parameters tuning of power system stabilizers using improved ant direction hybrid differential...

Electrical Power and Energy Systems 31 (2009) 34–42

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Parameters tuning of power system stabilizers using improvedant direction hybrid differential evolution

Sheng-Kuan Wang a,b,*, Ji-Pyng Chiou b, Chih-Wen Liu a

a Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROCb Electrical Engineering, Ming Chi University of Technology, 84 Gungjuan Rd., Taishan, Taipei 24301, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 15 February 2008Received in revised form 17 September2008Accepted 18 October 2008

Keywords:Hybrid differential evolutionPower system stabilizerDynamic stabilityOptimizationEigenvalue analysis

0142-0615/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.ijepes.2008.10.003

* Corresponding author. Address: Electrical EngineTechnology, 84 Gungjuan Rd., Taishan, Taipei 2430129089899; fax: +886 2 29084507.

E-mail address: [email protected] (S.-K. Wa

a b s t r a c t

The tuning of the PSS parameters for a multi-machine power system is usually formulated as an objectivefunction with constraints consisting of the damping factor and damping ratio. A novel mixed-integer antdirection hybrid differential evolution algorithm, called MIADHDE, is proposed to solve this kind of prob-lem. The MIADHDE is improved from ADHDE by the addition of accelerated phase and real variables. Theperformances of three different objective functions are compared to the MIADHDE in this paper. Bothlocal and remote feedback signals of machine speed deviation measurements can be selected as input sig-nals to the PSS controllers in the proposed objective function. The New England 10-unit 39-bus standardpower system, under various system configurations and loading conditions, is employed to illustrate theperformance of the proposed method with the three different objective functions. Eigenvalue analysisand nonlinear time domain simulation results demonstrate the effectiveness of the proposed algorithmand the objective function with a remote signal.

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1. Introduction

The dynamic stability of power systems is an important factorfor secure system operation. Low-frequency oscillation modeshave been observed when power systems are interconnected byweak tie lines [1,2]. The low-frequency oscillation mode, whichhas poor damping in a power system, is also called the electrome-chanical oscillation mode and usually occurs in the frequencyrange of 0.1–2.0 Hz [3–6]. The power system stabilizer (PSS) hasbeen widely used for mitigating the effects of low-frequency oscil-lation modes [7]. The construct and parameters of PSS have beendiscussed in many studies [3–6]. Currently, many plants prefer toemploy conventional lead-lag structure PSSs, due to the ease of on-line tuning and reliability [7–9]. Over the last two decades, variousparameter tuning schemes of PSS have been developed and appliedto solve the problem of dynamic instability in a power system. Theparameter tuning of PSSs in the power system has two majormethods, sequential tuning and simultaneous tuning [8]. In orderto obtain the set of optimal PSS parameters under various operat-ing conditions, the tuning and testing of PSS parameters must berepeated under various system operating conditions. Therefore, ifthe sequential tuning method is applied to tuning PSS parameters,

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ng).

the parameters tuning will become more complicated, and thetuned result maybe not a locally or globally optimal solution. Onthe other hand, in the case that the simultaneous tuning methodis employed in tuning of PSS parameters, which can simulta-neously relocate and coordinate the eigenvalues of various oscilla-tion modes under different operating conditions, the set of PSSparameters solutions can quite close to the globally optimal solu-tion. However, the drawback of the simultaneous tuning methodis the longer computation time required for large power systems.The simultaneous tuning of PSS parameters is usually formulatedas a very large scale nonlinear non-differentiable optimizationproblem. This kind of optimization problem is very hard, if notimpossible, to solve using traditionally differentiable optimizationalgorithms. Instead, we propose a stochastic optimizationapproach.

Abdel-Magid and Abido [8,10–13] have employed the tabusearch (TS), simulated annealing (SA), particle swarm optimization(PSO), evolutionary programming (EP), and genetic algorithm (GA)to optimize the parameters of the PSSs in the New England ten-ma-chine system. In the system, all PSSs are simultaneously designedto take into account mutual interactions. The objective functionis devised to optimize the desired damping factor (r) and/or thedesired damping ratio (n) of the lightly damped and undampedmodes. In this way, only the unstable or lightly damped oscillationmodes are relocated. Hongesombut et al. [9] used hierarchical andparallel micro genetic algorithms in a multi-machine system. DoBomfim et al. [14] used genetic algorithms to simultaneously tune

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Nomenclature

Generalr damping factorDx speed deviationT1, T2, T3, T4 lead/lag time constants of PSSn amount- weight constantX real variables#i;v random numbersi pheromone informationobject objective function valuegi heuristic informationa influence factor of pheromone information~Ijk difference vector between individual j and k for integer

variablesj scaling factor�IGþ1

ih hth gene of ith trial individual vector of (G+1)th genera-tion

e1 desired tolerance for the population diversityð�IGþ1

b ; �XGþ1b Þ best individual

n damping ratiod rotor angleKS gain of the PSSr0 expected damping factorf0 expected damping ratioI integer variablesNP number of individualsPH proportion constant of pheromone

q constant coefficientpi probability of choosing a mutation operatorb influence factor of heuristic informationIGþ1i perturbed individual vector of ith individual for integer

variablesCr crossover factorarg min the argument of the minimume2 desired tolerance for the gene diversity�a step size

SubscriptsZ=1,2,...,nZ index of local machinesRZ=0,1,...,nZ index of remote machinesq=1,2,...nq index of eigenvalues in the systempresent present generationh = 1,...,n index of integer variableN Nth PSSy=1,2,...,ny

index of system operating conditionsi=1,2,...,NP index of individualsj index of the genesg = (n + 1),...,m index of real variable

Superscripts0 initial valueG index of the generationbest best value

S.-K. Wang et al. / Electrical Power and Energy Systems 31 (2009) 34–42 35

multiple power system damping controllers with the objectivefunction of the sum of the spectrum damping ratios for all operat-ing conditions. These studies did not consider the remote feedbacksignals, which are available from synchronized phasor measure-ment units [15–17]. Kamwa et al. [18,19] used a decentralized/hierarchical control system with two global signals and one localsignal as input of PSSs. The global signals obtained the highest con-trollability of these oscillation modes. Hasanovic and Feliachi [20]used a genetic algorithm to accomplish simultaneous tuning ofmultiple power system damping controllers. Both local and remotemeasurement signals have been considered as input signals to thedamping controllers. However, the remote measurements are con-sidered only in a 4-machine system, and that remote signal is notdetermined from the computational results. Su and Lee [21,22]used improved mixed-integer hybrid differential evolution to solv-ing network reconfiguration and capacitor placement problems ofa distribution system, but those problems only contain integerdecision variables.

Chiou et al. [23] developed the ant direction hybrid differentialevolution (ADHDE) method, which utilizes the concept of an antcolony search to find a suitable mutation strategy in the HDEmethod [24,25] to accelerate the search for the global solution.The ADHDE has proved that it can obtain excellent computation re-sults to the optimization problems of integer variables in thecapacitor placement problem [23], but it has not been investigatedin terms optimization problems consisting of both integer and realvariables. In order to rapidly find a global solution by ADHDE, theaccelerated phase is added in this algorithm. In this study, the ma-chine numbers of remote feedback signals are regarded as the inte-ger variable. Therefore, the proposed optimization algorithm musthave the ability to search out the set of optimal solutions of integerand real variables at the same time. The optimization problems ofboth real and integer variables are generally called mixed-integernonlinear programming problems. The original ADHDE algorithm

gives it the ability to search out a set optimal solution of integerand real variables at the same time, so the algorithm is called themixed-integer ant direction hybrid differential evolution algorithm(MIADHDE). The MIADHDE algorithm in this study is proposed todetermine the optimal gain, time constants, and machine numbersof remote feedback signals of PSSs for the multi-machine systemby three different objective functions. The three objective functionswill be compared in terms of performance by the degree of systemdamped.

2. Problem formulation

2.1. Power system model

In this study, each generator is modeled as a two-axis model,which is a six-order model. The state vector of the generator is gi-ven as ½Dx; d;/fd;/1d;/1q;/2q�, where Dx and d are the speed devi-ation and rotor angle, respectively, and /fd;/1d;/1q and /2q are thecontribution to the rotor flux linkage as a result of field winding,one d-axis and two q-axis amortisseur circuits, respectively [26].These device models of a power system include generators, PSSs,and excitation systems, and can be formulated by:

_x ¼ f ðx; r; tÞ ð1Þ

where x is the vector of all state variables, and r is the vector of in-put variables. In the PSS design, the power system is usually linear-ized in terms of a perturbed value in order to perform the smallsignal analysis. Therefore, Eq. (1) can be represented as:

D _x ¼ ADxþ BDv

Du ¼ CDxþ DDv

(ð2Þ

where A is the power system state matrix, B is the input matrix, C isthe output matrix, D is the feedforward matrix, v is the vector of the

36 S.-K. Wang et al. / Electrical Power and Energy Systems 31 (2009) 34–42

network bus voltages, and u is the current vector of the injectioninto the network from the device [26].

The PSSs with a lead-lag structure of speed deviation input areconsidered in this study, and the transfer function of the Nth PSS isgiven by Eq. (3). It contains a remote feedback signal of speed devi-ation, DxRZ .

VSNðsÞ ¼ KSNsTWN

ð1þ sTWNÞð1þ sT1NÞð1þ sT2NÞ

ð1þ sT3NÞð1þ sT4NÞ

ðDxZ � DxRZÞ ð3Þ

where T1N, T2N, T3N, T4N, and KSN are the time constants and gain ofthe Nth PSS, Z = 1, 2,...,nZ is the index of machines, nZ is the numberof machines, DxZ is the speed deviation of the Zth local machine,DxRZ is the speed deviation of the RZth remote machine, RZ = 0, 1,2,...,nZ is the index of remote machines, and RZ = 0 represents thatthe PSS has no remote feedback signal, DxRZ ¼ 0. However, the con-servative assumption that the effect of communication delay can beneglected is made. The time constant TWN in the washout filter isusually kept constant during the optimization. Which typical valueis between 10 and 20 so that obtains well phase compensation per-formance. However, the washout time constant is fixed at 10 in thisstudy.

2.2. Objective function

When a disturbance occurs, the decaying rate of the power sys-tem oscillation is dominated by the maximum damping factor inthe system, and the amplitude of the each oscillation mode isdetermined by its damping ratio. Thus, the objective functions nat-urally include the damping factor and the damping ratio in the for-mulation for the tuning of PSS parameters. In this paper, threeobjective functions are considered.

(1) The first objective function for the optimal PSS parametersmay be formulated as follows:

MinF1 ¼Xny

y¼1

Xrq;yPr0

ðrq;y � r0Þ2 þ- �Xny

y¼1

Xfq;y6f0

ðf0 � fq;yÞ ð4Þ

subject toKSN;min 6 KSN 6 KSN;max

T1N;min 6 T1N 6 T1N;max




8>>>>>><>>>>>>:

ð5Þ

where y = 1, 2, 3,...,ny is the index of system operating conditionsconsidered in this design process, q = 1, 2,...,nq, is the index of eigen-values in the system, rq;y is the damping factor of the qth eigenvalueof the yth operating condition, r0 is a constant value of the expecteddamping factor, - is a weight for combining both damping factorsand damping ratios, fq;y is the damping ratio of the qth eigenvalue ofthe yth operating condition, and finally, f0 is a constant value of theexpected damping ratio. The constraint set is made up of bounds ofPSS parameters, which can be formulated as Eq. (5), and the speeddeviation measurement signals of the remote machine are ne-glected; DxRZ ¼ 0 in this formulation. The objective function onlyforces the unstable or lightly damped electromechanical oscillationmodes to be relocated, an objective function similar to the best onein the literature [8]. In the Eq. (4), the damping ratio difference isless than 1; if we take each square for this term, then the objectivefunction is unable to determine the optimum solution. The bestobjective value is zero for this function.

(2) The second objective function for tuning PSS parametersmay be chosen to minimize the following function.

MinF2 ¼Xny

y¼1

ðmax16q6nq

rq � r0Þy þ-Xny

y¼1

ðf0 � min16q6nq

fqÞy ð6Þ

where max rq and min fq are the maximum real part of the eigen-values and the minimum of the damping ratio, respectively, in theyth system operating condition. The objective function not only relo-cates the unstable or lightly damped oscillation modes to the leftside of the s-plane but also shifts other oscillation modes more tothe left side of the s-plane. The same constraints as Eq. (5) are used,and the speed deviation measurement signals of the remote ma-chine are neglected, so DxRZ ¼ 0 in this formulation.

3) The third objective function is the same as Eq. (6) but has dif-ferent constraints, as follows.

MinF3 ¼Xny

y¼1

ðmax16q6nq

rq � r0Þy þ-Xny

y¼1

ðf0 � min16q6nq

fqÞy ð7Þ

The constraints add the speed deviation measurement signals ofremote machines for the input to PSSs in Eq. (8), except the sameconstraints as Eq. (5) are used.

0 6 RZ 6 nZ ; where RZ–Z ð8Þ

3. MIADHDE algorithm

The ADHDE algorithm developed by Chiou et al. can be found inthe literature [23]. The main idea of ADHDE is to use the ant colonysearch system to find the proper mutation operator to acceleratethe search for the global solution. The MIADHDE algorithm im-proves the ADHDE to deal with the optimization problem of simul-taneously including integer and real variables and to add theaccelerated phase with the aim of faster convergence. The MIAD-HDE algorithm is briefly described below.

Step 1. InitializationThe initial population and load system data are created. Some

initial populations are randomly chosen, and they endeavor to cov-er the complete parameter space uniformly. The elements of everyrandom variable ðI0

i ;X0i Þ are assumed as follows

ðI0i ;X

0i Þ ¼ ðIi;min;Xi;minÞ þ ðINTð#iðIi;max � Ii;minÞÞ;#iðXi;max � Xi;minÞÞ; i ¼ 1; . . . ;Np

ð9Þ

where #i 2 ð0;1� is a random number, and INT operator is repre-sented to take the nearest integer. The two variables, X and I, arereal and integer variables, respectively. The initial process can cre-ate Np individuals of ðI0

i ;X0i Þ randomly.

Step 2. Ant direction searchIn every generation, each of Np ants selects a mutation operator

according to heuristic information and pheromone information.Different from [27–32], the difference between the objectivefunction value in the next generation and the best objective func-tion value of the present generation constructs a fluctuant phero-mone quantity. The fluctuant pheromone quantity is defined asfollows:

Dsi ¼Q

PH ; if the proper mutation operator is chosen0; otherwise

(; i¼ 1; . . . ;Np

ð10Þ

In Eq. (10), the proper mutation operator is chosen when theobjective function value of the next generation is better than thebest objective function value of the present generation. Where Qis a constant, PH is defined as follows:

PH ¼ objectnew

objectbestpresent � objectnew

�� ð11Þ


where objectbestpresent expresses the best objective function value of the

present generation and objectnew expresses the objective functionvalue of the next generation. The pheromone updating applies thefollowing rule.

sNewi ¼ q � sOld

i þ Dsi ð12Þ

The coefficient q must be fixed to a value <1 to avoid an unlim-ited accumulation of trace. The magnitude of the new pheromoneis updated by vaporizing ð1� qÞ percent of the pheromones fromthe previous iteration and the fluctuant pheromone quantity inthe current iteration. It is observed that the probability of select-ing a mutation operator is proportional to the pheromone quan-tity si and the information gi:. The information gi is defined asfollows:

gi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn

j¼1

ððIGij � IG

bjÞ=IGijÞ

2 þXm

j¼nþ1

ððXGij � XG

bjÞ=XGijÞ

2

vuut ð13Þ

where ðIGij ;X

Gij Þ denotes the jth gene of the ith individual in a popula-

tion in the Gth generation, and ðIGbj;X

GbjÞ denotes the jth gene of the

best individual in a population in the Gth generation. Hence, theprobability of choosing a mutation operator is defined as follows:

piðtÞ ¼sa

kðtÞ � gbkPNp

k¼1sakðtÞ � g

bk

ð14Þ

where a and b are parameters to regulate the influence of si and gi;,respectively. At the same time, the probabilities are arranged inascending order. The initial value of probability for choosing amutation operator is first defined as P = [p1, p2, p3, p4, p5] = [0.2,0.4, 0.6, 0.8, 1.0]. Subsequently, the five integers 1, 2, 3, 4, 5 are ran-domly arranged. For example, if the resultant arrangement is [1, 3,5, 2, 4], then by relating this arrangement to P, we have the proba-bilities of choosing mutation operators for strategy 1 = 0 � 0.2,strategy 3 = 0.2 � 0.4, strategy 5 = 0.4 � 0.6, strategy 2 = 0.6 � 0.8,and strategy 4 = 0.8 � 1.0.

Step 3. Mutation operationAccording to the probability of choosing a mutation strategy,

the corresponding mutation operation and a random numberv 2 ½0;1� determine the proper mutation strategy. Assume thatthe probability of choosing a mutation operator is P = [0.313,0.641, 0.647, 0.813, 0.981], and the corresponding mutation oper-ation is [1, 3, 5, 2, 4]. Suppose that the random number v is equalto 0.567, which is between 0.313 and 0.641; then the third muta-tion strategy is determined. This process is repeated for everyindividual.

Five mutation strategies have been introduced by [33–34]. Theessential ingredient in the mutation operation is the differencevector. Each individual pair in a population at the Gth generationdefines a difference vector ð~Ijk; ~XjkÞ as

~Ijk ¼ IGj � IG

k

~Xjk ¼ XGj � XG

k

(ð15Þ

The mutation process at the Gth generation begins by randomlyselecting either two or four population individualsðIG

j ; XGj Þ; ðI

Gk ; XG

k Þ; ðIGl ; XG

l Þ and ðIGm; XG

mÞ for any j, k, l and m, accordingto five mutation strategies. These four individuals are then com-bined to form a difference vector ð~Ijklm; ~XjklmÞ as

~Ijklm ¼ ~Ijk þ~Ilm ¼ ðIGj � IG

k Þ þ ðIGl � IG

mÞ~Xjklm ¼ ~Xjk þ ~Xlm ¼ ðXG

j � XGk Þ þ ðX

Gl � XG

mÞ

(ð16Þ

A perturbed individual is then generated based on the presentindividual in the mutation process by

ðIGþ1i ; XGþ1

i Þ ¼ ðIGp ;X

Gp Þ þ ðINTðj �~IjklmÞ;j � ~XjklmÞ ð17Þ

where the scaling factor, j; is a constant, and j, k, l and m are ran-domly selected.

The perturbed individual in Eq. (17) is essentially a noisy replicaof ðIG

p ; XGp Þ: Herein, the parent individual ðIG

p ; XGp Þ depends on the

circumstance in which the type of the mutation operations isemployed.

Step 4. Crossover operationIn order to extend the diversity of further individuals at the next

generation, the perturbed individual of ðXGþ1i ; IGþ1

i Þ and the presentindividual of ðXG

i ; IGi Þ are chosen by a binomial distribution to pro-

gress the crossover operation to generate the offspring. Each geneof the ith individual is reproduced from the perturbed individualðIGþ1

i ; XGþ1i Þ ¼ ½IGþ1

i1 ; IGþ1i2 ; . . . ; IGþ1

in ; XGþ1iðnþ1Þ; X

Gþ1iðnþ2Þ; . . . ; XGþ1

im � and thepresent individual ðIG

i ; XGi Þ ¼ ½I

Gi1; I

Gi2; . . . ; Iin;X

Giðnþ1Þ;X

Giðnþ2Þ; . . . ;XG

im; �:That is,

�IGþ1ih ¼

IGih; if a random number > Cr

IGþ1ih ; otherwise

(

�XGþ1ig ¼

XGig ; if a random number > Cr

XGþ1ig ; otherwise

(8>>>>><>>>>>:

ð18Þ

where i ¼ 1; . . . ;Np; h ¼ 1; . . . ; n; g ¼ ðnþ 1Þ; . . . ;m, and the cross-over factor Cr 2 ½0;1� is assigned by the user.

Step 5. Estimation and selectionThe evaluation function of a child is one-to-one competed to that

of its parent. This competition means that the parent is replaced byits child if the fitness of the child is better than that of its parent. Onthe other hand, the parent is retained in the next generation if thefitness of the child is worse than that of its parent, i.e.

ðIGþ1i ;XGþ1

i Þ ¼ arg minfFðIGi ;X

Gi Þ; Fð�IGþ1

i ; �XGþ1i Þg ð19Þ

ðIGþ1b ;XGþ1

b Þ ¼ arg minfFðIGþ1i ;XGþ1

i Þg ð20Þ

where arg min means the argument of the minimum.Step 6. Migrating operation if necessaryIn order to effectively enhance the investigation of the search

space and reduce the choice pressure of a small population, amigration phase is introduced to regenerate a new diverse popula-tion of individuals. The best individual ðIGþ1

b ;XGþ1b Þis selected to cre-

ate the new population. The hth or gth gene of the ith individual is asin equation Eq. (21) and where #ih; #ig and k are randomly gener-ated numbers uniformly distributed in the range of [0,1].

IGþ1ih ¼ IGþ1

bh þ INTð#ih � ðIh min � IGþ1bh ÞÞ; if k <

IGþ1bh�Ih min

Ih max�Ih min

IGþ1bh þ INTð#ih � ðIh max � IGþ1

bh ÞÞ; otherwise

8<:

XGþ1ig ¼

XGþ1bg þ #ig � ðXg min � XGþ1

bg Þ; if k <XGþ1

bg�Xg min

Xg max�Xg min

XGþ1bg þ #ig � ðXg max � XGþ1

bg Þ; otherwise

8<:

8>>>>>>><>>>>>>>:

ð21Þ

The migrating operation is executed only if a measure fails tomatch the desired tolerance of population diversity. The measureis defined as Eq. (22).

e ¼XNp

i¼1i–b

Xn

j¼1

gI þXm

j¼nþ1

gX

!,m � ðNp � 1Þ� �

< e1 ð22Þ

where

gI ¼0; if IGþ1

ij ¼ IGþ1ib

1; otherwise

(

gX ¼0; if

XGþ1ij�XGþ1

ib

XGþ1ib

�� < e2

1; otherwise

8<:

8>>>>>><>>>>>>:

ð23Þ


Parameters e1; e2 2 ½0;1� express the desired tolerance for thepopulation diversity and the gene diversity with respect to the bestindividual, and gI and gX are the scale indices. From Eqs. (22) and(23), it can be seen that the value e is in the range of ½0;1�. If e issmaller than e1; then the migrating operation is executed to gener-ate a new population to escape the local point; otherwise, themigrating operation is turned off.

Step 7. Accelerated operation if necessaryWhen the best individual at the present generation is not im-

proved any longer by the mutation and crossover operations, a de-cent method is then employed to push the present best individual

Fig. 1. Main calculation procedures of the proposed method.

toward attaining a better point. Thus, the accelerated phase is ex-pressed as follows.

ð�IGþ1b ; �XGþ1

b Þ ¼ðIGþ1

b ;XGþ1b Þ; if FðIGþ1

b ;XGþ1b Þ < FðIG

b ;XGb Þ

ðIGþ1b ;XGþ1

b Þ � aOFðI;XÞ�� I¼IGþ1

b

X¼XGþ1b

;otherwise

8><>: ð24Þ

Where ð�IGþ1b ; �XGþ1

b Þ denotes the best individual, as obtained from Eq.(20). The gradient of the objective function, OF, can be approxi-mately calculated by finite difference. The step size in Eq. (24) isdetermined by the descent property. Initially, a is set to one toobtain the new individual.

Step 8Steps 2 to 7 are repeated until the maximum iteration quantity

or the desired fitness value is reached

Fig. 2. New England 10-machine, 39-bus system.

Table 1Searched gains and time constants of PSSs by the MIADHDE.

Objective function Unit KSZ T1Z T2Z T3Z T4Z DxRZ

Objective functionwith F1

G2 22.879 0.9478 1.0000 0.5638 0.0855 –G3 9.9292 1.1294 0.8259 1.4010 0.1363 –G4 19.441 0.7037 0.0010 0.3081 1.0000 –G5 4.6526 0.4137 0.0034 0.4088 0.2991 –G6 25.983 0.8504 0.0680 0.6585 1.0000 –G7 25.337 1.3027 0.4465 1.8283 0.5807 –G8 12.609 1.3954 0.1049 0.5036 0.5714 –G9 1.9189 1.4208 0.0357 1.2558 0.8001 –G10 5.2901 0.0520 0.6492 0.0274 0.5501 –


G2 12.081 0.9993 0.8381 1.9811 0.1036 –G3 25.806 0.9094 0.4710 0.5889 0.1607 –G4 1.6731 1.1985 0.9523 1.6601 0.0514 –G5 15.362 1.4677 0.1265 1.1015 0.9676 –G6 2.5105 1.3828 0.0773 0.9510 0.2391 –G7 27.314 0.7836 0.9998 1.5500 0.2323 –G8 13.351 0.4623 0.0960 0.5360 0.2294 –G9 40.825 0.9267 0.9697 0.3875 0.0248 –G10 35.040 1.9999 0.0621 1.1086 0.6482 –


G2 5.6913 1.9998 0.0226 0.6646 0.5861 –G3 5.3500 1.2156 0.7453 0.8590 0.0489 Dx8

G4 18.907 0.7508 0.9990 0.6400 0.1054 Dx8

G5 29.716 1.1765 0.1375 0.7536 0.6921 –G6 20.367 0.1622 0.0167 0.7440 0.3808 –G7 10.896 1.9996 0.1486 0.7937 0.9121 –G8 39.123 0.7129 0.0117 0.1653 0.2576 –G9 10.212 0.7475 0.1184 1.5247 0.6918 Dx2

G10 13.134 1.1229 0.0228 0.8460 0.4242 Dx7


These computational procedures search configurations withvarious gain and time constants of PSS parameters so as to succes-sively reduce the fitness function value. The solution proceduresstart off with performing calculations of power flows for variousoperating conditions, and then small signal analysis calculatesthe eigenvalues, oscillation frequencies, and damping ratios forvarious operating conditions of the complete system. This compu-tational process of the MIADHDE algorithm for finding the optimalsmall signal stability solution is stated using the flowchart shownin Fig. 1.

4. Simulation results

The New England 10-machine, 39-bus system is used to illus-trate the performance of the proposed method [35–36]. Fig. 2 pre-sents the one-line diagram of the system. This system, which isunstable without PSS, has been widely used as a benchmark sys-tem for PSS parameter tuning problems. In the test system, exceptthat the first bus is an infinite bus, other all generators are

Table 2Eigenvalues, oscillation frequencies, and damping ratios with and without PSSs for four ca

equipped with PSSs, and generator parameters are modified toadd sub-transient parameters. There are four operating conditionsconsidered as follows:

� Case 0: Base case.� Case 1: The tie line 16–31 is out-of-service.� Case 2: Tie lines 14–34 and 36, 37 are out-of-service.� Case 3: Lines 16–31 and 21, 22 are out-of-service, loads at buses

21 and 36 are increased by 30% and power generation of G1 isincreased by 30%.

In the tuning process, the constraints on PSS time constants T1N,T2N, T3N, and T4N are set with the lower limit at 0.001 s and theupper limit at 2.0 s, and the gain KSN of PSS ranges from 1 to 50.There are 45 real variables and 9 integer variables to be optimizedin this paper. Parameters r0 and f0 are set to be �1.0% and 15%,respectively. The target values of the maximum damping factorand the minimum damping ratio are assumed the desired limitvalue of all eigenvalues in the study system [9]. The weight -

ses.

Fig. 3. The searched eigenvalues with objective function F1 for four cases.



parameter is set to be 1.35, which is derived from the experiencesof many experiments conducted on this problem. Also, in theMIADHDE algorithm, there are nine parameters to be set up firstbefore carrying out the optimization process. The first sevenparameters include population size NP, crossover factor Cr, maxi-mum iterations itermax, tolerance of gene diversity, a, b, Q set tobe NP = 5, Cr = 0.6, itermax = 5000, tolerance of gene diversity = 0for integer variable, and tolerance of gene diversity = 0.1 for realvariables, a = 2, b = 1, and Q = 1. The other two parameters, scalingfactor j and tolerance of population diversity e1, are relatively sen-sitive in this optimization process. Therefore, the two parametersare set according to the experiences of many experiments on theproblem of tuning PSS parameters. The best values of parametersj and e1 are 0.3 and 0.1, respectively. The study system using threedifferent objective functions are performed 20 times by the MIAD-HDE algorithm, and then the best solution is presented. The soft-ware was written in Matlab and executed on a Pentium4 2.0GHzwith 512MB of RAM. The average CPU time for with and withoutremote signals are 71267 and 10271 seconds, respectively.

Table 1 presents the PSS parameters searched by the MIADHDEmethod. Table 2 presents the principal eigenvalues, oscillation fre-quencies, and damping ratios. In Table 2, the boldface value repre-sents the biggest damping factor, and the value enclosed in asquare frame represents the smallest damping ratio. All dampingfactors are smaller than �1.0 and all damping ratios are greaterthan 0.15 in Table 2. The results of the objective function F2 showthe minimum damping ratio and the maximum damping factorunder all cases, which is better than the results of objective func-tion F1. Also, the results of F3 are better than those of F2. It meansthat the addition of remote feedback signals can increase the sys-tem dynamic stability. The principal eigenvalues are drawn inthe s-planes, which are shown in Figs. 3–5, respectively. In Fig. 5,the scale of the damp factor axis is twice as large as in Figs. 3and 4. The results reveal that the proposed MIADHDE can not onlyrelocate the unstable or lightly damped oscillation modes but alsoshift other oscillation modes more to the left in the s-plane. How-ever, if the remote feedback signals are lost, then the system stabil-ity is slightly degraded. Clearly, the system performance obtainedusing the objective function F3 is better than both objective func-tions F1 and F2.

A number of time domain simulations were performed todemonstrate the effectiveness of tuning parameters of PSSs usingthe proposed MIADHDE method. In these tests, a four-cyclethree-phase fault is applied at bus 29. From Figs. 6–9, line pow-ers from bus 29 to bus 26 are shown for illustration. The objec-tive function F1 is similar to best objective function in theliterature [8], which is sustained oscillation over six secondsfor any system operation case in Figs. 6–9. However, the objec-tive function F3 achieves a steady state the fastest of the threeobjective functions under any system operation conditions.These time domain simulations agree well with the results ofeigenvalue analysis.


5. Conclusions

A novel mixed-integer ant direction hybrid differential evolu-tion (MIADHDE) algorithm has been proposed to determine theoptimal tuning of power system stabilizers with three differentobjective functions. The MIADHDE algorithm searches for the opti-mal PSS parameters, which can include remote feedback of speeddeviation measurement signals. Eigenvalue analysis and nonlineartime domain simulation results demonstrate the effectiveness ofthe proposed algorithm. We believe the MIADHDE can serve as abetter alternative to the problem of optimal tuning of power sys-tem stabilizers using the stochastic optimization approach.

Fig. 6. The line powers from bus 29 to 26 for case0.





Acknowledgment

The authors would like to express their sincere gratitude tofinancial support of this work by the National Science Council ofthe ROC under contract number NSC-97-2221-E-131-024.

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