Probabilistic power system stabilizer design with consideration of optimal siting using recursive...

Probabilistic power system stabilizer design with considerationof optimal siting using recursive Genetic Algorithmz

Z. Wang1,2*,y, C.Y. Chung2, K.P. Wong2,3, Deqiang Gan1 and Y. Xue4

1College of Electrical Engineering, Zhejiang University, Hangzhou, China2Computational Intelligence Applications Research Laboratory (CIARLab), Department of Electrical

Engineering, The Hong Kong Polytechnic University, Hong Kong, China3School of Electrical, Electronic and Computer Engineering, The University of Western Australia,

Perth Australia4State Grid Electric Power Research Institute, Nanjing, China

SUMMARY

This paper proposes an approach for the probabilistic power system stabilizer (PSS) design problem withconsideration of optimal siting of the PSSs under multiple operating conditions. The design problem is firstformulated as a combinational optimization problem which contains discrete and continuous variables. Thepaper then develops a recursive Genetic Algorithm (GA) to solve the design problem. An integer-binarymixed coding scheme and a partially matched crossover (PMX) operator are applied for the recursive GA forperformance enhancement. The effectiveness of the proposed recursive GA approach for probabilistic PSSdesign scheme is demonstrated on two test systems. Copyright # 2010 John Wiley & Sons, Ltd.

key words: Genetic Algorithm; partially matched crossover; power system stabilizer (PSS); optimalsiting; probabilistic theory

1. INTRODUCTION

In large interconnected power systems, electromechanical oscillations between interconnected

synchronous generators can lead to insecure power system operation and loss of power supply. The

stability of such low frequency oscillations is subject to critical investigations and power system

stabilizer (PSS) has been developed to enhance the damping of oscillations so as to improve the

dynamic stability of power systems.

One of the main issues in PSS designs is the tuning of the parameters of the PSS [1–10].

Conventional PSSs (CPSSs) with lead/lag structures and fixed parameters have been widely used in

power systems [1–4]. Appropriate selection of the CPSS parameters would result in satisfactory

performance during system disturbance. The robustness of the CPSS for a wide range of operating

conditions has been a concern [3]. The robustness issue has motivated the development of some

applications of on-line tuning techniques to PSS design, such as neural network techniques [5–7]. To

consider complex system operating conditions, fuzzy-based PSSs were developed to solve the stability

problem by using fuzzy sets, fuzzy relation matrix, and fuzzy operations [8–10].

Another important issue in PSS design is to determine the optimal location to install the PSSs for the

best performance. Many approaches or indexes based on open-loop system model have been proposed

and successfully used to select proper PSS sites, such as participation factor analysis [11], residue

method [12], damping torque analysis [1], and sensitivity coefficients [13]. These methods can provide

fast indication of good locations for the PSSs. Some of above methods have the following features: (i)

PSS sites and parameters are decided by a sequential method, which considers the damping

EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2011;21:1409–1424Published online 7 October 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.508

*Correspondence to: Z. Wang, College of Electrical Engineering, Zhejiang University, Hangzhou, China.yE-mail: [email protected] article was published online on 07 October 2010. Errors were subsequently identified. This notice is included in theonline and print versions to indicate that both have been corrected 20 April 2011.

Copyright # 2010 John Wiley & Sons, Ltd.

enhancement of just one critical electromechanical mode at a time; (ii) the eigenanalysis is performed

on the open-loop system, which only considers the open-loop eigenvalues and eigenvectors to

determine the PSS sites; (iii) PSS site identification is generally independent of the changes of power

system operating conditions.

The above methods have the following issues: (i) eigenanalysis of the open-loop system can only

convey part of the information on how the control input affects the modes of the system. To gain the full

knowledge of the effects of the PSSs on the modes, it is desirable to know the effectiveness of PSSs in

changing the closed-loop eigenvalues associated with the selected modes [14]; (ii) it is often desirable

to identify sites for installing PSSs so that several modes can be damped out simultaneously in an

effective way [15]; (iii) a wide range of operating conditions and uncertainties will be considered so

that the PSS design will not be limited to a deterministic condition with a particular load level.; (iv) the

minimum number of PSSs needed is determined to meet the requirement of power system damping

criteria among a given group of PSS candidates so that the cost and complexity of controller design can

be decreased.

Traditionally, typical intra-area low-frequency oscillation problems in a power system frequently

occur under those stressed load conditions. Due to the increasing needs for interconnection between

areas of generations and the introduction of competition in power markets, power flow patterns become

more and more complicated and the low frequency oscillations do not always occur at peak load

conditions simultaneously in different areas. Hence, it is desired to achieve a robust PSS design for a

wide range of operating conditions.

To consider a wide range of operating conditions, conventional eigenanalysis has been extended to

probabilistic environment and probabilistic damping controller design has been developed for the

conventional PSS [16,17]. With nodal voltages regarded as basic random variables and determined by

probabilistic load flow calculation, the probabilistic distribution of each eigenvalue was obtained from

the probabilistic attributes of the nodal voltages, and described by its expectation and variance under

the assumption of a normal distribution. In this approach, probabilistic sensitivity indices (PSIs)

are proposed to facilitate ‘robust’ PSS siting [16]. Further, a coordinated synthesis model of PSS

parameters is developed and a quasi-Newton nonlinear programming is used to solve the probabilistic

PSS design problem [17]. In addition, the differential evolution (DE) method in evolutionary

computation (EC) has been applied in Reference [18] to alleviate the problem of initial conditions

dependence and the problem of local minimum trapping, which exist in conventional nonlinear

optimization methods. Despite DE’s simple implementation and parametric robustness in Reference

[18], the real coding technique is embedded in the differential operator and the coding scheme other

than real coding has to be derived from the former, which is not flexible and sufficient for solving the

problem proposed in this paper.

The purpose of this paper is to propose a novel method based on Genetic Algorithm (GA) to solve

the optimal PSS design problem, which can consider both optimal siting and parameter tuning

simultaneously under multiple operating conditions. The problem will be first described as a

combinational optimization problem, which can determine the optimal PSS siting and PSS parameters

among a group of PSS candidates so that the design criteria can be satisfied. A recursive GA based on a

mixed integer-binary coding and a partially matched crossover (PMX) operator is then developed to

solve the problem. Finally, the effectiveness and the potential of the proposed probabilistic PSS design

scheme are demonstrated on two test systems.

2. PROBABILISTIC EIGENVALUE ANALYSIS

Under the multi-operating conditions of a power system, all nodal injections, nodal voltages and

eigenvalues are regarded as random variables. Statistical attributes of nodal injections are determined

from system operating samples. Probabilistic distributions and stability probabilities of all eigenvalues

can be obtained by means of the probabilistic eigenvalue analysis [16–18].

In probabilistic eigenvalue analysis, the statistics characteristics of uncertainties can be app-

roximated with normal distribution assumption [19,20]. Hence, the statistical nature of an eigenvalue

can be described by its expectation and variance. For a particular eigenvaluelk ¼ak þ jbk, having an

Copyright # 2010 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2011;21:1409–1424DOI: 10.1002/etep

1410 Z. WANG ET AL.

expectationakand standard deviationsak , the distribution within f�1; ak þ ksakg with a distributionconstant k over [3.5, 4] has a probability from 0.99977 to 0.99997, which is very close to unity. Thus,

the upper limit of this distribution range a0k in Equation (1) can be regarded as an extended damping

coefficient from which the robust stability of multioperating conditions can be estimated.

Correspondingly, the damping ratio jk ¼ �ak=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2k þ b2

k

qwith expectation jk and standard deviation

sjk has an extended value j0k in Equation (2),

a0k ¼ ak þ k sak (1)

j0k ¼ jk�k sjk (2)

To ensure the system dynamic performance is satisfactory, all the eigenvalues need to satisfy the

requirement of damping constant and damping ratio in Equations (3) and (4) respectively. In other

words, all the eigenvalues should be located in a small-signal stable region, being defined as

a0k � aC (3)

j0k � jC (4)

where aC and jC are acceptable limits for damping constant and damping ratio, respectively.

3. PROBABILISTIC PSS DESIGN WITH OPTIMAL SITING

For description purpose, it is assumed that there are NPSS potential PSS sites (i.e. NPSS PSS

candidates) that are numbered in integer sequence 1; 2; . . .;NPSS, each number corresponds to a

generator index that could be a possible PSS location; among them KPSS sites (�NPSS) constitute a PSS

siting set F,

F ¼ fh1; h2; . . .; hi; . . .; hKPSSg (5)

where hi2f1; 2; . . .;NPSSg; each hi (i¼ 1, . . ., KPSS) represents a tentative PSS location; and there are

no two identical hi in the set F obviously.

3.1. PSS structure

A typical PSS structure with two lead/lag stages is adopted in this study as follows:

FiðsÞ ¼ Ki � pTw

1þ p Tw� 1þ p T1i

1þ p T2i� 1þ pT3i

1þ pT4i(6)

where i2F;Ki is a gain constant at ith PSS with positive value for speed input signal and negative value

for power input signal; Tw is washout time constant; T1i=T2i and T3i=T4i are lead/lag time constants. It

should be noted that the time constants T2i and T4i should not be less than 0.04 seconds to avoid

excessive amplification of input signal noise. The ranges of the PSS parameters are set as follows [21]:

[0.1p.u., 20p.u.] for Ki of PSS with speed input signal and [�20p.u.,�0.1p.u.] with power input signal,

[0.06–2.0 seconds] for T1i and T3i, [0.04s-0.2s] for T2iand T4i. In this study, Tw is fixed as 10 and

5 seconds for speed and power input signals respectively.

3.2. Parameter optimization

For optimization purpose, it is more convenient to introduce the standardized expectations of the

damping constant and damping ratio a�k and j�k , derived from Equations (1) and (2) and termed as ks


OPTIMAL-SITING PSS BY RECURSIVE GA 1411

criteria, as defined in the following:

a�k ¼ �ðak�aCÞ=sak � k (7)

j�k ¼ ð jk�jCÞ= sjk � k (8)

After the standardization in Equations (7) and (8), a�k and j

�k are per-unit variables and can be directly

compared. Therefore, the system can be regarded as robust stability when all normalized probabilistic

eigenvalues satisfy inequalities (7)–(8). Thus, an optimization problem is formulated in Equation (9) so

that the minimumKPSS, corresponding F, and their optimal parameters can be decided. Only those

‘weak’ eigenvalues (a�k < k or j�k < k) are included so that those unstable or poorly damped

electromechanical oscillation modes are relocated to a more stable region. If problem (9) is solvable

(i.e. a feasible solution exists), all the eigenvalues should satisfy the inequalities in Equations (7) and

(8) and the value of the objective function will be equal to zero; otherwise, it will be greater than zero.

MinfF;KPSSg[fKi;T1i;T2i;T3i;T4ig

f ðPÞ ¼Xa�k<k

ða�k�kÞ2 þ

Xj�k<k

ðj�k�kÞ2 (9)

s:t:Ki;min � Ki � Ki;max

T1;min � T1i � T1;max;T2;min � T2i � T2;max

T3;min � T3i � T3;max;T4;min � T4i � T4;max

where KPSS, F, Ki, T1i=T2i and T3i=T4i (i2F) are to be determined; P stands for the PSS parameter

vector; Ki;max=Ki;min, T1i;max=T1i;min, T2i;max=T2i;min, T3i;max=T3i;min and T4i;max/T4i;min are the limits of

PSS parameters and i2F.

To avoid the problem of dimensionality in solving the combinational optimization problem in

Equation (9), the PSS number (i.e. number of tentative PSS sites) is gradually increased and its effect is

evaluated recursively until the damping criteria are satisfied. This will be discussed further in the next

section. Also the problem in Equation (9) is a complex optimization problem characterized by an

implicit objective function of discrete-continuous variables. Besides, it is related to the evaluation of

probabilistic eigenvalues. Thus it is very difficult to resolve it using conventional methods, because the

continuity of the objective function does not exist and the Jacobian matrix of the objective function

cannot be easily obtained. In EC, GA is very powerful and flexible in dealing with optimization

problems with mixed discrete-continuous variables [22,23], which is beyond the capability of other

purely real-coded EC methods such as DE. When the probabilistic PSS design considers the optimal

siting in problem (9), it can be described as an optimization problem with mixed discrete-continuous

variables. Hence, a mixed integer-binary coded GA will be applied to solve this problem.

4. OPTIMIZATION BY GENETIC ALGORITHM

Heuristic algorithms obtain optimal or near-optimal solutions of a problem by searching over a

subspace. The most important advantage of heuristic algorithms is that they are not limited by

assumptions such as continuity, availability of derivative of objective function, etc. Techniques such as

tabu search [24], simulated annealing [25], and GA [26] have been applied to controller design

researches. GA is based on the mechanism of natural selection and it often produces high quality

solutions; those flexible coding techniques developed over the past 20 years further facilitate its

applications in various science and engineering fields [22,23].

The overall procedure for probabilistic PSS design with optimal-siting is illustrated in Figure 1 and

explained below. The inner loop of Figure 1 is the GA search process, which is mainly composed of

crossover, mutation and selection operation. To ascertain optimal PSS siting, the effect of a given

number of PSSs is to be evaluated recursively, as shown in the outer loop of Figure 1. The value of KPSS

(i.e. there are KPSS tentative PSS installed) is first initialized simply as one or some pre-specified

number (e.g. one third of total number). Proper selection of initial value of KPSS can reduce the number

of GA computation (i.e. the number of outer loop in Figure 1). Then the proposed GA is employed to


1412 Z. WANG ET AL.

solve the optimization problem. If the solution does not satisfy the pre-specified damping criteria, then

the number of PSSs will be increased and the GA search process will be restarted. The update of KPSS

and the GA search process, that is, the overall process, will stop under two conditions: the objective

function becomes zero (here the absolute value of the objective function below a very small tolerance,

say 10�8, is regarded as zero), i.e. damping criteria are satisfied; or the pre-set maximum number of

potential PSS locations is reached. For the GA search process in the inner loop, it can be stopped by one

of the following two independent stop criteria: (1) the specified maximum number of generations is

reached and (2) there occurs a severe stagnancy phenomenon, i.e. the best value of objective function

found so far has not changed over the past 20 iterations/generations.

4.1. GA coding scheme

A configuration of KPSS PSSs is composed of two types of parameters: the location index of PSSs

(integer number) and PSS parameters (real number). A particular GA coding scheme is developed to

deal with the two types of numbers as follows. There are two types of bits in the GA chromosome,

integer bits and binary bits, as shown in Figure 2(a)–(e). In Figure 2, the case of KPSS ¼ 4 and NPSS ¼ 7

is taken as an example.

Figure 1. Flowchart of the recursive GA for PSS design with optimal-siting.



There are two portions in the integer bits. The first portion is called PSS-site bits, as shown in the first

KPSS length parts of the chromosomes in Figure 2(a)–(e). PSS-site bits are composed of KPSS

length integers; each integer (�NPSS) represents a location index in set F that will install a PSS. The

second portion next to PSS-site bits is called dumb-PSS-site bits, as shown in the shadow parts in

Figure 2(a)–(e). Dumb-PSS-site bits are composed of (NPSS�KPSS) length integers; each integer

represents a potential location index that is not contained in set F. Each PSS-site bit as well as

dumb-PSS-site bit could appear at maximum once in the string. The order of integer bits in the string is

not important for a given configuration, but could have its importance when applying the operator of

crossover. The dumb-PSS-site bits do not have any effect on the configuration, rather they are

indispensable in keeping the diversity of GA population and assisting the crossover and mutation

operations. The binary bits in the remaining parts of chromosomes correspond to all potential PSS

parameters, which are coded in binary bits. There are also some dumb parameter bits in them, which

have no effect in the calculation of the value of objective function (9) temporarily; but only those binary

bits pertain to PSS-site bits will be involved in the calculation.

As a simple illustration in Figure 2(a), there is a pair of parents with four tentative PSS sites

(KPSS ¼ 4) among seven potential locations (NPSS ¼ 7). In the first parent, four PSS-site bits 3, 5, 7, 1

represent that the PSSs are tentatively installed at generator index 1, 3, 5 and 7. The three dumb-PSS-

site bits are 2, 4 and 6. The remaining 0–1 codes correspond to all potential PSS parameters arranged in

sequence.

4.2. GA operators

4.2.1. Crossover. The main crossover operator used here is a variation of the conventional two-point

crossover with the specified probability pc. For an ordinary two-point crossover of binary GA, two

crossover points are generated at random for the parents. Then the bits within the two positions of each

parent are swapped. In this paper, the PMX [22,23] will be utilized in view of the particularity of the

coding scheme discussed above. PMX is a bit-by-bit crossover operator, which can repair bit conflict

problems that are not easy to be handled.

The application of PMX is illustrated in Figure 2(a)–(e) and explained below. In these figures, each

GA string consists of KPSS PSS-site bits, (NPSS�KPSS) dumb-PSS-site bits and the remaining ‘ PSS-parameter bits. ‘ is the total length of binary string, which is dependent of the precision of data adopted[22,23]. Two crossover positions c1 and c2 are generated over [1,KPSS þ ‘] at random with

1 � c1 � c2 � KPSS þ ‘. The dumb-PSS-site bits do not participate but assist in the crossover

operation of PSS-site bits. According to different crossover positions, there are five crossover strategies

summarized as follows:

(i) If c1 < c2 � KPSS, as illustrated in Figure 2(a), since each PSS candidate could appear at

maximum once in a GA string, the PMX-based repair will be executed bit by bit in case there is

a bit conflict. In Figure 2(a), the integer bits within c1 and c2 of two parents are swapped:

specifically, when 7 is swapped with 5 and there is conflict happened with two 5 in the first

parent and two 7 in the second parent; for the first parent, the PMX repairs the conflict by

replacing the 5 before c1 with 7; for the second parent, the PMX repairs the conflict by

replacing the duplicate 7 in the dumb-PSS-site bits with 5. During this process, there is an

implicit bit swap happened between the dumb-PSS-site bits and the PSS-site bits. This

operation will be applied for remaining bits one by one until the swapping of all bits within

Figure 2. Cases of two-point crossover operation with KPSS ¼ 4 and NPSS ¼ 7.


1414 Z. WANG ET AL.

c1 and c2 is finished. In this case, the parameter bits would not be affected during the repairing

process.

(ii) If c1 � KPSS < c2, as illustrated in Figure 2(b), the first position c1 and an ‘implicit’ position

KPSS (i.e. automatically being a crossover position) are the two cutting positions for PMX

operation on the PSS-site bits. The second position c2 defines a cutting position for an ordinary

one-point crossover on the 0–1 bits of PSS parameters.

(iii) If KPSS < c1 < c2, as illustrated in Figure 2(c), there is no operation on any integer bits. An

ordinary two-point crossover is executed according to two cutting positions c1 and c2.

(iv) If c1 ¼ c2 � KPSS, as illustrated in Figure 2(d), the position c1 (c2) and an implicit position

KPSS define two cutting positions for PMX operation on the PSS-site bits.

(v) If KPSS < c1 ¼ c2, as illustrated in Figure 2(e), the position c1 (c2) defines a cutting position

for an ordinary one-point crossover operated on the PSS parameter bits.

4.2.2. Mutation. There are two types of mutation operators supported in the study: swap mutation

and bit-flipping mutation. A single-bit swap mutation is supported between PSS-site bits and dumb-

PSS-site bits with the specified probability pms. In Figure 3, one position x1 is selected at random in the

PSS-site bits; another x2 is selected at random in the dumb-PSS-site bits. The bits at two positions are

swapped so that new PSS siting configuration is produced. Another mutation operator is a bit-flipping

mutation performed on the binary bits [22] with the specified probability pmr. Different mutation

probability values are set for the two different types of mutation because they are independent of one

another.

4.2.3. Selection. In the study, a ranking selection strategy is employed. Individuals are sorted in

ascending order of their objective function values [23]. Besides, an elitismmechanism is adopted in the

GA procedure.

4.3. Design procedure

The problem in Equation (9) can be solved by the proposed GA according to the following steps:

Step 1 Initialization: Initialize NP chromosomes in the population in the following way:

Step 1.1 For each chromosome, make a random permutation of NPSS-length integer bits,

which represent NPSS site indexes.

Step 1.2 For each binary bit in the remaining ‘-length binary GA string, the bit is 0 when the

value of a random number generated between 0 and 1, rand [0,1], is less than 0.5,

otherwise, it is 1.

Step 2 Generate the next generation of NP chromosomes in the following way:

Step 2.1 Evaluate the objective function of the chromosomes in the current generation using

Equation (9). In the present work, the fittest chromosome in the current generation

is always retained in the next generation.

Step 2.2 Selection: Select two chromosomes as the parents by the ranking selection method.

Step 2.3 Crossover: Generate two crossover positions c1 and c2 at random, apply the PMX

to the two selected parents in the current generation when rand [0,1] is less than pc.

Otherwise, the two parents are retained and are taken as the child chromosomes in

the next generation. Repeat the selection step in Step 2.2 and the present step until

NP child chromosomes are formed in the next generation.

Figure 3. Swap mutation on the integer bits.



Step 2.4 Mutation: For each chromosome in the next generation, apply the swap mutation to

the integer bits of the chromosome when the swap mutation probability, pms, is

greater than rand [0,1]; otherwise, the integer bits will remain intact. Apply the bit-

flipping mutation to every binary bit one by one when the bit-flipping mutation

probability pmr is greater than rand [0,1]; otherwise, the binary bit will remain intact.

Step 3 The next generation formed in Step 2 is now taken to be the current generation. New

generations are produced by repeating the solution process starting from Step 2 until the

specified maximum number of generations is reached; or the solution stagnancy happens; or

stability criteria are satisfied.

5. CASE STUDIES

In this section, two test systems in Reference [18] are employed to demonstrate the effectiveness of the

proposed method. In the studies, the criteria for the damping ratio and damping constant are chosen as

jC ¼ 0:1 and aC ¼ �0:1 for both systems. The distribution constants k for the two systems are set to 4.0

and 3.5, respectively. The influence of algorithm parameters on the convergence performance of the

recursive GA including the different combinations of crossover probability pc, the bit-flipping mutation

probability pmr and swap mutation probability pms are studied. The software implementing the proposed

algorithm in the studies was developed using FORTRAN language and executed on an Intel P4 2.66GHz

CPU and 1G RAM computer. The pre-specified parameters for two test systems are listed in Table I.

5.1. Three-machine Power System (system I)

The three-machine nine-bus test system is shown in Figure 4, where G3 represents an equivalent

lumped interconnection system and is regarded as the slack bus generator. All machines are

represented as fourth-order models and equipped with fast-acting static exciter. The loads are modelled

as constant impedances. All dynamic parameters can be found in Reference [21]. Normal operation

values of nodal powers and PV bus voltages shown in Figure 4 are regarded as their expectations. Each

nodal power and PV bus voltage is assigned with standardized daily operating curves [16]. From these

curves, 480 operating samples are created and covariances of nodal injections are determined. The

statistical characteristics can be captured by the probabilistic eigenvalue analysis. Four hundred and

eighty system operating samples are created and the worst scenario is only marginally stable with

j < 0:005. The probabilistic eigenvalue analysis is performed and the worst damped modes are listed

in Table II. The corresponding modes have the standardized expectations of both a�k and j

�k less than 4,

which is regarded as inadequate for robust stability.

Table I. Pre-specified parameters.

Parameters System I System II

Maximum generation 100 100Population size 50 200Initial KPSS 1 3NPSS 2 7

Figure 4. Three-machine power system.


1416 Z. WANG ET AL.

5.1.1. Determining parameter settings. To improve the system damping, G1 and G2 in the system in

Figure 4 are two potential sites for PSS installation. Since crossover and mutation probability have

great impacts on the performance of canonical GA [22], several experiments on investigating the

influence of different combinations of fpc; pmr; pmsg on the convergence performance of the proposed

method were carried out first. The values of the parameters are set first at: pc2{0.50, 0.75, 0.90},pmr2{0.01, 0.05, 0.1, 0.3} and pms2 {0.001, 0.005, 0.01, 0.05}. The parameters were investigated for

50 trial simulations in succession according to three scenarios: (a) altering pc with pmr ¼ 0.01 and

pms ¼ 0.005; (b) altering pmr with pc ¼ 0.9 and pms ¼ 0.005; (c) altering pms with pc ¼ 0.9 and

pmr ¼ 0.05. The reasons for above arrangement are to be explained hereinafter.

The summary of the experiment results is given in Table III and the convergence curves of average

objective function are shown in Figure 5(a)–(c). In Table III, different cases are compared according to

the number of iterations taken (i.e. the evaluation number), with the maximum, the minimum and the

average statistics of total simulations given. Apparently, the less the average evaluation number, the

faster does the algorithm converge. Statistics information on the proportion of the proposed method

that can converge in one PSS (KPSS ¼ 1) and in two PSSs (KPSS ¼ 2) are also given in Table III.

Similarly, the larger proportion in caseKPSS ¼ 1, the faster does the algorithm converge. In Figure 5(a)–

(c), the highlighted/bold curves are those cases that can converge in case KPSS ¼ 1, while those normal

curves are related to those that can only converge in case KPSS ¼ 2. In common, the simulation process

in case KPSS ¼ 2 runs faster than that in case KPSS ¼ 1 for the same configuration of fpc; pmr; pmsg, butthe convergence performance between different cases of KPSS ¼ 2 are trivial. So only the cases with

KPSS ¼ 1 are utilized to evaluate the algorithm performance in the study. The highlighted solid line in

each figure is the curve corresponding to a preferable parameter setting in each scenario.

Table II. Electromechanical modes of the open-loop system I.

No. a b sa a� Pa j sj j� Pj

1 �0.910 7.895 0.3161 2.56 0.995 0.1145 0.0402 0.36 0.6412 �1.298 9.905 0.7779 1.54 0.938 0.1299 0.0783 0.38 0.649

Expectations: l ¼ a� jb and j; Standard deviation: s; Standardized expectation: a� ¼ �ðaþ 0:1Þ=sa and j� ¼ ðj�0:1Þ=sj;Distribution probabilities: Pa ¼ Pfa < 0g and Pj ¼ Pfj > 0:1g. Bold values denote all unsatisfactory damping constants,damping ratios and their distribution probabilities.

Table III. The results of GA parameter experiments.

pc pmr ¼ 0:01, pms ¼ 0:005 Evaluation number Trial statistics (%)

Max Min Avg KPSS ¼ 1 KPSS ¼ 2

0.50 2374 88 494 88 120.75 1995 93 483 86 140.90 2474 96 476 88 12

pmr pc ¼ 0:90, pms ¼ 0:005 Evaluation number Trial statistics (%)

Max Min Avg Max Min

0.05 1615 98 235 98 20.10 2304 99 326 94 60.30 2991 99 382 92 8

pms pc ¼ 0:90, pmr ¼ 0:05 Evaluation number Trial statistics (%)

Max Min Avg Max Min

0.001 1807 99 339 90 100.01 2450 99 253 96 40.05 2055 99 363 92 8



From the experiment results in Table III and Figure 5(a)–(c), the following can be observed:

(i) For pmr ¼ 0.01 and pms ¼ 0.005, the best setting for pc is 0.9, which corresponds to the curve

that converges most rapidly in KPSS ¼ 1. In the remaining experiments of scenario (b) and (c),

pc ¼ 0.9 is adopted.

(ii) For pc ¼ 0.9 and pms ¼ 0.005, the best setting for pmr is 0.05. In the remaining experiments of

scenario (c), pmr ¼ 0.05 is adopted.

(iii) For pc ¼ 0.9 and pmr ¼ 0.05, the best setting for pms is 0.005.

(iv) From the above observations, the values of the parameters infpc; pmr; pmsg are set to {0.90,

0.05, 0.005} and applied to the studies hereinafter.

5.1.2. Resultant PSS settings. The resulting PSS settings are given in Table IV, and all close-loop

electromechanical modes as shown in Table V are adequate for robust stability. When compared with

the original values in Table II, the system stability is much improved. In Reference [18], PSSs are

installed on G1 and G2 and the system damping are satisfactorily improved. By comparison, only G1 is

adequate to meet the robust stability requirement of this system with the proposed method.

5.1.3. Transient performance. To complete the study, the performance of the proposed approach is

evaluated and compared with another two methods: a gradient-based conventional optimization

method (CPSS), which is originated in Reference [4] and applied in Reference [18] under the worst

Gen

10 20 30 40 50 60

Ave

rage

val

ue o

f F(x

)

0

10

20

30

40

50

60

70

pmr=0.01, KPSS=1pmr=0.01, KPSS=2pmr=0.05, KPSS=1pmr=0.05, KPSS=2pmr=0.10, KPSS=1pmr=0.10, KPSS=2pmr=0.30, KPSS=1pmr=0.30, KPSS=2

scenario (a)

Gen10 20 30 40 50 60

Ave

rage

val

ue o

f F(x

)

0

20

40

60

80

100

120

140

160

Pc=0.50, KPSS=1Pc=0.50, KPSS=2Pc=0.75, KPSS=1Pc=0.75, KPSS=2Pc=0.90, KPSS=1Pc=0.90, KPSS=2

scenario (b)

Gen10 20 30 40 50 60

Aver

age

valu

e of

F(x

)

0

10

20

30

40

50

pms=0.001, KPSS=1pms=0.001, KPSS=2pms=0.005, KPSS=1pms=0.005, KPSS=2pms=0.010, KPSS=1pms=0.010, KPSS=2pms=0.050, KPSS=1pms=0.050, KPSS=2

scenario (c)

Figure 5. The influence of different parameters on the recursive GA.

Table IV. Resulting PSS parameters for three machine system.

Site K T1 T2 T3 T4

G1 3.532 0.650 0.183 0.136 0.044


1418 Z. WANG ET AL.

scenario of 480 operating conditions; the DE method in Reference [18] denoted as DE-PPSS, which is

applied under the same multiple operating conditions. The original scheme of PSSs for these two

methods in Reference [18] are reserved all through the study. Besides, the same damping criteria are

adopted in three methods.

To simulate a large disturbance imposed on the system, the same three-phase-to-earth fault near bus

6 at t¼ 0.2 seconds is applied as shown in Figure 4, the fault is then cleared by line isolation without

reclosure. The study system with the large disturbance impulsion will be tested upon 24 operating

conditions of each hour in the operating curves with three methods independently applied. A nonlinear

time domain simulation will be conducted for each case. The output limits of field voltage are

set to �5 p.u. respectively. The output limits of PSSs are set to �0.1 p.u. The following transient

performance indices J1 and J2 are calculated, which can measure the averaged total variation (ATV) of

signals,

J1 ¼ 1

N

XNn¼1

Z t¼tsim

t¼0

D"ðtÞj jdt (10)

J2 ¼ 1

N

XNn¼1

Z t¼tsim

t¼0

t � D2"ðtÞdt (11)

where "ðtÞ represents the selected output in response to the fault and D"ðtÞ ¼ "ðtÞ�"ðt�1Þ; N is the

total number of samples and tsim is the total simulation time. It is obvious that the lower the values of

these indices, the smaller deviation of the signal will present in response to disturbance.

The ATV comparison results among three methods: the CPSS method, the DE-PPSS method and the

proposed method (denoted as OS-PPSS) are listed in Table VI, where the following physical variables

are investigated: (i) Rotor angle of G1, G2 (relative to G3) in degree; (ii) Field voltage of G1, G2 and

G3 in p.u.; (iii) Terminal voltage of G1, G2 and G3 in p.u. It is important to observe that the ATV values

of OS-PPSS here are slightly larger than or comparable to those of DE-PPSS, but much lower than

those of CPSS, the reason is that the two normalized damping ratio j� of DE-PPSS in Reference [18]

are 10.40 and 23.92 while the corresponding values of OS-PPSS (see Table V) are as low as 4.15 and

4.24, respectively. Compared with OS-PPSS and DE-PPSS, the CPSS is designed based only on a

linearized model under a stressed operating condition and its performance may be unsatisfactory when

the operating environment varies significantly because of large disturbances. On the whole, the robust

stability criteria are well satisfied in OS-PPSS and the number of PSSs has been reduced from 2 to 1.

Accordingly, the cost and complexity of the OS-PSS scheme is reduced due to the number of PSS

decreases. Typical transient response curves of rotor angles of G1 and G2 at light, medium and heavy

load conditions in Figure 6 validate these observations.

Table V. Electromechanical modes of the closed-loop system I.


1 �4.391 7.413 0.9290 4.62 1.00 0.5096 0.0967 4.24 1.002 �1.945 9.434 0.1337 13.80 1.00 0.2247 0.0300 4.15 1.00

Table VI. Performance indices of system I.

d1 d2 vf1 vf2 vf3 vt1 vt2 vt3

J1 M1 0.1226 0.1235 0.0565 0.0421 0.0253 0.0024 0.0032 0.0033M2 0.0492 0.0586 0.0390 0.0291 0.0206 0.0022 0.0030 0.0033M3 0.0711 0.0727 0.0710 0.0317 0.0251 0.0024 0.0031 0.0033

J2 M1 0.0202 0.0231 0.0024 0.00164 0.00059 0.00005 0.00012 0.00016M2 0.0018 0.0032 0.0014 0.00069 0.00045 0.00005 0.00012 0.00016M3 0.0074 0.0088 0.0088 0.00105 0.00065 0.00005 0.00012 0.00016

M1, M2 and M3 represent CPSS, DE-PPSS and OS-PPSS, respectively.



5.2. Eight-machine Power System (system II)

The second test system in Figure 7 is a three-area system consisting of eight machines and 24 buses. All

generators are represented as sixth-order models and equipped with IEEE-I exciters and speed

governors. Details of network parameters, nodal powers, generator parameters and operating curves

can be referred to in Reference [18]. Similarly, the statistical characteristics of the system can be

captured by the probabilistic eigenvalue analysis based on sampled 480 operating conditions. Modal

analysis shows that an inter-area mode involvingmachines in different areas is a lightly damping mode.

There are seven unsatisfactory electro-mechanical oscillation modes and the details of these modes are

given in Table VII.

5.2.1. Results of tuning. To improve the system damping, G1–G7 are seven potential sites for PSS

installation. In Reference [18], six power-signal PSSs are installed on G1, G2, G3, G5, G6 and G7, and

the system damping are satisfactorily improved. By the method of this paper, the resulting PSS

parameters and electro-mechanical modes are listed in Tables VIII and IX, with the system damping

Figure 6. Transient responses of system I under typical load conditions.


1420 Z. WANG ET AL.

effectively enhanced as well. By comparison, PSSs on G3, G5, G6 and G7 are adequate to meet the

robust stability conditions (7)–(8).

5.2.2. Transient performance. Here a six-cycle three-phase fault is applied to the tie-line 8-15 near

bus 15 at t¼ 0.2 seconds as shown in Figure 7. The fault is then cleared by line isolation without

reclosure, making the tie-line out of operation. The deviations of generators’ electrical power in p.u.

are investigated. The performance indices are presented in Table X; and the electrical power responses

Figure 7. Eight-machine power system.

Table VII. Electromechanical modes of the open-loop system II.


1 �1.925 15.400 0.080 22.95 1.00 0.124 0.007 3.40 0.99972 �0.773 10.754 0.070 9.61 1.00 0.072 0.006 �4.59 0.003 �0.590 9.705 0.039 12.50 1.00 0.061 0.003 �12.65 0.004 �0.604 7.888 0.024 20.80 1.00 0.076 0.004 �5.42 0.005 �0.600 7.381 0.083 6.00 1.00 0.081 0.010 �1.93 0.02686 �0.365 6.420 0.046 5.72 1.00 0.057 0.007 �6.05 0.007 �0.033 3.854 0.007 �9.76 0.00 0.009 0.002 �49.55 0.00

Bold values denote all unsatisfactory damping constants, damping ratios and their distribution probabilities.

Table VIII. PSS parameters determined for system II.

Site K T1 T2 T3 T4

G3 �1.299 0.102 0.068 0.075 0.180G5 �0.195 0.266 0.196 0.176 0.158G6 �0.020 0.161 0.162 1.959 0.075G7 �1.104 0.146 0.164 0.097 0.166



of G8 are plotted in Figure 8. Similarly to last case, the ATV values of OS-PPSS are slightly larger than

or comparable to those of DE-PPSS, but much lower than those of CPSS. The least damping ratio j� inTable IX is about 3.51; by comparison, the corresponding value in Reference [18] is close to 4.0. The

proposed OS-PPSS scheme has better performance compared with CPSS, while using a reduced

number of PSS from 6 to 4 compared with DE-PPSS in Reference [18].

5.2.3. Discussion on large-scale system application. In larger-scale power systems, when these

population-based EAs are applied to solve the optimal PSS design and optimal location problems,

their convergence performance mainly depends on the number of PSS candidates considered in

the optimization problem, which can be limited to those most effective generators or areas in

power systems. It is expected that the computing time of the proposed approach will increase with

the increment of PSS candidate number. Fortunately, it is very amenable to parallel

implementation by nature so that computation time can be greatly reduced. The same case

study has been performed on a master-slave PC-cluster, which consists of one control node and 30

working nodes with 61 processors. Each processor is configured with 1 G RAM and dual Intel

Xeon 2.66 GHz CPUs. It has been found that the time consumption of the GA for this eight-

Table IX. Electromechanical modes of the closed-loop system II.


1 �1.959 15.465 0.081 22.95 1.00 0.126 0.007 3.68 1.002 �1.317 10.850 0.063 19.32 1.00 0.121 0.005 4.01 1.003 �1.451 10.107 0.134 10.08 1.00 0.142 0.012 3.51 1.004 �0.939 7.448 0.027 31.07 1.00 0.125 0.003 8.22 1.005 �2.205 7.410 0.027 77.69 1.00 0.285 0.008 22.91 1.006 �7.449 5.474 0.129 57.19 1.00 0.806 0.009 81.29 1.007 �2.149 3.099 0.078 26.27 1.00 0.570 0.032 14.63 1.00

Table X. Performance indices of system II.

Pe1 Pe2 Pe3 Pe4 Pe5 Pe6 Pe7 Pe8

J1 M1 0.0770 0.0571 0.1414 0.0845 0.1208 0.0678 0.2051 0.4106M2 0.0759 0.0428 0.0765 0.0607 0.0777 0.0519 0.1087 0.1385M3 0.0802 0.0337 0.0718 0.0568 0.0930 0.0506 0.1194 0.2052

J2 M1 0.0066 0.0023 0.0146 0.0058 0.0124 0.0033 0.0175 0.1069M2 0.0068 0.0020 0.0096 0.0047 0.0093 0.0029 0.0051 0.0121M3 0.0069 0.0018 0.0093 0.0046 0.0096 0.0029 0.0053 0.0187

Figure 8. Transient responses of G8 of system II under typical operating conditions.


1422 Z. WANG ET AL.

machine test system can be reduced to less than 3 minutes, while the whole process lasts about

75minutes for single PC platform.

6. CONCLUSION

This paper has formulated the problem of PSS parameter tuning and the determination of the minimum

number of PSS installation under multiple operating conditions as a combinatorial optimization

problem. A recursive GA has been developed to solve this problem with the application of an integer-

binary mixed coding scheme and PMX-based crossover operator. The effects of the different

combinations of the probability of crossover, random mutation and swap mutation on the performance

of the proposed recursive GA have been examined. Two test systems by the probabilistic eigenvalue

analysis and nonlinear simulation show that the proposed method is powerful and the minimum PSSs

locations can be determined to satisfy adequately robust stability requirements.

7. LISTS OF SYMBOLS AND ABBREVIATIONS

7.1. Symbols

lk ¼ ak þ jbk the kth eigenvalue

ak;jk;sak ;sjk expectation and standard deviation of damping constant and damping ratio

a0k; j

0k extended expectations of damping constant and damping ratio

a�k ; j

�k standardized expectations of damping constant and damping ratio

aC; jC acceptable limits for damping constant and damping ratio

c1, c2 two crossover positions

k distribution constant

Ki; Tw; T1i=T2i; T3i=T4i parameters of PSS control loop

Kpss number of current tentative PSSs

l total length of binary string

N total number of samples

NPSS number of PSS candidates

NP number/size of GA population

x1, x2 two mutation positions

F PSS candidate sets

pc crossover probability

pmr bit-flipping mutation probability

pms swap mutation probability

tsim total simulation time

7.2. Abbreviations

ATV averaged total variation

CPSS conventional power system stabilizer

DE differential evolution

EC evolutionary computation

GA genetic algorithm

PMX partially matched crossover

PSI probabilistic sensitivity indices

PSS power system stabilizer

ACKNOWLEDGEMENTS

This work is jointly supported by Research Grants Council of Hong Kong (Project No.: PolyU 5154/08E), theChina Postdoctoral Science Foundation (Grant No. 20090461354) and the open foundation of State Key Lab. ofPower System in Tsinghua University (SKLD09KM14). The early research foundation of Professor K.W.Wang in



the Hong Kong Polytechnic University, who now is with the School of Electric Engineering, ZhengzhouUniversity, is greatly appreciated.

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