Energy Conversion and Management 52 (2011) 903–910

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Multi-objective optimal power flow with FACTS devices

M. Basu ⇑Department of Power Engineering, Jadavpur University, Kolkata, India

a r t i c l e i n f o

Article history:Received 10 December 2009Received in revised form 9 August 2010Accepted 15 August 2010Available online 6 September 2010

Keywords:Multi-objective optimal power flowMulti-objective differential evolutionFlexible ac transmission systems

0196-8904/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.enconman.2010.08.017

⇑ Tel.: +91 9836839673.E-mail address: mousumibasu@yahoo.com

a b s t r a c t

This paper presents multi-objective differential evolution to optimize cost of generation, emission andactive power transmission loss of flexible ac transmission systems (FACTS) device-equipped power sys-tems. In the proposed approach, optimal power flow problem is formulated as a multi-objective optimi-zation problem. FACTS devices considered include thyristor controlled series capacitor (TCSC) andthyristor controlled phase shifter (TCPS). The proposed approach has been examined and tested on themodified IEEE 30-bus and 57-bus test systems. The results obtained from the proposed approach havebeen compared with those obtained from nondominated sorting genetic algorithm-II, strength paretoevolutionary algorithm 2 and pareto differential evolution.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Optimal power flow (OPF) is a static nonlinear programmingproblem which optimizes a certain objective function while satis-fying a set of physical and operational constraints imposed byequipment limitations and security requirements. Over the lastthree decades, many successful methods have been developed[1,2] such as, generalized reduced gradient method, successive lin-ear programming, successive quadratic programming, Newtonmethod, P–Q decomposition, interior point method, genetic algo-rithm and evolutionary programming.

The economic and social developments particularly the world-wide movement toward deregulation demand new concepts andtechnologies to best exploit the existing power system resourceswithout compromising system security due to the environmentaland/or economical constraint to build new generation centersand transmission lines. Hence there is an interest in better utiliza-tion of existing power system capacities by installing flexible actransmission systems (FACTS) devices [3]. FACTS devices are usedto control power flow so that it flows through the designatedroutes, increase transmission line capability to its maximum ther-mal limit, and improve the security of transmission system andstability of the network with minimal infrastructure investment.FACTS concept makes it possible to control voltage magnitudeand phase angle at chosen bus and circuit reactance of a transmis-sion system.

The fossil fuel-fired generating station releases sulfur oxides(SOx), nitrogen oxides (NOx), and carbon dioxide (CO2) into atmo-sphere. Due to increasing concern over the environmental consid-

ll rights reserved.

eration, society demands adequate and secure electricity not onlyat the cheapest possible price, but also at minimum level of pollu-tion. Several strategies to reduce the atmospheric pollution havebeen proposed and discussed in [5]. These include installation ofpollutant cleaning, switching to low emission fuels, replacementof the aged fuel burners with cleaner ones, and emission dispatch-ing. The first three options require installation of new equipmentand/or modification of the existing ones that involve considerablecapital outlay and hence they can be considered as long-term op-tions. In recent years, the emission dispatching option in whichboth cost and emission are to be optimized, has received muchattention since it requires only small modification of the basic eco-nomic dispatch to include emission. Fuzzy multi-objective optimi-zation technique for economic environmental dispatch (EED)problem has been proposed in [6]. However, the solutions aresub-optimal and the algorithm does not provide systematic frame-work for directing the search toward pareto-optimal front. In [7]linear programming based optimization procedure has been pre-sented in which objectives are considered one at a time. As theEED problem is a highly nonlinear optimization problem, classicaloptimization methods are not able to locate the global optimum.Furthermore, this approach does not give any information regard-ing the trade-off involved. A multi-objective stochastic searchtechnique for the EED problem has been presented in [8]. However,this technique is computationally involved and time consuming.

Over the past few years, the studies on evolutionary algorithms,i.e. multi-objective evolutionary algorithm (MOEA) [9], strengthpareto evolutionary algorithm 2 (SPEA 2) [10], pareto diferentialevolution (PDE) [11], nondominated sorting genetic algorithm-II(NSGA-II) [12], multi-objective differential evolution (MODE)[13], etc. have shown that these methods can be efficiently usedto eliminate most of the difficulties of classical methods. Since they

Nomenclature

PGm, QGm active and reactive power generations at bus mPmin

Gm ; PmaxGm lower and upper active power generation limits at bus

mQmin

Gm ;QmaxGm lower and upper reactive power generation limits at

bus mPDm, QDm active and reactive power demands at bus mPD, QD total active and reactive power demandsPL, QL active and reactive power transmission lossesPms, Qms injected active and reactive powers at bus m due to

TCPSVm, dm voltage magnitude and phase angle at bus mVmin

m ;Vmaxm lower and upper voltage magnitude limits at bus m

Sm transmission capacity of line mSmax

m maximum transmission capacity of line m?mn, hmn magnitude and angle of mnth element in bus admit-

tance matrix

rmn, xmn resistance and reactance between bus m and bus ngmn, bmn conductance and susceptance between bus m and bus nxcm reactance of TCSC mum phase shift angle of TCPS m/min

m ;/maxm

lower and upper limits of phase shift angle of TCPS mam, bm, cm, dm, em cost coefficients of generator mam; bm; cm;gm; km emission coefficients of generator mNG set of generator bus indexesNQ set of bus indexes having reactive power sourceNB set of bus indexesNL set of transmission line indexesNTCPS set of TCPSNTCSC set of TCSC

904 M. Basu / Energy Conversion and Management 52 (2011) 903–910

are population-based techniques, multiple pareto-optimal solu-tions can be found in one single simulation run.

In Ref. [15] strength pareto evolutionary algorithm algorithmhas been presented to handle economic environmental power dis-patch problem. Abido [16] has applied three multi-objective evolu-tionary algorithms, i.e. nondominated sorting genetic algorithm,niched Pareto genetic algorithm, and strength pareto evolutionaryalgorithm to solve economic environmental power dispatch prob-lem. In Ref. [17] fuzzy clustering-based particle swarm algorithmhas been proposed for economic environmental dispatch problem.Hugang et al. [18] have presented multi-objective adaptive im-mune algorithm (MOAIA) to solve optimal reactive power flow(ORPF) problem incorporating static voltage stability. In Refs.[15–17], only cost and emission are treated as competing objec-tives and FACTS devices are not considered.

In this paper, multi-objective differential evolution (MODE) isproposed to solve multi-objective OPF. Here, cost of generation,emission and active power transmission loss are optimized simul-taneously in FACTS device-equipped power systems while main-taining power balance constraints, active and reactive powergeneration limits, voltage limits, transmission line limits, andphysical limits of FACTS devices. Thyristor controlled series capac-itor (TCSC) and thyristor controlled phase shifter (TCPS) are inte-grated in OPF by using the reactance model and the injectedpower model respectively. In order to demonstrate the effective-ness of the proposed approach, the modified IEEE 30-bus and 57-bus test systems are taken as examples. Results obtained fromthe proposed approach have been compared to those obtained byusing nondominated sorting genetic algorithm-II (NSGA-II),strength pareto evolutionary algorithm 2 (SPEA2) and pareto dif-ferential evolution (PDE) with TCSC and TCPS at the fixed locations.

m mnmn jxr + cjx− n

m mV δ∠ n nV δ∠

Fig. 1. Model of TCSC.

2. Modeling of FACTS devices

2.1. Modeling of TCSC

The effect of TCSC on the network can be seen as a controllablereactance inserted in the related transmission line. The model ofthe network with TCSC is shown in Fig. 1.

The power flow equations of the line can be derived as follows:

Pmn ¼ V2mgmn � VmVngmn cosðdm � dnÞ � VmVnbmn sinðdm � dnÞ ð1Þ

Q mn ¼ �V2mbmn � VmVngmn sinðdm � dnÞ þ VmVnbmn cosðdm � dnÞ

ð2Þ

Pnm ¼ V2ngmn � VmVngmn cosðdm � dnÞ þ VmVnbmn sinðdm � dnÞ ð3Þ

Qnm ¼ �V2nbmn þ VmVngmn sinðdm � dnÞ þ VmVnbmn cosðdm � dnÞ

ð4Þ

where

gmn ¼rmn

r2mn þ ðxmn � xcÞ2

; bmn ¼ �xmn � xc

r2mn þ ðxmn � xcÞ2

2.2. Modeling of TCPS

TCPS can be modeled by a phase shifting transformer [4] withcontrol parameter u. The model of TCPS is shown in Fig. 2.

The power flow equations of the line can be derived as follows:

Pmn ¼V2

mgmn

cos2 /� VmVn

cos /½gmn cosðdm � dn þ /Þ þ bmn sinðdm � dn þ /Þ�

ð5Þ

Qmn¼�V2

mbmn

cos2 /�VmVn

cos/½gmn sinðdm�dnþ/Þ�bmn cosðdm�dnþ/Þ�

ð6Þ

Pnm ¼ V2ngmn �

VmVn

cos /½gmn cosðdm � dn þ /Þ � bmn sinðdm � dn þuÞ�

ð7Þ

Qnm ¼ �V2nbmn þ

VmVn

cos /½gmn sinðdm � dn þ /Þ þ bmn cosðdm � dn þuÞ�

ð8Þ

The injected real and reactive power of TCPS at bus m and bus n areas follows:

m mV δ∠ φ∠1 :1 mnmn jbg − n nV δ∠

m msms jQ+Ρ nsns jQ+Ρ n

Fig. 2. Power injection model.

M. Basu / Energy Conversion and Management 52 (2011) 903–910 905

Pms ¼ �gmnV2m tan2 /� VmVn tan /½gmn sinðdm � dnÞ � bmn

� cosðdm � dnÞ� ð9Þ

Q ms ¼ bmnV2m tan2 /þ VmVn tan /½gmn cosðdm � dnÞ þ bmn

� sinðdm � dnÞ� ð10Þ

Pns ¼ �VmVn tan /½gmn sinðdm � dnÞ þ bmn cosðdm � dnÞ� ð11Þ

Q ns ¼ �VmVn tan /½gmn cosðdm � dnÞ � bmn sinðdm � dnÞ� ð12Þ

3. Problem formulation

The multi-objective OPF problem with TCSC and TCPS devicesoptimizes cost of generation, emission and active power transmis-sion loss while satisfying several equality and inequality con-straints. The problem is formulated as follows.

3.1. Objectives

3.1.1. Fuel costThe fuel cost function of each thermal generating unit, consid-

ering the valve-point effects [19], is expressed as the sum of a qua-dratic and a sinusoidal function. The total fuel cost in terms of realpower output can be expressed as

F ¼XNG

m¼1

am þ bmPGm þ cmP2Gm þ dm � sin em � Pmin

Gm � PGm

� �n o��� ���h i

ð13Þ

3.1.2. EmissionThe atmospheric pollutants such as sulfur oxides (SOx) and

nitrogen oxides (NOx) caused by thermal generating units can bemodeled separately. However, for comparison purposes, the totalemission of these pollutants which is the sum of a quadratic andan exponential function [20] can be expressed as

E ¼XNG

m¼1

am þ bmPGm þ cmP2Gm þ gm expðkmPGmÞ

h ið14Þ

3.1.3. LossSystem active power transmission loss can be expressed as

PL ¼Xm;n

gmn V2m þ V2

n � 2VmVn cosðdm � dnÞh i

ð15Þ

subject to

XNB

n¼1

VmVn!mn cosðhmn þ dm � dnÞ � PGm � Pms þ PDm ¼ 0

8m 2 NB ð16Þ

XNB

n¼1

VmVn!mn sinðhmn þ dm � dnÞ � Q Gm � Qms þ Q Dm

¼ 0 8m 2 NB ð17Þ

XNG

m¼1

PGm � PD � PL ¼ 0 ð18Þ

XNQ

m¼1

Q Gm � QD � QL ¼ 0 ð19Þ

PminGm 6 PGm 6 Pmax

Gm 8m 2 NG ð20Þ

QminGm 6 Q Gm 6 Q max

Gm 8m 2 NQ ð21Þ

Vminm 6 Vm 6 Vmax

m 8m 2 NB ð22Þ

jSmj 6 Smaxm 8m 2 NL ð23Þ

uminm 6 /m 6 /max

m 8m 2 NTCSC ð24Þ

xmincm 6 xcm 6 xmax

cm 8m 2 NTCPS ð25Þ

4. Principle of multi-objective optimization

Most of the real-world problems involve simultaneous optimi-zation of several objective functions. These functions are noncom-mensurable and conflicting and often competing objectives. Multi-objective optimization having such conflicting objective functionsgives rise to a set of optimal solutions, instead of one optimal solu-tion because no solution can be considered to be better than anyother with respect to all objective functions. These optimal solu-tions are known as pareto-optimal solutions.

Generally, multi-objective optimization problem consisting of anumber of objectives and several equality and inequality con-straints can be formulated as follows:

Minimize f iðxÞ i ¼ 1; . . . ;Nobj ð26Þ

subject togkðxÞ ¼ 0 k ¼ 1; . . . ;K

hlðxÞ 6 0 l ¼ 1; . . . ; L

�ð27Þ

where fi is the ith objective function, x is a decision vector.

5. Multi-objective differential evolution

Differential evolution (DE) is a type of evolutionary algorithm[14] for optimization problems over a continuous domain. DE isexceptionally simple and robust. The basic idea of DE is to adaptthe search during the evolutionary process. At the start of the evo-lution, the perturbations are large since parent populations are faraway from each other. As the evolutionary process matures, thepopulation converges to a small region and the perturbationsadaptively become small. As a result, the evolutionary algorithmperforms a global exploratory search during the early stages ofthe evolutionary process and local exploitation during the maturestage of the search. In DE the fittest of an offspring competes one-to-one with that of corresponding parent which is different fromother evolutionary algorithms. This one-to-one competition givesrise to faster convergence rate. In multi-objective differential evo-lution (MODE) [13], a pareto-based approach is introduced toimplement the selection of the best individuals. Firstly, a popula-tion of size, NP, is generated randomly and objective functionsare evaluated. At a given generation of the evolutionary search,the population is sorted into several ranks based on nondomina-tion. Secondly, DE operations are carried out over the individualsof the population. Trial vectors of size NP are generated and objec-tive functions are evaluated. Both the parent vectors and trial vec-tors are combined to form a population of size 2NP. Then, the

906 M. Basu / Energy Conversion and Management 52 (2011) 903–910

ranking of the combined population is carried out followed by thecrowding distance calculation. The best NP individuals are selectedbased on its ranking and crowding distance. These individuals actas the parent vectors for the next generation.

6. Multi-objective differential evolution for OPF with FACTSdevices

The proposed algorithm can be described in the following steps:

Step 1: Generate box, R, of size NP. Parent vectors of size NP israndomly generated and kept in box R. Elements in a parentvector are real power generation of the generating units exclud-ing slack bus, voltage magnitude and phase angle of the busesexcluding slack bus, series capacitors of TCSC and phase shiftangle of TCPS. The ith parent vector is as follows:

pi ¼ ½PiG1; . . . ; Pi

Gm; . . . ; PiGNG

;Vi1; . . . ;Vi

m; . . . ;ViNB; di

1; . . . ; dim; . . . ; di

NB;

� xic1; . . . ; xi

cm; . . . ; xicNTCSC

;/i1; . . . ;/i

m; . . . ;/iNTCPS�T ; i ¼ 1;2; . . . ;NP

The components of pi are generated as PiGm � U Pmin

Gm ; PmaxGm

� �, Vi

m �

U Vminm ;Vmax

m

� �, di

m � U dminm ; dmax

m

� �, xi

cm � U xmincm ; xmax

cm

� �, /i

m � U

/minm ;/max

m

� �. U(a, b) denotes a uniform random variable ranging over

[a, b].

Table 1Results obtained from minimum cost, minimum emission and minimum loss for test syst

DE RCG

Minimum cost Minimum emission Minimum loss Min

PG1 (MW) 199.13 63.50 74.59 198PG2 (MW) 38.32 67.92 67.30 38PG5 (MW) 20.17 50.00 50.00 19PG8 (MW) 11.43 35.00 34.85 10PG11 (MW) 10.43 30.00 27.04 13PG13 (MW) 12.66 40.00 32.36 12Total PG (MW) 292.14 286.42 286.14 293xc3–4 (p.u.) 0.0123 0.0187 0.0084 0xc19–20 (p.u.) 0.0250 0.0251 0.0045 0u5–7 (�) �0.1891 �0.5478 �0.5329 �0u10–22 (�) 0.2177 0.0293 �0.4526 �0Cost ($/h) 826.54 1015.10 992.30 831Emission (ton/h) 0.4383 0.2048 0.2109 0Ploss (MW) 8.74 3.02 2.74 9CPU time (s) 505.6 511.3 497.4 714

Table 2Results obtained from case-1, case-2, and case-3 for test system-1.

MODE NSGA-II

Case-1 Case-2 Case-1 Case-2

PG1 (MW) 106.78 113.85 109.08 92.65PG2 (MW) 62.21 32.82 60.59 73.16PG5 (MW) 26.88 49.93 29.42 49.05PG8 (MW) 35.00 34.56 35.00 29.83PG11 (MW) 28.35 24.00 26.00 23.88PG13 (MW) 29.11 32.93 28.13 19.49Total PG (MW) 288.33 288.09 288.22 288.06xc3–4 (p.u.) 0.0101 0.0087 0.0082 0.0174xc19–20 (p.u.) 0.0156 0.0199 0.0227 0.0392u5–7 (�) 0.4698 0.2807 0.2005 �0.7219u10–22 (�) �0.0172 �0.3724 �0.7162 �0.6704Cost ($/h) 909.11 955.01 905.12 963.56Emission (ton/h) 0.2361 0.2395 0.2382 0.2386Ploss (MW) 4.93 4.69 4.82 4.67CPU time (s) 634.7 712.8 838.7 889.4

Newton–Raphson load flow is run for each parent vector pi.Three objective functions, i.e. cost of generation, emission andactive power transmission loss are calculated for each parent vec-tor pi.

Step 2: Classify these vectors into fronts based on nondomina-tion [13] as follows:

(a) Create new empty box R/ of size NP.(b) Compare each vector with all other vectors in R.(c) Start with i = 1.(d) If ith vector is not dominated by any other vector with

respect to all the objectives in R, keep ith vector in R/

and go to (f).(e) If ith vector is dominated by any other vector in R, go to

(f).(f) Increment i by one. If i 6 NP, go to (d) otherwise go to (g).(g) R/ now contains a sub-box (of size 6 NP) of nondominat-

ed vectors, referred to as the first front or sub-box. Assignit a rank number equal to one (Irank = 1). Remove all thenondominated vectors from R.

(h) Create subsequent fronts or sub-boxes of R/ with the vec-tors remaining in R and assign these Irank = 2, 3, . . .

Finally, all NP vectors are in R/ into one or more fronts.Step 3: To calculate the crowding distance, Ii,dist, for the ith vec-tor in any front, F, of R/ sort all the vectors in front, F, accordingto each objective function value in ascending order of magni-tude. The crowding distance of the ith vector in its front F isthe average side-length of the cuboid formed by using the

em-1.

A DE

imum cost Minimum emission Minimum loss Case 4 Case 5

.81 63.98 77.58 180.26 192.46

.96 67.75 69.58 49.32 48.38

.16 50.00 49.98 20.82 19.54

.64 35.00 34.96 17.61 11.60

.56 29.96 23.69 11.05 10.00

.03 40.00 30.43 12.69 12.00

.16 286.69 286.22 291.75 294.00

.0185 0.0192 0.0193 0.0190 0.0200

.0247 0.0246 0.0239 0.0243 0.0200

.5713 �0.5518 �0.5347 �0.5558 1.9137

.0281 �0.0288 �0.0292 �0.0286 0.8251

.03 1015.80 985.21 797.29 803.84

.4366 0.2049 0.2144 0.3756 –

.76 3.29 2.82 8.35 10.60

.8 707.6 711.7 487.3 265.8

PDE SPEA2 DE

Case-1 Case-2 Case-1 Case-2 Case-3

109.02 115.92 100.94 109.88 107.9859.45 30.75 69.13 38.57 58.5728.78 46.37 25.35 49.09 32.3835.00 32.22 32.82 31.56 27.6125.35 29.42 26.54 22.40 29.5130.86 33.51 33.83 36.62 33.27

288.46 288.19 288.61 288.12 289.320.0163 0.0152 0.0169 0.0080 0.00240.0051 0.0157 0.0051 0.0217 0.01700.8365 �0.4068 0.8363 �0.7448 0.61310.8537 �0.6532 0.8538 �0.9683 �0.0745

907.57 946.49 920.33 958.50 922.360.2375 0.2409 0.2327 0.2375 0.23645.06 4.79 5.21 4.72 5.92

641.3 723.5 850.2 903.5 521.9

850 900 950 10000.2

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

0.3

Emis

sion

(ton

/h)

Cost ($/h)

MODENSGA-IIPDESPEA2

Fig. 3. Pareto-optimal front of test system-1 for case-1.

850

900

950

1000

0.2

0.25

0.3

0.35

0.44

5

6

7

8

9

10

Cost ($/h)Emission (ton/h)

Loss

(M

W)

MODE

NSGA-II

PDE

SPEA2

Fig. 4. Pareto-optimal front of test system-1 for case-2.

M. Basu / Energy Conversion and Management 52 (2011) 903–910 907

nearest neighbors as the vertices. Assign large values of crowd-ing distance Idist to the boundary vectors (vectors with smallestand largest function values). Sort vectors of all fronts indescending order of crowding distance. Vector i is better thanvector j (i) if Ii,rank < Ij,rank or (ii) if Ii,rank = Ij,rank and Ii,dist > Ij,dist.Step 4: Perform DE operations over NP vectors in R/ to generateNP trial vectors and store these vectors in R// as follows:

(a) Create new empty box R// of size NP.(b) Select a target vector, i in R/.(c) Start with i = 1.(d) For each vector pi, a noisy random vector pr is generated

according to pr = pr1 + F � (pr2 � pr3) where three vectorspr1, pr2 and pr3 are randomly selected from NP vectorsin R/. These three vectors are mutually different and dif-ferent from pi. The scaling factor F e [0, 1.2] controls thespeed and robustness of the search. If any variable ofthe noisy random vector pr exceeds its upper limit it isset to its upper limit and if the variable reaches belowthe lower limit it is set to its lower limit. Newton–Raph-son load flow is run for the noisy random vector pr andcost of generation, emission and active power transmis-sion loss are calculated.

(e) Perform crossover between the target vector and noisyrandom vector to find the trial vector. This is carriedout by generating a random number and if random num-ber > CR (crossover factor), copy the target vector into thetrial vector else copy the noisy random vector into thetrial vector and put it in box R//.

(f) Increment i by one. If i 6 NP, go to (d) otherwise go toStep 5.

Step 5: Copy all NP parent vectors from R/ and all NP trial vectorsfrom R// into box R///. Box R/// has 2NP vectors.

(a) Classify these 2NP vectors into fronts based on nondom-ination and calculate the crowding distance of each vec-tor. Take the best NP vectors from Box R/// and put intoBox R////.

(b) This completes one generation. If generation number isequal to maximum number of generation go to Step 6.Else copy NP vectors from Box R//// to the starting box Rand go to Step 2.

Step 6: First vector of the first front obtained in the last gener-ation is the solution.

7. Numerical results

In this paper the performance of the proposed MODE-based OPFis evaluated by using two test systems, modified IEEE 30-bus and57-bus test systems whose generator characteristics are given inTables A-1 and A-2 respectively in the appendix. In order to showthe effectiveness of the proposed MODE algorithm, NSGA-II, SPEA2, and PDE have been applied to the same two test systems. Allthe algorithms, i.e. DE, RCGA, MODE, NSGA-II, SPEA 2 and PDE usedin this paper for solving OPF problem are implemented by usingMATLAB 7.0 on a PC (Pentium-IV, 80 GB, 3.0 GHz).

For both test systems, limits of phase shift angle of TCPS are ta-ken �5� < u < 5�. The series capacitor limits are taken in such amanner that the ratio of maximum series capacitor limit to linereactor is equal or more than 50%.

Three objective functions, i.e. fuel cost, emission and activepower transmission loss have been considered here. Two caseshave been considered. In case-1, economic environmental OPFproblem has been considered where both cost and emission objec-tives have been treated simultaneously as competing objectives. Incase-2, cost, emission and active power transmission loss havebeen treated simultaneously as competing objectives. The trade-off among competing objectives is obtained by emphasizing

nondominated solutions and also simultaneously emphasizing toget a well-distributed set of solutions. These two features are al-lowed to find a set of near pareto-optimal solutions.

In case-3, economic environmental OPF problem is convertedinto a single objective optimization problem by introducing pricepenalty factor h and can be expressed as

Minimize w ¼ F þ h� E ð28Þ

7.1. Test system-1

Test system-1 is IEEE 30-bus system. In this work, two lines,(3, 4) and (19, 20) are installed with TCSC and two lines, (5, 7)and (10, 22) are installed with TCPS. Here, voltage magnitude lim-its and phase angle limits of buses are set to 0.95p.u. 6 V 6 1.1p.u.and �14� < d < 0� respectively.

At first fuel cost, emission and active power transmission losshave been minimized individually by using differential evolution(DE). To verify the results obtained from DE, fuel cost, emission

908 M. Basu / Energy Conversion and Management 52 (2011) 903–910

and active power transmission loss have been minimized individ-ually by using real-coded genetic algorithm (RCGA). In case-4, costobjective without valve point effect has been optimized by usingDE and this result is compared with those obtained from hybridTS/SA approach [4] which is denoted by case-5. All the resultsare shown in Table 1. It is seen from Table 1 that for case-4 fuelcost is 797.29 $/h which is less than 803.84 $/h obtained fromcase-5.

MODE has been applied to solve case-1 and case-2. Case-3 isminimized by using DE. It is seen from Table 1 that fuel cost is826.54 $/h under cost minimization criterion, but it increases to1015.10 $/h and 992.30 $/h for the cases of the emission minimiza-tion and active power transmission loss minimization criteria byusing DE. Both cost and emission objectives have been optimizedsimultaneously in case-1 where fuel cost is 909.11 $/h which ismore than 826.54 $/h and less than 1015.10 $/h and emission is0.2361 ton/h which is less than 0.4383 ton/h and more than

Table 3Results obtained from minimum cost, minimum emission and minimum loss for test syst

DE

Minimum cost Minimum emission Minimum

PG1 (MW) 520.09 298.12 318.58PG2 (MW) 0 0 0PG3 (MW) 103.74 83.24 45.90PG6 (MW) 0 0 0PG8 (MW) 175.63 413.63 407.65PG9 (MW) 0 0 0PG12 (MW) 485.23 474.14 495.03Total PG (MW) 1284.69 1269.13 1267.16xc18–19 (p.u.) 0.0604 0.0830 0.0100xc31–32 (p.u.) 0.0199 0.0672 0.0004xc34–32 (p.u.) 0.0015 0.0009 0.0079xc40–56 (p.u.) 0.0932 0.0437 0.0819xc39–57 (p.u.) 0.0466 0.0772 0.0841u4–5 (�) �0.6131 �0.8995 �0.0745u5–6 (�) �0.6188 0.4297 �0.2807u26–27 (�) �0.4698 �0.8079 �0.9798u41–43 (�) 0.5099 �0.1375 �0.9053u53–54 (�) �0.1146 �1.0313 0.9798Cost ($/h) 8305.00 15903.00 15696.00Emission (ton/h) 2.4333 1.8589 1.9671Ploss (MW) 33.89 18.33 16.36CPU time (s) 689.9 694.2 701.7

Table 4Results obtained from case-1, case-2, and case-3 for test system-2.

MODE NSGA-II

Case-1 Case-2 Case-1 Case-2

PG1 (MW) 430.43 440.40 435.37 451.17PG2 (MW) 0 0 0 0PG3 (MW) 100.03 67.24 102.49 68.13PG6 (MW) 0 0 0 0PG8 (MW) 248.29 309.97 252.41 306.99PG9 (MW) 0 0 0 0PG12 (MW) 500.00 457.82 488.76 450.12Total PG (MW) 1278.75 1275.43 1279.03 1276.41xc18–19 (p.u.) 0.0607 0.0017 0.0607 0.0537xc31–32 (p.u.) 0.0891 0.0377 0.0891 0.0345xc34–32 (p.u.) 0.0762 0.0116 0.0762 0.0793xc40–56 (p.u.) 0.0821 0.0079 0.0821 0.0058xc39–57 (p.u.) 0.0445 0.0715 0.0445 0.0553u4–5 (�) 0.9282 �0.7735 0.9282 �0.0573u5–6 (�) �0.5558 0.9053 �0.5558 �0.8824u26–27 (�) �0.0286 �0.4813 �0.0286 �0.9339u41–43 (�) �0.0917 0.9740 �0.0917 0.3782u53–54 (�) �0.9912 0.9282 �0.9912 �0.0286Cost ($/h) 9941.6 11507.1 9992.0 11367.6Emission (ton/h) 2.0763 2.1216 2.0762 2.1502Ploss (MW) 27.94 24.64 28.23 25.60CPU time (s) 843.3 901.2 1023.1 1107.9

0.2048 ton/h. Active power transmission loss is 4.93 MW. Incase-2, where cost, emission and active power transmission lossobjectives have been optimized simultaneously, fuel cost andemission are 955.01 $/h and 0.2395 ton/h which are more thancase-1 and active power transmission loss is 4.69 MW which is lessthan 4.93 MW.

The results obtained from MODE are compared with those ob-tained from NSGA-II, PDE, SPEA2, and case-3 which are summa-rized in Table 2.

The distributions of 100 nondominated solutions obtained inthe last generation of MODE, NSGA-II, PDE, and SPEA2 for case-1of this test system are shown in Fig. 3. It is seen from Fig. 3 thatno objective (i.e. cost or emission) can be further improved withoutdegrading the other. Fig. 4 shows the distributions of 100 nondom-inated solutions obtained in the last generation of MODE, NSGA-II,PDE, and SPEA2 for case-2 of this test system. From Fig. 4, it is seenthat no solution can be considered to be better than any other with

em-2.

RCGA

loss Minimum cost Minimum emission Minimum loss

517.45 341.91 303.240 0 094.81 91.90 63.190 0 0181.75 419.25 400.750 0 0489.77 418.45 500.001283.78 1271.51 1267.180.0572 0.0830 0.05930.0832 0.0672 0.01790.0203 0.0009 0.01890.0480 0.0437 0.06410.0624 0.0772 0.0055�0.7678 �0.8995 �0.6532�0.7620 0.4297 �0.0917�0.3438 �0.8079 �0.7620�0.3953 �0.1375 0.6933�0.4011 �1.0313 0.24068410.50 15861.00 15427.002.4331 1.8894 1.906832.98 20.71 16.38874.9 878.7 881.3

PDE SPEA2 DE

Case-1 Case-2 Case-1 Case-2 Case-3

431.12 435.25 435.45 449.48 475.680 0 0 0 098.34 50.75 88.05 42.82 80.640 0 0 0 0253.31 306.75 262.55 301.61 276.030 0 0 0 0495.96 482.63 491.40 482.25 447.201278.73 1275.38 1277.45 1276.16 1279.550.0607 0.0096 0.0127 0.0457 0.00770.0891 0.0885 0.0632 0.0331 0.03600.0762 0.0139 0.0098 0.0173 0.08320.0821 0.0923 0.0958 0.0955 0.02210.0445 0.0993 0.0965 0.0047 0.05210.9282 0.7506 0.6474 �0.5214 0.8308�0.5558 0.6990 0.8365 0.0859 �0.4526�0.0286 �0.0344 0.8537 0.3495 �0.5500�0.0917 �0.4584 �0.4583 �0.7334 �0.7277�0.9912 0.3724 0.0974 �1.0199 0.813610045.5 11539.4 10240.7 11366.0 10398.112.0758 2.1572 2.0911 2.2187 2.211827.93 24.58 26.65 25.35 28.75852.5 911.6 1026.4 1122.2 702.9

0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25

x 104

1.85

1.9

1.95

2

2.05

2.1

2.15

2.2

2.25

2.3

2.35

Emis

sion

(ton

/h)

Cost ($/h)

MODENSGA-IIPDESPEA2

Fig. 5. Pareto-optimal front of test system-2 for case-1.

M. Basu / Energy Conversion and Management 52 (2011) 903–910 909

respect to all objectives, i.e. cost, emission and active power trans-mission loss.

7.2. Test system-2

Test system-2 is IEEE 57-bus system. In this work, five lines,(18, 19), (31, 32), (34, 32), (40, 56) and (39, 57) are installed withTCSC and five lines, (4, 5), (5, 6), (26, 27), (41, 43) and (53, 54) areinstalled with TCPS. Here, voltage magnitude limits of generatorbuses and load buses are set to 0.95p.u. 6 V 6 1.1p.u. and0.93p.u. 6 V 6 1.1p.u. respectively. Voltage angle limits are set to�20� < d < 0�.

Fuel cost, emission and active power transmission loss havebeen minimized individually by using DE. These results are

1.111.9

22.1

2.22.3

2.42.518

20

22

24

26

28

30

32

34

Emission (ton/h)

Loss

(MW

)

Fig. 6. Pareto-optimal front of

compared with those obtained from RCGA. All the results are pro-vided in Table 3.

MODE has been applied to solve case-1 and case-2. Case-3 isminimized by using DE. It is seen Table 3 that fuel cost is8305.00 $/h under cost minimization criterion, but it increases to15903.0 $/h and 15696.00 $/h for the cases of the emission minimi-zation and active power transmission loss minimization criteria byusing DE. In case-1, fuel cost is 9941.60 $/h which is more than8305.00 $/h and less than 15903.00 $/h and emission is 2.0763ton/h which is less than 2.4333 ton/h and more than 1.8589 ton/h. Active power transmission loss is 27.94 MW. In case-2, fuel costand emission are 11507.06 $/h and 2.1216 ton/h which are morethan case-1 and active power transmission loss is 24.64 MW whichis less than 27.94 MW.

The results obtained from MODE are compared with those ob-tained from NSGA-II, PDE, SPEA2, and case-3 which are shown inTable 4.

Fig. 5 shows the distributions of 100 nondominated solutionsobtained in the last generation of MODE, NSGA-II, PDE, and SPEA2for case-1 of this test system. It is seen from Fig. 5 that cost oremission can not be further improved without degrading theother. The distributions of 100 nondominated solutions obtainedin the last generation of MODE, NSGA-II, PDE, and SPEA2 forcase-2 of this test system are shown in Fig. 6. It is seen fromFig. 6, that no solution can be considered better than any otherwith respect all objectives, i.e. cost, emission and active powertransmission loss.

It is seen from the Tables that for both test systems, more CPUtime is required for case-1 than to single objective optimization be-cause more computations are needed for case-1 than single objec-tive optimization. Similarly, for both test systems, more CPU timeis required for case-2 than case-1 as more computations are re-quired for case-2 than case-1.

For both test systems, in DE, MODE, and PDE the populationsize, scaling factor, crossover constant and maximum generation

.151.2

1.251.3

1.351.4

1.451.5

1.55

x 104

Cost ($/h)

MODE

NSGA-II

PDE

SPEA2

test system-2 for case-2.

Table A-1Generator Data of IEEE 30-bus system.

Bus no. PminG

PmaxG a b c d e a b c g k

(MW) (MW) ($/h) ($/MW h) ($/MW2 h) ($/h) (rad/MW) (ton/h) (ton/MW h) (ton/MW2 h) (ton/h) (1/MW)

1 50 200 0 2.00 0.00375 18.00 0.0370 4.091E�2 �5.554E�4 6.490E�6 2.0E�4 2.857E�22 20 80 0 1.75 0.01750 16.00 0.0380 2.543E�2 �6.047E�4 5.638E�6 5.0E�4 3.333E�25 15 50 0 1.00 0.06250 14.00 0.0400 4.258E�2 �5.094E�4 4.586E�6 1.0E�6 8.000E�28 10 35 0 3.25 0.00834 12.00 0.0450 5.326E�2 �3.550E�4 3.380E�6 2.0E�3 2.000E�2

11 10 30 0 3.00 0.02500 13.00 0.0420 4.258E�2 �5.094E�4 4.586E�6 1.0E�6 8.000E�213 12 40 0 3.00 0.02500 13.50 0.0410 6.131E�2 �5.555E�4 5.151E�6 1.0E�5 6.667E�2

Table A-2Generator data of IEEE 57-bus system.

Bus no. PminG

PmaxG a b c d e a b c g k

(MW) (MW) ($/h) ($/MW h) ($/MW2 h) ($/h) (rad/MW) (ton/h) (ton/MWh) (ton/MW2 h) (ton/h) (1/MW)

1 0 600 0 2.00 0.00375 18.00 0.0370 4.091E�2 �5.554E�4 6.490E�6 2.0E�4 2.857E�32 0 500 0 1.75 0.01750 16.00 0.0380 2.543E�2 �6.047E�4 5.638E�6 5.0E�4 3.333E�33 0 500 0 3.00 0.02500 13.50 0.0410 6.131E�2 �5.555E�4 5.151E�6 1.0E�5 6.667E�36 0 500 0 2.00 0.00375 18.00 0.0370 3.491E�2 �5.754E�4 6.390E�6 3.0E�4 2.657E�38 0 650 0 1.00 0.06250 14.00 0.0400 4.258E�2 �5.094E�4 4.586E�6 1.0E�6 8.000E�39 0 500 0 1.75 0.01950 15.00 0.0390 2.754E�2 �5.847E�4 5.238E�6 4.0E�4 2.875E�3

12 0 500 0 3.25 0.00834 12.00 0.0450 5.326E�2 �3.555E�4 3.380E�6 2.0E�3 2.000E�3

910 M. Basu / Energy Conversion and Management 52 (2011) 903–910

number have been selected as 100, 1.0, 1.0 and 100 respectivelyand in RCGA, NSGA-II, and SPEA2 population size, crossover, muta-tion probabilities and maximum generation number have been se-lected as 100, 0.9, 0.2 and 100 respectively.

8. Conclusion

This paper has presented multi-objective differential evolutionto deal with optimal power flow problem of FACTS device-equipped power systems having multiple objectives such as fuelcost, emission and active power transmission loss. The results ob-tained from the proposed approach have been compared withthose obtained from nondominated sorting genetic algorithm-II,strength pareto evolutionary algorithm 2 and pareto differentialevolution on the modified IEEE 30-bus and 57-bus test systemswith TCSC and TCPS at the fixed locations. It is seen from the com-parison that the proposed approach provides a competitive perfor-mance in terms solution as well as computation time.

Appendix A

Tables A-1 and A-2.

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