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Chapter 01

Introduction

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1.1 OverviewObjective:

This Book presents a comprehensive descriptionof nonlinear control of electric power systemsusing nonlinear control theory.

Major Problem in Power System:

Out of many problems to be resolved and

improved in modern power systems, thereliability of power systems is main issue.Reliability consists of two different aspects.

Reliability

Static

Reliability

Dynamic

Reliability

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1.1 Overview Improving the dynamic reliability (stability) of

power system incorporates highly sophisticatedtechnology of control.

Initially the excitation control was employed forcontrol of power system, taking generator

terminal voltage as the single feedbackvariable.

deMello and Concordia proposed a controltechnique which, besides the deviation of theterminal voltage, took a supplementaryfeedback variable as another input, which couldbe the speed deviation w, the frequencydeviation f, or the deviation of active powerP

e.

Thus the excitation control of generators

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1.1 Overview

PSS model is established by linearizing the

nonlinear equations of power systems at a certainoperating point (a fixed equilibrium point X

e).

It is obvious that this approximately linearizedmodel is relatively accurate only when the actual

state X(t) is rather close to Xe.

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1.2 OUTLINE OF THE DEVELOPMENT OFCONTROL THEORY

Control is a general term for the theory andtechniques to change the dynamic performanceof a system by imposing certain inputs on thesystems, so as to satisfy certain requirements totheir best.

The classical control theory in complex variable

s=+j or frequency j domain.

ControlTheory

Classical Modern

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1.2 OUTLINE OF THE DEVELOPMENT OFCONTROL THEORY

Significance of Classical Theory:

The most significant feature of this theory lies inits own modeling method.

For a single input single-output linear time-invariant system, its dynamic behavior can be

described by the following constant-coefficientordinary differential equation

where, u(t) is the control input variable, x(t) theoutput variable.

( 1)

1 1 0( 1)

1 0

( ) ( ) ( )... ( )

( ) ( )... ( )

n n

n nn n

r

r r

d x t d x t dx t a a a a x t

dt dt dt

d u t du t b b b u t dt dt

+ + + +

= + + +

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1.2 OUTLINE OF THE DEVELOPMENT OFCONTROL THEORY

Perform the Laplace transformation on both sides of Eq.(1.2), with the zero initial state, we obtain

Defining the system's transfer function G(s) as the ratiobetween the Laplace transformations of the output andthe input of the system when the initial state is zero,

This equation, as a fraction of polynomials of s, is thegeneral form of transfer functions, which is also thebasic form of mathematical model in the classical

( ) ( )

( ) ( )

1

1 1 0

1

1 1 0

...

...

n n

n n

r r

r r

a s a s a s a X s

b s b s b s b U s

+ + + +

= + + + +

( )( )

( )

1

1 1 0

11 1 0

...

...

r r

r r

n n

n n

X s b s b s b s bG s

U s a s a s a s a

+ + + += =

+ + + +

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1.2 OUTLINE OF THE DEVELOPMENT OFCONTROL THEORY

Application Scope of Classical Theory:

The mathematical tools used are fairly simple,which are mainly Laplace transform method andalgebraic polynomials.

The systems that the transfer functions can modelare only linear constant control systems.`

This theory or method is only applicable to single-

input single-output systems. It conceals the internal dynamic behavior of the

system

1 2 OUTLINE OF THE DEVELOPMENT OF

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1.2 OUTLINE OF THE DEVELOPMENT OFCONTROL THEORY

Modern Control Theory

The most significant feature of this theory isthat it is applicable to multi-input, multi-output dynamic systems.

It is an integration of linear algebra andmodeling theory.

It is well known that, an nth-order lineardynamic system can be modeled as an nth-order constant-coefficient ordinary differentialequation.

State Space Modeling:

we define a set of variables {x, (t), x2 (t), , x,,(t)} or a vector X(t) = [x

1(t) x

2(t) ... x

n(t)]T as

the system's state variable set or state vector