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International Journal of Emerging Electric Power Systems

Volume 9, Issue 4 2008 Article 5

Effect of Control Parameters on DifferentialEvolution based Combined Economic

Emission Dispatch with Valve-Point Loadingand Transmission Loss

Kamal K. Mandal N. Chakraborty

Jadavpur University, [email protected] University, chakraborty [email protected]

Copyright c 2008 The Berkeley Electronic Press. All rights reserved.


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Effect of Control Parameters on DifferentialEvolution based Combined Economic

Emission Dispatch with Valve-Point Loadingand Transmission Loss

Kamal K. Mandal and N. Chakraborty

Abstract

Differential evolution (DE) has been proved to be a powerful evolutionary algorithm for globaloptimization in many engineering problems. The performance of this type of evolutionary algo-rithms is heavily dependent on the setting of control parameters. Proper selection of the controlparameters is very important for the success of the algorithm. Optimal settings of control param-eters of differential evolution depend on the specic problem under consideration. In this paper, astudy of control parameters on differential evolution based combined economic emission dispatchwith valve-point loading and transmission loss is conducted empirically. The problem is formu-lated considering equality constraints on power balance and inequality constraints on generationcapacity limits as well as the transmission losses and effects of valve point loadings. The feasi-bility of the proposed method is demonstrated on a fourteen-generator system. The results of theeffect of the variation of different parameters are presented systematically and it is observed thatthe search algorithm may fail in nding the optimal value if the parameter selection is not donewith proper attention.

KEYWORDS: control parameters, differential evolution (DE), combined economic emission dis-patch (CEED), price penalty factor

We would like to acknowledge and thank Jadavpur University, Kolkata, India for providing allthe necessary help to carry out this work.


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1 Introduction

The Differential Evolution (DE) as proposed by Storn and Price [1] is a powerfuloptimization technique designed for global optimization. Besides its goodconvergence properties, main advantage of DE lies with its conceptual simplicity;ease of use and less number of control parameters. Like any other evolutionaryalgorithm, the success of DE is heavily dependent on setting of controlparameters. It has three control parameters: (1) the population size P N (2) themutation factor m f , which is a real-valued factor that controls the amplification of differential variation and (3) the crossover factor RC , which is also a real-valuedfactor that controls the crossover operation. One of the main problems inevolution strategies of DE is to choose the control parameters such that it exhibitsgood behavior i.e. it does not prematurely converge to a point that is not globally

optimal or stagnate and has an acceptable rate of convergence toward the globaloptimum. Premature convergence may occur under different situations: thepopulation has converged to local optimum of the objective function or thepopulation has lost its diversity or the search algorithm proceeds slowly or doesnot proceed at all [2]. It is seen that DE may sometimes stops proceeding towardsglobal optimum and stagnation occurs. Stagnation may occur under varioussituations: the population has not converge to a local optimum or any other pointor the population is still remaining diverse and occasionally even if the newindividuals may enter in to the population, the searching algorithm does notprogress towards any better solutions [2].

The effectiveness, efficiency and robustness of the DE algorithm are

sensitive to the setting of control parameters. The best setting for the controlparameters depends on the problem in hand and requirement of computation timeand accuracy [3]. Although, as reported in the literature, control parameters of DE are not difficult to choose [1], but rules for choosing control parameters arenot general [1]. On the other hand, it is also reported that choosing proper controlparameters for DE is more difficult than expected [4]. It is important to selectoptimal parameters for each problem separately and carefully to avoid prematureconvergence or even stagnation [2], [5], [6]. Brest et al. [7] assessed the selectionof control parameter and reported that efficiency and robustness of DE algorithmare much more sensitive to the setting of mutation factor m f and crossoverratio RC than to the setting of the population size P N . Zaharie [8] discussed the

relationship between control parameters of DE and the evolution of populationvariance and reported critical interval for control parameters of DE. Teo proposeda method of self-adapting population size in addition to self-adapting mutationfactor and crossover ratio [9].

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Mandal and Chakraborty: Effect of Control Parameters

Published by The Berkeley Electronic Press, 2008


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Setting of control parameters in evolutionary algorithms like DE canbe classified into two categories: Parameter tuning and parameter control.Parameter tuning is the commonly practiced approach that is used in findingproper values for the parameters before the actual run of the algorithm. Then thealgorithm is tested using these values, which remain fixed during the run.Generally, electric power plants are operated on the basis of least fuel coststrategies without considering the pollutants produced. However presentlyenvironmental management has topped the list of utility management concern[10] and it has become almost compulsory for electric utilities to reduce thepollution level below certain limit. By proper load allocation among the variousgenerating units of the plants, the harmful effects of the emission of gaseouspollutants from power stations, particularly from thermal power stations, can bereduced. A revised power dispatch program is required in such cases that willconsider both the generation cost and emission cost as the objective functions.One of the major complications in the above considerations is that the cost andemission functions are conflicting to each other. In other words, minimizingpollution increases cost and vice versa. Since the ideas of minimum cost andminimum emission dispatch are conflicting to each other, therefore the problem of choosing the least cost generation schedule with environmental objectivesbecomes much more complicated. In this paper, tests for various parametersettings of DE are conducted for the combined economic emission dispatch withgenerator constraints, valve-point loading and transmission loss. The influence of parameter settings of DE is tested on a fourteen-generator system and results arepresented. Based on the results obtained, recommendations are suggested for thesuitable range of control parameters of Differential Evolution for the presentproblem.

2 Problem formulation2.1 Economy

The pure Economic Load Dispatch (ELD) problem is one of the major problemsin power system operation and planning. The classical ELD problem may bedescribed by minimizing the total fuel cost of the generating units under severaloperating constraints. The fuel cost curve for any unit is assumed to beapproximated by segments of quadratic functions of the active power output of the generator. For a given power system network, the problem may be describedas optimization (minimization) of total fuel cost as defined by (1) under a set of operating constraints;

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International Journal of Emerging Electric Power Systems, Vol. 9 [2008], Iss. 4, Art. 5

http://www.bepress.com/ijeeps/vol9/iss4/art5


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( ) ( )1)( 21

iiii

n

iig cPbPaPFC ++=

=

where )P(FC g is the total fuel cost of generation in the system ($/hr), iii c ,b ,a are

the fuel cost coefficients of the i th generating unit, iP is power generated by thei th unit and n is the number of thermal units. The coefficients ii b ,a and ic aregenerally obtained by curve fitting [11], [12].

However, for more practical and accurate modeling of fuel costfunction, the above expression is to be modified suitably. Modern thermal powerplants consist of generating units have multi-valve steam turbines in order toincorporate flexible operational facilities. The generating units with multi-valveturbines have very different cost curve compared with that defined by (1) and

exhibit a greater variation in the fuel cost curves. Typically, ripples are introducedin to the fuel cost curve as each steam valve starts to operate. The valve-pointeffect may be considered by adding a sinusoidal function [13], [14] to thequadratic cost function described above. Hence, the problem described by (1) isrevised as follows:

( )( )( ) ( )=

+++=n

iiiiiiiiiigv PP f ecPbPaPFC

1min,

2 2|sin|)(

where )P(FC gv is the total fuel cost of generation ($/hr) including valve point

loading, ii f ,e are the fuel cost coefficients of the i th generating unit reflectingthe valve-point effect.

The cost is minimized with the following generator capacities andactive power balance constraints as;

( )3max,min, iii PPP

( )41

L D

n

ii PPP +=

= where, min ,iP and max ,iP are the minimum and maximum power generation by i th

unit respectively, DP is the total power demand and LP is the total transmissionloss.

The transmission loss LP can be calculated by using B matrixtechnique and is defined by (5) as

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defined by (3) and (4).Now, for a trade off between fuel cost and emission cost, (7) can be revised as (8)as

Minimize( )8)(*)( 21 ggvt P EC hwPFC w +=

subjected to the constraints defined by (3) and (4). The weight factors 1w and 2w have many implications. For 11 =w and 02 =w the solution will yield results forpure economic dispatch. For 01 =w and 12 =w results for pure emission dispatchand for 121 == ww results for combined economic emission dispatch can beobtained.

The price penalty factor h can be found out by a practical method asdiscussed in [15]. The following steps can be used to find out the price penaltyfactor for a particular load.

1) Find out the average cost of each generator at maximum power output.2) Find out the average emission of each generator at its maximum output3) Divide the average cost of each generator by its average emission and thus

hi is given as

( )9 / $,,.........1)()(

)()(

max,max,

max,max,lbnih

PP EC

PPFC i

ii

iiv ==

4) Arrange the values of price penalty factor in ascending order.5) Add the maximum capacity of each unit ( max ,iP ) one at a time starting

from the smallest ih unit until Dmax ,i PP is realized.6) At this stage, ih associated with last unit in the process is the price penalty

factor h for the given load.From the above description, it is clear that the value of the price

penalty factor h is dependent on the total power demand and hence it will havedifferent values for different power demand. It is also important to note that thevalue of the price penalty factor h will be the same for ELD, EED and CEED aslong as the power demand is the same.

With all these functions, the main objective of this study is to find theoptimal values of control parameters for Differential Evolution (DE) basedcombined economic emission dispatch with above said constraints including thevalve-point loading and transmission loss.

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3 Differential Evolution (DE)

DE or Differential Evolution belongs to the class of evolutionary algorithms [1],[16] that include Evolution Strategies (ES) and conventional Genetic Algorithms(GA). DE differs from the conventional genetic algorithms in its use of perturbingvectors, which are the difference between two randomly chosen vectors. DE is ascheme by which it generates the trial vectors from a set of initial populations. Ineach step, DE mutates vectors by adding weighted random vector differentials tothem. If the fitness of the trial vector is better than that of the target vector, thetarget vector is replaced by the trial vector in the next generation.

DE offers several strategies for optimization. They are classifiedaccording to the following notation such as DE/x/y/z , where x refers to the methodused for generating parent vector that will form the base for mutated vector, y indicates the number of difference vector used in mutation process and z is thecrossover scheme used in the cross over operation to create the offspringpopulation [16], [17]. The symbol x can be rand (randomly chosen vector) orbest (the best vector found so far). The symbol y, number of difference vector,is normally set to be 1 or 2. For cross over operation, a binomial (notation: bin)or exponential (notation: exp) operation is used.

The version used here is the DE/ rand /1/ bin, which is described by thefollowing steps.

3.1 Initialization

The optimization process in DE is carried with four basic operations:initialization, mutation, crossover and selection. The algorithm starts by creating apopulation vector P of size N P composed of individuals that evolve over G generation. Each individual X i is a vector that contains as many as elements as theproblem decision variable. The population size N P is an algorithm controlparameter selected by the user. Thus,

( )10,........................................,......... )()()( G N G

iG

P X X P =

[ ]( )11.....,.........1,...............,......... )(

,

)(

,1

)(

P

T G

i D

G

i

G

i N i X X X ==

The initial population is chosen randomly in order to cover the entire searchingregion uniformly. A uniform probability distribution for all random variables inassumed in the following as;

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( ) ( )12minmaxmin)0(, j j j ji j X X X X +=

where P N ........... ,.........i 1= and D........... ,......... j 1= ; D is the number of decision or control variables, min j X and

max j X are the lower

and upper limits of the j the decision variables and [ ]1,0 j is a uniformly

distributed random number generated anew for each value of j. )( i , j X 0 is the j th

parameter of the i th individual of the initial population.

3.2 Mutation Operation

Several strategies of mutation have been introduced in the literature of DE [16].The essential ingredient in the mutation operation is the vector difference. Themutation operator creates mutant vectors ( )iV by perturbing a randomly selectedvector ( )k X with the difference of two other randomly selected vectors( )ml X and X according to;

( )13)()()()( GmGlmG

k G

i X X f X V +=

where k X , l X and m X are randomly chosen vectors [ ]P N ...,.......... ,.........1 and .imlk In other words, the indices are mutually different and also fromthe running index i. The mutation factor m f that lies within [0, 2] is a userchosen parameter used to control the perturbation size in the mutation operatorand to avoid search stagnation.

3.3 Crossover Operation

In order to extend further diversity in the searching process, crossover operation isperformed. The crossover operation generates trial vectors ( )iU by mixing theparameter of the mutant vectors with the target vectors. For each mutant vector,an index [ ]P N ........ ,.........q 1 is chosen randomly using a uniform distributionand trial vectors are generated according to;

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( )14,)(

,

,)(,)(

,

==

otherwiseGi j X

q jor RC jif Gi jV

G

i jU

where P N ........., ,.........i 1= and D........, ,......... j 1= ; j is a uniformlydistributed random number within [0, 1] generated anew for each value of j. Thecrossover factor [ ]10 ,C R is a user chosen parameter that controls the diversityof the population. )G( i , j X ,

)G(i , jV and

)G(i , jU are the j th parameter of the i th target

vector, mutant vector and trial vector at G generation respectively.

3.4 Selection Operation

Selection is the operation through which better offspring are generated. Theevaluation (fitness) function of an offspring is compared to that of its parent. Theparent is replaced by its offspring if the fitness of the offspring is better than thatof its parent, while the parent is retained in the next generation if the fitness of theoffspring is worse than that of its parent. Thus, if f denotes the cost (fitness)function under optimization (minimization), then

( ) ( )( )

=+ 15

,

,

)(

)()()(

)1(

otherwise X

X f U f if U X

Gi

Gi

Gi

Gi

G

i

The optimization process is repeated for several generations. This allowsindividuals to improve their fitness while exploring the solution space for optimalvalues. The iterative process of mutation, crossover and selection on thepopulation will continue until a user-specified stopping criterion, normally, themaximum number of generations allowed, is met. The other type of stoppingcriterion, convergence to the global optimum, is possible if the global optimum of the problem is available.

3.5 Constraint Handling

The success of any optimization method, such as DE, lies in the fact that howdoes it handle the constraints relating to the problem. Most of the evolutionaryalgorithms were originally developed to solve unconstrained optimization

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problems. However, over the last few decades, several methods have beenproposed to handle constraints in evolutionary algorithms. Michalewicz et alpresented a complete review of constrained optimization problems in evolutionaryalgorithms [18] and these methods of constraints handling can be grouped intofour categories: methods based on preserving feasibility of solutions, methodsbased on penalty functions, methods which make a clear distinction betweenfeasible and unfeasible solution and hybrid methods.

The present work is based on strategy to generate and keep controlvariable in the feasible region [19]. This can be achieved as follows:

( )

= 16

)(,

max,

)(,

max,

min,

)(,

min,

)(,

otherwise X

X X if X

X X if X

X

Gi j

i jGi ji j

i jGi ji j

Gi j

where, P N .,.......... ,.........i 1= and D ,.......... ,......... j 1= .

In other words, when the generated value is less than or equal to the minimumgenerating limit, it is set at the minimum. On the other hand, if the generatedvalue is greater than or equal to the maximum generating limit, it is set at the

maximum; otherwise the generated value is retained.

4 Results of Parameter Study

Combined economic emission dispatch (CEED) problems considering generatorconstraints, valve point effects and transmission loss have been used here forstudying the effect of different parameter settings of differential evolution (DE).The test system considered in this paper consists of fourteen generators. The costcoefficients, generation limits and emission coefficients, valve point coefficientsare derived from [20], [21] with modifications to incorporate valve-point loadingand transmission losses. Those are shown in Table 1. The problem is solved as a

combined economic emission dispatch one with 11 =w and 12 =w . Transmissionloss is calculated using B-loss coefficient matrix according to (5). The maincontrol parameters considered here are the mutation factor m f , the crossover ratio

RC and the population size P N .

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Table 1Unit Data for Fourteen-Generator System with Valve Point Loading; a, b, c, e andf are Fuel cost coefficients. , and are Emission coefficients

Unit Pmin Pmax a b c e f 1 150 455 0.0050 1.89 150 300 0.035 0.016 -1.500 23.3332 150 455 0.0055 2.00 115 200 0.042 0.031 -1.820 21.0223 20 130 0.0060 3.50 40 200 0.042 0.013 -1.249 22.0504 20 130 0.0050 3.15 122 150 0.063 0.012 -1.355 22.9835 150 470 0.0050 3.05 125 150 0.063 0.020 -1.900 21.3136 135 460 0.0070 2.75 120 150 0..063 0.007 0.805 21.9007 135 465 0.0070 3.45 70 150 0.063 0.015 -1.400 23.0018 60 300 0.0070 3.45 70 150 0.063 0.018 -1.800 24.0039 25 162 0.0050 2.45 130 150 0.063 0.019 -2.000 25.12110 25 160 0.0050 2.45 130 100 0.084 0.012 -1.360 22.99011 20 80 0.0055 2.35 135 100 0.084 0.033 -2.100 27.01012 20 80 0.0045 1.60 200 100 0.084 0.018 -1.800 25.10113 25 85 0.0070 3.45 70 100 0.084 0.018 -1.810 24.31314 15 55 0.0060 3.89 45 100 0.084 0.030 -1.921 27.119

The study on the effect of different parameter settings of differentialevolution (DE) is performed for a demand of 2000 MW.

4.1 Effect of varying Mutation factor - m f

To check the effect of the mutation factor m f , it is varied from 0.95 to 0.05 insteps of 0.05. The crossover factor RC and the population size are set at 0.90 and

100 respectively following the recommendation from the literature [1], [4], [22],[23]. RC is set to a relatively higher value in order to have higher diversity in thepopulation. It means that on an average 90% of the elements of the trial vectorsare identical to the mutant vector that implies a high diversity for the presentsetting of RC . One hundred (100) independent runs are performed for everyparameter combinations. Maximum number of iteration is set at 300. Table 2shows the minimum cost, maximum cost, average cost, minimum emission,maximum emission, average emission and the average computation time. It isclearly seen that from Table 2 that both cost and emission increases for lowervalues as well as higher values of m f . For example, with m f = 0.1, the minimumcost and the minimum emission are found to be $10037.00 and 3738.30 lbsrespectively, while the corresponding values with m f = 0.90 are $9740.00 and3311.20 lbs respectively. Several tests were performed with higher number of iterations (up to 1000 in step of 100) for lower values of m f ( 0.15), but noimprovement in results were observed. It seems that the vector could not reach the

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minimum value and got stuck somewhere on their way to the minimum withlower values and higher values of m f . This may be due to the fact that during the

searching process the difference vector for the perturbation has decreased for toolow values of m f . The values in the range of 750450 . f . m yields better resultsin terms of cost, emission and computation time. Also, it is observed that forhigher values of m f ( 0.75), computation time increases to some extent.Optimum result is obtained with m f = 0.50, when minimum cost, minimumemission and computation time are found to be $ 9592.40, 3232.20 lbs and16.2781 seconds respectively.

Table 2Effect of mutation factor m f on Combined Economic Emission Dispatch (CEED)

with P N = 100 and RC = 0.90 for a demand of 2000 MW.Valueof f m

Fuel Cost ($) Emission (lb) AverageCPU Time

(Sec.)Minimum Maximum Average Minimum Maximum Average

0.95 9701.20 10917.00 10007.00 3310.10 4117.90 3436.20 37.28750.90 9740.00 10215.00 9903.10 3311.20 3566.20 3405.30 35.27660.85 9693.60 1.0066.00 9869.10 3399.00 3872.40 3401.80 34.62660.80 9697.10 9994.20 9817.60 3380.10 3556.80 3416.60 31.77960.75 9648.70 9840.20 9756.30 3385.10 3439.90 3394.10 25.79370.70 9628.70 9872.90 9720.30 3287.90 3341.00 3295.40 23.36870.65 9598.10 9817.80 9714.20 3285.70 3338.60 3219.30 20.07810.60 9589.70 9856.40 9691.30 3285.50 3693.10 3398.20 18.9016

0.55 9596.50 9995.50 9727.70 3284.60 3534.90 3249.60 17.02650.50 9592.40 9845.90 9644.80 3232.20 3430.80 3351.50 16.27810.45 9615.50 10142.00 9840.10 3252.20 3700.50 3376.20 15.38910.40 9684.80 10090.00 9860.30 3425.70 3839.50 3618.60 14.60780.35 9764.50 10153.00 9998.10 3432.50 4632.10 3801.70 14.71870.30 9877.10 10659.00 10208.00 3405.70 4039.10 3804.50 14.48130.25 9935.50 10604.00 10218.00 3644.30 4422.80 4040.30 14.60940.20 9954.00 10452.00 10187.00 3777.90 4430.20 4159.60 14.45940.15 10137.00 10905.00 10562.00 3686.20 4692.50 4156.30 14.05000.10 10037.00 10962.00 10550.00 3738.30 4859.80 4213.00 14.15310.05 10159.00 11122.00 10603.00 3955.90 4584.90 4339.80 14.3125

Thus, it is seen that for the present problem, the value of m f should not besmaller than a certain value (0.45) in order to find the minimum value. It isobserved that smaller value of m f increases the chance of not finding theminimum at all.

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4.2 Effect of varying Crossover factor - RC

To find the influence of the crossover factor R

C , it is also varied from 0.95 to 0.05in steps of 0.05. The mutation factor m f and the population size are set at 0.50and 100 respectively following the result obtained above and recommendationfrom the literature. In this case also, one hundred (100) independent runs areperformed for every parameter combinations and maximum number of iteration isset at 300. The results for different combination of parameter are shown in Table3 in terms of the minimum cost, maximum cost, average cost and the averagecomputation time.

Table 3Effect of crossover ratio RC on Combined Economic Emission Dispatch (CEED)

with P N =100 and m f = 0.50 for a demand of 2000 MW.Valueof RC

Fuel Cost ($) Emission (lb) AverageCPUTime(Sec.)

Minimum Maximum Average Minimum Maximum Average0.95 9618.40 9783.80 9684.90 3262.90 3533.90 3310.30 16.29840.90 9592.40 9845.90 9644.80 3232.20 3430.80 3351.50 16.27810.85 9598.30 9821.80 9690.80 3287.10 3446.30 3392.50 16.27810.80 9596.90 10071.00 9770.10 3255.00 3911.90 3365.10 16.07650.75 9684.90 10188.00 9736.20 3261.05 3572.50 3358.40 16.12970.70 9614.80 10503.00 9792.40 3283.90 3664.50 3313.30 16.6781

0.65 9603.40 98710.00 9723.50 3289.90 3340.20 3366.80 16.41560.60 9639.10 10059.00 9791.00 3286.50 3887.70 3381.80 16.48900.55 9672.50 10058.00 9725.80 3291.50 3642.40 3315.40 16.56870.50 9677.70 10233.00 9844.20 3325.90 3816.80 3393.10 17.90940.45 9690.00 10235.00 9769.10 3382.90 3518.30 3318.70 16.97970.40 9615.70 10153.00 9812.60 3305.80 3580.60 3307.30 17.67500.35 9694.40 10136.00 9793.50 3293.90 3815.10 3341.50 17.66400.30 9660.50 10135.00 9833.60 3268.00 3768.90 3356.60 20.47970.25 9803.10 10320.00 9976.80 3288.60 3900.70 3361.30 18.77650.20 9740.40 10268.00 9952.80 3370.20 3452.40 3381.90 19.08900.15 9888.40 10528.00 10132.00 3350.00 3773.60 3414.10 23.63590.10 10028.00 10936.00 10217.00 3389.40 3898.30 3561.00 28.04370.05 10139.00 10492.00 10339.00 3442.40 3940.30 3642.90 28.4640

From the Table 3, it is seen that for lower values of RC , the minimum cost,minimum emission and the computation time increase. For example, with RC =0.1, the minimum cost, the minimum emission and the average computation timeare found to be $10028.00, 3389.40 lbs and 28.0437 seconds respectively. In this

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case also, several tests were performed with higher number of iterations (up to1000 in step of 100) for lower values of RC ( 0.10), but no improvement in

results were observed. The values in the range of 950300 .C . R yields betterresults in terms of cost, emission and computation time. The optimum result isobtained with RC = 0.90, when the minimum cost, minimum emission andcomputation time are found to be $9592.40, 3232.20 lbs and 16.2781 secondsrespectively.

4.3 Effect of varying Population size - P N

In order to obtain the effect of the population size P N , it is varied from 30 to 160in steps of 10 considering the higher dimension (for the present case it is 14,explain earlier) of the problem under consideration. The mutation factor

m f and

the crossover factor RC are set at 0.50 and 0.90 respectively based on the processdescribed in section 4.1 and 4.2. It this case also, one hundred (100) independentruns are performed for every parameter combinations and maximum number of iteration is set at 300.

Table 4Effect of Population size P N on Combined Economic Emission Dispatch (CEED)with RC = 0.90 and m f = 0.50 for a demand of 2000 MW.

Valueof

P N

Fuel Cost ($) Emission (lb) AverageCPU

Time(Sec.)

Minimum Maximum Average Minimum Maximum Average30 10030.00 10945.00 10341.00 3342.00 4750.20 4178.70 4.670340 9884.10 10279.00 10082.00 3571.40 4499.40 3945.50 6.560950 9810.30 10284.00 10073.00 3486.40 4041.80 3743.80 8.141060 9864.70 10421.00 10014.00 3339.70 3962.10 3593.30 10.672070 9764.00 9943.00 9814.50 3257.70 3845.80 3447.20 11.531080 9698.50 10040.00 9741.30 3237.70 3649.80 3385.30 13.182890 9594.40 10052.00 9742.60 3275.90 3466.40 3298.30 14.4984100 9592.40 9945.90 9644.80 3232.20 3430.80 3351.50 16.2781110 9591.40 9970.00 9688.40 3238.20 3724.60 3306.30 17.9687120 9592.40 9905.70 9695.50 3237.30 3691.10 3300.90 19.7750130 9588.00 9929.40 9727.60 3233.60 3561.40 3248.30 22.2343140 9593.70 9998.20 9762.30 3231.00 3717.30 3357.20 27.7953150 9593.30 10009.00 9702.20 3231.20 3338.60 3253.20 24.9750160 9592.70 9811.00 9659.10 3232.10 3360.10 3242.70 26.6390

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The results for different combinations are presented in Table 4. It is seenthat convergence rate is 100% above a value of P N = 80. However, computationtime increases gradually along with population size. The optimum result isobtained with P N = 100, when the minimum cost, emission and computation timeare found to be $9592.40, 3232.20 lbs and 16.2781 seconds respectively.

4.4. Results of Combined Economic Emission Dispatch (CEED)

Control parameters of DE were selected through the process as above and theoptimal parameters as obtained are P N =100, m f = 0.50 and RC = 0.9. Maximumiteration was set at 300 . Table 5 shows the results for optimized cost, generationschedule, penalty factors, losses, emission output and computation time forcombined economic emission dispatch (CEED) with a demand of 1500 MW,

2000 MW and 2500 MW.

Table 5Solution for Fourteen Generator system for Combined Economic EmissionDispatch (CEED) with P N = 100, RC = 0.90 and m f = 0.50

Demand (MW)Generation of Units (MW) 1500 2000 2500P1 150.20 239.76 329.53P2 150.23 150.01 219.54P3 96.51 95.21 129.99P4 120.01 119.75 120.06P5 150.10 199.86 249.75P6 135.25 284.59 384.25P7 135.07 234.86 284.43P8 60.59 159.73 209.56P9 125.10 124.89 161.74P10 139.09 137.32 159.89P11 58.77 66.65 79.95P12 79.97 79.95 79.93P13 64.07 84.97 84.84P14 52.95 52.43 52.65Total Generation (MW) 1517.88 2029.98 2546.11Losses (MW) 17.88 29.98 46.11Penalty Factor ( h) 1.2191 1.5299 1.5716CPU Time (Sec). 36.2812 16.2736 18.3542Iterations 300 300 300Fuel Cost ($/hr) 6869.90 9592.40 12871.00Emission (lb/hr) 1340.20 3232.20 6316.30

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It is important to note that the process takes longer time when the demandis near minimum or maximum limits. For example, the CPU time is 36.2812 sec.and 18.3542 sec. with a demand of 1500 MW and 2500 MW respectively,whereas it is 16.2736 sec. with a demand of 2000 MW as shown in Table 5.Thisis due to the fact that the algorithm takes more time for generating initial set of feasible solutions satisfying constraints near the lower and upper generationlimits .

5 Conclusion

The success of DE is heavily dependent on setting of control parameters. Optimalsettings of the control parameters of DE depend on the specific optimizationproblem in hand. In this paper, a parameter study of control parameters of differential evolution (DE) is conducted for combined economic emissiondispatch with generator constraints, valve-point loading and transmission loss.The effect of variation of all the three control parameters of DE i.e. N P , f m and C R are investigated and the results are presented. It is seen that prematureconvergence and even stagnation can occur due to wrong parameter selection.Based on the results obtained, recommendations are suggested for the range of control parameters of DE.

Appendix A: List of Symbols

)P(FC g : total fuel cost of generation in the system ($/hr)

iii c ,b ,a : fuel cost coefficients of the i th generating unit

iP : power generated by the i th unitn : number of thermal units

)P(FC gv : total fuel cost of generation in ($/hr) including valve point loading

ii f ,e : fuel cost coefficients of the i th generating unit reflecting valve-pointeffect

min ,iP : minimum power generation by i th unit

max ,iP : maximum power generation by i th unit

DP : total power demand

LP : total transmission loss

ij B : elements of loss coefficient matrix B

)P( EC g : total amount of emission (lb/hr)

iii , , : emission coefficients of the i th unit

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t : total operational cost of the system1w , 2w : weight factors

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Biographies

K. K. Mandal was born in Calcutta, India on 25th December, 1964. He receivedthe B.E Degree in Electrical Engineering from Jadavpur University, Kolkata,India in 1986 and M.E Degree from Allahabad University, Allahabad, India in1998. His employment experience includes Indian Telephone Industries, NationalInstitute of Technology, Durgapur, India. He is presently working as a Reader inthe Department of Power Engineering, Jadavpur University, Kolkata, India. Hispresent research interest includes Power Economics, Deregulated ElectricityIndustry and Power Electronics.

N. Chakraborty was born in Kolkata, India on 27 th August, 1964. He received hisBachelor of Electrical Engineering in 1986 and Masters of Electrical Engineeringin 1989 from Jadavpur University, Kolkata. He was awarded with the D.I.C fromImperial College, London, U.K. and Ph.D Degree from University of London in1999. At present he is a professor in the Department of Power Engineering,Jadavpur University, Kolkata. His fields of research interest include Power andEnergy Economics, Applied Superconductivity and Environmental Measurementsand Analysis.

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