Electric Power Systems Research 80 (2010) 1286–1292

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Electric Power Systems Research

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Differential evolution algorithm for emission constrainedeconomic power dispatch problem

A.A. Abou El Elaa, M.A. Abidob, S.R. Speaa,∗

a Electrical Engineering Department, Faculty of Engineering, Menoufiya University, Egyptb Electrical Engineering Department, King Fahd University of Petroleum and Minerals, Saudi Arabia

a r t i c l e i n f o

Article history:Received 27 July 2009Received in revised form 2 January 2010Accepted 29 April 2010Available online 1 June 2010

Keywords:

a b s t r a c t

In this paper, a differential evolution (DE) algorithm is developed to solve emission constrained economicpower dispatch (ECEPD) problem. Traditionally electric power systems are operated in such a way thatthe total fuel cost is minimized regardless of emissions produced. With increased requirements for envi-ronmental protection, alternative strategies are required. The proposed algorithm attempts to reduce theproduction of atmospheric emissions such as sulfur oxides and nitrogen oxides, caused by the operationof fossil-fueled thermal generation. Such reduction is achieved by including emissions as a constraintin the objective of the overall dispatching problem. A simple constraint approach to handle the system

Economic power dispatchDifferential evolution algorithmFuel cost minimizationEE

constraints is proposed. The performance of the proposed algorithm is tested on standard IEEE 30-bussystem and is compared with conventional methods. The results obtained demonstrate the effectiveness

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. Introduction

The basic objective of economic power dispatch (EPD) of electricower generation is to schedule the generation unit outputs so as toeet the load demand at minimum operating cost while satisfying

ll unit and system equality and inequality constraints [1,2]. Thisroblem has been tackled by many researchers in the past. The

iterature of the EPD problem and its solution methods are surveyedn [3].

The generation of electricity from fossil fuel releases severalontaminants, such as sulfur oxides, nitrogen oxides and carbonioxide, into the atmosphere. Recently the problem which hasttracted much attention is pollution minimization due to theressing public demand for clean air. Since the text of the Cleanir Act Amendments of 1990 and similar acts by European and

apanese governments, environmental constraints have topped theist of utility management concerns [4].

Several strategies to reduce the atmospheric emissions haveeen proposed and discussed. These include installation of pol-

utant cleaning equipment, switching to low emission fuels,

eplacement of the aged fuel-burners with cleaner ones, and emis-ion dispatching. The first three options require installation of newquipment and/or modification of the existing ones that involveonsiderable capital outlay and, hence, they can be considered as

∗ Corresponding author.E-mail address: shi spea@yahoo.com (S.R. Spea).

378-7796/$ – see front matter. Published by Elsevier B.V.oi:10.1016/j.epsr.2010.04.011

solving the emission constrained economic power dispatch problem.Published by Elsevier B.V.

long-term options. The emission dispatching option is an attractiveshort-term alternative in which both emission and fuel cost is to beminimized. In recent years, this option has received much atten-tion since it requires only small modification of the basic economicdispatch to include emissions [5,6].

Several methods have been used to represent emission levels.A summary of environmental/economic dispatch algorithms dat-ing back to 1970 using conventional optimization methods hasbeen provided in [7]. In [4], the environmentally constrained eco-nomic dispatch problem is solved using the Hopfield NN methodin which the energy function of the Hopfield Neural Networkcontains both the objective function, and equality and inequalityconstraints. Also, the emission is inserted as a constraint and theproblem was solved using Neural Network in [8]. Abido [1,9–11]tried to find the best compromise between the conflicting targets ofminimum cost and minimum emission by means of suitable Pareto-based multi-objective procedures. In other research direction, theemission/economic dispatch problem was converted to a singleobjective problem by linear combination of different objectives asa weighted sum [12].

A new evolutionary computation technique, called differen-tial evolution (DE) algorithm, has been proposed and introducedrecently [13–16]. The algorithm is inspired by biological and soci-

ological motivations and can take care of optimality on rough,discontinuous and multi-modal surfaces. The DE has three mainadvantages: it can find near optimal solution regardless the initialparameter values, its convergence is fast and it uses few numberof control parameters. In addition, DE is simple in coding and easy

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(a) Initialization

At the very beginning of a DE run, problem independent vari-ables are initialized in their feasible numerical range. Therefore,

A.A. Abou El Ela et al. / Electric Powe

o use. It can handle integer and discrete optimization problems13–16].

The performance of DE algorithm was compared to that of differ-nt heuristic techniques. It is found that, the convergence speed ofE algorithm is significantly better than that of genetic algorithms

GAs) [15]. In [17], the performance of DE algorithm was comparedo particle swarm optimization (PSO) and evolutionary algorithmsEAs). The comparison is performed on a suite of 34 widely usedenchmark problems. It is found that, DE algorithm is the best per-orming algorithm as it finds the lowest fitness value for most of theroblems considered in that study. Also, DE algorithm is robust; it

s able to reproduce the same results consistently over many tri-ls, whereas the performance of PSO is far more dependent on theandomized initialization of the individuals [17]. Therefore, the DElgorithm seems to be a promising approach for engineering opti-ization problems. It has successfully been applied and studied toany artificial and real optimization problems [18–21].In this paper, a differential evolution algorithm is used to

olve emission constrained economic power dispatch problem. Theroblem is formulated as a nonlinear optimization problem withquality and inequality constraints. A simple constraint approach toandle the system constraints is proposed. The performance of theroposed algorithm is tested on standard IEEE 30-bus system and

s compared with conventional methods. The potential and effec-iveness of the proposed approach are demonstrated. Additionally,he results are compared to those reported in the literature.

. Problem formulation

The solution of ECEPD problem aims to minimize operating fuelost while satisfying all system equality and inequality constraints.he solution of ECEPD problem attempts to reduce the productionf atmospheric emissions such as sulfur oxides SOx and nitrogenxides NOx, caused by the operation of fossil-fueled thermal gen-ration. Such reduction is achieved by including emissions as aonstraint in the objective of the overall dispatching problem.

.1. Problem objective

.1.1. Minimization of fuel costThe generator cost curves are represented by quadratic func-

ions. The total $/h fuel cost F(PG) can be expressed as:

(PG) =NG∑i=1

ai + biPGi + ciP2Gi (1)

here NG is the number of generators, ai, bi and ci are the costoefficients of the ith generator, and PGi is the real output power ofenerator i.

.2. System constraints

.2.1. Power balance constraintThe total power generation must cover the total load demand

D and the real power loss in transmission lines Ploss. Hence,

NG

i=1

PGi = PD + Ploss (2)

.2.2. Generation capacity constraintFor stable operation, real power output of each generator is

estricted by its lower and upper limits as follows:

minGi ≤ PGi ≤ Pmax

Gi , i = 1, . . . , NG (3)

ms Research 80 (2010) 1286–1292 1287

2.2.3. Security constraintsFor secure operation, the transmission line loading is restricted

by its upper limits as follows:

Sli ≤ Smaxli , i = 1, . . . , nl (4)

where Sl is transmission line loading (or line flow), and nl is thenumber of transmission lines.

2.2.4. Emission constraintThe atmospheric pollutants such as SOx and NOx, caused by

fossil-fueled thermal units can be modeled separately. However,for comparison purposes, the total ton/h emission (E(PGi)) of thesepollutants can be expressed as [1]:

E(PGi) =NG∑i=1

10−2(˛i + ˇiPGi + �iP2Gi) + �i exp(�iPGi) (5)

where ˛i, ˇi, � i, �i, and �i are coefficients of the ith generatoremission characteristics. Hence, the emission constraint can beexpressed as:

E(PGi) ≤ ˚ (6)

where ˚ is the emission tolerance.

3. Basics of differential evolution

In 1995, Storn and Price proposed a new floating point encodedevolutionary algorithm for global optimization and named it dif-ferential evolution (DE) algorithm owing to a special kind ofdifferential operator, which they invoked to create new off-springfrom parent chromosomes instead of classical crossover or muta-tion [13].

DE algorithm is a population based algorithm using three oper-ators; crossover, mutation and selection. Several optimizationparameters must also be tuned. These parameters have joinedtogether under the common name control parameters. In fact, thereare only three real control parameters in the algorithm, which aredifferentiation (or mutation) constant F, crossover constant CR, andsize of population NP. The rest of the parameters are dimension ofproblem D that scales the difficulty of the optimization task; max-imum number of generations (or iterations) GEN, which may serveas a stopping condition; and low and high boundary constraints ofvariables that limit the feasible area [13,14].

DE works through a simple cycle of stages, presented in Fig. 1.These stages can be cleared as follow:

Fig. 1. DE cycle of stages.

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288 A.A. Abou El Ela et al. / Electric Powe

f the jth variable of the given problem has its lower and upperound as xL

jand xu

j, respectively, then the jth component of the ith

opulation members may be initialized as,

i,j(0) = xLj + rand(0, 1) · (xu

j − xLj ) (7)

here rand(0, 1) is a uniformly distributed random numberetween 0 and 1.

b) Mutation

In each generation to change each population member �Xi(t), aonor vector �vi(t) is created. It is the method of creating this donorector, which demarcates between the various DE schemes. How-ver, in this paper, one such specific mutation strategy known asE/rand/1 is discussed.

To create a donor vector �vi(t) for each ith member, three param-ter vectors xr1, xr2 and xr3 are chosen randomly from the currentopulation and not coinciding with the current xi. Next, a scalarumber F scales the difference of any two of the three vectors andhe scaled difference is added to the third one whence the donorector �vi(t) is obtained. The usual choice for F is a number between.4 and 1.0. So, the process for the jth component of each vectoran be expressed as,

i,j(t + 1) = xr1,j(t) + F · (xr2,j(t) − xr3,j(t)) (8)

c) Crossover

To increase the diversity of the population, crossover operators carried out in which the donor vector exchanges its components

ith those of the current member �Xi(t).Two types of crossover schemes can be used with DE technique.

hese are exponential crossover and binomial crossover. Althoughhe exponential crossover was proposed in the original work oftorn and Price [13], the binomial variant was much more used inecent applications [17].

In exponential type, the crossover is performed on the D vari-bles in one loop as far as it is within the CR bound. The first timerandomly picked number between 0 and 1 goes beyond the CR

alue, no crossover is performed and the remaining variables areeft intact. In binomial type, the crossover is performed on all Dariables as far as a randomly picked number between 0 and 1 isithin the CR value. So for high values of CR, the exponential and

inomial crossovers yield similar results.Moreover, in the case of exponential crossover one has to be

ware of the fact that there is a small range of CR values (typically0.9, 1]) to which the DE is sensitive. This could explain the rule ofhumb derived for the original variant of DE. On the other hand, forhe same value of CR, the exponential variant needs a larger valueor the scaling parameter F in order to avoid premature convergence22].

In this paper, binomial crossover scheme is used which is per-ormed on all D variables and can be expressed as:

i,j(t) ={

vi,j(t) if rand(0, 1) < CRxi,j(t) else

(9)

here ui,j(t) represents the child that will compete with the parenti,j(t).

d) Selection

To keep the population size constant over subsequent genera-ions, the selection process is carried out to determine which one ofhe child and the parent will survive in the next generation, i.e., at

ems Research 80 (2010) 1286–1292

time t = t + 1. DE actually involves the Survival of the fittest principlein its selection process. The selection process can be expressed as,

�Xi(t + 1) ={ �Ui(t) if f ( �Ui(t)) ≤ f (�Xi(t))

�Xi(t) if f (�Xi(t)) < f ( �U(t))(10)

where f ( ) is the function to be minimized. So, if the child yieldsa better value of the fitness function, it replaces its parent in thenext generation; otherwise, the parent is retained in the popula-tion. Hence the population either gets better in terms of the fitnessfunction or remains constant but never deteriorates.

4. Implementation of the proposed algorithm

The proposed algorithm is developed and implemented to solveECEPD problem as follows.

4.1. Settings of the proposed approach

The proposed DE-based approach has been developed andimplemented using the MATLAB software. Initially, several runshave been done with different values of DE key parameters suchas differentiation (or mutation) constant F, crossover constant CR,size of population NP, and maximum number of generations GENwhich is used here as a stopping criteria to choose the best suitablevalues of key parameters. In this paper, the following values areselected:

F = 0.9; CR = 0.5; NP = 50; GEN = 200

4.2. DE-based ECEPD implementation

The main features of DE-based ECEPD can be summarized asfollow:

4.2.1. InitializationThe first step in the algorithm is creating an initial population.

All the independent variables (PGi except for the slack bus) haveto be generated according to formula (7), where each independentparameter of each individual in the population is assigned a valueinside the given feasible region of the generator. This creates parentvectors of independent variables for the first generation.

4.2.2. Constraint handling

1. Since the independent variables are created within their limits,they readily satisfy the corresponding inequality constraints.

2. After, finding the independent variables, dependent variableswill be calculated from load flow solution, consequently thepower balance constraint represents by Eq. (3) is satisfied. How-ever, the value of PG1 may violate its limits. So, it is checked if itis within limits or not. If PG1 violates its limits, set PG1 = e, where,e is a very high value. So, its corresponding fuel cost value willbe very high and will not be chosen.

4.2.3. Evaluation (fitness assignment)Calculate the fitness values of the different individuals. The indi-

viduals in the current population are evaluated in the objectivespace and then assigned a scalar value known as fitness. Depend-ing on the fitness values, individuals will be selected to form the

new population. Individuals which have a low fitness value havethe chance to be selected.

It is worth mentioning that the constraint-handling approachimplemented in this study is that the unfeasible solutions are penal-ized by assigning a very high value for their fitness.

A.A. Abou El Ela et al. / Electric Power Systems Research 80 (2010) 1286–1292 1289

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Table 2Best solutions for cost and emission optimized individually for Case 1.

Best cost Best emission

PG1 10.9774 40.5993PG2 29.9880 45.9152PG5 52.3821 53.8250PG8 101.6245 38.2705PG11 52.4363 53.7882PG13 35.9918 51.0018Fuel cost 600.1114 638.2907Emission 0.2221 0.1942

genetic algorithm (NPGA) [9]. The comparison results are given inTables 3 and 4. From the comparison, it is noticed that the proposedalgorithm gives more reduction in cost compared with LP and thesame results as in Ref. [9]. But the number of function evaluations

Fig. 2. Single line diagram of IEEE 30-bus test system.

. Results and discussion

The proposed DE-based ECEPD algorithm has been tested on thetandard IEEE 6-generator 30-bus test system shown in Fig. 2. Theystem data is given in Appendix A [10,23]. The values of fuel costnd emission coefficients are given in Table 1 [10].

To demonstrate the effectiveness of the proposed algorithm,hree different cases have been considered as follows:

Case 1: For the purpose of comparison with the reported results,the system is considered as lossless and the security constraintis released. Therefore, the problem constraints are the powerbalance constraint without Ploss and the generation capacity con-straint only.Case 2: Ploss is considered in the power balance constraint, andthe generation capacity constraint is also considered.Case 3: All constraints are considered.

.1. Case 1 (only generation capacity constraint is considered)

To investigate the effectiveness of the proposed algorithm theystem is considered as lossless and the security constraint iseleased. Also, emission is considered as an objective not a con-

able 1uel cost and emission coefficients.

G1 G2 G3 G4 G5 G6

Costa 10 10 20 10 20 10b 200 150 180 100 180 150c 100 120 40 60 40 100

Emission˛ 4.091 2.543 4.258 5.326 4.258 6.131ˇ −5.554 −6.047 −5.094 −3.550 −5.094 −5.555� 6.490 5.638 4.586 3.380 4.586 5.151� 2.0E−4 5.0E−4 1.0E−6 2.0E−3 1.0E−6 1.0E−5� 2.857 3.333 8.000 2.000 8.000 6.667

Fig. 3. Convergence of fuel cost objective for Case 1.

straint for the purpose of comparison with the reported results andto find the optimal value for emission, which will used as a guide toassume the emission tolerance value. In this case, fuel cost repre-sented by Eq. (1) and emission represented by Eq. (5) are optimizedindividually. The results of this case are given in Table 2. Conver-gence of fuel cost and emission objectives are shown in Figs. 3 and 4.The results of the proposed algorithm were compared to thosereported using linear programming (LP) [23], and niched Pareto

Fig. 4. Convergence of emission objective for Case 1.

1290 A.A. Abou El Ela et al. / Electric Power Systems Research 80 (2010) 1286–1292

Table 3Results of best cost comparison.

LP [23] NPGA [9] Proposed algorithm

PG1 15 10.954 10.9774PG2 30 29.967 29.9880PG5 55 52.447 52.3821PG8 105 101.601 101.6245PG11 46 52.469 52.4363PG13 35 35.963 35.9918Best cost 606.3140 600.1140 600.1114Correspon. emission 0.2233 0.2221 0.2221

Table 4Results of best emission comparison.

LP [23] NPGA [9] Proposed algorithm

PG1 40 40.584 40.5993PG2 45 45.915 45.9152PG5 55 53.797 53.8250PG8 40 38.30 38.2705PG11 55 53.791 53.7882PG13 50 51.012 51.0018Best emission 0.1942 0.1942 0.1942Correspon. cost 639.6000 638.2600 638.2907

Table 5Best solutions for cost and emission optimized individually for Case 2.

Best cost Best emission

PG1 30.6964 46.2961PG2 60.1386 54.3618PG5 97.1934 38.9249PG8 51.7076 54.4023PG11 35.5532 51.4505PG13 11.5287 41.0047

au

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Fig. 6. Convergence of emission objective for Case 2.

Table 6Best fuel cost results of Case 3 considering different values of emission tolerance.

˚ 0.1942 0.2 0.21 0.22

PG1 46.2961 38.7728 22.6546 19.2541PG2 54.3618 52.0112 51.3624 54.7384PG5 38.9249 25.3813 78.6511 37.8667PG8 54.4023 72.8142 59.5027 58.4694PG11 51.4505 46.5518 44.8452 36.8925PG13 41.0047 51.0066 29.7755 79.6738

Fuel cost 608.0658 645.0850Emission 0.2193 0.1942Ploss 3.4180 3.0403

re less with the proposed algorithm as the population size of 50 issed compared with 200 for NPGA.

.2. Case 2 (the power balance constraint is also considered)

In this Case, Ploss is considered in the power balance constraint.

lso, cost and emission are considered as separate objectives andptimized individually. Emission is considered as an objective inhis case for the purpose of comparison with the reported resultsnd to find the optimal value for emission, which will be used as a

Fig. 5. Convergence of fuel cost objective for Case 2.

Fuel cost 645.0850 626.0345 612.3914 609.2920Emission 0.1942 0.1979 0.2063 0.2106Ploss 3.0403 3.1379 3.3914 3.4949

guide to assume the emission tolerance value in case of consideringlosses. The results of this case are given in Table 5. Convergence offuel cost and emission objectives are shown in Figs. 5 and 6.

5.3. Case 3 (all constraints are considered)

In this case, all constraints are considered. Emission is consid-ered as a constraint. From Case 1 and Case 2, it is observed thatthe min. value for emission obtained by the proposed algorithm is

0.1942 ton/h. This value will be considered as a guide for choosingemission tolerance (˚). Several runs are carried out using differentvalues for ˚. The results of this case are given in Table 6.

Table A.1Load data.

Bus no. Load Bus no. Load

P (p.u.) Q (p.u.) P (p.u.) Q (p.u.)

1 0.000 0.000 16 0.035 0.0182 0.217 0.127 17 0.090 0.0583 0.024 0.012 18 0.032 0.0094 0.076 0.016 19 0.095 0.0345 0.942 0.190 20 0.022 0.0076 0.000 0.000 21 0.175 0.1127 0.228 0.109 22 0.000 0.0008 0.300 0.300 23 0.032 0.0169 0.000 0.000 24 0.087 0.067

10 0.058 0.020 25 0.000 0.00011 0.000 0.000 26 0.035 0.02312 0.112 0.075 27 0.000 0.00013 0.000 0.000 28 0.000 0.00014 0.062 0.016 29 0.024 0.00915 0.082 0.025 30 0.106 0.019

r Systems Research 80 (2010) 1286–1292 1291

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Table A.3The minimum and maximum limits for generators real power with the initialsettings.

Min Max Initial

PG1 5 150 10PG2 5 150 25PG5 5 150 20PG8 5 150 135PG11 5 150 36

A.A. Abou El Ela et al. / Electric Powe

From the table, it is observed that increasing emission toleranceives more reduction in fuel cost. The system operator can choosehe suitable choice of ˚ according to his interest and system needs.

From the above cases, the results obtained show the effective-ess of the proposed algorithm. Also, from the comparison withPGA, it is noticed that, the same reduction in cost can be obtainedith less number of function evaluations with the proposed algo-

ithm as the population size of 50 is used compared to 200 forPGA.

. Conclusions

In this paper, a differential evolution optimization algorithm haseen proposed, developed, and successfully applied to solve emis-ion constrained economic power dispatch (ECEPD) problem. Theroposed approach has been tested and examined on the standard

EEE 30-bus test system. Several cases have been considered, theimulation results demonstrate the effectiveness and robustnessf the proposed algorithm to solve ECEPD problem. Moreover, theesults of the proposed DE algorithm have been compared to those

eported in the literature. The comparison confirms the effective-ess and the superiority of the proposed DE approach over the otherechniques in terms of solution quality.

able A.2ine data.

Line no. From bus To bus Line impedance

R (p.u.) X (p.u.)

1 1 2 0.0192 0.05752 1 3 0.0452 0.18523 2 4 0.0570 0.17374 3 4 0.0132 0.03795 2 5 0.0472 0.19836 2 6 0.0581 0.17637 4 6 0.0119 0.04148 5 7 0.0460 0.11609 6 7 0.0267 0.0820

10 6 8 0.0120 0.042011 6 9 0.0000 0.208012 6 10 0.0000 0.556013 9 11 0.0000 0.208014 9 10 0.0000 0.110015 4 12 0.0000 0.256016 12 13 0.0000 0.140017 12 14 0.1231 0.255918 12 15 0.0662 0.130419 12 16 0.0945 0.198720 14 15 0.2210 0.199721 16 17 0.0824 0.193222 15 18 0.1070 0.218523 18 19 0.0639 0.129224 19 20 0.0340 0.068025 10 20 0.0936 0.209026 10 17 0.0324 0.084527 10 21 0.0348 0.074928 10 22 0.0727 0.149929 21 22 0.0116 0.023630 15 23 0.1000 0.202031 22 24 0.1150 0.179032 23 24 0.1320 0.270033 24 25 0.1885 0.329234 25 26 0.2544 0.380035 25 27 0.1093 0.208736 28 27 0.0000 0.396037 27 29 0.2198 0.415338 27 30 0.3202 0.602739 29 30 0.2399 0.453340 8 28 0.6360 0.200041 6 28 0.0169 0.0599

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PG13 5 150 60Fuel cost ($/h) 704.96Emission 0.2904

Acknowledgement

Dr. M.A. Abido would like to acknowledge the support of KingFahd University of Petroleum and Minerals, Saudi Arabia.

Appendix A. System data

Data for IEEE 30-bus test system (100 MVA base) are given inTables A.1–A.3.

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