Comparison and Study of Minimizing Rotor Angle

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Procedia Engineering

Procedia Engineering 00 (2011) 000–000

www.elsevier.com/locate/procedia

International Conference on Modeling Optimisation and Computing

Comparison and Study of minimizing rotor angle instability Using GSA and FF Algorithm

Karthikeyan. K a*,Arun Annamalai. Ab ,Chandrabose. B. M.c, Azeezur Rahman. Ad

a Assistant Professor,Tagore Engineering College, Chennai,India,b UG student, Chennai,India cUG student, Chennai,India ,dAssistantProfessor, SMIT, Chennai, India

Abstract

This paper presents a study and comparison of Gravitational Search (GS) and Firefly (FF) algorithms for the enhancement of rotor angle stability for a wide range of operating conditions. A systematic approach for optimizing the parameters of Power System Stabilizer (PSS) using Integral Square Error (ISE) technique has been implemented in this paper. A single machine infinite bus power system with system parametric uncertainties is considered as a case study and the proposed methods are evaluated against one another at this test system. The simulation results clearly indicate the effectiveness and validity of the proposed methods. © 2011 Published by Elsevier Ltd.

Keywords: GS algorithm; Firefly algorithm; Low Frequency Oscillations; PSS Design; Rotor angle Stability; SMIB system.

1. Introduction

Power system stability is defined as the ability of an electric power system, for a given initial operating condition, to regain state of operating equilibrium after being subjected to a physical disturbance, with most system variables bounded so that practically the entire system remains intact [1]. Modern bio inspired algorithms include a wide variety of population based algorithms which can be applied to various power system optimization problems. The Phenomenon of stability of modern interconnected power systems has received a great deal of attention in recent years. One problem that faces power systems nowadays is the low frequency oscillations arising from interconnected power Systems [2]. Sometimes, these oscillations sustain for minutes and grow to cause system separation, if adequate damping is not provided. A cost efficient and satisfactory solution to the problem of low

*K.Karthikeyan .Tel:+044-42020951; 9445146423 E-mail address: [email protected]

Karthikeyan. Ka* ,Arun Annamalai. Ab ,Chandrabose. B. M.c, Azzezur Rahman. Ad

frequency oscillations is to provide damping by implementing Power System Stabilizers (PSS). In recent years, several approaches based on Modern control theory have been applied to PSS design problem. These include optimal control, adaptive control, variable structure control and intelligent control [5-6]. Unfortunately, the conventional techniques are time consuming, as they are iterative and require heavy computation burden and slow convergence.

Recently Gravitational Search (GSA) is developed to optimize the economic dispatch problems and the

way to solve those problems was explained clearly in [7] and [9]. This paper presents a method for stability enhancement of a Single Machine Infinite Bus (SMIB) system using PSS whose parameters are tuned through FF and GSA. The performance of the stabilizer has been tested for various disturbances. The simulation has been implemented in MATLAB/SIMULINK.

This paper is organized as follows: Section II presents the modeling of SMIB system. The problem

formulation is described in Section III. A short overview of GSA and FF algorithm are presented in Section IV and V respectively. Simulation results obtained using GSA and FF algorithm is provided and discussed in Section VI. Conclusions are given in Section VII. Nomenclature

Pe Electrical output power Et Generator terminal voltage Pm Generator input power ω Angular speed ω0 Base angular speed δ Rotor angular position Td0 ’ d axis open circuit time constant wr Governor reference angular speed Tq0 ’ q axis open circuit time constant D Damping coefficient Efd d axis field voltage Eq’ q axis transient voltage M Inertia constant of generator KA Exciter gain Ra Armature resistance TA Exciter time constant re Equivalent resistance of transmission lines xe Equivalent reactance of voltage transmission lines xd Synchronous reactance xd ’ Transient reactance xq q axis reactance of generator Eb Infinitive bus voltage Vref Reference value of the terminal voltage Tw Washout block time constant ∆ Change from nominal values s Laplacian operator


2. Modeling of Power System

A typical power system comprises a number of generating units connected to a set of loads through transmission lines. A mathematical model of the power system is generally built up from models for each of the generating units, together with a model of the network and loads.

2.1. Single Machine Infinite Bus (SMIB) system

A SMIB system is considered for investigations by using its parameters and its diagram are shown in Fig. 1. Infinite Bus is a system with constant voltage and constant frequency, which is the rest of the power system

Fig. 1.Single Line diagram of SMIB system.

Fig.2 Philips-Heffron linearized model

1

KAsTA

3

1 3 0

K'sK Td

2K

0s

4K

6K

5K

1K

mT

'qE

sv

1

Ms D

PSS

eT

Efd

G

Et Ebre xe

1v


The synchronous machine described in Fig.1 is a fourth order model. Fig.2 shows the well-known Philips-Heffron block diagram of linearized model of the system. The interaction among these variables is expressed in terms of the six constants K1-K6. These constants with the exception of K3, which is only a function of the ratio of impedance, are function of the operating real and reactive loading as well as the excitation levels in the machine [3]. The linearized equations describing the system of Fig. 2 are given below:

.

0= (1)

.

'

1 2

1 qK D K E

M (2)

0

'.'

4'

3

1

d

q

q fd

EE K E

T K

(3)

.

1 1

1t

R

v E vT

(4)

Eqns. (1-4) gives the state space form of the above model.

2.2. Power System Stabilizer (PSS)

The structure of PSS is shown in Fig. 3. It consists of a gain block with gain Kstab, a signal washout block and single first-order phase lead-lag compensation blocks. The input signal of the PSS is the speed deviation (Δω) and the output is the stabilizing signal ΔVS which is added to the reference voltage of the excitation system. The signal washout block serves as a high -pass filter, with the time constant TW, high enough to allow signals associated with oscillations in input signal to pass unchanged. From the view point of the washout function, the value of TW is not critical and may be in the range of 1 to 20 seconds [3].

Fig.3. Block Diagram of PSS

1

w

w

sT

sT 1

2

1

1

sT

sT

stabK

Gain Washout

Phase Compensation

sv


3. Objective Function

The primary goal is to minimize this objective function in order to improve the system stability. The design problem can be formulated as following optimization problem. In this paper, Integral Square Error (ISE) and Integral Time Absolute Error (ITAE) of the rotor angle deviation (∆δ) is taken as the objective function [10]. In this paper the parameters of PSS are tuned by using ISE method, for the comparison ITAE value is calculated by using the optimized parameters obtained by ISE for various conditions.

min max

min max1 1 1

min max2 2 2

stab stab stab

Minimize

J ISE

Subject to

K K K

T T T

T T T

(9)

The expression of ISE and ITAE are described as follows;

2

0

2

0

( )

( )

t tsim

t

t tsim

t

ISE t dt

ITAE t t dt

(10)

where tsim : Total simulation time period Kstab : Stabilizer gain T1 and T2 : Lead-Lag time constants Kp, Ki and Kd : Proportional, Integral and Derivative constants.

4. Gravitational Search Algorithm

Rashedi et al. proposed one of the newest heuristic algorithms, namely Gravitational Search Algorithm

(GSA) in 2009. GSA is based on the physical law of gravity and the law of motion [7]. The gravitational force between two particles is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. GSA a set of agents called masses has been proposed to find the optimum solution by simulation of Newtonian laws of gravity and motion.

In the GSA, consider a system with m masses in which position of the ith mass is defined as follows:

1 1 2 d ni i i iX x ,...,x ,...,x , i , ,...,m (11)

where dix is position of the ith mass in the dth dimension and n is dimension of the search space. At the

specific time ‘t’ a gravitational force from mass ‘j’ acts on mass ‘i’, and is defined as follows:


pi ajd d dij j i

ij

M t xM tF t G t x t x t

R t

(12)

where Mi is the mass of the object i, Mj is the mass of the object j, G(t) is the gravitational constant at time t, Rij (t) is the Euclidian distance between the two objects i and j, and ε is a small constant.

The total force acting on agent i in the dimension d is calculated as follows:

d di j ij

mF ( t ) rand F t

j i j i

(13)

According to the law of motion, the acceleration of the agent i, at time t, in the dth dimension, dia t is

given as follows:

dd ii

ii

F ( t )a t

M t (14)

Furthermore, the next velocity of an agent is a function of its current velocity added to its current acceleration. Therefore, the next position and the next velocity of an agent can be calculated as follows:

1

1 1

d d di i i i

d d di i i

v t rand xv t a t

x t x t v t

(15)

where randi is a uniform random variable in the interval [0, 1]. The gravitational constant, G, is initialized at the beginning and will be decreased with time to control

the search accuracy. In other words, G is a function of the initial value (G0) and time (t):

o

t

To

G( t ) G( G ,t )

G( t ) G e

(16)

The masses of the agents are calculated using fitness evaluation. A heavier mass means a more efficient agent. This means that better agents have higher attractions and moves more slowly. Supposing the equality of the gravitational and inertia mass, the values of masses is calculated using the map of fitness. The gravitational and inertial masses are updating by the following equations:

ii

fit t worst tm t

best t worst t

(17)

1

i

j

m tiM tm

m tj

(18)

where fiti(t) represents the fitness value of the agent i at time t, and the best(t) and worst(t) in the population respectively indicate the strongest and the weakest agent according to their fitness route. For a minimization problem:

1

1

jj ,...,m

jj ,...,m

best t min fit t

worst t max fit t

(19)


4.1. GSA Algorithm for Tuning the PSS parameters

GSA algorithm has been employed for tuning the PSS parameters [8].

1. Search space identification. 2. Randomized initialization. 3. Fitness evaluation of agents. 4. Update G(t), best(t), worst(t) and Mi(t) for 5. i = 1,2,..,N. 6. Calculation of the total force in different directions. 7. Calculation of acceleration and velocity. 8. Updating agents’ position. 9. Repeat steps 3 to 7 until the stop criterion is reached. 10. End.

The optimal values of the PSS parameters Kstab ,T1 and T2 using GSA are found in MATLAB software

and the objective function is GSA parameters selected for tuning the PSS parameters are given in Table.1.

Table 1. Parameters used for PSO algorithm

GSA parameters Value

Number of agents 50

Maximum number of iterations 1000

ElitistCheck 1

power of 'R' 1

The ranges of the optimized parameters of the PSS are [0.1-50] for Kstab and [0.1-2] for T1, and T2 [2].


5. Firefly Algorithm

By idealize some of the flashing characteristics of fireflies we can develop firefly-inspired algorithm.

The three idealized rules are All fireflies are unisex so that one firefly will be attracted to other fireflies regardless of their

sex. Attraction is proportional to their brightness, thus for any two flashing fireflies, the less bright

one will move towards the brighter one. The attractiveness is proportional to the brightness and they both decrease as their distance increases. If there is no brighter one than a particular firefly, it will move randomly.

The brightness of a firefly is determined or affected by the landscape of the objective function.

5.1. Light Intensity and Attractiveness:

In the firefly algorithm there are two important issues: The variation of light intensity and formulation of the attractiveness. The brightness is associated with the encoded objective function it will determine the attractiveness of firefly. In maximum optimization problem the brightness I at a location x is given as I x f x . The attractiveness must be judged by other fireflies as it vary with the distance

ijr between the firefly i and firefly j as the distance from source increases the light is absorbed by the media. So the light intensity decreases so we should allow the attractiveness to vary with the degree of absorption.

According to the inverse square law, the light intensity I r is given by

(20)

where I s is the intensity at source, For a given medium with a fixed light absorption coefficient , the light

intensity I varies with the distance r, that is (21)

0

rI I e

where

0I is original light intensity. In Gaussian form (22)

2

0

rI r I e

As a firefly’s attractiveness is proportional to the light intensity, the attractiveness of a firefly is

(23)

2

sII r

r

2

0

re


Hence, 21 r

The above two equations gives the characteristic distance as 1

. In an actual implementation, the

attractiveness function r can be denoted as

(24)

For a fixed , the characteristic length becomes,

(25)

The above equation can be rewritten as ,

1m

(26)

The distance between the fireflies i and j at ix and jx , respectively, is the Cartesian distance

(27)

where i ,kx is the kth component of the spatial coordinate ix of the ith firefly. In 2-D case, we have

(28)

The movement of a firefly i is attracted to another more attractive firefly j is determined by

2

0ij

r

i i j i ix x e x x

,

(29) where the second term is due to the attraction and the third term is due to randomization.

0

mrr e

1

1m

2

1

d

ij i j i ,k j ,kk

r x x x x

2 2

ij i j i jr x x y y


5.2. Firefly Algorithm for Tuning the PSS parameters

The inputs to excitation system are the terminal voltage VT, reference voltage Vref and the signal from PSS. Voltage transducer effect is neglected.

Pseudo code of the firefly algorithm is: Objective function f x , 1

T

dx x ,......,x Generate initial population of fireflies 1 2ix i , ,......,n Light intensity iI at ix is determined by if x Define light absorption coefficient While (t<Max Generation) For i=1: n all n fireflies

For j=1: n all n fireflies If i jI I , Move firefly I towards j; end if

Vary attractiveness with distance via exp r Evaluate new solutions and update light intensity

End for j End for i Rank the fireflies and find the current global best g End while Post process results and visualization.

TV

refV

Efd Efd

Optimal PSS parameters Kw, T1and T2

Fig.4. Excitation system with PSS

1

K AsTA

sv

stabK 1

w

w

sT

sT 31

2 4

11

1 1

sTsT

sT sT

Fire-Fly and GSA Algorithm


6. Simulation Results

To check the adequacy of stabilizers, simulation of the SMIB model of the power system is carried out by the procedure given below [4].

Case 1: 10% change in voltage setting Vref of the excitation system. After the simulation, the optimal parameters of the PSS by using GSA and FF are given in the

Table.1.

Table 1. Optimal Values of the PSS parameters.

Table 2. Comparison of ISE and ITAE values for Case 1: GSA-PSS and FF-PSS

Fig. 5. Time response plot of Rotor Angle Deviation – Case: 1 Fig. 6. Time response plot of Rotor Speed Deviation - Case: 1

PSS parameters Kstab T1 T2

Optimal values – FF

Optimal values - GSA

46.6191

43.2851

1.5215

1.0643

0.2811

0.1826

Parameters ISE ITAE

FFA-PSS 93.7841 659.5690

GSA-PSS 94.7556 657.2390


6.1. Robustness of FF and GSA

Case 2: 10% step change in power input ∆Tm of the system and also a three phase to ground fault is

applied at t=1sec and removed t=1.5sec in the transmission line, by disconnecting the faulted line and reclosing the same line. Table 2 and 3 show the main objective function ISE and ITAE index value of the FF-PSS is less compared with that of the GSA-PSS for different case studies of the system.

Fig.7. Time response plot of Rotor Angle Deviation – Case: 2 Fig.8. Time response plot of Rotor Speed Deviation – Case: 2

Table 3. Comparison of ISE and ITAE values for Case 2 : GSA-PSS and FF-PSS

7. Conclusion

This paper has shown an application of a GSA and FF algorithm to determine the optimal parameters of power system stabilizer. The design problem of PSS’s parameters selection is converted into an optimization problem which is solved by using GSA and FF technique. Simulation shows that the oscillations of synchronous machine can be rapidly damped with the proposed PSS over a wide range of conditions. This indicates the efficiency of the proposed GSA and FF algorithm in tuning PSS and

Parameters ISE ITAE

FFA-PSS 0.0226 5.2333

GSA-PSS 0.0580 4.9586


stabilizing the system under low frequency oscillations. By using comparative study, FF-PSS shows the superiority over GSA-PSS for rotor angle stability enhancement study.

References

[1] Kundur, P., Paserba, J., Ajjarapu, V., Andersson, G.,Bose, A., Canizares, C.., Hatziargyriou, N., Hill, D.,Stankovic, A.,Taylor, C. Van Cutsem, T., and Vittal, V.; , “Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions,”IEEE Transaction on Power Systems, vol. 19, no. 2, 2004, pp. 1387–1401.

[2] Liu W, Venayagamoorthy GK, Wunsch DC. “A heuristic dynamic programming based power system stabilizer for a turbo generator in a single machine power system”. IEEE Transactions on Industrial Applications, vol.41, no.5, pp.1377-1385, 2005.

[3] P. Kundur, Power System Stability and Control, McGraw-Hill, Inc. 1993, pp.774-835. [4] P. Lakshmi and M. Abdullah Khan, “Stability enhancement of a multimachine power system using fuzzy logic based power

system stabilizer tuned through genetic algorithm”, Electric Power and Energy Systems, No.22, 2000, pp.137–145. [5] M.A. Abido, Simulated annealing based approach to PSS and FACTS based stabilizer tuning, Electrical Power and Energy

Systems, 22, 2000, pp. 247-258. [6] M.A. Abido, Parameter optimization of multimachine power system stabilizers using genetic local search, Electrical Power and Energy Systems, 23, 2001, pp. 785-794. [7] Esmat Rashedi, Hossein Nezamabadi-pour and Saeid Saryazd, “GSA: A Gravitational Search Algorithm”, Information

Sciences, vol. 179, 2009, pp. 2232–2248. [8] Dan Simon, “Biogeography-Based Optimization”, IEEE Transactions on evolutionary computation, vol. 12, no. 6, 2008, pp

702-713. [9] S. Duman, U. Güvenç, and N. Yörükeren “Gravitational Search Algorithm for Economic Dispatch with Valve-Point Effects”,

International Review of Electrical Engineering, ,2008, vol. 5, no. 6, pp. 2890-2895. [10] Serhat Duman and Ali Öztürk, Robust Design of PID Controller for Power System Stabilization by Using Real Coded Genetic

Algorithm.International Review of Electrical Engineering, vol. 4, no. 5, Part B, 2009, pp. 925-931.

Appendix A.

Machine parameters (p.u)

xd = 0.973 , xq = 0.55 , xd’ = 0.19 , M = 9.26

Td0’= 7.76 s , D=0 , ω0 = 377 rad / s

Exciter parameters KA= 50 , TA= 0.05

Transmission Line parameters (p.u) re= 0 , xe= 0.997

Operating point (p.u) Vt0= 1

Documents

Comparison and Study of Minimizing Rotor Angle