comparison and improvement of evolutionary programming techniques for power system optimal reactive...

Comparison and improvement of evolutionaryprogramming techniques for power systemoptimal reactive power flow

C.H. Liang, C.Y. Chung, K.P. Wong and X.Z. Duan

Abstract: Some recently developed evolutionary programming (EP) techniques for the optimalreactive power flow (ORPF) problem are investigated. By comparing the performances of fourcontrol schemes of the strategy parameter (CSSPs), two principles for designing effective CSSPs areidentified and are further used to develop two improved CSSPs. The effectiveness of the L!evydistribution based technique of adaptive fast EP is also studied. The investigations are conductedwith simulations on the ORPF problems of the IEEE 14-, 30- and 118-bus standard test systems.

1 Introduction

Optimal reactive power flow (ORPF) is of significantinfluence on the economic and secure operation of powersystems. It is, however, a complex combinatorial optimisa-tion problem involving a nonlinear objective function,nonlinear constraints and discrete variables. Since the late1960s, many conventional gradient-based algorithms havebeen used to solve ORPF. Their common disadvantages are(a) requiring continuous and differentiable objective func-tions, (b) easy to be trapped into local minima due to theheavy dependence of the ultimate solution on the singlestarting point, and (c) difficult to handle discrete controlvariables, but simple rounding off may cause significanterrors and even infeasible solutions. To overcome thesedisadvantages, more flexible and robust evolutionaryalgorithms (EA), including genetic algorithm (GA), evolu-tionary strategy (ES), evolutionary programming (EP) andparticle swarm optimisation (PSO) have been applied to theORPF problems since the mid-1990s [1–9]. The superiorityof EA to conventional methods has been verified [2, 10] interms of searching the near global optimal solution, and thepotential of solving ORPF by EA is affirmed although it isstill suffering some problems, such as premature conver-gence and long computing time.

As a branch of EA, EP uses only the mutation operatorto generate new individuals. In general, mutation hasgreater exploratory power than crossover [11], so EP isless likely to fall into local minima [10]. However, standardEPs converge slowly and may not be directly applicableto practical problems. So, during the last decade, new

techniques of EP have emerged to speed up convergenceand improve solution quality.

One class of new EP techniques is new control schemes ofthe strategy parameter (CSSPs). This paper first comparesfour typical CSSPs (named CSSP1-CSSP4) developed in theliterature on the ORPF problems of the IEEE 14- and30-bus standard systems. After analysing the comparisonresults, two principles for designing effective CSSPs areidentified. Based on them, two improved CSSPs (namedICSSP1 and ICSSP2) are then proposed and theirperformances are verified on the IEEE 14-, 30- and 118-bussystems.

Another class of new EP techniques is to use new randomdistributions other than Gaussian distribution in themutation formula. Yao [12] proposed the fast EP (FEP)and improved fast EP (IFEP) techniques based on theCauchy distribution, and Lee [13] further generalised theminto a L!evy distribution-based adaptive fast EP (AFEP)technique. This paper also investigates the effectiveness ofAFEP for the ORPF problem.

2 EP-based ORPF

The goal of ORPF is to minimise real power loss by settingthe control variables u, including generator voltages (VG),transformer tap ratios (T) and the susceptances of shuntcompensators (Bsh), while maintaining the dependentvariables x, including voltages of PQ-buses and reactivepower generations of generators, within limits:

min FL ð1Þ

s:t: hðu; xÞ ¼ 0 ð2Þ

umin � u � umax ð3Þ

xmin � x � xmax ð4Þwhere FL denotes real power loss, (2) is the load flowequations, and (3) and (4) are constraints on control anddependent variables. Among the three types of controlvariables, VG is continuous, whereas T and Bsh are discrete.

In EP-based ORPF, an initial population of N (popula-tion size) candidate solutions or individuals is first generatedrandomly and is used as the parent population of the firstiteration or generation. Each individual of the parent

C.H. Liang and X.Z. Duan are with the College of Electrical and ElectronicEngineering, Huazhong University of Science and Technology, Wuhan,People’s Republic of China

C.Y. Chung and K.P. Wong are with the Computational IntelligenceApplications Research Laboratory (CIARLab), Department of ElectricalEngineering, The Hong Kong Polytechnic University, Hong Kong, People’sRepublic of China

E-mail: [email protected]

r IEE, 2006

IEE Proceedings online no. 20050081

doi:10.1049/ip-gtd:20050081

Paper first received 14th March 2005 and in final revised form 21st June 2005

228 IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 2, March 2006

population is then perturbed (mutated) according to amutation scheme to generate a same-sized offspringpopulation. Each individual of the parent and offspringpopulation is evaluated by a load flow program and isassigned a fitness value according to a fitness function toreflect its superiority. The higher its fitness value is, thebetter the individual. Then, from the 2N individuals of theparent and offspring population, the N better ones areselected according to a selection scheme to form the parentpopulation of the next generation. This process repeats untila stopping criterion, which is a maximum number ofgenerations in this paper, is fulfilled. The flow chart of EP-based ORPF is shown in Fig. 1 with details describedbelow.

2.1 InitialisationGenerally, the initial population is randomly generatedaccording to:

ui;j½0� ¼ Uðuminj ; umax

j Þ ð5Þwhere ui,j[k] denotes the value of the jth control variable ofthe ith individual in the kth generation and k¼ 0 representsthe initial population. Uðumin

j ; umaxj Þ generates uniform

random values between and uminj and umax

j , which denote,

respectively, the lower and upper limits of control variable j.For discrete variables of type T or Bsh, the randomlygenerated values are rounded off to their nearest validvalues. In this paper, one individual of the initial populationis given as the initial operating state, i.e. its control variablesare set to the values they take in the initial operating state ofthe power system. Other individuals are randomly gener-ated. This arrangement can enhance the overall perfor-mance because the best individual of the initial population isat least as good as the initial operating state. This can avoidthe case that the best individual of the initial populationbecomes very poor if the whole population is generatedrandomly.

2.2 MutationThe value of each control variable in the individuals of theoffspring population is obtained by perturbing the corre-sponding variable in the individuals of the parent popula-tion according to:

u0i;j½k� ¼ ui;j½k� þ si;j½k�Ni;jð0; 1Þ ð6Þ

where si,j[k] denotes the corresponding strategy parameterof ui,j[k] and Ni,j(0,1) is a Gaussian random value generatedanew at each time of mutation. If the u0i;j½k� generated by (6)

is outside the range of ðuminj ; umax

j Þ, it is fixed to umaxj or umin

j .

For discrete variables of type T or Bsh, the u0i;j½k� is roundedoff to the nearest valid value.

2.3 Load flow and fitness evaluationFor each individual in the parent and offspring population,load flow is calculated and a fitness value fi is assigned to itaccording to:

f1 ¼1

FLi þ KV FVi þ KQFQið7Þ

FVi ¼XnPQ

l¼1

jVPQil � VPQil limjVPQilmax � VPQilmin

FQi ¼XnG

l¼1

jQGil � QGil limjQGilmax � VGilmin

where FVi and FQi denote the sum of the normalisedviolation of PQ-bus voltages and generator reactive powergenerations of individual i; VPQil lim and QGil lim denote theviolated upper or lower limit; KV and KQ are the penaltycoefficients. In this paper, KV and KQ of the kth generationare dynamically set according to:

KV ðkÞ ¼ KQðkÞ ¼ Kmin þ kKmax � Kmin

Gen � maxð8Þ

where Gen�max is the maximum number of generations,and Kmax and Kmin are set to 1.0 and 0.005, respectively.

2.4 SelectionThe q-tournament selection scheme [14] is adopted in thispaper. Each individual is assigned a score si according to:

si ¼Xq

l¼1

1 if fi4fl

0 if fiofl

�ð9Þ

where fi is the fitness of individual i, fl is the fitness of anopponent individual randomly selected from the whole 2Nindividuals, and q is called the tournament size. The Nindividuals with higher scores are selected to form theparent population of the next generation. Tournament sizeq is set to 0.9N in this paper.

3 Formulation of CSSP1–CSSP4

It is obvious in (6) that the performance of an EP algorithmis to a large extent determined by the control scheme of thestrategy parameter (CSSP). Four typical CSSPs (CSSP1–CSSP4), which are mainly used in the literature of thepower engineering field, are studied in this paper.

3.1 CSSP1CSSP1 is the classical EP (CEP) [13, 15]. It is actuallyborrowed from ES and is widely used as a general mutationscheme in many problem contexts [16]. It can act as areference base for comparison:

si;j½k� ¼ si;j½k � 1� expftNið0; 1Þ þ t0Ni;jð0; 1Þg ð10Þ

t ¼ 1ffiffiffiffiffiffiffi2Mp and t0 ¼ 1ffiffiffiffiffiffiffiffiffiffiffi

2ffiffiffiffiffiMpp

where Ni(0,1) denotes a Gaussian random value, which isfixed for individual i, and while Ni,j(0, 1) is generated anewat each time of mutation for the jth control variable. M isthe total number of control variables. si,j[0] should be

initialisation

yes

stop

start

load flow and fitness evaluation

selection

terminate?

mutation

no

Fig. 1 Flowchart of EP-based ORPF

IEE Proc.-Gener. Transm. Distrib., Vol. 153, No. 2, March 2006 229

properly set so that u0i;j½k� in (6) will generally fall into the

range of ðuminj ; umax

j Þ. After test, it is set to 0.1 in this paper.

3.2 CSSP2CSSP2 was proposed by Lai and Ma [17, 18] and was usedin [4, 10]

si;j½k� ¼ b½k� fi½k�fmax½k�

ðumaxj � umin

j Þ ð11Þ

b½k� ¼b½k � 1� � bstep; if fmin½k� � fmin½k � 1�b½k � 1�; if fmin½k�ofmin½k � 1�bfinal; if b½k � 1� � bstepobfinal

8<:

ð12Þwhere fi[k] denotes the fitness value of the ith individual inthe kth generation; fmax[k] and fmin[k] denote the maximumand minimum fitness in the kth generation. b[0] should beset to around 1 and bfinal to 0.005. bstep ranges between 0.001and 0.01, which depends on the total number of genera-tions. Accordingly, b[0] is set to 0.995, bfinal to 0.005,and bstep to the reciprocal of the maximum number ofgenerations in this paper. It should be pointed out that, in[17] and [18], fitness is defined according to (7 0) and is to beminimised, and the selection scheme used is (90):

fi ¼ FLi þ KV FVi þ KQFQi ð70Þ

si ¼Xq

l¼1

1 if fiofl

0 if fi4fl

�ð90Þ

3.3 CSSP3CSSP3 was proposed by Wong and Yuryevich in [19, 20].

si;j½k� ¼fmax½k� � fi½k�

fmax½k�þ ak

� �� ðumax

j � uminj Þ ð13Þ

where a is a positive constant slightly less than unity andshould be selected according to the total number ofgenerations.

3.4 CSSP4CSSP4 was proposed by Shi and Xu in [21, 22], and wasused in [8]:

si;j½k� ¼ ai;j½k� exp bi;j½k�fmin½k�fi½k�

� �ð14Þ

ai;j½k� ¼ Y ðN þ kÞX12l¼1

Xl � 6

��

��

!,

bi;j½k� ¼ ln ðumaxj � umin

j Þ 2ai;j½k�X12l¼1

Zl � 6

��

��

!, !

where X, Y, ZBU(0, 1) are uniform random numbers.

4 Comparison of CSSP1–CSSP4

4.1 Scenarios and parametersThe performance of CSSP1–CSSP4 is compared on theORPF problems of the IEEE 14 - and 30-bus systems [23].The range of reactive power generation of the slack busgenerator is set to �250–250MVAr for the 14-bus systemand �20–200MVAr for the 30-bus system. Voltage limitsof PQ-buses are set to 0.95–1.05p.u. Totally there are 10and 12 control variables for the 14- and 30-bus systems,respectively, and their settings are listed in Tables 1 and 2.In the original system state, all compensators are notcommitted and the initial power loss is 13.5507MW for the

14-bus system and 17.8984MW for the 30-bus system. TheNewton-Raphson algorithm is used for load flow evalua-tion and the precision is set to a 0.001p.u. on a 100MVAbase. Violation tolerances of 0.001 and 0.005p.u. respec-tively, are set, for PQ-bus voltages and reactive powergenerations of generators.

CSSP1–CSSP4 are implemented in C codes. Each ofthem is tried for 20 trials to solve the ORPF problems, andthe comparison is based on statistical performances. More-over, to eliminate the influence of initialisation, each trial ofeach EP algorithm starts from an identical initial populationthat is generated once. The population size N is set to 80 forthe 14-bus system and 160 for the 30-bus system. Themaximum number of generations is set to 200. Parameter ain CSSP3 is set to 0.97 after test.

4.2 Simulation resultsFor the 14-bus system, the convergence curves of CSSP1–CSSP4, which reflect how the average loss of the bestindividual of each generation over the 20 trials decreaseswith generation number k, are shown in Fig. 2. The

Table 1: Control variable settings of IEEE 14-bus system(p.u.)

Max Min Step

VG 1.1 0.9 Continuous

T 1.1 0.9 0.01

Bsh-9 0.18 0.0 0.06

Bsh-14 0.18 0.0 0.06

Table 2: Control variable settings of IEEE 30-bus system(p.u.)

Max Min Step

VG 1.1 0.9 Continuous

T 1.1 0.9 0.02

Bsh-10 0.2 0.0 0.05

Bsh-24 0.04 0.0 0.01

CSSP1CSSP2CSSP3CSSP4

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

13.2

13.3

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW

Fig. 2 Convergence curves of CSSP1–CSSP4 (14-bus system)


statistics of the final solutions are shown in Table 3. Theresults of the 30-bus system are shown in Fig. 3 andTable 4. It can be observed that:

(a) CSSP1, i.e. the classical EP, performs clearly worse thanthe other three CSSPs with respect not only to the quality ofthe final solution, but also to the convergence character-istics.

(b) CSSP2 and CSSP3 can reach acceptable solutions, buttheir convergence characteristics are bad. The convergencecurve of CSSP2 is a near linear one, and the convergencecurve of CSSP3 has an apparent multiphase feature.

(c) CSSP4 performs best among CSSP1–CSSP4; itreaches the best solution and has a smooth convergencecurve. But it also converges slowly, the convergencecurve does not level off until the generation numberapproaches 200.

4.3 Analysis of comparison resultsFrom the above results we see that CSSP2–CSSP4 aregenerally better and more robust than CSSP1. Underliningthis are two principles embodied in (11)–(14).

The first principle is that problem specific informationshould be incorporated into the evolution scheme. The termðumax

j � uminj Þ in (11)–(14) acts for this function. It can

induce the mutated value of the control variables to fallgenerally into their respective range. Otherwise, as inCSSP1, it is difficult to find a proper single si,j[0] for allcontrol variables that has different ranges of possible value.

The second principle is that the strategy parameter, i.e.the magnitude of mutation, should generally start with arelatively large value so that the whole search space can befully explored, then decrease dynamically or adaptivelyaccording to the feedback information of the evolutionprocess, and finally should reach a small enough value asthe evolution process approaches the end so that theevolution can converge stably.

The second principle is well reflected by CSSP3.Expression (fmax[k]–fi[k])/fmax[k] in (13) means that themutation magnitude of an individual is adaptively deter-mined by the relative distance between its fitness and that ofthe best individual in the current population. The larger isthe relative distance, the larger is the mutation magnitude.But it is insufficient to use the expression (fmax[k]�fi[k])/fmax[k] only because fi[k] tends to approach fmax[k] quicklyand the mutation magnitude will soon decrease to nearzero, so the mutation operation will become ineffective.This handicaps the evolution process to improve thesolution as exemplified by Fig. 4. To solve this problem,the term ak is used in (13). It provides an additionalmutation offset, which dynamically decreases as generationnumber k increases.

However, the convergence characteristic of CSSP3 is badbecause the mutation offset is incorporated in an improperway. Since the expression (fmax[k]–fi[k])/fmax[k] and the termak are combined by summation in (13), the total mutationmagnitude is too big at the early stage, so the early flatphase of the convergence curve results. To avoid thissituation, an improved version of CSSP3 is proposed andwill be discussed in Section 6.

For CSSP2, a7k7 in (11) adaptively decreases accordingto the information of the evolution process, but since thefitness is to be minimised here, (12) means that, if thesolution in generation k is better than that in generationk�1, then the current value of b is maintained, otherwisereduced by a small step. However, by the second principlewe suggest that the solution becomes better and better as

Table 3: Final solution statistics of CSSP1–CSSP4 for 14-bus system

Ave loss (MW) Std. dev. Min loss (MW) Max loss (MW) NIS*

CSSP1 12.7242 0.1376 12.5277 13.0706 1

CSSP2 12.5114 0.0283 12.4641 12.5751 0

CSSP3 12.4648 0.0478 12.3868 12.5644 0

CSSP4 12.4393 0.0228 12.4087 12.4974 0

*NIS: number of infeasible solutions in 20 trials

CSSP1CSSP2CSSP3CSSP4

generation k0 20

16.4

16.6

16.8

17.0

17.2

17.4

17.6

17.8

18.0

18.2

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW

Fig. 3 Convergence curves of CSSP1–CSSP4 (30-bus system)

Table 4: Final solution statistics of CSSP1–CSSP4 for 30-bus system

Ave loss (MW) Std. dev. Min loss (MW) Max loss (MW) NIS

CSSP1 17.5479 0.3274 16.7272 17.8144 9

CSSP2 16.5335 0.0383 16.4791 16.6128 0

CSSP3 16.6358 0.3330 16.3941 17.8131 0

CSSP4 16.4148 0.0208 16.3861 16.4807 0


generation number increases, and the magnitude ofmutation should also be reduced gradually. Moreover, theexpression fi[k]/fmax[k] in (11) also contradicts the secondprinciple, the average value of which becomes larger as theevolution process progresses. Based on these observations,an improved version of CSSP2 is proposed and will bediscussed in Section 6.

For CSSP4, ai,j[k] in (14) dynamically decreases as kincreases, but as the evolution process progresses, fi[k] andfmin[k] will become very close so that the right side of (14)will be approximately equal to:

ðumaxj � umin

j Þ 2X12l¼1

Zl � 6

��

��

!,ð15Þ

which means that the mutation magnitude is kept as arandom value related to the range of the control variables,even if the generation number is very large. This can achieve‘global’ search ability as stated in [22], but the convergencecharacteristics may be sacrificed.

5 Effectiveness of adaptive fast EP technique

5.1 Adaptive fast EP (AFEP)From (6) we know that, besides the strategy parameter,another component that affects the performance of EP isthe random distribution used. Generally, Gaussian distribu-tion is used. But it has been recently found that, undercertain situations, using the Cauchy distribution insteadof Gaussian distribution can achieve better performance[5, 12]. The density function of the Cauchy distribution is(t40):

ft ¼1

pt

t2 þ x2; �1oxo1 ð16Þ

The Cauchy distribution has longer tail than the Gaussiandistribution, i.e. it allows larger mutations to take place at ahigher probability. This characteristic can help to jump outlocal minima. It is appropriate to say that the Cauchydistribution is good at coarse global search, while theGaussian distribution is good at fine local search. Tocombine the advantages of both the Cauchy and Gaussiandistributions, an improved fast EP (IFEP) is proposed in[12]. It generates two intermediate offspring from a parentindividual using the Cauchy and the Gaussian distributions,

respectively, and the better one is selected as the finaloffspring. This scheme allows the program to determineadaptively which distribution is better at different stages ofthe evolution process. The Cauchy and the Gaussiandistributions are special cases of the L!evy distribution,which has the following density function:

La;gðxÞ ¼1

p

Z 10

e�gqa cosðqxÞdq; x 2 R ð17Þ

where g40 is a scaling factor which can be set to 1 withoutloss of generality, and 0oao2 is the parameter thatcontrols the shape of L!evy distribution. When a¼ 1, itbecomes the Cauchy distribution and when a-2, itbecomes the Gaussian distribution. The smaller theparameter a, the longer tail the distribution has.

Based on the L!evy distribution, Lee [13] proposed theadaptive fast EP (AFEP) technique. It generates fourintermediate offspring from a single parent individual usingfour different L!evy distributions (parameter a takes fourdifferent values), and the best one is selected as the finaloffspring.

5.2 Effectiveness of AFEPTo check its effectiveness, AFEP is combined with all theabove mentioned CSSPs to solve both 14- and 30-bussystems. Since 0oao2, four evenly located parametera¼ 0.5, 1.0, 1.5 and 2.0 are used to form the four differentL!evy distributions that are used in AFEP. Moreover, tomake a fair comparison base with respect to bothcomputation burden and performance, the population sizeis set to 20 for the 14-bus system and 40 for the 30-bussystem, which is one-quarter of that for the Gaussiandistribution. This puts AFEP in a slightly adverse situationsince in total seven candidate solutions are tried, socompensation is made by adopting the best quarter of theinitial population used for Gaussian distribution as theinitial population for AFEP.

Tables 5 and 6 compare the final solution statistics of theGaussian distribution or AFEP combined with the fourCSSPs. Convergence curves of the 14-bus system are shownin Figs. 5–8, in which ‘G’ denotes Gaussian and ‘A’ denotes

Table 5: Final solution statistics of Gaussian distributionand AFEP combined with CSSP1–CSSP4 (14-bus system)

Gaussian AFEP

CSSP Ave. loss (MW) NIS Ave. loss (MW) NIS

CSSP1 12.7242 1 12.6853 1

CSSP2 12.5114 1 12.5229 0

CSSP3 12.4648 0 12.4247 0

CSSP4 12.4393 0 12.4724 0

Table 6: Final solution statistics of Gaussian distributionand AFEP combined with CSSP1–CSSP4 (30-bus system)

Gaussian AFEP

CSSP Ave. loss (MW) NIS Ave. loss (MW) NIS

CSSP1 17.5479 9 17.4625 5

CSSP2 16.5335 0 16.5382 0

CSSP3 16.6358 0 16.3830 0

CSSP4 16.4148 0 16.4493 0

withwithout

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

13.2

13.3

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW

Fig. 4 Convergence curves of CSSP3 with and without ak (14-bussystem)


AFEP. The situation of the 30-bus system is very close tothat of the 14-bus system, so illustrations for the 30-bussystem are not given for space consideration. It is shownthat:

(a) The effectiveness of AFEP first depends on what CSSPit is combined with: AFEP performs better than theGaussian distribution when combined with CSSP1 andCSSP3, but worse when combined with CSSP2 and CSSP4.

(b) AFEP mainly affects the final solution quality of EP,and has minor influence on convergence rate.

These observations are reasonable since in (6) the CSSPsdetermine how big the mutation range should be, whereasthe random distribution only determines how much of thisrange should be utilised. Therefore, CSSPs dominate theoverall performance of EPs. In general, the AFEP techniquemay be of small attraction for ORPF problems of practicalpower systems because additional computational time isneeded to generate the random values of the L!evydistribution.

6 Two improved CSSPs

According to the analysis in Section 4, two improvedCSSPs, namely ICSSP1 and ICSSP2, are proposed in thefollowing.

The disadvantage of CSSP3 in (13) is that the mutationmagnitude is dominated by the value of the expression(fmax[k]�fi) [k]/fmax[k] in the early stage of the evolutionaryprocess and it is only at the late stage that the mutationmagnitude will then be dominated by the ak term. To avoidthis disadvantage, ICSSP1 is formed based on CSSP3 and(13) is now replaced by (18):

si;j½k� ¼ ak fmax½k� � fi½k�fmax½k�

þ ð1� akÞak

� �ðumax

j � uminj Þ

ð18Þ

ICSSP2 is based on CSSP2, with (11) and (12) replaced by(19) and (20):

ai;j½k� ¼ b½k� fmax½k� � fi½k�fmax½k� � fmin½k�

ðumaxj � umin

j Þ ð19Þ

generation k0 20

12.65

12.70

12.75

12.80

12.85

12.90

12.95

13.00

13.05

13.10

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW

CSSP1+ ACSSP1+ G

Fig. 5 Convergence curves of CSSP1 combined with Gaussiandistribution and AFEP (14-bus system)

CSSP2+ACSSP2+ G

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW


CSSP3+ GCSSP3+ A

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

13.2

13.3

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW


CSSP4+ GCSSP4+ A

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW



b½k� ¼b½k � 1� � bstep; if fmax½k� � fmax½k � 1�b½k � 1�; if fmax½k�ofmax½k � 1�bfinal; if b½k � 1� � bstepobfinal

8<:

ð20Þ

where b[0], bfinal and bstep are set the same as in CSSP2. Asall other mutation schemes except CSSP2, (7) is used as thefitness function and (9) as the selection scheme. So (20)means that, if the solution in generation k is as good as or isbetter than that in generation k�1, b is then reduced by asmall step. Otherwise, b is maintained at the current valueto do more exploration. Besides, the defect of fi[k]/fmax[k]in (11) is avoided here by replacing it with (fmax[k]–fi[k])/(fmax[k]–fmin[k]).

The performances of the ICSSP1 and ICSSP2 are verifiedon IEEE 14 -, 30- and 118-bus standard test systems, withcomparison to their corresponding original versions CSSP3and CSSP2. The IEEE 118-bus test system [23] has a totalof 77 control variables, the settings of which are quotedfrom [8]. The initial operating state has a power loss of132.8630MW. For the 118-bus system, the population sizeis set to 800 and the maximum number of generations to2000, the parameter a is set to 0.996 for CSSP3 and 0.995for ICSSP1 after test. Other conditions for calculation arethe same as in Section 4. Simulation results are listed inTable 7 and are shown in Figs. 9–14. These results showthat:

(a) ICSSP1 and ICSSP2 perform consistently better thantheir corresponding original versions, CSSP3 and CSSP2.

This holds not only for the final results, but also for theconvergence characteristics.

(b) As compared to CSSP4, the best one among CSSP1–CSSP4, ICSSP1 consistently reaches better final results than

Table 7: Average losses (MW) reached by ICSSP1, 2 withcomparison to original CSSPs

14-bus 30-bus 118-bus

ICSSP1 12.4256 16.4073 125.5901

CSSP3 12.4648 16.6358 126.8913

ICSSP2 12.4734 16.4574 129.5289

CSSP2 12.5114 16.5335 131.9859

CSSP4 12.4393 16.4148 128.2854

CSSP3

ICSSP1CSSP4

generation k0 20

12.4

12.5

12.6

12.7

12.8

12.9

13.0

13.1

13.2

13.3

40 60 80 100 120 140 160 180 200

pow

er lo

sses

, MW

Fig. 9 Convergence curves of ICSSP1 with comparison to CSSP3and CSSP4 (14-bus system)

generation k0 20

16.4

16.6

16.8

17.0

17.2

17.4

17.6

17.8

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Fig. 10 Convergence curves of ICSSP1 with comparison toCSSP3 and CSSP4 (30-bus system)

generation k0 200

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CSSP2

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CSSP4 does, and ICSSP2 has better convergence character-istics than CSSP4 does for the 30-bus system.

(c) For the 118-bus system, ICSSP1 performs much betterthan the others. The power loss of the final solutionobtained by ICSSP1 is 125.59MW, while it is 126.89MWfor CSSP3, 128.29MW for CSSP4, 131.98MW for CSSP2and 129.53MW for ICSSP2.

7 Conclusions

Several recently developed EP techniques in the literaturehave been investigated and have been compared andimproved in this paper. By analysing the comparison resultsof four control schemes of the strategy parameter (CSSPs),two principles for designing effective CSSPs are identified:(a) problem specific information should be incorporatedinto the evolution scheme; and (b) the strategy parametershould generally start with a relatively large value, whiledynamically and adaptively decreasing to a small enoughvalue as the evolution process approaches the end.According to the second principle, two improved CSSPs

have been proposed and their performance improvementsverified on the IEEE 14-, 30- and 118-bus standard testsystems. The effectiveness of the L!evy distribution basedtechnique of adaptive fast EP (AFEP) has also beenchecked. It is found that: (a) the effectiveness of AFEP firstdepends on what CSSP it is combined with, and CSSPdominates the overall performance of an EP algorithm; and(b) AFEP mainly affects the final solution quality of EP,and has minor influence on convergence rate. In general,AFEP is not very attractive for ORPF applications withconsideration of the additional computing time required bythis technique.

8 Acknowledgments

This work was supported by Research Grants Councilof Hong Kong (PolyU 5225/04E), the Department ofElectrical Engineering of The Hong Kong PolytechnicUniversity, the Natural Science Foundation of China grant59807004, and the Teaching and Research Award Programfor Outstanding Young Teachers of MOE, People’sRepublic of China.

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generation k0 20

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