)cosa- 3x,1,

(8.1)

yr

where, v is DC voltage of a rectifier, Vim, the AC (line-to-line) voltage of

one

the rectifier, I, the DC current, x, the leakage reactance of the converter transformer in the rectifier station, a the firing angle. When the current in DC line equals zero, the DC voltage equals 3 NF2

Vdro =

V,,(t)cosa

(8.2)

Jr

From Eq. (8.1) we know that as the DC current Idr increases, VV decreases. The magnitude of the decrease is proportional to both Idr and x, . Therefore, the DC voltage Vd, has a linear relation with the DC current I,,

if the firing angle a remains constant. But when a varies, the relation between DC voltage and DC current is shown in Fig. 8.1.

Rectifier voltage

'PA

Inverter current Id,

Figure 8.1

Vd,-Id, characteristic of rectifier with various firing angles

279


8.2.2

Voltage-Current Characteristics on Inverter Side

According to formula (5.82), the relation between DC voltage and advance angle 8 on the inverter side (at the receiving end) is Vd, =

3,5 Va,(t)cos,6+ 3 x,ld, 9

(8.3)

)r

v'.

where, Vd, is the DC voltage of the inverter; Id; the DC current; V., the AC voltage (line-to-line) of the inverter. Similarly, when DC current equals zero, the DC voltage of the inverter is = 3,12 VthO

a

Va, (t)cos fl

(8.4)

Eq. (8.3) shows that when DC current Id, increases, the DC voltage Vd, will increase proportionally if advance angle /3 remains constant. The voltage-current characteristics with different advance angles 6 are shown in Fig. 8.2.

!3z

tai

Inverter voltage Vd,

constant

Q3 r)

.

v,

-

(t) sin a + Ta cosa

VV(t)sina

Ad

3r2Va, (t) sin a

2la Vd,+(a-ao) Va,(t)sina

V

(8.28a)

Calculating the derivative of z3 in (8.26), substituting it into a3 = v2 and rewriting the result, we can obtain the constant voltage control law for the inverter as K_

(v)

of (t)=-

3' IVa; (t) sin /3

.Va; (t) sin fl

Va

pVa; (t))sin,B

(8.28b) 0000

TP x,

+

Tfl cos,6

v2+

Id+(Q-fin)

From Eq. (8.18a) and Eq. (8.18b), the control law of the linear system (8.27a) can be written as V1 _ -Z, - -13Z2

(8.29)

vz = -Z3 ONO

where the expressions of z, , z2 and z3 are given by Eq. (8.26)

X36

ono

Substituting Eq. (8.29) into Eq. (8.28) and using the coordinate transformation in Eq. (8.26), we have the nonlinear control law with constant current for the rectifier and constant voltage for the inverter as follows. The control law for the rectifier with constant current is rT.Ld_ tie

u«f (t)=- 3,F2 V_ (t) sin a +

-

(mod + (V S -

Va.

)mod

)rLd

Ta cos a Va,(t)sina

sx

YrTa

3,r2Va,(t)sina

(8.30)

Vd. +(a-ao)

The control law of the inverter with constant voltage is (V)

7Ta

Ta cos /3

V.,(t)sinfl ai

3-F2 Va,.(t)sin/3 Tfi x,

(8.31)

300


In order to test the control effect of these nonlinear control laws, we have made the following comparison by still using the case described in Fig. 8.8. Under the same conditions, the computer simulations have been carried out on the system with the conventional control method (as shown in Fig.

.nor,

8.5(a) and Fig. 8.7, where ka 30.0, kai=0.005s, Ta 0.02s, T,0.05s). The simulation results of different control modes are shown in Fig. 8.12 and Fig. 8.13.

120.0 661

angle

821 831

Rotor

20.0

0

0

04

-80.0

-180.0 .

0.0

I

I

I

1..0

2.0

3.0

Time (Sec.) (a)

AC voltage of the converter (p.u.)

1.2

V., 1.1

1.0

0.9

I

0.8 0

I

1.0

t

2.0 Time (Sec.) (b)

I

3.0

System dynamic response under conventional controllers with constant current at rectifier and constant voltage at inverter (a) Rotor angle dynamic response of No.2-No.6 generators (b) The AC voltage dynamic response of inverters

Figure 8.12


301

Rotor angle (degree)

120.0

20.0

-80.0

-180.01 0.0

I

I

1.0

2.0

I

3.0

4.0

AC voltage of the converter

Time (Sec.) (a)

V.,

0.81

1

0.0

1.0

1

1

I

2.0

3.0

4.0

1

.5.0

C/]

Time (Sec.) (b)

System dynamic response under nonlinear controllers with constant current at rectifier and constant voltage angle at inverter (a) Rotor angle dynamic response of No.2-No.6 generators (b) The AC voltage dynamic response of inverters

Figure 8.13

w-'

any

From the simulation results, it can be seen that the nonlinear control is superior over the linear conventional one not only in suppressing the inverter AC bus voltage fluctuation, but also in restraining the power oscillation in

AC/DC systems. Nonlinear control technology provides new promising ways to further improve the operation security and dynamic performance of the AC/DC power systems when it is used for DC converters.

302


NONLINEAR CONTROL OF DC SYSTEMS AND STABILITY OF AC/DC SYSTEMS

8.4

coo

In the previous section, we have discussed the conventional and the nonlinear control methods mainly from the viewpoint of stable operation of the AC/DC link itself. Although these control methods can damp the power

l<w

oscillations in some extent, they can hardly improve the stability of the

overall system. Thus, in what follows, we will concentrate on the improvement of the performance of the whole AC/DC power system. This can be achieved because the DC regulation system can change the power flow distribution quickly. By using the nonlinear system control theory presented in Chapter 3, the design method of the nonlinear supplementary control for DC converters will be discussed.

8.4.1

Modeling for Nonlinear Stabilizing Control Design of AC/DC Systems

The most typical AC/DC power system is shown in Fig. 8.14. In this diagram, PL, and PL2 are the local loads of the equivalent generators 1 and

2, respectively, which can be seen as two power systems of different capacities.

P A,. Pe, P.

3F

RE

I

P

Pe2

H

II equivalent machine

2' equivalent

machine

PL,

PL2

Figure 8.14

An AC/DC power system

The swing equation of the equivalent generator 1 is S, = w,

(8.32a)

tiw, = H1

(P., - Da w, - P11)

and the equation of the equivalent generator 2 is


303

8z =wz &2 =

wo

(Pm2

H2

D2

(8.32b)

C02 - Pet)

coo

The electromagnetic power Pe, and Pee of the equivalent generator 1 and 2 can be respectively represented as Pe, =PL, +P,+PP (8.33a) (8.33b)

Pee = PL2 - Pd, - P.

The meanings of the symbols in the above equations are shown in Fig. 8.14. Subtracting Eq. (8.32b) from Eq. (8.32a), we can obtain the swing equation expressed by relative rotor angle 812 and relative rotation speed 0)12 as

812 = wit QJp

(8.34)

Pn,2)-(DL C, - D2 wz) d),2 =(O Pmi H2 H, H, H2

-(°2 H,

PLI -

O PLZ)+( + H,

H2

" )P -( -9 + H2

H,

w0 H2

)Pdc

When the stability of the AC/DC power system is studied, the dynamic characteristic of the DC line is usually not taken into consideration, and the DC power regulation is regarded as a first-order inertia element. Thus the equation of the DC power can be written as 1

Pd, =

T Td

(-Pde +PdeI. +u&)

(8.35)

where P. is the reference value of the DC power, Td the equivalent time àte)

constant of the DC system, and u. the control variable of the DC system. As stated previously, the target of the nonlinear control studied here is to keep the whole system stable by regulating the transmission power on the DC line especially when the AC system is disturbed by a large disturbance. For the system shown in Fig. 8.14, we can evaluate the system stability by checking the

absolute value of the relative angle between the two equivalent generators within 180°. This relative angle is used as a stability index of the connected power systems. After a large disturbance, the smaller the relative angle is, the more stable the system will be. Therefore, we can take the relative rotor angle increment between the two equivalent generators as the output equation for the nonlinear controller design. That is

(8.36) y(t) =8,0) -81zo Combining Eqs.(8.34), (8.35), (8.37), we can obtain the state equations and the output equation of a AC/DC nonlinear control system

304

o,.

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS 512 = 6)12

612=( H,

-(0H,PL I 0.b

Pd. =

LoPml-L0Pm2)-(D6)1-D26)2) H2 H, H2 (00 0 -(Wo + 0 )Pd, + )PW - H2 H, H2 H, H2

.

(8.37a)

PL2+(u?0

P& + P&REF + ud,

T Td BCD

The output equation is y(t) = h(X(t)) = 512 (t) -8,0

8.4.2

(8.37b)

Nonlinear Control Design for Stabilizing AC/DC Systems

According to the design method of the SISO nonlinear system control discussed in Chapter 3, for the system in Eq. (8.37), we can calculate its Lie derivatives in every order, namely Lg h(X) = 0 LgL fh(X) = 0

(8.38)

6)o 6)o LgLfh(X)=--(---) Td H, H2 z

1

0

From the above results, the relative degree of the system in Eq. (8.37) equals 3, i.e. r = 3. Therefore, the coordinate transformation (D can be calculated as follows Z1 =

h(X) = 512 -'5120

3

-

= L2f h(X ) =

ro12

(1 PL, -

- ( H,0 P

°

HI

H2

6)0 Pm 2)

ml

H2

PL2)+(T 0 + H,

w° H2

- ( D' HI

6)

D2

1

H2

Ono

=2=Lfh(X)=6)12

(8.39)

6)) 2

'!Lo-)Pd,

)Pac -(Oj0 +

H2

H,

The Jacobian matrix of this coordinate transformation is az,

az, 6)P&

'

0

6)512

aCU12

aZ2

aZ2

aZ2

0

1

x512

x6)12 aZ3

a"d,

aZ3

a512

x6)12

aids

III

aZ3

CO° + cw°

- (H1

aPa.

H2) x512

+

Jm

aZ,

where we have made the following assumptions

0 0

- DH - ( Hl CO

+ 6)o H2

1

P

)( + 6)P&

305


D,IH,=D2/Hz=D/H The transmission power in AC line is the function of relative rotor angle 812 and the DC transmission power Pdc , i.e. Pa = P.(812, Pte) .

The determinant value of the Jacobian matrix J, can be easily figured out as det(J

-(

0 +'00 )(1+ apO2 )

H,

H2

aPdc

Obviously, if a? /aPQ is not equal to -1, the relations shown in Eq. (8.39) can be considered as a coordinate transformation of the system in Eq.

(8.37), which can transform the AC/DC nonlinear control system in Eq. (8.37) into a linear and controllable system in the subset S2={812,a12,PP I aPOClaPd t-

-1}.

The resultant linear system is z1 = z2

(8.40)

Z2 = Z3

Z3 = V

and the output equation is Y(t) = zl

Substituting the derivative of z3 in Eq. (8.39) into the Eq. (8.40) and

after some manipulation, we can obtain the nonlinear supplementary stabilizing control law for the converters as follows 3 udc=-Lfh(X)+v=

HI

H2

to°

a`0

H,

H2 Td

tea`

LBLfh(X)

D°12 +(Co +-0)Pn,+v H

+ Pdc-Pdc;,,.

(8.41)

V" _ -z1 - 2.29z2 - 2.1423 = -0812 - 2.29(012 - 2.14w12

_ -J w12 dt -2.29a12 -2.14r

O^0

G00

where the optimal control v` for the linear system (8.40) can be obtained from Example 3.2, which is (8.42)

Substituting the above equation into Eq. (8.41), we obtain the nonlinear stabilizing control law of the DC system as udc =

HI H2 7-d (f 1 a)12dt + 2.29ro12 + (2. l4 H1 + H2 WO °

D)[o12

H

-7 dP + P. - PP

CAD

Eq. (8.43) is the nonlinear control strategy that we are seeking for. Let us make some observation of Eq. (8.43)

(8.43)

306


.'301

CAD

CAD

"C7

o:.

(1) The nonlinear control law studied here consists of the proportion, integral and derivative elements of the AC system frequency difference between the two sides of the DC line with the derivative element of the active power P,, on the AC line in parallel with the DC line. The structure is not complicated and easy for implementation. (2) The nonlinear control law has no direct relations with the structure

and parameters of the network, but is related to the equivalent inertia

`CC

constants of the equivalent generators at the two ends (the sending end and receiving end) and the DC system equivalent time constant T,. Therefore, this controller has comparatively strong adaptability to the network structure and parameters. (3) If the two systems linked by the AC/DC parallel transmission lines are of large capacity and complex structure, w,Z in the control law (8.43) can be replaced by the difference between the AC bus voltage frequencies of the rectifier and the inverter. (4) If the AC lines are tripped off in parallel with DC lines, from the design method studied above, the nonlinear control law can be obtained as

8.4.3

H HZ

Td

a.;

ud, =

(f w12dt+2.29W12+(2.14- D)W12)+(Pk -P&REF) (8.44) H

Effects of Nonlinear Control for Stabilizing AC/DC Systems

Nod

In order to demonstrate the ability of the nonlinear stabilizing control of the DC system (in formula (8.44)) to improve the stability of the AC system,

the digital simulation is carried out on the system shown in Fig. 8.9. In (IQ

formula (8.44), the AC system frequencies of the converters are taken from those of the AC bus voltages at the converters. The equations are

(_f

1

dOr

1

dO,

Jr - fo + 2, dt

f =f0+21r

p (8.45)

di

where Br and 0, are respectively the phase angles of the AC bus voltage in the rectifier and the inverter and fo is the steady state frequency. The disturbance is assumed to occur by tripping off two 100MW generator units in the No.5 plant at O.Os and then putting the unit back into normal operation at 0.5s. This means about 23.6% of the total capacity of the AC system at the inverter side is affected directly. The nonlinear control with constant current and constant extinction angle (see Eqs. (8.19b) and (8.20)), or the nonlinear


Rotor angle (degree)

307

Power on

line (p.u.)

6.0

5.0

4.0

3.0

1

0.0

1.0

2.0

4.0

3.0

5.0

Time (Sec.) (b)

metro

Figure 8.15 System dynamic response under nonlinear controllers for AC/DC system stability (a) Rotor angle dynamic response of No.2 No.6 generators (b) The power dynamic response of the DC line (y1

'-'0

`ì

control with constant current and constant voltage (see Eqs. (8.30) and (8.31)), which have been discussed in Section 8.3, cannot keep the system

stable. The reason is that both controllers can not regulate the DC

0.p

transmission power in sufficient extent, and the performance index, constant current, constant extinction angle and constant voltage operation, can not be

satisfied under large disturbances. However, the nonlinear controller designed for AC/DC system stability improvement in this section can keep the system stable under the same disturbance. The simulation results are

308


shown in Fig. 8.15. From Fig. 8.15(b) we can easily observe that this nonlinear controller can regulate the amount of the power flow of the DC line sufficiently, so the stable operation of the AC/DC system is achieved.

8.5 REFERENCES A. E. Hammad, "Analysis of Power System Stability Enhancement by Static Var :7.

I.

°.r

G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis,

n~y

4.

.,e

3.

Compensators", IEEE PWRS, Vol.1, No.4, pp.222-227, 1986. CIGRE Working Group 14.01, "Use of Static or Synchronous Compensator on HVDC Systems", Electra, Vol. 91, pp.51-82, 1983. E. Uhlman, "Stabilization of an AC Link by a Parallel DC Link", Direct Current, No. 8,

2.

ova

1964.

McGraw-Hill, 1968. IEEE Committee Report, "Dynamic Performance Characteristics of North American HVDC Systems for Transient and Dynamic Stability Evaluations", IEEE PAS, Vol. 100, pp. 3356-3364, 1981. 6. IEEE Committee Report, "HVDC Control for System Dynamic Performance", IEEE PWRS, Vol. 6, No. 2, pp. 743-752, May, 1991. 7. J. F. Clifford, A. H. Schmidt, "Digital Representation of a DC Transmission System and Its Controls", IEEE PAS, Vol. 89, pp. 97-105, 1970. 8. K. R. Padiyar, HVDC Power Transmission System: Technology and System Interactions, John Wiley & Sons, 1990. 9. P. K. Dash, A. C. Liew and A. Routary, "High-Performance Controller for HVDC Transmission Links", IEE Proc.-Generation, Transmission and Distribution, Vol. 141, No. 5, pp.422-428, 1994. 10. R. L. Cresap, D. N. Scott, W. A. Mittelstadt and C. W. Taplor, "Operation Experience with Modulation of the Pacific HVDC Intertie", IEEE PAS, Vol. 97, No.4, 1978. 11. R. T. Byerly, D. T. Poznaniak and E. R. Taylor, "Static Reactive Compensation for Power Transmission System", IEEE Power Engineering Society Winter Meeting Paper 82 WM 179-0, New York, Jan. 31-Feb. 5, 1982. 12. S. Kapriclian, K. Clements and J Turi, "Application of exact linearization techniques for steady-state stability enhancement in a weak AC/DC system". IEEE PWRS, Vol. 7, No. 2, pp. 536-543, May, 1992. 5.

°".

-=W

ti'

0'O

con

t17

0

Sam

ono

F'>-

coo

`y°

EAU

.C7

`v°

J"'

°--

nab

0.U

O>e

T. Machida, "Improving Transient Stability of AC System by Joint Usage of DC

ion

13.

Systems", IEEE PAS, Vol. 85, No. 3, pp.226-232, 1966. T. Smed, G. Anderson, "Utilizing HVDC to Damp Power Oscillations", IEEE PWRS, Vol. 8, No. 2, pp.620-627, 1993. 15. X. Jiang, A. M. Gole, "An Energy Recovery Filter for HVDC Systems", IEEE PWRS, Vol. 9, No.1, pp.119-127, 1994. 16. Y. Z. Sun, Q. Lu and J. Gao, "A New Nonlinear Modulation Control for HVDC Power .-:

14.

°3'

coo

ti'

Transmission Systems", CSEE/IEEE International Conference on Power System 0+0

Technology, Beijing, Sept., 1991.

Chapter 9

Nonlinear Control of Static Var Systems

9.1

INTRODUCTION

Nowadays static reactive power compensator or Static Var System (SVS) has been widely used in power systems to improve the stability of the system

voltages as well as the stability under small and large disturbances. The primary function of SVS is to maintain a substantially constant voltage of its '-$)

installed bus by generating or absorbing the required amount of reactive power, which also benefits to the reactive power balance of the power system.

If a static Var compensator is installed at a midpoint of a long-distance transmission line, it is possible to hold the terminal voltage of its connection point essentially constant by continuous regulation of the reactive power. Then, the "electrical distance" of the transmission line can be regarded as being halved. Thus an increase in the stability limits of the power system can be expected. With its ability to continuously regulate the reactive power, high speed of response and simple configuration, static Var compensator has become more and more widely used [1-6]. This chapter will provide a detailed discussion on the theory and control

of the reactive power compensation. It starts with the basic concept of reactive power compensation and marches on to the configuration of the Static Var system and its modeling. Then, the fundamental principles and methods of nonlinear control system discussed previously will be applied to

analyze the performance of SVS, and to solve the control problem. Computer simulations are performed, and the results show the effectiveness (IQ

of the SVS with the nonlinear control for improving the dynamic performances and stability of the power system.

310


9.2

FUNDAMENTALS OF REACTIVE POWER COMPENSATION

9.2.1

Reactive Power Flow in a Transmission System

In this section we define some principle symbols and terms used with the purpose of describing the fundamentals of reactive power compensation. If a reactor or capacitor connected to a node of network is treated as a

load as shown in Fig. 9.1, then the, active and reactive power supplied at voltage V can be defined as (9.1)

P + jQ = VI

where V is the nodal voltage vector, and I the conjugate complex of the load current vector.

(b)

(a)

Figure 9.1

Reactor and capacitor treated as load (a) Reactor (b) Capacitor

From Eq. (9.1), the active and reactive power consumed by a pure reactor and a pure capacitor can be expressed respectively as Reactor:

PL+jQL=i

V2

(9.2a)

then QL=a- L

V2

(9.2b)

Pc +jQc = -jcoCV2

(9.3a)

PL = 0

Capacitor:

Nonlinear Control of Static Var System

311

then

Pc=0

(9.3b)

Qc=-wCV2

where L is the inductane, C the capacitance, w = 2nf , f the frequency of the bus voltage of the power network. We give the following definition. If Q > 0, then the load is said to be of

reactive characteristic, it absorbs reactive power from the system. And, if Q>°

voltage virtually constant under the variation of the power flow by providing

reactive power required by the system, and hence further enhance the transmission power limit (stability limit) of the line.

9.3

CONFIGURATION OF STATIC REACTIVE COMPENSATORS

In practice, there are many different types of static compensators for transmission application, and the main types will be briefly described in this section.

9.3.1

Thyristor-Controlled Reactor (TCR)

r..

The single-phase diagram of a thyristor-controlled reactor (TCR) is shown in Fig. 9.8. The thyristor controller consists of two anti-parallel thyristor valves, which conduct on alternate half-cycles of the supply frequency. And a reactor is connected in series with the thyristor controller. Each valve is further composed of many thyristors, as shown in Fig. 9.8, and connected in series and parallel to withstand high voltage and large current. If the supply voltage vs is

320

NONLINEAR CONTROL SYSTEMS AND POWER SYSTEM DYNAMICS L ITCR

Figure9.8

Single phase diagram of TCR and its waveforms

VS = V sin wt

(9.35)

From Fig. 9.8, the voltage equation of the circuit is given as

Ldi-VS.=0

(9.36)

NIA

Solving above equation for i we have V i(t) = L JVsdt +c =-cos Cot + C 1

wL

where c

is

(9.37)

the integration constant. And from the initial condition SIC

i(wt = a) = 0 (see Fig. 9.8), we get

---(cos a - cos wt) L

(9.38)

0(Q

where, a is the gating delay angle in rad. The waveforms of the current I,CR, anode-cathode voltage of thyristor V,cR, and the voltage at the reactor at different gating angle a are shown in

Fig. 9.9. The waveform of IrcR indicates that the TCR is operated as a continuously adjustable reactor. The effects of increasing the gating angle a are to reduce the fundamental component of the current I,cR. As the gating angle a approximates to 180°, I,cR approximates to zero. Full conduction is obtained with a gating angle of 90°, where the waveform of current is

Nonlinear Control of Static Var System

321

ITCR

N VTCR

V, 1

U \I/ a near 90°

Figure 9.9

a=120°

011-4

a=150°

1

a near 180°

--

Current and voltage waveforms at different gating angle

0

nearly sinusoidal, so its magnitude of fundamental component (elementary wave) reaches its maximum value as shown in Eq. (9.38). Fig. 9.9 shows that the current waveform involves harmonic components. However, we are mainly concerned with the fundamental component of the current in the design of compensators for a transmission system. According to the Fourier analysis, the fundamental component of the current can be written as (9.39)

i, (t) = a, cos cot + b, sin cot

r-.

From Eq. (9.38), current i(t) is an even function, i(t) = i(-t) , i.e. b, =0; and considering i(t + T /2) = -i(t), we can figure out the actual value of a, as a,

_

2,r-2a+sin2a

(9. 40)

In view of the following formula T12 0 =Tf

f(r ) cos 27rr d r

(9 . 41)

[r]

a,

where a is expressed in radian. Substituting Eq. (9.40) into Eq. (9.39), the magnitude of the fundamental component of current is

N-1

V 2,r - 2a + sin 2a '

wL

(9.42)

rr arc

Obviously, the fundamental component is a function of gating angle a. Dividing both sides of Eq. (9.42) with the magnitude of voltage, the

322


fundamental frequency susceptance BR of TCR will be I' V

=BTCR - BL

2,r-2a+sin2a

(9.43)

/T

However, if the reactor is represented as Z = jx, its susceptance is Y=1/Z= jlx = jBL, i.e.

BL =-1 =X

(9.44)

1

wL

C/]

Cep

!3.

where L is the inductance of the reactor. The relation between BrcR and gating angle a is shown in Fig. 9.10. Static compensator is also named as static reactive power compensator; as a continuously adjustable reactor, TCR is the simplest type of SVS.

X

X

a

2

Figure 9.10 The control characteristic of BTCR

SVS can be generally thought of as a device with adjustable parameters (susceptance), the relationship between its terminal voltage v,,, and current IS,,S is expressed as

(9.45)

Isvs = jBsvsV svs C%1

As aforementioned, for the simplest type of SVS, Bv,, = B,cR , if a shunt fixed capacitor bank (FC) is added to TCR, then BSUS =B,R +BB, where Bc = cvC , C is the capacitance. Such a type of static compensation PL,

4-i

system SVS is called TCR-FC type compensator in short. The configuration of TCR-FC type compensator is illustrated in Fig. 9.11. From Fig. 9.11, BSS in Eq. (9.45) will be B SVS

__

B (BC + BTCR

Bo+Bc+BTCR

(9.46)

323

Nonlinear Control of Static Par System

The configuration of TCR-FC type compensator C.7

Figure 9.1 1

Since the gating angle a can vary continuously between 180° and 90°, the value of B,r.R will change continuously between BL and 0. As a=180°, B,cR = 0, and the maximum value of susceptance of SVS becomes (9.47)

BQ Bc

B svs

= B+Bc ICJ

Similarly, as a=90°, B,r.R = BL, and the minimum susceptance of SVS given by

i

B

B (BC + BL)

(9.48)

svs=Bo+Be+BL

t17

Analyzing Eq. (9.46) we know that the susceptance B,, of static Var alb

system is a nonlinear function of B,,,. However, if susceptance B c / BL To

holds, and the function V(X) is called the storage function of the dissipative system.

0

How- to comprehend the above definition of a dissipative system? If the nonnegative storage function V(X) is taken as an energy function and S(U,Y) the energy supply rate of the system, i.e. the supply power, then the right side of (10.11) can be regarded as the supply energy gained by the system in the period of [TO,7 ] ; and the left side is the increment of storage energy in the same period of time. It is clear that for a period of time [To,71, if the increment of storage energy of the system is less than the supply energy gained by the -system, then there must be some energy dissipated in the period [7, T,] . Therefore, it is reasonable to assign to the system the

i:.

s"7

r'.

vii

'via/

attributes of dissipativity. After defining and explaining the definition of a dissipative system, let us consider a sort of dissipative system with a certain supply rate.

If system (10.10) is an asymptotically stable dissipative system at X(0) = 0, and the supply rate is (10.12)

S(U,Y) =7211U112 -1IYI12

then according to Definition 10.5 and the characteristic of V(X), i.e. V(X(T)) > 0 for T> 0 , and V(X(0)) = V(0) = 0 , the inequality 0 0

(10.13)

must be held, and formula (10.13) can be rewritten as 0:- f01'a (y211U(t)112

-IIY(t)112)dt,

y>0

(10.14)

From Proposition 10.1 in Section 10.2.1 we realize what the inequality

350


(10.14) really means is just that the L2 -gain G2 of the nonlinear system as shown in Eq. (10.10) is less than or equal to a prescribed number y. From the above an important proposition is therefore formed as follows.

Proposition 10.2 Consider a nonlinear system as shown in Eq.(10.10). If X(0) = 0 is an asymptotically stable equilibrium point of the system, and the system is dissipative with respect to the supply rate 2IIU(t)IIZ

then the L2 -gain G2 of the system from input U to output Y is less than or equal to the S=y

- IfY(t)II2, i.e.

to (y 2IIU(t)II2 - IIY(t)II2 )dt >- 0,

prescribed positive number y.

10.3

NONLINEAR ROBUST CONTROL

10.3.1 Description of Nonlinear Robust Control From the discussion in Section 10.1, we have comprehended the meaning of nonlinear robust control. The strict definition of nonlinear robust control problem will be given in this section. Considering a nonlinear system described by the differential equations of the form

X(t) = f(X(t))+g1(X(t))W(t)+g(X(t))U(t) Z(t) = h(X(t)) + K(X(t))U(t)

(10.15)

Cs.

where, X E R" is a state vector; U E Rm a control vector; W E R' a disturbance vector; f (X) , g, (X) , g2 (X) , h(X) and K(X) are smooth vector functions defined on the state space; and Z E R° is a penalty vector. The aim of the nonlinear robust control is twofold: firstly, to find a control strategy which makes the closed loop system (10.15) be asymptotically stable at X(O) = 0 when W = 0; secondly, the control strategy can make the L2 -gain G2 from disturbance W to penalty vector CS'

Z be less than or equal to a prescribed positive number y, that is (see Proposition 10.1 in 10.2.1) f0(y2IIWII2-IIZII2)dt - 0

VT>0

(10.16)

p.'

The first purpose is the prerequisite for a nonlinear robust control system - or for any control system. The controllers that satisfy the first

Nonlinear Robust Control of Power Systems

351

°,m

(IQ

condition are called admissible controllers. Our discussions in subsequent sections are all based on the admissible controllers. The second condition is the essence of robust control issue, which indicates that the control law sought for must have the ability of disturbance attenuation. The degree of the effect of the ability is characterized by the value of y, the smaller the value of y is, the stronger the effect of disturbance attenuation will be. If y reaches its minimum, then we can get the so-called optimal robust control. From the engineering design point of view, we do not always seek for the "optimal" robust control law, but seek for the appropriate or satisfactory one. Obviously, if the value of y prescribed is too small which is less than the actually existing minimum of itself, then the robust control issue has no solution. Why the problem of nonlinear robust control is sometimes also named by some literature as nonlinear H. control? This needs to be explained. We know that there are no transfer functions for a nonlinear control system, nor, therefore, H. norm of transfer function matrix. However from previous statements we know that a control is said to be a robust one of a nonlinear system, if it is such that the L2 -gain from disturbance W to penalty Z less than or equal to a given positive number y, which could be named in short form as L2 -gain control. Since from Proposition 10.1 in ..1

..o

.fl

Section 10.2.1 we know that in linear case the H. norm IITII. precisely equals

L2 -gain,

namely,

IITIL = G2 ,

some

borrowed

authors

this

4-+

terminology of "H control" when discussing robust control problem of a nonlinear system [2, 4, 16, 23, 25]. However we prefer to use the terminology of L2 -gain control, robust control or disturbance attenuation .fl

0

control for a nonlinear system [8].

.Fr

103.2 General Form of the Nonlinear Robust Control Law For the sake of completeness, let us recall that given a nonlinear system in the form as X(t) = f(X(t))+g1(X(t))W(t)+K(X(t))U(t)

(10.17a)

Z(t) = h(X(t)) + K(X(t))U(t)

(10.17b)

where, X E R" is a state vector; U E R' a control vector; W E R' a disturbance vector, which is a harmful input whose influence should be reduced;

L()

352


Z E R° is a penalty vector, which may include dynamic tracking errors and

the cost of the control U. So the L2-norm of Z(t) should be as small as o.,

possible. f(X), g,(X), g2(X), h(X) and K(X) are all smooth vector

functions defined on R", satisfying f(0) = 0, h(o)=O. The task of this section is to find a general form of state feedback U = U.(X)

such that as W = 0 the system is asymptotically stable at the equilibrium point X(0) = 0, and L2 -gain G2 of the system from the disturbance W to the penalty Z is less than or equal to a described positive number 7. In other words, according to Proposition 10.1, the system to be designed should satisfy the following condition (10.18)

dT > 0

.(U, W) =1o (IIZII2 - r 211W 112 )dt I, and a control strategy U = U' (X) , such that for V7>1. 2dr

VT >_ 0

holds and the closed-loop system is asymptotically stable provided W=O. Here it should be pointed out that studying the nonlinear robust control problem for the system in Eq. (10.49) corresponds to studying the same problem one for the system in Eq. (10.15) with the penalty output h(X

Z(t)

0`t))1 + CU(t)]

For the sake of convenience, here we study the feedback linearization of the system in Eq. (10.49) with X E R" , W E R2 , u e R , y e R . The

results achieved can be extended without much difficulty to more general affine nonlinear systems.

Denote by g,(X)=[g,i(X) 9,2(X)] and W=fw, w,]". Now we assume that the relative degree p of the system from control u to output y is equal to n, and the relative degree related to disturbance w, and w2 are p, and p2 , respectively, and denote ,u the minimum between p, and p2 , of course p S n . Thus, by means of coordinates transformation and state feedback v = a(X) + ,l3(X)u

(10.50) V'>

Z = T(X)

the nonlinear affine system X = f(X) + g2 (X)u

y = h(X)

can be transformed into a linear system in the form

(10.51)

362


Z= AZ+B2v Y. =CZ

(10.52)

where h(X) ...

Lf'h(X) = T(X) _ LJh(X) LJ''h(X)

Lf'h(X) a(X) = L'fh(X) /3(X) = Lg,L;'h(X)

L'Jh(X) denotes the r"' order Lie derivative of h(X) along f(X), 0:5 r:5 n, and A, B2, and C are Brunovsky normal form as discussed in Chapter 3 and Chapter 4 [3, 7, 20, 21]. Then the system in Eq. (10.49) can be written as

Z=AZ+B2v+ a(X)g'(X)W

(10.53)

YZ =CZ Set

Lg h(X)

aX) g, Mw

=

L g LJ%-2 h(X)

L,,, L'-2h(X) J

Lg LJ -1 h(X)

L,,

J

...

W=

Lgsh(X)

Lg Lf h(X)

L,,, L' h(X)

LguL-'h(X) J

L',2 L' j' h(X)

2

I

According to the definition of the relative degree, the above matrix becomes 0 0

W = Lg L''h(X)

0 0

Lg,L"7'h(X)

Lg Lfh(X)

Lg=Lfh(X)

Lg L'J'h(X)

Lg,L'J'h(X)

Then the system in Eq. (10.53) can be written as

(10.54)

363

Nonlinear Robust Control of Power Systems

Z = AZ+B,W +B2v Yz =CZ

where B, _

)x(w)

0

O(n-N+l

(10.55)

0<w l) (, N+n

p-n

'(n-k+i)x(n-N+l)]

(1;°o

Now, we can design a controller for the system in Eq. (10.55) according

to the linear robust control theory presented in Section 10.4. The above robust control problem has a solution if the following Riccati inequality ATP+PA + Y2 PB1B; P-PB2BTp+CTC 0, find U` that minimizes the performance index

J(U,W)=

2 2f (IlYll+IIU11-y2NWll)dt 2

(10.58)

under the worst possible disturbance W' maximizing the index as shown in (10.58), which is denoted by infsupJ(U,W) G.

185

HVDC , 277

.O.

Effective value, 170 Effects of the valve opening limit, 188 Electrical-hydraulic, 245 transducer, 187 Electromagnetic dynamic equation for field winding,

Hamiltonian function, 64 Hamilton-Jacobi-Isaacs (HJI), 356 High-pressure regulated valve or steam controlled valve, 188 High-pressure, medium-pressure and low-pressure turbines , 187

.fl

d, q coordinates of a synchronous generator 27 Damping effects, 177 windings, 174 DC control system, 196 transmission, 191 DDP, 109 Dead band, 245 Demagnetization, 177 Derived mapping, 36 Diffeomorphism, 25 Dissipative systems, 348 Disturbance decoupling, 61 Dynamic characteristic, 303 deviation, 93 equations, 247 performance, 205 response, 202

r=7

transformer, 191 circuit of inverter, 193 process, 192 Coordinate

via state feedback, 60 conditions, 87 design method, 59

Large-capacity steam turbine generator set, 187

Index

375

linearized normal form, 50 linearly independent vector fields, 152 Line-to-earth capacitance of DC transmission line, 192 Load compensator, 313 Local diffeomorphism, 30

c,y

Penalty function, 3428 Performance index, 2 Per-unit instantaneous output power, 170 Phase controller, 196 Physical dynamic simulations, 222 PID (proportional integral differential), 200 Power fraction, 187 modulation, 284 Power System Stabilizer (PSS), 205 .-.,

(LOEC), 222 quadratic regulation (LQR), 63

Oil-pressure control signals, 187 Oil-servomotor, 247 One inertia block, 189 One-machine, infinite-bus system, 17

poi

nee

Large-disturbance stability, 222 Liebracket, 25 derivative, 25 Limit block, 188 Linear multi-variable control, 205 Linear Optimal Excitation Controller

Prime movers, 245 Magnetic circuits, 170

q-axis transient electric potential, 178

conductivity, 170

Mathematical model of multi-machine power systems, 223

Reactive

Mechanical power, 209 Mechanical-hydraulic, 245

power flow, 310 Reactor, 310 Rectifier, 191 transformer, 193 ,O,

co.

MKS units, 170 Multiple inputs and multiple outputs (MIMO), 121

Reheater, 246 Relative degree, 26 Riccati algebraic equation, 65 p».

121

°F'

medium-pressure regulated valve or fast valve, 187 MIMO affine nonlinear control systems,

power compensation, 282

Robust control, 343 Robustness, 343 Rotor windings, 170

Natural power, 316

Salient-pole generator, 183

Nominal operation state, 197 Nonlinear excitation control law, 217 optimal excitation control (NOEL), 223 state feedback, 209 Normal affine nonlinear system, 209 Normal opened contact, 188

Self admittance, 182 Self inductance, 170 inductive reactance, 174

0'!

Mutual inductance, 170

chi

Servo mechanisms, 245 systems, 245

Signal amplifier, 196 SISO (single input single output), 59

C'7

Nonlinear Control Systems and Power System Dynamics

376

Variational method, 63

_a'

£,'

affine nonlinear system, 59 Small disturbance stability, 205 Smoothing reactors, 191

Vector

.lam

:;,

.G+

Speed resolution, 245 State feedback, 59 law, 94 Static reactive power compensator, 309 Static Var system (SVS), 309 Stator current, 170 windings, 170 Steady state, 196 Steam flow, 246 s..,

^-.

G..

.'+

turbine, 187 turbine with reheater, 187 valving control system, 189 Storage function, 348 Surge-impedance loading(SIL), 316 Swing equations, 166 Symmetric three-phase system, 170 Synchronous operation, 176 System compensator, 314 Terminal voltage, 170 Three-phase controllable bridge-type circuit, 192 short circuit fault, 240 Thyristor -switched capacitor (TSC), 325 -controlled reactor (TCR), 319 Time lags, 245 Time scale of electromechanical transient process of a power system, 189 Transformation matrix, 107 Transformer electric potential, 174 Transient .n»

a0.

«0.

.-+

...

C/)

gyp'

fl'

.fl

`;P

E--

state, 178 !-+

,f+

stability of power systems, 186

Two-phase short circuit fault, 256 Valve opening, 187

field, 26 field sets, 26 Voltage losses, 176 stability, 282 -current characteristics, 278

Water hammer, 245 Whole region of generator operation state, 189

Zero dynamics design method, 91

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