4 SMALL SIGNAL STABILITY ANALYSIS USING … SIGNAL...S.v.Padmavathi EEE Dept., GIT, Gitam University Visakhapatnam-530 045 Email: [email protected] ABSTRACT: Power system Stabilizers

International Journal of Electrical Engineering and Technology (IJEET),

ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 1, Number 1, May - June (2010), © IAEME

47

SMALL SIGNAL STABILITY ANALYSIS USING FUZZY

CONTROLLER AND ARTIFICIAL NEURAL NETWORK

STABILIZER

A.padmaja

EEE Dept., GIT, Gitam University

Visakhapatnam-530 045

Email: [email protected]

V.s.vakula




T.Padmavathi



Email:[email protected]

S.v.Padmavathi




ABSTRACT:

Power system Stabilizers are used in order to damp out the low frequency

oscillations which are due to disturbances. This paper attempts to investigate the

performance of Conventional Power System Stabilizer (CPSS), Fuzzy Logic Power

System Stabilizer (FLPSS) and Artificial Neural Network based Power System Stabilizer

(ANNPSS) under Prefault and Postfault conditions for different loadings. The parameters

of Conventional Power System Stabilizer (CPSS) is designed using Pole-Placement

Technique and the parameters of Fuzzy logic Power System Stabilizer (FLPSS) is tuned

to their optimal values in order to minimize the overshoot in step response of rotor angle

International Journal of Electrical Engineering

and Technology (IJEET), ISSN 0976 – 6545(Print)

ISSN 0976 – 6553(Online), Volume 1

Number 1, May - June (2010), pp. 47-70

© IAEME, http://www.iaeme.com/ijeet.html

IJEET © I A E M E



48

deviation using Particle Swarm Optimization (PSO) technique whereas Artificial Neural

network based Power System Stabilizer (ANNPSS) is trained by Linear Optimal Control

(LOC) theory. Designed power system stabilizers are applied to a single machine infinite

bus system and are tested in four operational conditions; normal load, heavy load, light

load and solid 3-phase fault occurrence in a transmission line. The simulation study

reveals that the performance of Linear Optimal Control (LOC) based Artificial Neural

Network Power System Stabilizer is much improved with Artificial Neural Network

based Power System Stabilizer under different operating conditions.

Key words: Power System Stabilizer, Fuzzy Logic Controller, Particle Swarm

Optimization, Artificial Neural network, Pole placement technique, Linear Optimal

Control.

1. INTRODUCTION

Power Systems experience low frequency oscillations which are in the range of

0.1 to 2.5Hz due to disturbances which may grow and lead to dynamic instability of the

system if the applied damping torque is insufficient and may limit the ability to transmit

power. Power System Stabilizer (PSS) provides required damping by producing an

electrical torque component in phase with rotor speed deviation to enhance the system

stability from low frequency oscillations by controlling the excitation using auxiliary

stabilizing signals. PSS should be provided with appropriate phase lead circuits to

compensate for the phase lag between the exciter input and electrical torque. In this

paper, the pole placement technique is considered to determine the parameters of

Conventional Power System Stabilizer (CPSS) in order to stabilize a single machine

connected infinite bus system. However, conventional PSS which is designed at certain

operating point does not provide satisfactory results for all operating conditions due to

approximation in modeling and variation of the system topology due to error occurrence.

Therefore, in recent years, soft computing methodologies like fuzzy logic and

neural networks have been investigated for designing the PSS. The fuzzy Logic based

Power System Stabilizers (FLPSS) have great potential in increasing the damping of

generator oscillations. The inputs to the FLPSS are ω∆ , ω∆ &and ω∆∫ . The first step in

the design of FLPSS is to design conventional PID (Proportional-Integral-Derivative)



49

power system stabilizer which is based on a linear approximation of a nonlinear power

plant around the operating point. The main disadvantage with the classical PID controller

is that it cannot successfully control a plant with strong non-linearities and various

operating conditions. During a major disturbance such as a fault, the operating point of a

power system drifts; conventional PID controllers do not work well under such

conditions. A nonlinear controller such as proposed Fuzzy controller will be more

effective to cover a wider range of operating conditions. Design of Fuzzy Logic Power

System Stabilizer is not an easy task. It is very important to appropriately tune the

parameters used in FLPSS. In this paper, Particle Swarm optimization (PSO) is used to

tune the parameters of FLPSS. The main advantage of PSO over other Global Search

techniques is its algorithmic simplicity as it uses few parameters and easy to implement.

Stabilizing method based on Linear Optimal Control (LOC), make a suitable damping at

different loading condition, but also this method needs determination of all system states

that seems to be very difficult. In recent years Artificial Neural Network (ANN)

methodology has widely used in power system engineering. In an attempt to cover a wide

range of operating conditions, Artificial Neural Network Power system stabilizer

(ANNPSS) have been proposed which is trained by Linear Optimal Control.

The efficacy of the stabilizer is tested on a single machine infinite bus bar system

under four operational conditions; normal load, heavy load and light load, and in the case

of fault occurrence in a transmission line. This paper is organized as follows. The

problem is formulated in section 2. A brief explanation about the design of different

stabilizers i.e the design of Conventional Power System Stabilizer (CPSS) using Pole-

placement technique, the design of Fuzzy Logic Power system stabilizer using Particle

Swarm Optimization (PSO) technique and design of Artificial Neural Network based

Power System Stabilizer using Linear Optimal Control (LOC-ANNPSS) is presented in

section 3. Finally, to evaluate the effectiveness of proposed LOC-ANNPSS, simulation

results are provided in section 4 for various operating conditions and conclusions are

given in section 5.



50

2. PROBLEM FORMULATION

Figure 1 shows the power system under study [1] is a Synchronous machine

connected to a large power system through a transformer and a double circuit

transmission lines. Figure 2 shows the single line schematic of the system including the

Conventional Power System Stabilizer (CPSS), Fuzzy Logic Power System Stabilizer

(FLPSS) and proposed Linear Optimal Control based Artificial Neural Network Power

System Stabilizer (LOC-ANNPSS), where they are discussed in the next sections. In this

paper, for the analysis and design of control system, the system may be linearized, since

the disturbance considered being small. During low frequency oscillations, the current

induced in a damper winding is negligibly small; hence damper windings are completely

ignored in the system model. The natural oscillating frequency for d-axis and q-axis

armature windings being extremely high, their eigen modes will not effect the low

frequency oscillations. Hence can be ignored. Finally the field winding which is directly

connected to excitation system, and has low eigen mode frequency is taken into account

in system modeling. The performance of CPSS, FLPSS and LOC-ANNPSS is evaluated

in four operational conditions; normal, heavy, light loads, and in the case of solid 3-phase

fault occurrence at bus 2 in second circuit.

Figure 1 Single machine infinite bus system



51

Figure 2 Schematic of the SMIB with different power system stabilizers

Figure 3 Linearized model of SMIB with Conventional PSS

The linearized state equations for single machine connected to infinite bus are given as:

x A x Bu= +&

y Cx Du= + (1)

Where T

q fdx E Eδ ω ′ ′ = ∆ ∆ ∆ ∆

The state matrix can be written as:



52

0

1 2

4

3

5 6

0 0 0

0 0

1 10

10

do do do

E E

E E E

k k

M M

kA

k

k k k k

T T T

ω

τ τ τ

− =

− ′ ′ ′

− −

0

0

0

E

E

B

k

T

=

, [ ]0 1 0 0C = , [1]D = (2)

Constants 1k to 6k represent the system parameters at certain operating condition

[2,3]. Analytical expressions for these parameters as function of loading (P, Q) are

derived in [4]. System data is given in APPENDIX

To cover multi-operating conditions of the machine under study normal, heavy

and light loading regimes are selected (pu): And also the performance of conventional

PSS for prefault and postfault conditions under all above loading conditions is analyzed.

The Constants k1 to k 6 for multioperating conditions are tabulate in Table.1

Table 1 k1 to k 6 for multioperating conditions

3. DESIGN OF POWER SYSTEM STABILIZERS (PSS)

3.1 Design of Conventional power System Stabilizer (CPSS) using Pole-

placement technique

Now, the problem is defined as follows:

Given system (1), the transfer function of the system can be obtained using transfer

matrix

BAsICsG 1)()( −−= (3)

The Linearised incremental model of synchronous machine with an exciter and power

system stabilizer is as shown in Fig 3. The technique for selection of stabilizer parameters

is explained by considering the (small Signal) transfer function (Fig 5) from the voltage

Constants Loading

condition

Fault

Condition 1k 2k 3k 4k 5k 6k

Prefault 1.146 1.1775 0.3361 1.7784 0.1175 0.516 Normal

load Postfault 0.6588 0.6722 0.38617 1.468 0.10711 0.6726

Prefault 1.3 1.135 0.3361 1.7146 0.12027 0.57176 Heavy

Load Postfault 1.116 0.6367 0.38617 1.4145 0.1087 0.73233

Prefault 1.14076 0.88164 0.3361 1.332 0.13756 0.566 Light

Load Postfault 1.027 0.7364 0.38617 1.111 0.1266 0.67113



53

regulator reference of the machine where the stabilizer is to be applied to the speed

deviation of that machine. Such a transfer function can be obtained from the linearised

state space equations of single machine infinite bus system.

Figure 4 Block diagram of system transfer function with out PSS

The state space equations include a reasonable representation of excitation system

dynamics. The poles of G(s) are exactly the eigen values of the linearized single machine

system without the present stabilizer. The effect of adding a stabilizer with transfer

function H(S) to the system can be seen by considering the block diagram shown in

Figure 5.

Figure 5 Block diagram of system transfer function with PSS

The modified transfer function now becomes the closed loop transfer function

( )

( )1 ( ) ( )

c

G sG s

G s H s=

− (4)

The Eigen values of the system including the Stabilizers on the machine are the

poles of this closed loop transfer function, and satisfy the closed loop characteristic

equation:

1 ( ) ( ) 0G s H s− = (5)

If we now specify a pair of desired complex eigen values (complex conjugates of

each other), we can substitute one of these into above equation which, upon separation

into real and imaginary parts, will yield two equations in the three unknown stabilizer

parameterspss

K , 1T and 2T in H(s).

These equations can then be solved to determine the stabilizer parameters. Since

there are two equations with three unknown parameters, the additional degree of freedom

can be used to control to some extent the locations of the eigen values other than the

G(s) ∆ω∆Vref

H(S)

G(s) ∆ω ∆Vref

- +



54

primary desired complex pair. The objective is to design a single stage PSS in the form

of

( ) refV

G sω

=∆

(6)

1

2

1( )

1 1

wpss

w

sT sTH s K

sT sT

+=

+ + (7)

This stabilizes the system by placing the poles at desired location so as to stabilize

the system at various operating conditions. The washout time constant w

T is taken as

10sec and the Time constant 2T is taken as 0.05 sec. using the pole-placement technique

the desired poles are -0.30868+j7.8516 and -0.30868-j7.8516.The variables pss

K and T1

are calculated and tabulated in Table.2.

Table 2 Parameters of Conventional power System Stabilizer

pssK 1T (sec) 2T (sec)

10.75 0.485 0.05

The Linearized model of the Single Machine Infinite Bus system with

Conventional power System Stabilizer is shown in Figure 3.

3.2 Design of Fuzzy Logic power System Stabilizer (FLPSS) using PSO

3.2.1. Fuzzy Logic Controller (FLC)

The change in load causes the variation of the generator dynamic characteristics

so that the different operating conditions are obtained. Power system stabilizer must be

capable of providing appropriate stabilization signals over a broad range of operating

conditions and disturbances. Traditional power system stabilizers rely on linear design

methods. On the other hand, a conventional PSS is designed for a linear model

representing the generator at a certain operating point and it does not cover wide range of

operating conditions effectively. During a major disturbance such as a fault, the operating

point of a power system drifts; conventional controllers do not work well under such

conditions whereas a non-linear controller can perform well under such situations. Hence

a Fuzzy Logic Controller (FLC), which has non-linear structure, is proposed in order

improve the system stability by minimizing the maximum overshoot. A simple Fuzzy



55

logic Controller can be depicted using the block diagram shown in Figure 6. The steps

followed in designing Fuzzy Logic Controller (FLC) are summarized [12] below:

1. Identify the variables (inputs, states and output) of the plant.

2. Partition the universe of discourse or the interval spanned by each variable into a

number of fuzzy subsets, assigning a linguistic label.

3. Determine or assign a membership function for each fuzzy subset.

4. Form the rule base i.e., assigning the fuzzy relationships between the inputs’ or states’

fuzzy subsets on the one hand and the outputs’ fuzzy subsets on the other hand.

5. Choose appropriate scaling factors for the input and output variables in order to

normalize the variables to the [0, 1] or the [-1, 1] interval.

6. Fuzzify the inputs to the controller

7. Use fuzzy approximate reasoning to infer the output contributed from each rule.

8. Aggregate the fuzzy outputs recommended by each rule.

9. Apply defuzzification to form a crisp output.

Figure 6 Structure of Fuzzy Logic Controller

The implementation of fuzzy controller in a PSS structure is shown in Fig.6 and

its illustrations can be explained as the following steps [5]:

Step 1: In the scheme, ω∆ , ω∆ & and ω∆∫ are used as the input signals and s

V as

output signal of the controller. The coefficients pK , i

K and d

K are used to keep the

signals within allowable limit. These coefficients are known as scaling factors which



56

transform the real value scale to required value in decision limit. The output signal (s

V ) is

injected at the summing point of the exciter as the supplementary signal.

Step 2: Identical membership functions have been chosen for each of input and

out put variables in order to achieve normalization on the physical variables.

Normalization allows the controller to associate equitable weight to each of the rules and

to calculate the stabilizing signal correctly.

Step 3: Each of the input fuzzy variables i

x is assigned three linguistic fuzzy

subsets such as Negative (N), Zero (Z), Positive (P) and output fuzzy variables, is

assigned seven linguistic fuzzy subsets such as Large Negative (LN), Medium Negative

(MN), Small Negative (SN), Zero (Z), Small Positive (SP), Medium Positive (MP) and

Large Positive (LP). Symmetrical and normalized triangular membership functions for

input and output fuzzy variables are shown in Fig.4.14 and 4.15

Figure 7 Membership functions for input variables

Figure 8 Membership functions for output variables.

Step 4: The range of each fuzzy variable is normalized between -1 to 1 by

introducing a scaling factor to represent the actual signal.

Step 5: Formation of rule base i.e., the set of decision rules which gives all

possible combinations of inputs and outputs is based on the previous experience in the



57

field. The decision table made up of (3 X 3 X 3 =) 27 rules expressed using the same

linguistic labels as those of the inputs is shown in Table.3.

Table.3 Rule base for three input Fuzzy Controller

1 N N N PL

2 N N Z P

3 N N P P

4 N Z N P

5 N Z Z P

6 N Z P Z

7 N P N P

8 N P Z Z

9 N P P N

10 Z N N P

11 Z N Z P

12 Z N P Z

13 Z Z N P

14 Z Z Z Z

15 Z Z P N

16 Z P N Z

17 Z P Z N

18 Z P P N

19 P N N P

20 P N Z Z

21 P N P N

22 P Z N Z

23 P Z Z N

24 P Z P N

25 P P N N

26 P P Z N

27 P P P NL

Step 6: Centroid method of defuzzification is applied to convert fuzzy output to

crisp out put signal s

V which is added at the summing junction of the exciter as the

supplementary signal.

Figure 9 depicts the implementation of fuzzy controller in a Linearized model of SMIB



58

Figure 9 Linearized model of SMIB with FLPSS

3.2.2. Overview of PSO

Particle swarm optimization (PSO) is an evolutionary computation technique

developed by Dr. Eberhart and Dr.Kennedy in 1995, inspired by social behavior of bird

flocking or fish schooling. PSO is a population based optimization tool. As shown in

Fig.10, the system is initialized with a population of random solutions and searches for

optima by updating generations. All the particles have fitness values, which are evaluated

by the fitness function to be optimized, and have velocities, which direct the flying of the

particles. The particles are “flown” through the problem space by following the current

optimum particles. PSO is initialized with a group of random particles (solutions) and

then searches for optima by updating generations. In every iteration, each particle is

updated by following two “best” values. The first one is the best solution (fitness) it has

achieved so far. (The fitness value is also stored.) This value is called pbest. Another

“best” value that is tracked by the particle swarm optimizer is the best value, obtained so

far by any particle in the population. This best value is a global best and called gbest.

When a particle takes part of the population as its topological neighbors, the nbest value

is a local best and is called lbest. After finding the two best values, the particle updates its

velocity and positions with following equations:

* 1* () * ( ) 2 * () * ( )id id id id id id

V W V c rand P X c rand P X= + − + − ----------(8)



59

i d i d i dX X V= + ------------------------------------------------(9)

Vid is the particle velocity; Xid is the current particle (solution). Pid and Pgd are

pbest and gbest. rand ( ) is a random number between (0, 1). 1c , 2c are learning factors.

Now, the problem is defined as follows:

Given system (1), the objective is to design a single stage PSS in the form of

( )

( )

( )

( )

is U c p d

Ic P D

KV K G s K K s

s

KG s K K s

s

ω= + + ∆

= + +

----------------------------(10)

WhereP

K , D

K and I

K are the proportional, derivative and Integral gains,

p

K , i

K , d

K and U

K are the input and out put gains or scaling factors

Figure 10 Flow Chart for PSO



60

The input gainsp

K , i

K , d

K and output gainU

K are tuned to their optimal values

by solving the min-max optimization problem defined in (11) using Particle Swarm

optimization Technique.

Minimize J = ( )max

maxss ss

tδ δ δ ∆ − ∆ ∆ ∀ selected regimes ------------------(11)

Where J represents worst overshoot over selected regimes, ( )max

tδ∆ and ss

δ∆ represents

respectively the maximum and steady state values of torque angle deviation. Table.4

gives the PSO Optimized parameters for FLPSS

Table 4 PSO Optimized parameters of FLPSS

PK

IK

DK

UK

0.35 -52 0.012 1.025

3.3 Design of Artificial Neural Network Power System Stabilizer

(ANNPSS)

The use of Artificial Neural Networks (ANNs) is the most powerful approach in

Artificial Intelligence. ANNs are information processing structures which emulate the

architecture and operational mode of the biological nervous tissue. Any ANN is a system

made up of several basic entities (named neurons) which are interconnected and operate

in parallel transmitting signals to one another in order to achieve a certain processing

task. One of the most outstanding features of ANNs is their capability to simulate the

learning process. They are supplied with pairs of input and output signals from which

general rules are automatically derived so that the ANN will be capable of generating the

correct output for a signal that was not previously used.

3.3.1 Linear Optimal Control (LOC)

Stabilizing method based on linear optimal control [6], make a suitable damping

at different loading condition, but also this method needs determination of all system

states that seems to be very difficult. Using of ANN to self tuning of stabilizers

parameters is introduced. In the proposed PSS designed in this work, the ANN is trained

by LOC-PSS. This stabilizer uses the linear optimal control theory to design a linear state

feedback control loop.



61

There are two important steps in the successful design of such a controller. One is

to have a precise mathematical model of the controlled system. Another is to select a

suitable weighting matrix in the performance index.

The stabilization problem is to design a stabilizer which provides a

supplementary stabilizing signal to increase the damping torque of the system. The

design linear optimal stabilizer is based on the theory of linear optimal regulator. In order

to formulate the problem of stabilization using linear optimal control theory, a set of state

variables must be first selected. Then the state equation for the system is written in the

vector matrix differential equation form

BuAxx +=& (12)

Where x is the state n vector, u is the control m vector, A and B are constant

matrices of dimensions n x n and n x m respectively. Suppose that the performance index

is to minimize the integral

( )dtRuuQxxJTT

∫∞

+=0

2

1 (13)

Where Q is the positive semi-definite matrix and R is a positive definite matrix,

then the optimal control for the system with performance index is given by

Kxu −= (14)

Where PBRKT1−= (15)

and P is the solution of the algebraic matrix Riccati equation

01 =+−+ −QPBPBRPAPA

TT (16)

Riccati equation is the key to the design of the linear optimal control systems.

Once the matrices Q and R are known, the matrix P can be obtained by solving above

equation and then the optimal control signal U is calculated. The traditional way to select

Q, R matrices is to use trial and error method. Many simulation studies have to be done in

time domain with different weighting matrices to choose the ones that provide desired

performance. Because of the complexity, the matrices Q and R are commonly chosen as

diagonal matrices.

The Single Machine Connected to Infinite Bus with Linear Optimal Controlled

ANNPSS is shown in Figure 11.



62

Figure 11 Linearized model of SMIB with LOC-ANNPSS

3.3.2 Design of Artificial Neural Network Power System Stabilizer

(ANNPSS) using Linear Optimal Control (LOC)

In the proposed PSS, the ANN is trained by using Linear Optimal Control theory

to make a suitable damping at different loading condition. Taking active power (P),

reactive power (Q),Terminal Voltage(Vt) and the line reactance (Xe) as input layers and

the optimal feedback gain matrix K = [ K1 K2 K3 K4 ] as output layer the training

data is obtained for 180 operating conditions. Using this data a neural network is trained

with different algorithms and found that the cascaded forward feed back propagation

learning algorithm is capable of giving the best results with least error.

The ANN is trained using Neural Networks tool box GUI/MATLAB environment

and the trained network is shown in Figure 12.



63

Figure 12 Trained Neural network

The same network has been simulated to obtain state feedback gains for prefault

and postfault conditions under multi-operating conditions i.e., heavy, normal and light

loads. ANNPSS is designed using the feed-back gains from all states and stabilizes the

Single Machine Infinite Bus System as shown in Figure 11.

Table 5 shows the PSS parameters computed off-line and with ANN for different

operating conditions.

Inputs(p.u) State feed-back gains

computed with LOC

State feed-back gains

computed with ANN

Loadi

ng

condit

ion

Fault

Condi

tion P Q V Xe K1 K2 K3 K4 K1 K2 K3 K4

Prefau

lt 0.

7

0.

3 1

0.4

75

1.721

261

-

328.

04

6.22

697

0.028

655

1.754

567

-

327.

14

6.242

084

0.017

435

Light

Postfa

ult 0.

7

0.

3 1

0.6

5

1.693

238

-

346.

36

5.62

658

0.026

433

1.573

387

-

346.

58

5.670

416

0.018

584

Prefau

lt

1 0 1

0.4

75

1.446

592

-

179.

89

5.21

851

0.024

879

1.409

395

-

180.

94

5.613

755

0.014

357

Norm

al

Postfa

ult

1 0 1

0.6

5

1.192

353

-

202.

66

4.75

860

0.023

082

1.099

083

-

203.

00

4.936

068

0.015

271

Prefau

lt 1.

2

0.

2 1

0.4

75

1.651

396

-

308.

40

7.06

922

0.031

656

1.665

965

-

315.

25

7.217

78

0.016

549

Heav

y

Postfa

ult 1.

2

0.

2 1

0.6

5

1.388

625

-

335.

62

6.37

813

0.029

203

1.359

244

-

345.

43

6.791

326

0.018

064

Table 5 PSS parameters computed off-line and with ANN for different operating

conditions.



64

4. RESULTS & DISCUSSIONS

The performance of CPSS, FLPSS and LOC-ANNPSS for Prefault and Postfault

conditions is investigated under three different loading conditions [3] shown in Table 6

at a unit step disturbance of. ± 5 %.

Table 6 Loading Conditions

Loading Conditions P Q

Normal 1.0 0.0

Heavy 1.2 0.2

Light 0.7 0.3

The Constants 1k to 6k constants for multi-operating conditions tabulated in Table 7.

Table 7 1k to 6k constants for multi-operating conditions

Constants Loading

condition

Fault

Condition 1k 2k 3k 4k 5k 6k

Prefault 1.146 1.1775 0.3361 1.7784 0.1175 0.516 Normal load

Postfault 0.6588 0.6722 0.38617 1.468 0.10711 0.6726

Prefault 1.3 1.135 0.3361 1.7146 0.12027 0.57176 Heavy Load

Postfault 1.116 0.6367 0.38617 1.4145 0.1087 0.73233

Prefault 1.14076 0.88164 0.3361 1.332 0.13756 0.566 Light Load

Postfault 1.027 0.7364 0.38617 1.111 0.1266 0.67113

The performance of CPSS, FLPSS and LOC-ANNPSS is evaluated in

MATLAB/SIMULINK for Prefault i.e., before the occurrence of a solid 3-phase fault at

bus 2 and postfault i.e., after the clearance of fault by line outage. The efficacy of PSS

can be studied only after the clearance of fault by opening the circuit breakers at both

ends simultaneously. These simulation studies are carried out under above loading

conditions.

Figure 13 to Figure 18 shows the rotor angle deviation response curves for a

disturbance of +5% in mT∆ under Prefault and Postfault conditions for a solid three-phase

fault at bus 2 when the system is considered at multi-operating conditions.

It can be observed that Without Power System Stabilizer (WOPSS), the system in

unstable and the overshoot is minimized to negligible value and also settling time is

significantly reduced with Artificial Neural Network Power System Stabilizer (ANNPSS)

where as Conventional Power System Stabilizer (CPSS) and Fuzzy Logic Power System

Stabilizer (FLPSS) are stabilizing the system with small oscillations and overshoot.



65

Table 8 and Table 9 shows the comparison between WOPSS, CPSS, FLPSS and

LOC-ANNPSS in view of percentage peak over shoot and settling time respectively for

multi-operating conditions.

Table 8 Comparision between CPSS, FLPSS and LOC-ANNPSS (Percentage Peak

Overshoot)

Peak Overshoot (%) Loading Fault condition

CPSS FLPSS LOC-ANNPSS

Prefault 37.54 20.32 10.80 Light

Postfault 50.43 25.95 18.51

Prefault 31.22 21.18 10.30 Heavy

Postfault 46.08 28.785 9.49

Prefault 17.51 17.51 16.70 Normal

Postfault 34.33 27.66 15.40

Table.9 Comparision between CPSS, FLPSS and LOC-ANNPSS (Settling Time)

Settling Time (in sec.) Loading Fault condition

CPSS FLPSS LOC-ANNPSS

Prefault 8.6071 6.1801 1.0803 Light

Postfault 8.4227 4.8365 1.4974

Prefault 8.6135 7.8021 1.0491 Heavy

Postfault 6.0333 5.8179 1.2797

Prefault 6.7973 6.7973 1.6678 Normal

Postfault 6.06 5.7933 1.7354

From the comparision table, it can be observed that during postfault condition, the

electrical power transfer is more than the prefault condition and therefore the peak over

shoot is high compared to prefault condition.

The aim of the proposed work is to minimize the percentage peak over shoot and

settling time of the step response of generator rotor angle by optimally tuning the

parameters of ANNPSS using LOC. The performance of LOC-ANNPSS is compared

with CPSS and FLPSS. It can be observed from the presented results that the LOC-

ANNPSS minimizes the percentage peak overshoot and settling time under multi-

operating conditions compared with CPSS and FLPSS.

The eigen values of the system without any stabilizer, with CPSS and with Linear

Optimal Controller are tabulated in Table.10.



66

Table 10 Comparison of the Eigen values without stabilizer, with CPSS and with Linear

Optimal Controller

Loading

Conditi

on

Fault

condition

Without stabilizer With CPSS

(pss

K =10.67,

1T =0.485sec,

2T =0.05sec)

With Linear

Optimal

Controller

(LOC)

Normal

(P=1.0,

Q=0.0)

Prefault

(e

X =0.475)

-0.3008+ 7.8847i

-0.3008 - 7.8847i

-1.4997,-18.2672

-1.1853 + 9.0430i

-1.1853 - 9.0430i

-11.1119, -1.4510

-25.4352

-3.3561 +10.0684i

-3.3561 -10.0684i

-18.1229, -7.9728

Post fault

(e

X =0.65)

-0.2664 + 7.1732i

-0.2664 - 7.1732i

-2.1621,-17.6289

-1.1914 + 8.0558i

-1.1914 - 8.0558i

-10.9766, -2.1769

-24.7873

-3.2754+ 9.3815i

-3.2754 - 9.3815i

-17.4974 , -7.8163

Light

(P=0.7,

Q=0.3)

Prefault

(e

X =0.475)

-0.2289 + 7.8522i

-0.2289 - 7.8522i

-1.8612,-18.0496

-0.9343 + 8.6599i

-0.9343 - 8.6599i

-12.0411, -1.8439

-24.6150

-3.7982 +10.6229i

-3.7982 -10.6229i

-17.7740, -9.3256

Post fault

(e

X =0.65)

-0.1923 + 7.4352i

-0.1923 - 7.4352i

-2.3117,-17.6275

-0.8604 + 8.0619i

-0.8604 - 8.0619i

-12.1704,-2.3390

-24.0936

-3.6296 +10.0587i

-3.6296 -10.0587i

-17.3978, -8.8831

Heavy

(P=1.2,

Q=0.2)

Prefault

(e

X =0.475)

-0.2556 + 8.3825i

-0.2556 - 8.3825i

-1.8094,-18.0479

-0.9343 + 8.6599i

-0.9343 - 8.6599i

-12.0411, -1.8439

-24.6150

-4.0992 +11.4597i

-4.0992 -11.4597i

-17.5834,-10.4148

Post fault

(e

X =0.65)

-0.2178 + 7.7581i

-0.2178 - 7.7581i

-2.5131,-17.3749

-0.8604 + 8.0619i

-0.8604 - 8.0619i

-12.1704, -2.3390

-24.0936

-3.9559 +10.7688i

-3.9559 -10.7688i

-16.9487,-10.0649

5. CONCLUSIONS

From the results listed in Table 8, Table 9 and Table.10, it is evident that a

Conventional Power System Stabilizer (CPSS) designed for certain operating point does

not cover all operating conditions effectively. Under this situation, a nonlinear controller

such as Fuzzy Logic Power System Stabilizer (FLPSS) can control the plant to some

extent but with small percentage overshoot in its response whereas Linear optimal

Control (LOC) based Artificial Neural Network Power System Stabilizer (LOC-

ANNPSS) can perform very well with least overshoot and negligible settling time.



67

The simulation studies reveal that the proposed Linear Optimal Control based

Artificial Neural Network Power System Stabilizer (LOC-ANNPSS) is very effective to

fault conditions and minimizes the overshoot and settling time.

From the Fig 13 to 18, it is evident that the dynamic performance of the proposed

Linear Optimal Control based Artificial Neural Network Power System Stabilizer (LOC-

ANNPSS) is also superior to that of CPSS and FLPSS.

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time, t in sec ----->

Roto

r A

ngle

Devia

tion ----->

WOPSS

CPSS

FLPSS

LOCANN-PSS

Figure 13 Rotor Angle Deviation for a disturbance of +5% in

mT∆ under normal loading

for prefault condition

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

Time t in sec ---->

Roto

r A

ngle

Devia

tion ----->

WOPSS

CPSS

FLPSS

LOCANN-PSS


mT∆ under normal loading

for postfault condition



68

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time, t in sec ---->

Roto

r A

ngle

Devia

tion -

--->

WOPSS

CPSS

FLPSS

LOC-ANNPSS


mT∆ under light loading for

prefault condition

0 1 2 3 4 5 6 7 8 9 100

0.02

0.04

0.06

0.08

0.1

0.12

Time, t in sec ---->

Roto

r A

ngle

De

via

tion -

--->

WOPSS

CPSS

FLPSS

LOC-ANNPSS


mT∆ under light loading for

postfault condition



69

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Time,t in sec ---->

Roto

r A

ngle

Devia

tion

---

->

WOPSS

CPSS

FLPSS

LOC-ANNPSS


mT∆ under heavy for

prefault condition

0 1 2 3 4 5 6 7 8 9 100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time,t in sec ---->

Roto

r A

ngle

Devia

tion ---->

WOPSS

CPSS

FLPSS

LOC-ANNPSS


mT∆ under heavy for

postfault condition

APPENDIX

The System under study is a thermal generating station consisting of four

555MVA, 24KV, 60Hz units. The network reactances shown in Figure 1 are in p.u. on

2220MVA, 24KV base(referred to LT side of step-up transformer). Resistances are

assumed to be negligible.

Equivalent generator parameters in p.u:

dX = 1.81,

dX ′= 0.3,

qX = 1.76,

doT ′ = 8sec, H = 3.5MJ/MVA, Vt = 1.0

Exciter: Ek = 25, E

T = 0.05Sec.



70

REFERENCES

[1] P. Kundur, Power System Stability and Control. New York: McGraw-Hill, 1964.

[2] SAUER, P. W.—PAI, M. A, Power System Dynamics and Stability, Prentice Hall,

1997.

[3] Hisham M.Soliman-Ehab H.E.Bayoumi, Mohamed F.Hassan (2008), PSO-based

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[4] H.M.Soliman, M.M.F.Sakr (1988), Wide-Range power system pole placer’ IEE

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[5] R.Hooshmand, M.Ataei (2006), An Auto Tuning of Fuzzy Logic PSS Design Under

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systems 5-1.

[6] J. Ahmadian, M. Jalali, R. Pouaghababa, M. Nouhi (2008), Power System Stabilizers

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[8] Francisco P.Demello (1969), Concepts of Synchronous Machine Stability as Affected

by Excitation Control, IEEE transactions on Power apparatus and systems,

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[9] Wah-Chun Chan, Yuan-Yih Hsu (1983), An optimal variable structure stabilizer for

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Documents

4 SMALL SIGNAL STABILITY ANALYSIS USING … SIGNAL...S.v.Padmavathi EEE Dept., GIT, Gitam University Visakhapatnam-530 045 Email: [email protected] ABSTRACT: Power system Stabilizers