26

CHAPTER2

POWER SYSTEM STABILIZER

2.1 INTRODUCTION

This chapter explains the necessity of power system stabilizers

along with AVR and exciter. The structure of lead-lag power system stabilizer

is explained with phase compensation blocks, wash out filter and stabilizer

gain block. The effect of PSS is included in the state space model.

The selection of PSS parameters and its limitations are discussed.

2.2 NEED FOR PSS

High performance excitation systems are essential for maintaining

steady state and transient stability of modern synchronous generators, apart

from providing fast control of the terminal voltage. Bus fed static exciters

with thyristor controllers are increasingly used for both hydraulic and thermal

units. They are characterized by high initial response and increased reliability

due to advances in thyristor controllers.

Well established fast acting exciters with high gain AVR can

contribute to oscillatory instability in power systems. This type of instability

is characterized by low frequency (0.2 to 2.0 Hz) oscillations which can

persist for no apparent reason. There are several instances of such occurrences

which have been recorded and studied. This type of instability can endanger

27

system security and limit power transfer (Milanovic 2002). The major factors

that contribute instability are,

(i) Loading of the generator or tie line

(ii) Power transfer capability of transmission lines

(iii) Power factor of the generator (leading power factor is more

problematic than lagging power factor operation)

(iv) AVR gain

A cost efficient and satisfactory solution to the problem of

oscillatory instability is to provide damping for generator rotor oscillations.

This is conveniently done by providing Power System Stabilizers (PSS)

which are supplementary controllers in excitation system. The input for PSS

is derived from rotor velocity, frequency, electrical power or a combination of

these variables whereas, output is controlled voltage SV ( Anderson and Fouad

1977, DeMellow et al 1978, Ohtsuka et al 1986, Samarassinghe et al 1997).

The objective of designing PSS is to provide additional damping torque

without affecting the synchronizing torque at critical oscillation frequencies.

2.3 BASIC CONCEPTS OF PSS

The basic function of a PSS is to extend the angular stability of a

power system. This is done by providing supplemental damping to the

oscillation of synchronous machine rotors through the generator excitation

(Asgharian 1994). This damping is provided by an electric torque applied to

the rotor that is in phase with the speed variations. The oscillations of concern

typically occur in the frequency range of 0.2 to 3.0 Hz, and insufficient

damping of these oscillations may limit ability to transmit power. In practical

system, the various modes (of oscillation) can be grouped into three broad

categories.

28

(i) Intra-plant modes (generator 1G swings against 2G ) in which

only the generators within a power plant participate. The

oscillation frequencies are generally high in the range of 1.5 to

3.0 Hz.

(ii) Local modes in which several generators ( 1G and 2G swing

together against 3G ) in an area participate. The frequencies of

oscillations are in the range of 0.8 to 1.8 Hz.

(iii) Inter area modes in which generators (generators 1G to

3G swing against 4G ) over an extensive area participate. The

oscillation frequencies are low and in the range of 0.2 to 0.5

Hz.

The above categorization can be illustrated with the help of a

system consisting of two areas connected by weak AC tie as shown in

Figure 2.1. Area 2 is represented by a single generator 4G . The area 1

contains 3 generators 1G , 2G and 3G .

1G

4G2G

3G

Figure 2.1 A sample power system

29

The distinction between local modes and inter area modes applies

mainly for those systems which can be divided into distinct areas which are

separated by long distances. For systems in which the generating stations are

distributed uniformly over a geographic area, it would be difficult to

distinguish between local and inter area modes from physical considerations.

However, a common observation is that the inter area modes have the lowest

frequency and participation from most of the generators in the system spread

over a wide geographic area. The PSSs are designed mainly to stabilize local

and inter area modes (Feliachi and Zhang 1988, Hui Ni et al 2002).

The main objective of providing PSS is to increase the power

transfer in the network, which would otherwise be limited by oscillatory

instability. The PSS also must function properly when system is subjected to

large disturbances. PSS can extend power transfer stability limits which are

characterized by lightly damped or spontaneously growing oscillations in the

0.2 to 3.0 Hz frequency range. This is accomplished via excitation control to

contribute damping to the system modes of oscillations (Abe and Doi 1983,

Zhou et al 1991).

Consequently, it is the stabilizer’s ability to enhance damping under

the least stable conditions is important. Additional damping is primarily

required under the conditions of weak transmission and heavy load which

may occur, while attempting to transmit power over long transmission lines

from the remote generating plants or relatively weak tie between systems

(Boukarim et al 2000). Contingencies, such as line outage, often rashed such

conditions. Hence system normally have adequate damping can often benefit

from stabilizers during such conditions.

30

2.4 OVERVIEW OF POWER SYSTEM STABILITY

CONTROLLERS

Fleming et al (1990) proposed three novel approaches to improve a

conventional PSS in a SMIB system. These improved stabilizers used the

conventional PSS in the usual manner plus modification of the terminal

voltage feedback signal to the excitation system as a function of the

accelerating power on the unit. The nonlinear action increased the power

system stability greatly. It was concluded that these three kinds of improved

stabilizers can improve power system stability much more than the

conventional PSS which has been used widely in power systems since the

1970's. Compared to the three kinds of improved stabilizers, the improved

PSS is the best one since it is effective for both small and large disturbances,

and is also effective to improve both overshoot and settling time of rotor

speed deviations.

Chen et al (1993) discussed a new self-optimizing pole shifting

control strategy for an adaptive PSS. Based on an identified model of the

system, the control was computed by an algorithm which shifted the closed-

loop poles of the system to some optimal locations inside the unit circle in the

z-domain to minimize a given performance criterion. With the self

optimization property, outside intervention in the controller design procedure

was minimized and simplified the tuning procedure during commissioning.

Also, a new method of calculating the variable forgetting factor in real-time

parameter identification was discussed. For real-time control, a low order

system model could be used to represent the controlled system. The proposed

control strategy based on a pole-shifting approach combined the advantages

of pole assignment control algorithm and minimum variance control

algorithm. The closed loop pole locations were optimally calculated by the

control algorithm in order to minimize the given performance index. Test

31

results for various conditions showed that the proposed adaptive stabilizer

could provide good damping over a wide frequency range and increased the

dynamic and transient stability margins.

Khaladi et al (1993) developed an optimization approach using the

modal performance measure for the selection of PSS parameters in multi-

machine power systems. The goal of the optimization problem was to damp

out the sustained low frequency oscillations in the outputs of a linearized

power system. This paper also considered bounded and unbounded PSS

parameters and compared the effects of bounds on the end results. He showed

that the performance measure was not a convex function in the PSS

parameters. That is, there exist many local minima and possibly a global

minimum. System responses were obtained to have very satisfactory low

frequency oscillation for different sets of “optimal” PSS parameter values.

Furthermore, bounds on the parameters played an important role in

ascertaining the stability of the system. The system model was derived from a

fixed operating condition, and based on this model an optimal set of PSS

parameters was obtained. For implementation purposes, optimal set of PSS

parameters could be computed, based on the proposed technique, for a

number of operating regions. If the system underwent any abnormal

conditions, then the selected PSS parameters would be switched to the

optimal values of the new operating condition.

Kothari et al (1993) discussed a comprehensive approach for the

design of discrete mode conventional PSS considering a SMIB system, using

the ISE technique. Investigations revealed that the sampler and zero order

hold (ZOH) were modeled for sampling frequencies less than 20 times the

Nyquist frequency. A comprehensive sensitivity analysis revealed that the

discrete mode conventional PSS was quite robust and its nominal optimum

parameters need not be reset following ± 20% changes in inertia constant H,

32

field open circuit time constant 0dT , line reactance ex , AVR gain AK or loading

P and Q from their nominal values.

Kothari et al (1995) designed a self-tuning PSS using the pole-

shifting technique. The controller used a state-feedback law, whose gains

were evaluated from the pole-shifting factor. The proposed method was

simple and computationally efficient. The dynamic performance of the

proposed PSS was quite satisfactory and the PSS adapted quickly to varying

operating conditions. The method used a model formulation which obviated

the need for state observers and the output was directly used to derive the

feedback control signal. It combined this with a simple pole-shifting control

technique in this framework to achieve quite satisfactory dynamic

performances. The control calculations are simple and require less

computational effort.

Yuan-Yih Hsu et al (1996) discussed the identification and tuning

of exciter constants for a generating unit at the Second Nuclear Power Plant

of Taiwan Power Company. Field test was first performed on the excitation

system with the generator open-circuited. Since the field test results differed

from the computer simulation results using manufacturer's constants, he

modified the manufacturer's constants based on previous experience to reach a

preliminary set of parameters for the excitation system. Then a hybrid

nonlinear simulation-sensitivity matrix method was developed to further

refine the excitation system parameters. The exciter constants were tuned in

order to give better dynamic response. Field tests were conducted in order to

compare the dynamic response of the generator without and with PSS.

Simoes Costa et al (1997) proposed a method to design the power

system controllers in order to damp electromechanical oscillations. It could be

applied to the design of both PSS for synchronous generators and

33

supplementary signals associated to other damping sources. Some attractive

features of the method were: the parameters of all controllers were jointly

determined, there was no restriction on the type of supplementary signals to

be used, and controller structures were compatible with those nowadays

employed in electric utilities. The control problem solution exploits sparsity,

combining a system description which preserves a sparse structure with an

adequate mathematical formulation of the optimal design approach. The

results were validated by using both eigenvalue analysis and nonlinear

simulation. A method based on structurally constrained optimal controller

design for the determination of controller settings in multimachine power

systems was introduced. The settings for all controllers in a multimachine

system could be simultaneously determined through an integrated procedure

which takes into account all dynamic interactions. The results, were given

both in terms of eigenvalues and nonlinear simulation curves illustrate the

applicability of the method to realistic power systems.

Pourbeik and Gibbard (1998) developed a method for the

simultaneous coordination of PSSs and FACTS device stabilizers (FDSs) by

using the concept of induced torque coefficients. The proposed coordination

scheme employed linear programming. A case study was given which

illustrated the coordination of PSSs and FDSs in a three-area system with 29

stations, 3 SVCs and 400 states. A two-stage method was described for the

coordination of PSSs and FDSs in a multimachine system for the purpose of

improving dynamic performance. The first stage involved the determination

of stabilizer transfer functions to affect a left-shift in the rotor modes of

oscillation. The second stage involved solving a linear programming (LP)

problem to calculate the minimum values of stabilizer gains to satisfy

specified left-shifts in the modes. If zero was the minimum allowable

stabilizer gain, the algorithm chosen the minimum number of stabilizers

34

required to meet the LP constraints. This constituted the optimal location of

PSSs and FDSs on existing generators and FACTS devices.

Choi and Jia (2000) discussed the inherent dynamical relationship

between the under-excitation limiter (UEL) and the PSS control loops in

synchronous generators using the frequency response technique. It was shown

that the limiters should be designed to effect a much slower response

characteristics as their main function was to prevent excessive stator end-core

heating. The analysis also showed that a reduction in the values of the slope

of the boundary curves, which prescribe the operating region of the limiters,

was accompanied by a decrease in the damping level of the closed-loop

excitation control systems. It was shown that the tuning of the UEL and the

PSS could be carried out separately without considering the interaction

between the two control loops. Analysis of the power system model showed

that the damping level due to the UEL increased along with the slope of

limiter boundary curve.

Soliman et al (2000) designed a simple robust PSS that could

properly function over a wide range of operating conditions. The lead

compensator design was achieved by drawing the root loci for a finite number

of extreme characteristic polynomials. Such polynomials were obtained, using

the Kharitonov theorem, to reflect wide loading conditions on characteristic

equation coefficients. For this purpose the explicit analytical forms for the

coefficients of the system transfer functions were derived. Simulation results

illustrated the effectiveness of the proposed stabilizer as it was applied to the

original nonlinear differential equations describing system dynamics under

wide loading conditions at lagging and leading power factors.

Boukarim et al (2000) discussed several control design techniques,

namely, the classical phase compensation approach, the -synthesis, and a

35

linear matrix inequality (LMI) technique to coordinate two PSSs in order to

stabilize a 5-machine equivalent of the South/Southeast Brazilian system. The

open-loop system had an unstable inter area mode and could not be stabilized

using only one conventional power system stabilizer. Both centralized and

decentralized controllers were considered. The different designs were

compared and several observations were provided. To test the performance of

the controllers, a disturbance was applied to the closed-loop systems using the

PacDyn Program. The disturbance was a simultaneous application of a

positive perturbation in the mechanical power of a particular generator and a

negative perturbation in the mechanical power of the southeast equivalent

machine. The speed variations were higher for the centralized controllers than

for the decentralized controllers. This was due to higher gains of decentralized

controllers. The centralized -synthesis design produced more transients than

the centralized LMI controller design. The centralized controllers required

much low gain to achieve the same amount of damping enhancement than the

decentralized controllers. However, the centralized controllers had less

disturbance rejection capabilities and required fast communication links to

implement.

Milanovic (2002) investigated dynamic interactions among various

controllers used for stabilizing a synchronous generator. The effectiveness of

a PSS connected to the exciter and/or governor in damping electromechanical

oscillations of isolated synchronous generator was examined. The interactions

among PSSs connected to the exciter and/or the governor loop, automatic

voltage regulator, governor and multi-stage double-reheat turbine and

dynamic load were considered. It was shown that depending on the type and

number of controllers used and dynamics modeled, interactions could result in

unstable operation of the system for a range of operating conditions. It was

also shown that the PSS connected to the governor loop provides better

damping of low-frequency oscillations and better robustness of the generator

36

to a change in operating conditions than the PSS connected to the exciter

loop. The paper further showed that a properly tuned PSS connected to

governor loop could provide better overall damping of the system oscillations.

Gupta et al (2003) designed PSS for a SMIB using periodic output

feedback. The non-linear model of a machine was linearised at different

operating points and 16 linear plant models were obtained. For each of these

plants an output injection gain was obtained using the LQR technique. A

robust periodic output feedback gain which realized these output gains was

obtained using an LMI approach. This robust periodic output control was

applied to a non-linear plant model of the machine at different operating

points. This method did not require the complete set of states of the system

for feedback and was easily implementable. The slip signal was taken as

output and the periodic output feedback control was applied at an appropriate

sampling rate. This method was more general in nature than the static output

feedback method, and also required small magnitudes of the control inputs for

these plants. It was found that the robust controller designed provided good

damping enhancement for various operating points of a single-machine

system connected to an infinite bus.

Chow et al (2004) discussed the practical experience in assigning

PSS projects to provide the designer with a challenging design problem using

three different techniques. The design of PSS projects using root-locus,

frequency domain, and state-space methods were provided. The projects

provided the designer with some realistic and challenging design experience

and exposed them to a well-known power system design problem. A

saturation block could be added to the output of the PSS to limit its

contribution in the voltage regulator input.

Yee et al (2006) proposed a simple method that directly optimizes

the nonlinear system by using system time response to large disturbances. The

37

properties of the method and its application specifics were discussed in detail.

The effectiveness of the method in the coordinated tuning of multiple power

system stabilizers was illustrated on a multimachine test power system. The

proposed graphical optimization could be used to directly optimize the system

based on the time responses and to obtain controller settings to successfully

damp the oscillations in power output. The time response of the output could

also be directly influenced through the inclusion of suitable additional

constraints. The major drawbacks of this methodology were the time required

for the optimization and the sensitivity of the optimization process to the

initial starting point.

Hui Ni et al (2002) proposed a supervisory level PSS (SPSS) using

wide area measurement. The robustness of the proposed controller was

capable of compensating for the nonlinear dynamic operation of power

systems and uncertain disturbances. The coordination of the robust SPSSs and

local PSSs was implemented based on the principles of multi agent system

theory. This theory was an active branch of applications in distributed

artificial intelligence (DAI). The performance of the robust controller as a

power system stability agent was studied using a 29-machine 179-bus power

system. Using wide area measurements, the robust controller was a

supervisory level controller that could track system interarea dynamics online.

An LMI-based method was applied to design controllers. Based on the

concept of multi agent systems, the robust controllers were embedded into a

system-intelligent agent, which was coordinated with local agents to increase

system damping. Based on limited testing, the simulation results showed that

the proposed robust controller could effectively damp system oscillations

under wide range of operating conditions.

Elices et al (2004) discussed a physical interpretation of two state

feedback controllers for damping power system electromechanical

38

oscillations. They had been developed by Electricité de France (EDF). The

first one was called the desensitized four loop regulator (DFLR) and it was

designed to damp local electromechanical oscillations. It was a robust

controller which offerd good performance despite the variations of the

generator operating conditions. The second controller was called the extended

desensitized four loop regulator (EDFLR) and it was designed to address both

local and interarea oscillations. The physical interpretation was accomplished

by converting the state feedback scheme to the standard structure formed with

an AVR plus a PSS. Two widely used PSS design methods based on

eigenvalue sensitivities and frequency response were reviewed to obtain the

interpretation. The DFLR could be interpreted as a controller which provided

the suitable phase compensation according to these two PSS design methods

over a wider frequency range. The EDFLR could be interpreted as a controller

which maximized its robustness under uncertainties at both PSS output and

the input of the plant. The EDFLR achieved a better compromise between the

damping ratio of the local and interarea modes, and it was robust not only

under uncertainties at the output of the PSS but also under uncertainties at the

input of the plant.

Brown et al (1970) discussed the relative effect on transient

stability limits of several options available to the system planner, including

transmission strength, excitation system response, and early valving. Early

valving offered attractive stability improvement in those cases where such

improvement was needed. A 10-cycle three-phase fault stuck-breaker criterion

was a very difficult criterion to meet with normal system designs and it was

impossible to live with for the lighter machines. Other measures, such as

series braking resistors, would had to be adopted. For three-phase faults

cleared in primary relaying times, such as 3 cycles, early valving was quite

effective even for low inertia machines, but was usually not needed unless the

post-fault reactance exceeded 60 percent which indicated a weak system. The

39

excitation systems could not greatly improve performance from a first swing

stability point of view; they could play a very important role in providing

damping and assuring stability through subsequent swings.

De Mello and Concordia (1969) discussed the phenomena of

stability of synchronous machines under small perturbations by examining the

case of a SMIB through external reactance. The analysis developed insights

into effects of thyristor-type excitation systems and established understanding

of the stabilizing requirements for such systems. These stabilizing

requirements included the voltage regulator gain parameters as well as the

transfer function characteristics for a machine speed derived signal along with

the voltage regulator reference for providing damping of machine oscillations.

The study had explored a variety of machine loadings, machine inertias, and

system external impedances with a determination of the oscillation and

damping characteristics of voltage or speed following a small disturbance in

mechanical torque. An attempt had been made to develop some unifying

concepts that explain the stability phenomena of concern, and to predict

desirable phase and magnitude characteristics of stabilizing functions.

A wide variety of controllers are discussed to damp inter-area

oscillations. The PSS with AVR is considered for the present study.

2.5 PSS WITH AVR AND EXCITER

The basic function of a PSS is to add damping to the generator rotor

oscillations by controlling its excitation using auxiliary stabilizing signal(s)

(Demello and Charles Concordia 1969). To provide damping, the stabilizer

must produce a component of electrical torque in phase with the rotor speed

deviations (Abdel-Magid and Swift 1976, Hsu 1986).

40

The theoretical basis for a PSS may be illustrated with the aid of the

block diagram shown in Figure 2.2 for a SMIB system.

3

3

1 sTK

1K

s0

4K

eTfd

2KDKHs2

1

mT

fdE

r)(sGex

6K

5KRsT1

1

refV

tE

1v

)(sGPSSr

sv

Figure 2.2 Block diagram representation with AVR and PSS

Since the purpose of a PSS is to introduce a damping torque

component. A logical signal to use for controlling generator excitation is the

speed deviation r .

If the exciter transfer function )(sGex and the generator transfer

function between fdE and, eT were pure gains, a direct feedback of r ,

would result in a damping torque component. However, in practice both the

generator and the exciter (depending on its type) exhibit frequency dependent

gain and phase characteristics. Therefore, the PSS transfer function, )(sGPSS ,

should have appropriate phase compensation circuits to compensate for the

phase lag between the exciter input and the electrical torque (Khaldi et al

1993). In the ideal case, with the phase characteristic of )(sGPSS being an exact

inverse of the exciter and generator phase characteristics to be compensated,

the PSS would result in a pure damping torque at all oscillating frequencies.

41

2.6 STRUCTURE OF PSS

Boukarim et al (2000) designed PSS using classical phase

compensation approach, µ-synthesis and LMI techniques. By comparing the

above techniques the centralized µ-synthesis design produced more transients

than centralized LMI controller. But classical phase compensation design had

additional initial transients due to lightly damped control mode and it was

giving better performance than µ-synthesis and LMI techniques. Herbert

Werner et al (2003) also discussed about LMI techniques. But due to the

foresaid disadvantage LMI technique can not be implemented for effective

design of PSS.

Sanchez-Gasca and Chow (1996), Kothari et al (1996), Simoes

Costa et al (1997), Soliman et al (2000) and Milanovic (2002) used phase

compensation block along with washout filter and gain blocks for PSS design.

The classical phase compensation approach is used in this thesis.

The block diagram of classical lead-lag power system stabilizer is

shown in Figure 2.3. It consists of a gain block, washout circuit, phase

compensation block and a limiter.

W

W

sTsT1STABK

r sv

2

1

11

sTsT

4

3

11

sTsT

Figure 2.3 Lead-Lag power system stabilizer

The major objective of providing PSS is to increase the power

transfer in the network, which would otherwise limited by oscillatory

instability. The PSS must also function properly, when the system is subjected

to large disturbances (Parniani and Lesani 1994).

42

2.6.1 Washout Block

The washout circuit is provided to eliminate steady-state bias in the

output of PSS which will modify the generator terminal voltage. The PSS is

expected to respond only to transient variations in the input signal (say rotor

speed) and not to the dc offsets in the signal. This is achieved by subtracting

from it the low frequency components of the signal obtained by passing the

signal through a low pass filter.

The washout circuit acts essentially as a high pass filter and it must

pass all frequencies that are of interest. If only the local modes are of interest,

the time constant WT can be chosen in the range of 1 to 2 seconds. However,

if inter area modes are also to be damped, then WT must be chosen in the

range of 10 to 20 seconds. The higher value of WT also improved the overall

terminal voltage response during system islanding conditions.

2.6.2 Phase Compensation Block

It provides the appropriate phase-lead characteristic to compensate

for the phase lag between the exciter input and the generator electrical (air-

gap) torque. The Figure 2.4 shows a single first-order block. In practice two

or more first-order blocks may be used to achieve the desired phase

compensation. In some cases, second-order blocks with complex roots have

been used.

Normally, the frequency range of interest is 0.1 to 3.0 Hz, and the

phase-lead network should provide compensation over this entire frequency

range. The phase characteristic to be compensated changes with system

conditions; therefore, a compromise is made and a characteristic acceptable

for different system conditions is selected. Generally some under

compensation is desirable so that the PSS, in addition to significantly

43

increasing the damping torque, results in a slight increase of the

synchronizing torque.

W

W

sTsT1

AK

2

1

11

sTsT

fdE

STABK

RsT11

refV

tE

2vr sv

1v

Figure 2.4 Thyristor excitation system with AVR and PSS

2.6.3 Gain Block

The stabilizer gain STABK determines the amount of damping

introduced by the PSS. Ideally, the gain should be set at a value

corresponding to maximum damping; however, it is often limited by other

considerations. The stabilizer gain STABK is chosen by examining the effect for

a wide range of values. Ideally, the stabilizer gain should be set at a value

corresponding to maximum damping. Gain is set to a value which results in

satisfactory damping of the critical system mode(s) without compromising the

stability of other modes, or transient stability, and which does not cause

excessive amplification of stabilizer input signal noise.

2.6.4 Stabilizer Output Limiter

In order to restrict the level of generator terminal voltage

fluctuation during transient conditions, limits are imposed on the PSS output.

44

The effect of the two limits is to allow maximum forcing capability while

maintaining the terminal voltage within the desired limits (Choi et al 2000).

The input for PSS is change in rotor speed and the output is control

voltage for the exciter.

2.7 STATE SPACE MODEL WITH PSS

Chow et al (2004) designed PSSs using root locus design,

frequency-response design and state space design. In root locus method a PI

controller was applied as a voltage regulator. As the proportional gain

increased, the closed loop step response became faster and steady state error

became smaller, but the oscillation due to the swing mode became less

damped. In frequency-response method, a phase lag-lead controller was used

to plot the compensated system frequency-response and to find the gain and

phase margins. In state space design full-state feedback laws and observers

derived from pole placement were used to design the voltage regulator and

PSS. Choi et al (2000), Hui Ni et al (2002), Guptha et al (2003, 2005) and

Elices et al (2004) used state space design for SMIB and multimachine

systems.

By considering the above drawbacks, the state space method is used

in this thesis.

2.7.1 Single Machine Connected to Infinite Bus (SMIB) System

A general system configuration for the synchronous machine

connected to the large system is shown in Figure 2.5. This general system is

used for the study of small signal stability study (Kundur 1994).

45

Figure 2.5 General configuration of a single machine power system

The general system configuration can be reduced to the Thevenin’s

equivalent circuit shown in Figure 2.6.

Figure 2.6 Equivalent circuit of a single machine power system

For any given system condition, the magnitude of the infinite bus

voltage EB remain constant when machine is perturbed.

The state space model of the system with PSS can be obtained as

follows by using field circuit dynamics and effects of AVR (Kundur 1994).

From block 4 of Figure 2.4, using perturbed values,

rSTABW

W KpT

pTv

12 (2.1a)

Hence,

46

221 v

TpKvp

WrSTAB

The expression for 2vp can be written in terms of the state

variables as follows,

213121121

21 v

TT

HaaaKvp

WmfdrSTAB

(2.1b)

mSTAB

fdr TH

Kvaaaavp 22555352512 (2.1c)

where

W

STAB

STABSTAB

TaaKa

aKaaKa1

551353

12521151

(2.2)

Since 2vp is not a function of 1v and 3v , 05654 aa . From

block 5 of Figure 2.4,

2

12 1

1pTpT

vvs (2.2a)

Hence

ss vT

vT

vpTTvp

22

22

2

1 11 (2.2b)

Substituting for 2vp , given by equation (2.1b), then

mSTAB

sfdrs TH

KTTvavavaaaavp

22

166265164636261

(2.3)

47

where

266

255

2

16553

2

163

522

16251

2

161

1

1

Ta

Ta

TTaa

TTa

aTTaa

TTa

(2.4)

From block 2 of Figure 2.4,

1vvKE sAfd

The field circuit equation, with PSS included becomes,

sfdfd vavaaap 361343332 (2.5)

where

Aadu

fd KL

Ra 0

36

(2.6)

The complete state space model including PSS can be written as,

s

fd

r

s

fd

r

vvv

aaaaaaaaa

aaaaaaa

aaaa

vvv

2

1

6665636261

55535251

444342

36343332

21

131211

2

1

000000

0000000000

(2.7)

48

where,

R

R

R

Aadu

fdA

adsfd

ads

fd

fd

adsfd

fd

D

Ta

TK

a

TK

a

a

KL

RKba

LmLL

LR

a

LmLR

a

faH

Ka

HKa

HKa

1

0

1

22

2

2

44

643

542

41

03234

20

33

10

32

0021

213

112

11

(2.8)

The K-constants can be written as follows (Kundur 1994).

2220

0222

0

06

1110

0111

0

05

32

324

33

323

00020022

0010011

1 mL

LmLnREe

nLnLmREeK

mLmLnREe

nLnLmREe

K

baK

abK

iLL

iLmiLnK

iLmiLnK

fdadsla

t

qaqsla

t

d

adslat

qaqsla

t

d

qfd

adsqadsaqdaqsad

qadsaqdaqsad

(2.9)

49

The above state space model is used in this thesis. The generator and transmission line data are given in Appendix 1.

2.7.2 Multimachine Power System

Analysis of practical power system involves the simultaneous

solution of equations representing the following:

Synchronous machines, the associated excitation system and

prime movers

Interconnecting transmission network

Static and dynamic (motor) load

Other devices such as HVDC converters, static VAR

compensators

The dynamics of the machine rotor circuits, excitation systems,

prime mover and other devices are represented by differential equations. The

result is that the complete system model consists of large number of ordinary

differential and algebraic equations (Hsu and Chen 1987, Tse G.T. Tso 1993).

Figure 2.7 Structure of the complete power system model

50

The general structure of complete model is shown in Figure 2.7.

The formulation of the state equations for small-signal analysis involves the

development of linearized equations about an operating point and elimination

of all variables other than the state variables (Lim and Elangovan 1985).

The multimachine system consists of extensive transmission

networks, loads, a variety of excitation systems and prime mover models,

HVDC links, and static var compensators (Rogers 2000, Simoes Costa et al

1997). Therefore the state equations are formulated by treating wide range of

devices.

The linearized model of each dynamic device is expressed in the

following form,

vBxAx iiii (2.10)

vYxCi iiii (2.11)

where

ix are the perturbed values of the individual device state variables

ii is the current injection into the network from the device

v is the vector of network bus voltage

In equations (2.10) and (2.11), iB and iY have non-zero elements

corresponding only to the terminal voltage of the device and any remote bus

voltages used to control the device. The current vector ii has two elements

corresponding to the real and imaginary components. Similarly, the voltage

vector v has two elements per bus associated with the device. Such state

equations for all the dynamic devices in the system may be combined into the

form,

51

vBxAx DD (2.12)

vYxCi DiD (2.13)

where x is the state vector of the complete system, and DA and DC are block

diagonal matrices composed of iA and iC associated with the individual

devices.

The interconnecting transmission network is represented

transmission network is represented by the node equation,

vYi N (2.14)

The elements of NY include the effects of nonlinear static loads.

Equating equation (2.13) associated with the devices and Equation (2.14)

associated with the network, then

vYvYxC NDD (2.15)

Hence,

xCYYv DDN1)( (2.16)

Substituting the above expression for v in equation (2.12) yields

the overall system state equation,

xCYYBxAx DDNDD1)( (2.17)

Axx

where the state matrix A of the complete system is given by,

DDNDD CYYBAA 1)( (2.18)

52

Figure 2.8 Single line diagram of 4-Machine, 10-Bus System

Figure 2.9 Single line diagram of 10-Machine, 39-Bus System

53

The single line diagram for 4-Machine, 10-Bus System and

10-Machine, 39-Bus System are shown in Figures 2.8 and 2.9 respectively.

The system data are given Appendix 2 and 3 respectively. Figure 10

represents the overall excitation system including PSS.

mTeT

mSqE

qd ii ,

fdEsV

tV

refV

Figure 2.10 Block diagram including excitation system and PSS

2.8 SELECTION OF PSS PARAMETERS

The overall excitation control system is designed so as to:

(i) Maximize the damping of the local plant mode as well as

inter-area mode oscillations without compromising the

stability of other modes;

(ii) Enhance system transient stability;

(iii) Not adversely affect system performance during major system

upsets which cause large frequency excursions; and

(iv) Minimize the consequences of excitation system malfunction

due to component failures.

54

The block diagram of the PSS used to achieve the desired

performance objectives is shown in Figure 2.2. The final stage in stabilizer

design involves the evaluation of its effect on the overall system performance.

First, the effect of the stabilizer on various modes of system oscillations is

determined over a wide range of system conditions by using a small-signal

stability program (Fleming et al 1981). This includes analysis of the effects of

the PSS on local plant modes, inter area modes, and control modes

(Arredondo 1997). In particular, it is important to ensure that there are no

adverse interactions with the controls of other nearby generating units and

devices.

The excitation control systems, designed and describe above,

provide effective decentralized controllers for the damping of

electromechanical oscillations in power systems (Fleming et al 1990).

Generally, the resulting design is much more robust that can be achieved

through use of other methods. The overall approach is used on

acknowledgement of the physical aspects of the power system stabilization

problem. The method used for establishing the characteristics of the PSS is

simple and required only the dynamic characteristics of the concerned

machines to be modeled in detail. Detailed analysis of the performance of the

power system is used to establish other parameters and to ensure adequacy of

the overall performance of the excitation control. The result is a control that

enhances the overall stability of the system under different operating

conditions (Gibbard 1991).

Since the PSS is tuned to increase the damping torque component for wide range of frequencies, it contributes to the damping of all system modes in which the respective generator has a high participation. This includes any new mode that may emerge as a result of changing system conditions. It is possible to satisfy the requirements for a wide range of system conditions with fixed parameters. Here, the effects of stabilizer on

55

various modes of oscillations are determined for a wide range of system conditions using eigen value programs. Also the effect of stabilizer was determined using discrete domain.

In this study single-machine-infinite-bus power system is considered initially. The supplementary stabilizing signal considered is one proportional to speed. A widely used conventional PSS is considered throughout the study.

The transfer function of PSS can be written as shown below from Figure 2.2,

4

3

2

1

11

11

1 sTsT

sTsT

sTsTKv

W

WSTAB

r

s

(2.19)

The first term is stabilizer gain. The second term is washout term with a time lag WT . The third term is a lead compensation to improve the

phase lag through the system. The numerical values of WT , 2T , 4T and system

data are given in Appendix 1. The remaining parameters namely STABK , 1T and

3T are assumed to be adjustable parameters.

The optimization problem is selection of these PSS parameters easily and accurately. The PSS parameters can be optimized by using any one of the following techniques.

(i) Artificial Neural networks (Chan and Hsu 1983, Chaturvedi et al 1999, Chaturvedi et al 2004a, Chaturvedi et al 2004b and Chaturvedi and Malik 2005, Guan et al 1996, He et al 1997, Liu et al 2003, Ramakrishna and Malik 2004, Ravi Segal et al 2000, Shamsollahi and Malik 1997, Shamsollahi and Malik 1997 1999, Yuan-Tih and Chao-Rong 1991, Zhang et al 1993 and 1995)

56

(ii) Fuzzy logic (El-Metwally et al 1996, Hariri and Malik 1996,

Hassan et al 1991, Hiyama and Sameshima 1991, Hoang and

Tomsovic 1996, Hosseinzadeh and Kalam 1999 a,

Hosseinzadeh and Kalam 1999 b, Lu et al 2000, Ortmeyer

and Hiyama 1995, Wen et al 1998.

(iii) Artificial neuro-fuzzy technique (Abido and Abdel-Magid

et al 1998, Linkens and Nyongesa 1995, Ruhua You et al

2003, You et al 2003)

(iv) Genetic algorithms (Abdel-Magid et al 1999, Abdel-Magid

and Abido 2003, Chiang 2005, Potts et al 1994, Taranto and

Falcao 1998)

(v) Evolutionary programming (Dechanupaprittha et al 2004,

Fogel 1995).

(vi) Ant colony optimization

(vii) H∞ Technique (Antal Soos and Malik et al 2002, Asgharian

1994, Chen et al 1995, Chuanjiang Zhu et al 2003,

Kwakernaak 1993, Yang 1997).

(viii) Differential evolution

(ix) Simulated annealing (Abido 2000).

(x) Tabu search (Abdel-Magid et al 2000, Abido and Abdel-

Magid 2000).

(xi) Particle swarm optimization (Abido 2002, Eberhart and Shi

2001, Esmin et al 2005, Gaing 2003).

In this thesis the PSS parameters are selected by solving the

objective function using Artificial Intelligence (AI) techniques like SA, PSO

57

and TS which were already handled by Abido (2000 and 2002) and

Abdel-Magid et al (2000) for various systems. For a given operating point, the

power system is linearized around the operating point, the eigen values of the

closed-loop system are computed, and the objective function is evaluated. It is

worth mentioning that only the system electromechanical modes are

incorporated in the objective function. The bounds on the parameters used in

the AI techniques are given in Appendixes 1,2 and 3.

2.9 CONCLUSION

In this chapter the importance and structure of lead-lag PSS was

studied. Chow et al (2004) implemented root locus method, frequency

response method and state space method. In root locus method, the steady

state error was small and the oscillation due to swing mode was less damped.

In frequency response method, the frequency response was plotted to find

gain and phase margin to analyze stability of the system. But in state space

design full state feedback laws and observers derived from pole placement

were used to design PSS. The state space design was better than root locus

and frequency response methods and is used in the present study. The

elements of state vector were obtained by using field circuit dynamics and

effects of AVR. The generalized model of multimachine system was obtained

by considering transmission networks, loads, variety of excitation systems,

prime mover models, HVDC links and static var compensators. The selection

of PSS parameters to maximize the damping of the local plant mode as well

as inter-area mode of oscillations was explained. The various optimization

techniques for tuning of PSS parameters were summarized.