Optimal placement and parameter setting of SVC and TCSC using PSO to mitigate small signal stability problem

Electrical Power and Energy Systems 42 (2012) 334–340

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Optimal placement and parameter setting of SVC and TCSC using PSO to mitigate small signal stability problem D. Mondal a,⇑, A. Chakrabarti b, A. Sengupta b a b

Department of Electronics and Instrumentation Engineering, Haldia Institute of Technology, Purba Medinipur, Haldia 721 657, India Department of Electrical Engineering, Bengal Engineering and Science University, Howrah 711 103, India

a r t i c l e

i n f o

Article history: Received 1 November 2010 Received in revised form 7 May 2011 Accepted 10 April 2012

Keywords: Particle Swarm Optimization (PSO) Small signal stability Static Var Compensator Thyristor Controlled Series Compensator

a b s t r a c t This paper aims to select the optimal location and setting parameters of SVC (Static Var Compensator) and TCSC (Thyristor Controlled Series Compensator) controllers using PSO (Particle Swarm Optimization) to mitigate small signal oscillations in a multimachine power system. Though Power System Stabilizers (PSSs) associated with generators are mandatory requirements for damping of oscillations in the power system, its performance still gets affected by changes in network configurations, load variations, etc. Hence installations of FACTS devices have been suggested in this paper to achieve appreciable damping of system oscillations. However the performance of FACTS devices highly depends upon its parameters and suitable location in the power network. In this paper the PSO based technique is used to investigate this problem in order to improve the small signal stability. An attempt has also been made to compare the performance of the TCSC controller with SVC in mitigating the small signal stability problem. To show the validity of the proposed techniques, simulations are carried out in a multimachine system for two common contingencies, e.g., load increase and transmission line outage. The results of small signal stability analysis have been represented employing eigenvalue as well as time domain response. It has been observed that the TCSC controller is more effective than SVC even during higher loading in mitigating the small signal stability problem. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Low frequency power oscillations are the challenging problem in interconnected power systems. These oscillations may sustain and grow to cause system separation if no adequate damping is available [1]. Conventionally, additional damping in system is introduced by the application of PSS [2,3]. The Power electronic controlled devices, such SVC, HVDC link have also been used for many years [4]. In [5] it has been reported the better effectiveness of a SVC than PSS in damping power system oscillations. An optimal power flow (OPF) and transmission loss minimization model with SVC has been developed in [6] to improve the system stability and security of a practical power network. The development of FACTS [7] has generated much attention of the researchers to not only improve the damping of the oscillation of the electromechanical modes but also to enhance the system power transfer capability. In [8] Unified power flow controller (UPFC), a modern FACTS device has been used to provide adequate damping in power system network with changing system conditions. Thyristor Controlled Series Compensator (TCSC), a series controlled FACTS ⇑ Corresponding author. E-mail addresses: [email protected] (D. Mondal), a_chakraborti55@yahoo. com (A. Chakrabarti), [email protected] (A. Sengupta). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.04.017

device is increasingly applied for the purpose of improvement of damping of the electromechanical oscillations in modern power systems [9,10]. The optimal placement of FACTS controller in power system networks has been reported in literatures based on different aspects. A method to obtain optimal location of TCSC has been suggested in [11] based on real power performance index and reduction of system VAR loss. In [12] optimal allocation of SVC using Genetic Algorithm (GA) has been introduced to achieve the optimal power flow (OPF) with lowest cost generation in power system. But the optimal allocations of SVC and TCSC controllers using PSO to investigate the small signal stability problem have not been discussed in existing literature. In this paper this fact has been taken into consideration as well as a PSO based technique is proposed to place the SVC and TCSC controllers separately in a multimachine system in order to damp the small signal oscillations. It is a well known fact that optimal parameter setting of power system analysis controller is a complex exercise. The conventional techniques reported in the literatures [13,14] are time consuming, require heavy computation burden and they have slow convergence rate too. Many stochastic search methods have been developed for global optimization problems, such as artificial neural network, genetic algorithm and evolutionary programming [15–17].

D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

Recently, Particle Swarm Optimization (PSO) method, developed by Kennedy and Eberhart [18] has appeared as a promising algorithm for handling the optimization problems. PSO is a robust, non-linear and population based stochastic optimization technique which can generate high-quality solutions within shorter calculation time and has more stable convergence characteristics than other stochastic methods. Though PSO has been employed in several research papers [19,20] for the design of optimal FACTS controllers, the applications are mostly limited to the case of single machine infinite bus system. In this paper, the PSO has been used to search the best location and the parameters of both SVC and TCSC controller separately. The application is extended to study the small signal oscillation problem in case of a multimachine power system considering all network bus dynamics. The paper is organized as follows: Section 2 describes the small signal modeling of the multimachine system working with SVC and TCSC controllers. The general theory of PSO and the proposed parameter optimization algorithm has been discussed in Section 3. In Section 4 the SVC and the TCSC controller parameters and their optimal locations are identified using PSO algorithm and subsequently the PSO based SVC and TCSC controllers are installed separately in a standard test system. To verify the proposed method, three cases have been considered. Case 1: original test system without SVC and TCSC controllers following disturbances (load increase and transmission line outage). Case 2: system with TCSC and PSO based optimal location and parameters. Case 3: system with SVC and PSO based optimal location and parameters. In all cases the small signal performance of the system has been investigated.

335

SVC incorporating an auxiliary controller [21] has been shown in Fig. 1b. The voltage input, DVscc of the SVC controller is measured from the SVC bus. The machine speed, Dm (=Dx/xs) is taken as the control input to the auxiliary controller. The firing angle (a) of the thyristors determines how much susceptance is included in the network. The SVC equivalent susceptance, Bsvsc at fundamental frequency is given by [22]

Bsv c ¼ 

X L  XpC ð2ðp  aÞ þ sinð2aÞÞ XC XL

ð1Þ

while its profile as a function of firing angle corresponding to a capacitive reactance, XC = 1.1708 pu and inductive reactance, XL = 0.4925 pu has been shown in Fig. 2. Setting KP is zero, the linearized state equations of the SVC controller can be represented as

DV_ s ¼ 

    1 K sv c 1 K sv c T 1 _ DV s þ Dx þ Dx T2 xs T 2 xs T 2

Da_ ¼ K I DV s þ K I DV sv c  K I DV ref DB_ sv c ¼ 

1 1 Da  DBsv c T sv c T sv c

ð2Þ ð3Þ ð4Þ

where Tsvc is the time delay of the SVC module and a = (p  r) is the firing angle of the thyristor. Ksvc, T1 and T2 are the gain, lead and lag time constant of the auxiliary controller respectively. DBsvc is the linearized equivalent succeptance of the SVC and can be obtained from Eq. (1) as

DBsv c ¼

2ðcosð2aÞ  1Þ Da XL

ð5Þ

2. System modeling 2.2. Modeling of TCSC 2.1. Modeling of SVC The most popular configuration of this type of shunt connected device is a parallel combination of fixed capacitor C with a thyristor controlled reactor (TCR) (Fig. 1a). The block diagram of a basic

(a)

node n

node n

Vn

ISVC

Vn

ISVC Bsvc

The basic TCSC module and the transfer function model of a TCSC controller [23] have been shown in Fig. 3a and b, respectively. This simple model utilizes the concept of a variable series reactance which is adjusted through appropriate variation of the firing angle (a). The controller comprises of a gain block, a signal washout block and a phase compensator block. The input signal is the normalized speed deviation (Dm), and output signal is the stabilizing signal (i.e. deviation in conduction angle, Dr). Neglecting washout stage, the TCSC controller model can be represented by the following state equations;

Da_ ¼ 

C

    1 K tcsc 1 K tcsc T 1 _ Da  Dx  Dx T2 xs T 2 xs T 2

L

(b)

ΔVref

ΔVsvc − ΔV s

+ KP +

+

σ0

Firing angle regulator

KI s

Δσ

K svc (1+ sT1 ) (1+ sT2 )

+

1 1 + sTsvc

ΔBsvc

SVC internal delay

Δν

Auxiliary controller Fig. 1. Model of SVC controller (a) SVC module and (b) block diagram of SVC controller.

Fig. 2. Bsvsc as function of firing angle a.

ð6Þ

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D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

C

(a)

S

ΔPst Rst

t Xst L Max

(b)

Δν

K tcsc Gain

sTW 1 + sTW

1 + sT1 1 + sT2

Washout

Lead-Lag stage

Δσ

1 1 + sTtcsc

+

σ0

Min

ΔXtcsc

TCSC internal delay

Fig. 3. Model of TCSC controller (a) TCSC module and (b) structure of TCSC based controller.

DX_ tcsc ¼ 

1 T tcsc

Da 

1 T tcsc

DX tcsc

ð7Þ

0 ¼ C 2 DX þ D3 DIg þ D4 DV g þ D5 DV l

ð12Þ

0 ¼ D6 DV g þ D7 DV l

ð13Þ

Here Eqs. (10) and (11) represent the linearized differential equations and linearized stator algebraic equations of the machine while Eqs. (12) and (13) correspond to the linearized network equations pertaining to the generator buses and the load buses. Different state variables of the multimachine model are shown in Appendix A.1. The inclusion of a SVC controller in this multimachine model results in addition of state variables, Dxsv c ¼ ½DV s Da DBsv c T corresponding to the SVC controller in Eqs. (10)–(12) and the SVC reactive power output equation in the network Eq. (13). The SVC linearized reactive power injection at the bus n can be obtained from the following equation

DQ n ¼

@Q n @Q n @Q n Dhn þ DV n þ Da @hn @V n @a

ð14Þ

The steady-state relationship between the firing angle a and the equivalent TCSC reactance, Xtcsc is described by the following relationship [24]

where

X tcsc ¼ X C þ C 1 ð2ðp  aÞ þ sinð2ðp  aÞÞÞ  C 2 cos2 ðp  aÞ

The multimachine linearized model with TCSC controller can be formulated separately by adding the state variables Dxtcsc ¼ ½Da DX tcsc T corresponding to the TCSC controller in Eqs. (10)– (12) and the TCSC power flow equation in the network Eq. (13). The TCSC linearized real power flow equation between bus s and t can be obtained from the following equation:

 ðp  aÞÞ  tanðp  aÞÞ  tanðx  ðx where X LC ¼

XC XL , X C X L

C1 ¼

X C þX LC

p

and C 2 ¼

ð8Þ

4X 2LC pX L .

The linearized TCSC equivalent reactance can be obtained from (8) is

DX tcsc ¼ 2C 1 ð1 þ cosð2aÞÞ þ C 2 sinð2aÞð- tanð-ðp  aÞÞ   cos2 ðp  aÞ  tan aÞ þ C 2 -2  1 Da 2 cos ð-ðp  aÞÞ

Q n ¼ Bsv c V 2n

DPst ¼ ð9Þ

@Pst @Pst @Pst @Pst @Pst Dh s þ DV s þ Dh t þ DV t þ Da @hs @V s @ht @V t @a

ð15Þ

ð16Þ

where

The TCSC equivalent reactance (Xtcsc) as a function of the firing angle (a) (with XC = 5.75XL X at a base frequency of 50 Hz) has been shown in Fig. 4.

Pst ¼ V 2s g st  V s V t ðg st cos dst þ bst sin dst Þ

2.3. Multimachine small signal stability model with SVC and TCSC

Y st ¼

ð17Þ

and

1 Rst þ jðX st þ X tcsc Þ ¼ g st  jbst ¼ Rst þ jðX st  X tcsc Þ R2st þ ðX st þ X tcsc Þ2

ð18Þ

The small signal stability model of a multimachine system can be described by the following state space equations [25]

with

DX_ ¼ A1 DX þ B1 DIg þ B2 DV g þ E1 DU

ð10Þ

@Ps @Pst @ @ @ ¼ ¼ V 2s g þ V s V t ðcos dst g þ sin dst bst Þ @ a st @ a st @a @a @a

ð19Þ

0 ¼ C 1 DX þ D1 DIg þ D2 DVg

ð11Þ

Again

@Pst @Ps ¼ @hs @hs and

@Pst @P s ¼ : @V s @V s The linearized reactive power flow balance equations can be obtained by similar methods. The matrix form of the power flow equations of TCSC at node s and t are given in the Appendix A.2. Eliminating DIg from the respective Eqs. (10)–(13), the overall system matrix with SVC and with TCSC controller can be obtained as

½Asv c ð7mþ3Þð7mþ3Þ ¼ ½A0   ½B0 ½D0 1 ½C 0 

ð20Þ

and

½Atcsc ð7mþ2Þð7mþ2Þ ¼ ½A0   ½B0 ½D0 1 ½C 0  Fig. 4. Variation of TCSC reactance (Xtcsc) with firing angle (a).

respectively.

ð21Þ

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D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

3. Problem formulation

3.3. PSO algorithm

3.1. Theory of Particle Swarm Optimization (PSO)

The implementation of the PSO algorithm has been described here with its flow chart being shown in Appendix A (Fig. A.1). To optimize Eq. (24), routines from PSO toolbox [26] are used. The

Particle Swarm Optimization was first developed in 1995 by Eberhart and Kennedy [18]. The PSO algorithm begins by initializing a random swarm of M particles, each having R unknown parameters to be optimized. In each iteration, the fitness of each particle is evaluated according to the selected fitness function. The algorithm stores and progressively replaces the best fit parameters of each particle (pbesti, i = 1, 2, 3, . . . , M) as well as a single most fit particle (gbest) among all the particles in the group. The trajectory of each particle is influenced in a direction determined by the previous velocity and the location of gbest and pbesti. Each particle’s previous position (pbesti) and the swarm’s overall best position (gbest) are meant to represent the notion of individual experience memory and group knowledge of a ‘‘leader’’, respectively. The parameters of each particle (pi) in the swarm are updated in each iteration (n) according to the following equations:

v eli ðnÞ ¼ w  v eli ðn  1Þ þ acc1  rand1  ðgbest  pi ðn  1ÞÞ þ acc2  rand2  ðpbest i  pi ðn  1ÞÞ pi ðnÞ ¼ pi ðn  1Þ þ v eli ðnÞ

ð22Þ ð23Þ

where v eli (n) is the velocity vector of particle i. acc1, acc2 are the acceleration coefficients that pull each particle towards gbest and pbesti positions respectively and are often set to be 2.0. w is the inertia weight of values e (0, 1). rand1 and rand2 are two uniformly distributed random numbers in the ranges [0, 1].

Fig. 5. IEEE-14 bus system with the application of SVC and TCSC.

3.2. Objective function and optimization problem

(a)

The optimization problem represented here is to search the optimal location and the parameter set of the SVC and TCSC controller using PSO algorithm. This results in minimization of the critical damping index (CDI) given by

Location Lead-lag Time Gain

CDI ¼ J ¼ ð1  jfi jÞ

ð24Þ

(b)

ri þxi

The objective of the optimization is to maximize the damping ratio (f) as much as possible. There are four tuning parameters of the SVC and TCSC controller; the controllers gain (K), lead time constant (T1), lag time constant (T2) and the location number (Nloc). These parameters are to be optimized by minimizing the objective function J given by the Eq. (24). With the change of locations and parameters of the SVC and TCSC controller the damping ratio (f) as well as J varies. The problem constraints are the bounds on the possible locations and parameters of the SVC and TCSC controller. The optimization problem can then be formulated as:

½Given by ð24Þ

1

13

8

Gain

12 14 0.01 0.25 0.1 0.4

1

Nloc T2 T1

20

Ksvc Maximum range

Location

Lead-lag Time

14

0.2 0.5 0.7 1.0

Minimum range

ri Here, fi ¼ pffiffiffiffiffiffiffiffiffiffiffi is the damping ratio of the ith critical swing mode. 2 2

Minimize J

10

6 9 0.01 0.09 0.1 0.3

16

17

10

20

0.3 0.5 0.65 1.0 20

Nloc T2 T1

Ktcsc Maximum range

Minimum range

Fig. 6. Particle configurations (a) SVC controller and (b) TCSC controller.

ð25Þ Table 1 Swing modes without SVC and TCSC.

Subjectto

#

K min 6 K 6 K max T min 6 T 1 6 T max 1 1

ð27Þ

6 T 2 6 T max T min 2 2

ð28Þ

max Nmin loc 6 N loc 6 N loc

ð29Þ

1 2 3 4

Load increased at bus #9 (PL = 0.339 pu, QL = 0.190 pu)

Transmission line (#4–13) outage

Swing modes

Damping ratio

Swing modes

Damping ratio

1.5446 ± j7.5274 1.4244 ± j6.5313 1.1590 ± j6.1460 0.8831 ± j5.8324

0.20101 0.21308 0.18531 0.14972

1.5482 ± j7.5222 1.4291 ± j6.5339 1.1501 ± j6.1659 0.8845 ± j5.8336

0.20159 0.21367 0.18337 0.14992

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D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

Table 2 PSO based SVC and TCSC controller parameters and locations. SVC parameter Ksvc = 20.0 T1 = 1.0 T2 = 0.15

SVC location (# bus)

TCSC parameter

TCSC location (# line)

#10

Ktcsc = 16.809 T1 = 1.0 T2 = 0.2264

#16

objective function corresponding to each particle is evaluated by small signal analysis program of the proposed test system (Fig. 5). 3.3.1. Particle configuration The particle is defined as a vector which contains the SVC and TCSC controller parameters and the location number: K, T1, T2 and Nloc as shown in Eq. (30)

Particle : ½K

T1

T2

Nloc

ð30Þ

Here, K stands for the respective gains of SVC and the TCSC controller and are termed as Ksvc and Ktcsc, respectively. The initial population is generated randomly for each particle and is kept within a typical range. The particle configurations corresponding to the SVC and TCSC controller are shown in Fig 6a and b, respectively. All the load buses (bus #6, 7, 8, 9, 10, 11, 12, 13 and 14) of the test system (Fig. 5) are proposed here for possible locations of the SVC and therefore, bus #6 and bus #14 are considered as N min loc and N max loc (Fig. 6a), respectively. Similarly, the network branches (line #12, 13, 14, 15, 16, 17, 18, 19 and 20) between two load buses are chosen for installing locations of the TCSC and therefore, the max line #12 and line #20 are assigned for N min (Fig. 6b), loc and N loc respectively.

4. Results and performance study 4.1. Application of PSO in the test system

Fig. 7. Convergence rate of the objective function for gbest (a) with SVC and (b) with TCSC.

The validity of the proposed PSO algorithm has been tested here on an IEEE-14 bus system (Fig. 5). This system has also been used widely in the literature [27] for small signal stability analysis. In order to study the small signal performance of the system the simulation is carried out for two independent types of disturbances: (i) real and reactive load increased at a particular bus #9 (15% more than nominal case) (ii) outage of a transmission line (#4–13). The swing modes of the system without SVC and TCSC dynamics are listed in Table 1. It has been observed that the mode #4 is the critical one as the damping ratio of this mode is smallest compared to other modes. Therefore, stabilization of this mode is essential in order to improve small signal stability. The PSO algorithm generates the best set of parameters as well as the best location (Table 2) corresponding to both the SVC and TCSC controllers by minimizing the objective function J. The convergence rate of the objective function J towards gbest with number of particles 15 and generations 200 has been shown in Fig. 7a and b. The damping ratio of the critical swing mode with the application of PSO based SVC and the PSO based TCSC controller has been represented in Table 3. It has been observed that the TCSC controller adds more enhancement of damping with respect to the SVC controller. The time response analysis (Fig. 8) of rotor speed deviation (Dx) of the Machine #1 relative to the machine #5 implies that the TCSC controller imparted better settling time to the system compared to the SVC controller.

Table 3 Application of PSO based SVC and TCSC controller. With SVC

Load increase (15% more than nominal value) Line outage (#4–13)

With TCSC

Critical swing mode

Damping ratio

Critical swing mode

Damping ratio

0.98121 ± j6.0070

0.16121

1.0611 ± j5.7341

0.18196

0.98224 ± j6.0568

0.16008

1.0602 ± j5.7519

0.18127

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D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

value PL = 0.295 pu, QL = 0.166 pu in steps. In each case eigenvalues of the system matrix are checked for stability. It has been observed that at load PL = 2.60 pu, QL = 0.166 pu Hopf bifurcation [25] takes place for the critical swing mode #4 and led to low-frequency oscillatory instability of the system. When the SVC and TCSC are installed at bus #10 and line #16 separately there is no Hopf bifurcation of swing modes. The swing modes of the system without and with SVC and TCSC controllers have been represented in Tables 4 and 5, respectively. This implies that the inclusion of SVC and TCSC can put off the Hopf bifurcation until further increase of load levels. 5. Conclusions A novel stochastic method for tuning and optimal placement of SVC and TCSC controllers in order to mitigate the small signal oscillation problem in a multimachine system has been presented in this paper. PSO algorithm has been implemented for optimal parameter setting and identification of optimal site of SVC and the TCSC controller. The enhancement of small signal stability has been achieved by minimizing a desired objective function. The performance of the PSO based SVC and the TCSC controller has also been compared with the application in a standard multimachine power system considering all network bus dynamics. The nature of critical eigenvalue and time response analysis reveal that the TCSC controller is more superior than SVC to improve the small signal oscillation problem even during critical loading. The present approach of PSO based optimization technique seems to have good accuracy, faster convergence rate and is free from computational complexity. Appendix A A.1. State vectors of the multimachine model Fig. 8. Rotor speed deviation response of machine #1 relative to machine #5 (a) Load increase at bus #9 and (b) Line outage #4–13.

X ¼ ½X T1

X T2



X Tm T ;

Ig ¼ ½Id1

Iq1

Id2

Iq2



Idm

Iqm T ;

V g ¼ ½h1

V1

h2

V2



hm

V m T ;

4.2. Implication of SVC and TCSC controller on critical loading In order to study the effect of loading on system stability, the real load (constant power) at bus #9 is increased form its nominal

#

1 2 3 4

V mþ1

V l ¼ ½hmþ1

Table 4 Effect of critical loading without SVC and TCSC. Nominal load (PL = 0.295 pu, QL = 0.166 pu)

Hopf bifurcation load at bus #9 (PL = 2.60 pu, QL = 0.166 pu)

Swing modes

Damping ratio

Swing modes

Damping ratio

1.6071 ± j7.5211 1.4987 ± j6.5328 1.2074 ± j6.1633 0.9461 ± j5.8552

0.20896 0.2236 0.19225 0.15953

1.1190 ± j7.7098 1.6357 ± j5.9069 0.9230 ± j2.5144 0.0072 ± j4.6175

0.14363 0.26687 0.34461 0.00156

U ¼ ½U T1

U T2

#

1 2 3 4

SVC at bus #10 Damping ratio

Swing modes

Damping ratio

1.2337 ± j7.5753 1.4921 ± j6.0160 1.776 ± j2.7131 0.9712 ± j3.4139

0.16074 0.24072 0.54768 0.27363

1.1172 ± j7.6664 1.5937 ± j5.7222 3.194 ± j2.4392 1.1218 ± j3.6820

0.1442 0.2683 0.7947 0.2914

hn

for U i ¼ ½T Mi

V n T ; V refi T

For node s and node t

2 @Ps @hs

@P s @V s

@P st @hs

@Q s @V s @Pst @V s

@ht

Swing modes



A.2. The linearized power flow equations of TCSC in matrix form

2 @Pt

TCSC in line #16

U Tm T

V mþ2

where i = 1, 2, . . . , m (number of machines) and i = m + 1, m + 2, . . . , n (number of load buses).

6 @Q s 0¼6 4 @hs Table 5 Application of SVC and TCSC with Hopf bifurcation load.



hmþ2

xi E0qi E0di Efdi V Ri RFi T ;

X i ¼ ½di

6 @Q t 0¼6 4 @ht

@P st @ht

@Pt @V t @Q t @V t @Pst @V t

ðA:1Þ

@a

32

3 Dh t 7 7 @Q s 76 DV t 5 @ a 54 @P st Da @P t @a

@a

A.3. PSO algorithm Fig. A.1.

32

3 Dh s 7 7 @Q s 76 DV s 5 @ a 54 @P st Da @P s @a

ðA:2Þ

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D. Mondal et al. / Electrical Power and Energy Systems 42 (2012) 334–340

Start

Specify parameters for PSO: initial velocity, no. of particles, max iteration

Generate initial population Iter.=1 Run small signal stability and eigenvalue analysis program

Compute objective function for each particle in the current population

Iter.= Iter. + 1

For all particles determine and store pbest, gbest

Iter. > Max. Iter. ?

Update velocity and particle position No

Yes Output Result

Stop

Fig. A.1. Flow chart of the implemented PSO.

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