hybrid differential evolution with biogeography-based optimization for solution of economic load...

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 4, NOVEMBER 2010 1955

Hybrid Differential Evolution With Biogeography-Based Optimization for Solution of Economic

Load DispatchAniruddha Bhattacharya, Member, IEEE, and Pranab Kumar Chattopadhyay

Abstract—This paper presents a hybrid technique combiningdifferential evolution with biogeography-based optimization(DE/BBO) algorithm to solve both convex and nonconvex eco-nomic load dispatch (ELD) problems of thermal power unitsconsidering transmission losses, and constraints such as ramprate limits, valve-point loading and prohibited operating zones.Differential evolution (DE) is one of the very fast and robust evo-lutionary algorithms for global optimization. Biogeography-basedoptimization (BBO) is a relatively new optimization. Mathe-matical models of biogeography describe how a species arises,migrates from one habitat (Island) to another, or gets extinct. Thisalgorithm searches for the global optimum mainly through twosteps: migration and mutation. This paper presents combinationof DE and BBO (DE/BBO) to improve the quality of solutionand convergence speed. DE/BBO improves the searching abilityof DE utilizing BBO algorithm effectively and can generate thepromising candidate solutions. The effectiveness of the proposedalgorithm has been verified on four different test systems, bothsmall and large. Considering the quality of the solution andconvergence speed obtained, this method seems to be a promisingalternative approach for solving the ELD problems in practicalpower system.

Index Terms—Biogeography-based optimization, differentialevolution, economic load dispatch, prohibited operating zone,ramp rate limits, valve-point loading.

I. INTRODUCTION

E CONOMIC load dispatch (ELD) allocates generationamong the committed generating units in the most eco-

nomical manner subject to different operational constraints.Various investigations on ELD have been undertaken to date asbetter solutions would result in significant saving in operatingcost. The fuel cost characteristics of modern generating unitsare highly nonlinear with demand for solution techniqueshaving no restrictions on to the shape of the fuel cost curves.The calculus-based methods [1] fail in solving these typesof problems. The dynamic programming approach, proposedby Wood and Wollenberg [2], though does not impose any

Manuscript received November 04, 2009; revised January 06, 2010. First pub-lished March 15, 2010; current version published October 20, 2010. This workwas supported by the Electrical Engineering Department, Jadavpur University,Kolkata, India. Paper no. TPWRS-00862-2009.

The authors are with the Department of Electrical Engineering, Jadavpur Uni-versity, Kolkata, West Bengal 700 032, India (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TPWRS.2010.2043270

restriction on the nature of the cost curves, but suffers fromthe curse of dimensionality and larger simulation time. Modernmeta-heuristic algorithms are a promising alternative for solu-tion of complex ELD problems. Genetic algorithm [3], artificialneural networks [4], simulated annealing [5], tabu search,evolutionary programming [6], particle swarm optimization,[7], ant colony optimization [8], differential evolution (DE)[9], artificial immune system (AIS) [10], bacterial foragingalgorithm (BFA) [11], etc., that have been developed so far andapplied successfully to ELD problems, belong to this class.Although these methods do not always guarantee global bestsolutions, they often achieve a fast and near global optimalsolution. Recently, different hybridization and modification ofGA, EP, PSO, methods like IFEP [12], IGAMU, RQEA, DSGA,UHGA, PSOSQP, NPSO-LRS [13], APSO [14], SOH-PSO[15], SA-PSO, ESO, etc. are being proposed for solving ELDin search of better quality and fast solution.

DE, invented by Price and Storn in 1995 [16], is a popu-lation-based stochastic parallel search evolutionary algorithmfor minimizing nonlinear, nondifferentiable, continuous spacefunctions. Differential evolution enriches a population of can-didate solutions over several generations using three basic op-erations, namely, mutation, crossover, and selection operators.DE has been found to yield better and faster solution, satisfyingall the constraints, both for uni-modal and multi-modal ELDsystem, using its different crossover strategies. It has been foundthat, in DE, initially the solutions move very fast towards theoptimal point but at later stages when fine tuning operation isrequired, DE fails to perform satisfactorily in ELD problems.Many modifications on the basic DE algorithm, like differentialevolution, in combination with sequential quadratic program-ming (DEC-SQP), hybrid differential evolution (HDE), vari-able scaling hybrid differential evolution (VSHDE), self-tuningHDE (STHDE), etc., have been proposed to remedy the situa-tion.

Very recently,ba new optimization concept, based on Bio-geography, has been proposed by Simon [17]. Biogeography ismodeled after the process of natural immigration and emigrationof species between islands in search of more friendly habitats. Ahabitat is any Island (area) that is geographically isolated fromother Islands. Migration of some species out of one habitat intoanother is known as emigration. When some species enters intoone habitat from some other habitat, the process is known as im-migration. Areas that are well suited as residences for biologicalspecies are said to have a high habitat suitability index (HSI).The variables that characterize habitability are called suitability

0885-8950/$26.00 © 2010 IEEE

1956 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 25, NO. 4, NOVEMBER 2010

index variables (SIVs). can be considered as the inde-pendent variables of the habitat, and can be calculatedusing these variables. BBO works on two mechanisms: Migra-tion and Mutation. Optimization technique based on this con-cept is known as biogeography-based optimization (BBO) [17].BBO has already proved its effectiveness as a commendable op-timization technique. In BBO, poor solutions accept a lot of newfeatures from good ones which helps to improve the quality ofthose solutions. This is a unique feature of BBO. The superiorityof the performance of BBO, compared to other EAs, has alreadybeen established in [17]–[19] (on some benchmark functions).Present authors also have applied BBO successfully for solutionof ELD problem [20] and consider it as a more promising ap-proach compared to other already established EAs.

In order to exploit the exploration and the exploitation capa-bilities of DE and BBO, a hybrid technique combining DE withBBO, referred to as DE/BBO, is being proposed for the solutionof ELD problems. In this paper, the migration operator of BBOalong with mutation, crossover and selection operators of DEhave been combined together to effectively utilize the goodnessof both DE and BBO to enhance the convergence property andto improve quality of solution.

The authors have applied this newly developed algorithm tosolve non-convex complex ELD problems. This paper considersfour types of non-convex ELD problems. These are 1) ELD withquadratic cost function, prohibited operating zones and Ramprate limits (ELDPOZRR):-3 Generator System, 2) ELD withquadratic cost function and transmission loss (ELDQCTL):-38Generator system, 3) ELD with valve-point loading effects andwithout transmission loss (ELDVPL):-40Generator system, and4) ELD with combined valve-point loading effects and multi-fuel options (ELDVPLMF).

Section II of the paper provides a brief description and math-ematical formulation of different types of ELD problems. TheDE and BBO approach are described in Sections III and IV,respectively. Section V describes the proposed DE/BBO algo-rithm along with a short description of the algorithm used inthese test systems. Simulation studies are presented and dis-cussed in Section VI. The conclusion is in Section VII.

II. ECONOMIC LOAD DISPATCH PROBLEMS

The ELD may be formulated as a nonlinear constrained opti-mization problem. Both convex and non-convex ELD problemshave been considered here. Four different types of ELD prob-lems have been formulated and solved by DE/BBO approach.

A. ELDQCTL

The objective function of ELD problem may be written as

(1)

where is cost function of the th generator, and is usuallyexpressed as a quadratic polynomial; , , and are the costcoefficients of the th generator; is the number of committedgenerators; is the power output of the th generator. TheELD problem consists in minimizing subject to followingconstraints.

Real Power Balance Constraint:

(2)

The transmission loss may be expressed using B-coefficientsas

(3)

Generator Capacity Constraints: The power generated byeach generator shall be within their lower limit and upperlimit . So that

(4)

B. ELDPOZRR

The objective function of this type of ELD problem issame as mentioned in (1). Here the objective func-tion is to be minimized subject to the constraints of (2), (4), andramp-rate limits as mentioned below.

Ramp Rate Limit Constraints: The power generated, ,by the th generator in certain interval may not exceed that ofprevious interval by more than a certain amount , theup-ramp limit and neither may it be less than that of the previousinterval by more than some amount , the down-ramp limitof the generator. These give rise to the following constraints.

As generation increases

(5)

As generation decreases

(6)

and

(7)Prohibited Operating Zone: The prohibited operating zones

are the range of output power of a generator where the operationcauses undue vibration of the turbine shaft. Normally operationis avoided in such regions. Hence mathematically the feasibleoperating zones of unit can be described as follows:

(8)

where represents the number of prohibited operating zones ofunit . is the upper limit and is the lower limit of theth prohibited operating zone of the th unit. Total number of

prohibited operating zone of the th unit is .

C. ELDVPL

In ELD with “Valve point loadings”, the objective functionis represented by a more complex formula, given as is

BHATTACHARYA AND CHATTOPADHYAY: HYBRID DIFFERENTIAL EVOLUTION 1957

given by

(9)

The objective of ELDVPL is to minimize of (9) subject to theconstraints given in (2) and (4) as in . Transmissionloss is not considered. Here is zero.

D. ELDVPLMF

For a power system with generators and fuel optionsfor each unit, the cost function of the generator with valve-pointloading is expressed as

for fuel option

(10)

where and are the minimum and maximum powergeneration limits of the th generator with fuel option , respec-tively; , , , , and are the fuel-cost coefficients ofgenerator for fuel option .

The objective function is to be minimized subject to the sameconstraints as mentioned in (2)–(4).

Calculation for Slack Generator: Let N committed gen-erating units deliver their power output subject to the powerbalance constraint (2) and the respective capacity constraints(4). Assuming the power loadings of first (N-1) generatorsare known, the power level of th generator (called slackgenerator) is given by

(11)

The transmission loss is a function of all the generator out-puts including the slack generator and it is given by

(12)

Expanding and rearranging, (10) becomes

(13)

The loading of the dependent generator (i.e., th) can then befound by solving (13) using standard algebraic method.

The above equation can be simplified as

(14)

where

The positive roots of the equation are obtained as

where (15)

To satisfy the equality constraint (11), the positive root of (15)is chosen as output of the th generator.

If the positive root of quadratic equation violates operationlimit constraint of (4) at the initialization process of the algo-rithm, then Generation value of first (N-1) generators is reini-tialized until the positive root satisfies the operation limit andother constraints (if any).

If the positive root of quadratic equation violates operationlimit constraint of (4) at the later stage of the algorithm thatmeans when the modified generation value is obtained afterapplying necessary steps of the algorithm (which is used forsolving ELD problems; as it is DE/BBO in this case), then thatmodified generation set is discarded and different steps of thealgorithm is reapplied on its old value until it satisfies the oper-ation limit and other constraints (if any).

III. DIFFERENTIAL EVOLUTION (DE)

Differential evolution (DE) is technically population basedEvolutionary Algorithm, capable of handling non-differen-tiable, nonlinear and multi-modal objective functions. DEgenerates new offspring by forming a trial vector of eachparent individual of the population. The population is improvediteratively, by three basic operators: mutation, crossover, andselection. A brief description of different steps of DE algorithmis given below.

A. Initialization

The population is initialized by randomly generating individ-uals within the boundary constraints

(16)

where “ ” function generates random values uniformly inthe interval [0, 1]; is the size of the population; is thenumber of decision variables. and are the lowerand upper bound of the th decision variable, respectively.


B. Mutation

As a step of generating offspring, the operations of “Muta-tion” are applied. “Mutation” occupies quite an important rolein the reproduction cycle. The mutation operation creates mu-tant vectors by perturbing a randomly selected vectorwith the difference of two other randomly selected vectorsand at the th iteration as per the following equation:

(17)

, , and are randomly chosen vectors at the th itera-tion and . , ,and are selected anew for each parent vector. isknown as “Scaling factor” used to control the amount of pertur-bation in the mutation process and improve convergence.

C. Crossover

Crossover represents a typical case of a “genes” exchange.The trial one inherits genes with some probability. The parentvector is mixed with the mutated vector to create a trial vector,according to the following equation:

if orotherwise

(18)

where ; .are the th individual of

target vector, mutant vector, and trial vector at iteration,respectively. is a randomly chosen indexthat guarantees that the trial vector gets at least one parameterfrom the mutant vector even if . isthe “Crossover constant” that controls the diversity of thepopulation and aids the algorithm to escape from local optima.

D. Selection

Selection procedure is used among the set of trial vector andthe updated target vector to choose the best. Selection is realizedby comparing the cost function values of target vector and trialvector. Selection operation is performed as per the followingequation:

if

otherwise(19)

The pseudo-code of the DE algorithm is shown in Algorithm1.

Algorithm 1: The DE algorithm with Strategy 1

1) Generate the initial population P

2) Evaluate the fitness for each individual in P

3) while the termination criterion is not satisfied

4) for to

5) Select uniform randomly

6)

7) for to D

8) if or

9)

10) else

11)

12) end

13) end

14) end

15) for to

16) Evaluate the offspring

17) if is better than

18)

19) end

20) end

21) end

where is the number of decision variables. is the size ofthe parent population . is the th variable of the solution

. is the offspring. randint(1,D) is a uniformly distributedrandom integer number between 1 and . Many schemes of cre-ation of a candidate are possible. Here Strategy 1 has been men-tioned in the algorithm. More details on different DE schemescan be found in [21] and [22].

IV. BIOGEOGRAPHY-BASED OPTIMIZATION (BBO)

Biogeography describes how species migrate from one islandto another, how new species arise, and how species become ex-tinct. A habitat is any Island (area) that is geographically iso-lated from other Islands. Habitats with a high tend to havea large number of species, while those with a low havea small number of species. Habitats with a high have alow species immigration rate because they are already nearlysaturated with species. By the same token high habitatshave a high emigration rate. Habitats with a low have ahigh species immigration rate because of their sparse popula-tions. The emigration rate works similarly. Emigration in BBOdoes not mean that the emigrating island loses a feature. Theworst solution is assumed to have the worst features; thus, it hasa very low emigration rate and a low chance of sharing its fea-tures. The solution that has the best features also has the highestprobability of sharing them. This approach is known as biogeog-raphy-based optimization [17].

Mathematically the concept of emigration and immigrationcan be represented by a probabilistic model. Let us consider theprobability that the habitat contains exactly species at .

changes from time to time as follows:

(20)


where and are the immigration and emigration rates whenthere are species in the habitat. This equation holds becausein order to have species at time , one of the followingconditions must hold:

1) there were species at time , and no immigration or em-igration occurred between and ;

2) there were species at time , one species immi-grated;

3) there were species at time , one species emigrated.If time is small enough so that the probability of more

than one immigration or emigration can be ignored then takingthe limit of (19) as gives the following equation:

(21)For the straight-line graph of migration [17], the equation foremigration rate and immigration rate for number ofspecies can be written as per the following way:

(22)

(23)

When the value of , then combining (22) and (23)

(24)

Algorithm 2: Habitat migration

for to

Select a habitat with probability proportional to

if

for to

Select another habitat withprobability proportional to

if

Randomly select anfrom habitat

Replace a random inwith that selected SIV

of

end

end

end

end

In BBO, there are two main operators, the migration and themutation. One option for implementing the migration operatoris described in Algorithm 2. Where rand (0,1) is a uniformly dis-tributed real random number in (0, 1) and is the th SIV set

of habitat . With the migration operator, BBO can share theinformation among solutions. Especially, poor solutions tendto accept more useful information from good solutions. Thismakes BBO be good at exploiting the information of the cur-rent population. Details about the two operators can be found in[17].

V. DE/BBO APPROACH

DE has been found to yield better and faster solution, sat-isfying all the constraints, both for uni-modal and multi-modalsystem, using its different crossover strategies. But when systemcomplexity and size increases, DE method is unable to mapits entire unknown variables together in a better way. Due topresence of crossover operation in Evolutionary based algo-rithms, many solutions whose fitness are initially good, some-times loose their quality in later stage of the process. In BBOthere is no crossover-like operation; solutions get fine tunedgradually as the process goes on through migration operation.This gives an edge to BBO over techniques mentioned above.In a nut shell, DE has good exploration ability in finding the re-gion of global minimum. Similarly, BBO has good exploitationability in global optimization problem. In order to utilize boththe properties of DE and BBO for solution of complex optimiza-tion problems, a hybrid technique called DE/BBO has been de-veloped [23]. Proposed DE/BBO approach is described below:

A. Hybrid Migration Operator

Hybrid migration operator is most important step in DE/BBOalgorithm. In this algorithm child population takes new fea-tures from different sides. These are mutation operation of DE,migration operation of BBO and corresponding parents ofoffspring. The core idea of the proposed hybrid migration oper-ator is based on two considerations. Here, due to this hybridiza-tion good solutions would be less destroyed, while poor solu-tions can accept a lot of new features from good solutions. Inthis sense, the current population can be exploited sufficiently.In Algorithm 3 the “DE/rand/1” mutation operator is illustrated.

Algorithm 3: Hybrid Migration operator of DE/BBO

1) for to

2) Select uniform randomly

3)

4) for to D

5) if

6) if or

7)

8) else

9) Select with probability

10)

11) end

12) else


13)

14) end

15) end

16) end

B. Main Procedure of DE/BBO

Introducing previously mentioned hybrid migration operatorof BBO into DE has developed one new algorithm knownas DE/BBO. The structure of proposed DE/BBO is also verysimple. Detail hybrid method is given in Algorithm 4. For moreinformation, refer to [23].

Algorithm 4: The Main DE/BBO algorithm

1) Generate the initial population P

2) Evaluate the fitness for each individual in P

3) while the termination criterion is not satisfied

4) For each individual calculate species count probability

5) For each individual calculate Immigration rate andemigration rate

6) Modify the updated population with the Hybrid

Migration

operation of Algorithm 3

7) for to

8) Evaluate the offspring

9) if is better than

10)

11) end

12) end

13) end

C. DE/BBO Algorithm for ELD Problem

In this section, a new approach, DE/BBO algorithm is de-scribed for solving the ELD problems.Step 1) For initialization, choose number of Generator units

, population size . Specify maximum and min-imum capacity of each generator, power demand,B-coefficients matrix for calculation of transmissionloss. Initialize DE parameters like Crossover Proba-bility , Scaling Factor . Also initialize the BBOparameters like max immigration rate , max emi-gration rate , lower bound for immigration proba-bility per gene, upper bound for immigration proba-bility per gene, etc. Set maximum number of Itera-tion.

Step 2) Initialize the Population . Since the decision vari-ables for the ELD problems are real power gener-ations, they are used to represent each element of

TABLE ICOMPARISON AMONG DIFFERENT METHODS AFTER 50 TRIALS

(THREE-GENERATOR SYSTEM, �� )

a given population set. Each element of the Popu-lation matrix is initialized randomly within the ef-fective real power operating limits. Each populationset of the population matrix should satisfy equalityconstraint (2) using the concept of slack generator asmentioned in Section II. Each individual populationset of the population matrix represents a potentialsolution to the given problem.

Step 3) Calculate the fitness value for each population setof the total population for given emigration rate

, immigration rate . Fitness value represents thefuel cost of the generators in the power system for aparticular power demand.

Step 4) Probabilistically perform hybrid migration operationon those elements of population matrix , whichare selected for migration. Perform hybrid migrationoperation on the Population set as per Algorithm 3.

Step 5) After migration operation, new offspring populationset is generated. In ELD problems these rep-resent new modified generation values of generators

. Equality constraint (2) is satisfied using con-cept of slack generator as mentioned in Section II.

Step 6) Fitness value of each newly generated population set(offspring matrix ) is recomputed, i.e., fuel costof each power generation set.

Step 7) Perform selection operation between parent popula-tion and newly generated offspring basedon their fitness values as per (19).

Step 8) Go to step 3 for the next iteration. This loop can beterminated after a predefined number of iterations.

VI. NUMERICAL EXAMPLE AND SIMULATION RESULT

Proposed DE/BBO algorithm has been applied to solve ELDproblems in four different test cases for verifying its feasibility.The software has been written in MATLAB-7 language andexecuted on a 2.3-GHz Pentium IV personal computer with512-MB RAM.

A. Description of the Test Systems

1) Test Case 1: A 3 generators system with ramp rate limitand prohibited operating zone is considered here. The inputdata have been adapted from [3]. The load demand is 300MW. Results obtained from proposed DE/BBO, BBO, andother methods have been presented in Table I. The convergencecharacteristic is shown in Fig. 1.


Fig. 1. Convergence characteristic of three-generator system �� .

TABLE IIBEST POWER OUTPUT FOR 38-GENERATOR SYSTEM ��

2) Test Case 2: A system with 38 generators is taken here.Fuel cost characteristics are quadratic. The input data of thesystem are from [24]. The load demand is 6000 MW. The resultobtained using proposed DE/BBO algorithm has been comparedwith BBO, PSO_TVAC [25], and other methods and is shownin Table II. Its convergence characteristic is shown in Fig. 2.

Fig. 2. Convergence characteristic of 38-generator system �� .

3) Test Case 3: A system with 40 generators with valve pointloading is used here. The input data are given in [12]. The loaddemand is 10 500 MW. Transmission loss has not been consid-ered here. The result obtained from proposed DE/BBO methodhas been compared with BBO, ICA_PSO [26], SOH_PSO [15],and other methods. Their best solutions are shown in Table III.Its convergence characteristic is shown in Fig. 3.

4) Test Case 4: A simple system with 10 thermal units isconsidered here. The input data are taken from [27]. The loaddemand is 2700 MW. Transmission loss has not been consideredhere. The result obtained from the proposed DE/BBO, BBO,different PSO techniques [13] and different GA [27] methodsare shown in Table IX. Convergence characteristic of the ten-generator systems in case of BBO algorithm is shown in Fig. 4.

B. Determination of Parameters for DE/BBO Algorithm

To get optimal solution using the DE/BBO algorithm, thefollowing procedures have been applied to calculate optimumvalue of Scaling Factor and Crossover probability .

1) The Population Size is fixed at 80.2) Crossover Probability, Scaling Factor is increased from 0.1

to 0.9 in suitable steps as shown in Tables VI and VII.Performance of DE/BBO algorithm in ELDVPL system is

calculated for all the above mentioned combination. Fifty inde-pendent trials have been made with 1000 iterations per trial. Theminimum generation costs for this system reported so far are121 413.20 $/h (actual 121 422.1684 $/h) [26], 121 501.14 $/h[15], 121 664.4308 $/h [13], and 121 704.7391 $/h [13]. In caseof DE/BBO algorithm, based on the simulation results obtainedfor different value of parameters given in Tables VI and VII,Crossover Probability CR 0.2 and Scaling Factor 0.7 givesoptimal generation cost 121 420.8963 $/h and it is more consis-tent.

C. Effect of Population Size on DE/BBO Algorithm

Change in population size, affects the performance of theDE/BBO algorithm. The optimum population size is found tobe related to the problem dimension and complexity. Table VIIIshows the performance of the DE/BBO algorithm for differentpopulation sizes. Tests were carried out 50 times each for the


TABLE IIIBEST POWER OUTPUT FOR 40-GENERATOR SYSTEM ��

Fig. 3. Convergence characteristic of 40-generator system (with valve pointloading, �� ).

40-unit system. A population size of 80 resulted in achievingglobal solutions more consistently for the test system.

After a number of careful experimentation, followingoptimum values of DE/BBO parameters have finally

Fig. 4. Convergence characteristic of ten-generator system (with multi-fuel op-tions, �� ).

been settled: , DE Parame-ters— , Scaling Factor .

The maximum migration rates ( and ) are relative quanti-ties. That is, if they all change by the same percentage, then thebehavior of BBO not changes [17]. This is because if andchange, then the migration rates and change by the same rel-ative amount for each solution. Same happens in DE/BBO alsoas total migration step of BBO is incorporated in DE/BBO inthe same manner. So value of following parameters of DE/BBOalgorithm is same as mentioned in BBO [20].

BBO parameters—

,.

D. Comparative Study

1) Solution Quality: From the Tables I, II, III, and IX, itis clear that the minimum costs achieved by DE/BBO are lessthan those reported in recent literature [14], [20], [25], [27].Tables I, IV, and X also show that the average costs achievedby DE/BBO are the least of all other methods, emphasizing itssuperior quality of solution. Test Cases 1, 3, and 4 indicate thatthe average costs obtained by DE/BBO are less for both convexand non-convex ELD problems.

2) Computational Efficiency: Tables I, II, III, and IX showthat the minimum cost achieved by DE/BBO are 3619.7565 $/h,9 417 235.786391673 $/h, 121 420.8948 $/h, and 605.6230127$/h for Test Cases 1, 2, 3, and 4, respectively. Those are less ascompared to the reported results in recent literature. Again fromTables I, IV, and IX, it is also clear that the DE/BBO approachis also efficient as far as computational time is concerned. Timerequirement is quite less and either comparable or better thanother mentioned methods. So as a whole it can be said that theDE/BBO method is computationally efficient than previouslymentioned methods.

3) Robustness: Since initialization of population is per-formed using random numbers in case of stochastic simulationtechniques, so randomness is inherent property of these tech-niques. Hence the performances of stochastic search algorithmsare to be judged over a number of trials. So many trials withdifferent initial population have been carried out to test the


TABLE IVCOMPARISON AMONG DIFFERENT METHODS AFTER

50 TRIALS (40-GENERATOR SYSTEM)

Exact generation cost from the above schedule is 121 422.1684 $/h whichis higher than that reported in [26].

TABLE VFREQUENCY OF CONVERGENCE, 40-GENERATOR SYSTEM OUT OF 50 TRIALS

TABLE VIINFLUENCE OF PARAMETERS ON DE/BBO PERFORMANCE

TABLE VIIINFLUENCE OF PARAMETERS ON DE/BBO PERFORMANCE

consistency of the DE/BBO algorithm. Table V shows thefrequency of attaining minimum cost within different rangesfor Test Case 3 out of 50 independent trials. It can be seen thatDE/BBO is robust as it reaches the minimum cost 50 times outof 50 times compared to 38 times in BBO.

TABLE VIIIEFFECT OF POPULATION SIZE ON 40-GENERATOR SYSTEM (F:-0.7, CR:-0.2)

TABLE IXBEST POWER OUTPUT FOR TEN-GENERATOR SYSTEM ��

TABLE XCOMPARISON AMONG DIFFERENT METHODS AFTER 100 TRIALS

(TEN-GENERATOR SYSTEM)

VII. CONCLUSION

In this paper, the DE/BBO method has been successfully im-plemented to solve different convex and non-convex ELD prob-lems. It has been observed that the DE/BBO has the ability to


converge to a better quality solution and possesses better con-vergence characteristics and robustness than ordinary BBO. Itis also clear from the results obtained by different trials that theproposed DE/BBO method can avoid the shortcoming of pre-mature convergence exhibited by other optimization techniques.Due to these properties, the DE/BBO method in future can betried for solution of complex power system optimization prob-lems.

ACKNOWLEDGMENT

The author would like to thank D. Simon, Associate Professorin the Electrical and Computer Engineering Department, Cleve-land State University, whose paper on BBO has helped a lot tocarry out this work. The authors are also thankful to JadavpurUniversity for providing infrastructural facilities to conduct theresearch work.

REFERENCES

[1] A. A. El-Keib, H. Ma, and J. L. Hart, “Environmentally constrainedeconomic dispatch using the Lagrangian relaxation method,” IEEETrans. Power Syst., vol. 9, no. 4, pp. 1723–1729, Nov. 1994.

[2] J. Wood and B. F. Wollenberg, Power Generation, Operation, and Con-trol, 2nd ed. New York: Wiley, 1984.

[3] P. H. Chen and H. C. Chang, “Large-scale economic dispatch bygenetic algorithm,” IEEE Trans. Power Syst., vol. 10, no. 4, pp.1919–1926, Nov. 1995.

[4] C.-T. Su and C.-T. Lin, “New approach with a Hopfield modelingframework to economic dispatch,” IEEE Trans. Power Syst., vol. 15,no. 2, p. 541, May 2000.

[5] K. P. Wong and C. C. Fung, “Simulated annealing based economicdispatch algorithm,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib.,vol. 140, no. 6, pp. 509–515, Nov. 1993.

[6] T. Jayabarathi, G. Sadasivam, and V. Ramachandran, “Evolutionaryprogramming based economic dispatch of generators with prohibitedoperating zones,” Elect. Power Syst. Res., vol. 52, pp. 261–266, 1999.

[7] Z.-L. Gaing, “Particle swarm optimization to solving the economic dis-patch considering the generator constraints,” IEEE Trans. Power Syst.,vol. 18, no. 3, pp. 1187–1195, Aug. 2003.

[8] Y. H. Hou, Y. W. Wu, L. J. Lu, and X. Y. Xiong, “Generalized antcolony optimization for economic dispatch of power systems,” in Proc.Int. Conf. Power System Technology, Power-Con 2002, Oct. 13–17,2002, vol. 1, pp. 225–229.

[9] N. Nomana and H. Iba, “Differential evolution for economic load dis-patch problems,” Elect. Power Syst. Res., vol. 78, no. 3, pp. 1322–1331,2008.

[10] B. K. Panigrahi, S. R. Yadav, S. Agrawal, and M. K. Tiwari, “A clonalalgorithm to solve economic load dispatch,” Elect. Power Syst. Res.,vol. 77, no. 10, pp. 1381–1389, 2007.

[11] B. K. Panigrahi and V. R. Pandi, “Bacterial foraging optimization:Nelder-Mead hybrid algorithm for economic load dispatch,” IET Gen.,Transm., Distrib., vol. 2, no. 4, pp. 556–565, 2008.

[12] N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionary pro-gramming techniques for economic load dispatch,” IEEE Trans. Evol.Comput., vol. 7, no. 1, pp. 83–94, Feb. 2003.

[13] I. Selvakumar and K. Thanushkodi, “A new particle swarm optimiza-tion solution to nonconvex economic dispatch problems,” IEEE Trans.Power Syst., vol. 22, no. 1, pp. 42–51, Feb. 2007.

[14] B. K. Panigrahi, V. R. Pandi, and S. Das, “Adaptive particle swarmoptimization approach for static and dynamic economic load dispatch,”Energy Convers. Manage., vol. 49, no. 6, pp. 1407–1415, 2008.

[15] K. T. Chaturvedi, M. Pandit, and L. Srivastava, “Self-organizing hierar-chical particle swarm optimization for nonconvex economic dispatch,”IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1079–1087, Aug. 2008.

[16] R. Storn and K. Price, Differential Evolution—A Simple And EfficientAdaptive Scheme For Global Optimization Over Continuous Spaces,Int. Comput. Sci. Inst., Berkeley, CA, Tech. Rep. TR-95-012, Mar.1995.

[17] D. Simon, “Biogeography-based optimization,” IEEE Trans. Evol.Comput., vol. 12, no. 6, pp. 702–713, Dec. 2008.

[18] D. Simon, M. Ergezer, and D. Du, Markov Analysis of Biogeography-Based Optimization. [Online]. Available: http://academic.csuohio.edu/simond/bbo/markov/.

[19] D. Simon, M. Ergezer, and D. Du, Population Distributions in Biogeog-raphy-Based Optimization With Elitism. [Online]. Available: http://academic.csuohio.edu/simond/bbo/markov/.

[20] A. Bhattacharya and P. K. Chattopadhyay, “Biogeography-basedoptimization for different economic load dispatch problems,” IEEETrans. Power Syst. [Online]. Available: http://ieeexplore.ieee.org/xpl/tocpreprint.jsp?isnumber=4374138&punumber=59.

[21] R. Storn and K. Price, Home Page of Differential Evolution, 2008. [On-line]. Available: http://www.ICSI.Berkeley.edu/~storn/code.html.

[22] K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Prac-tical Approach to Global Optimization. Berlin, Germany: Springer-Verlag, 2005.

[23] W. Gong, Z. Cai, and C. X. Ling, DE/BBO: A Hybrid Differ-ential Evolution With Biogeography-Based Optimization forGlobal Numerical Optimization. [Online]. Available: http://em-beddedlab.csuohio.edu/BBO/.

[24] M. Sydulu, “A very fast and effective non-iterative “Lamda LogicBased” algorithm for economic dispatch of thermal units,” in Proc.IEEE Region 10 Conf. TENCON, Sep. 15–17, 1999, vol. 2, pp.1434–1437.

[25] K. T. Chaturvedi, M. Pandit, and L. Srivastava, “Particle swarm op-timization with time varying acceleration coefficients for non-convexeconomic power dispatch,” Int. J. Elect. Power Energy Syst., vol. 31,no. 6, pp. 249–257, Jul. 2009.

[26] J. K. Vlachogiannis and K. Y. Lee, “Economic load dispatch—A com-parative study on heuristic optimization techniques with an improvedcoordinated aggregation-based PSO,” IEEE Trans. Power Syst., vol. 24,no. 2, pp. 991–1001, May 2009.

[27] C.-L. Chiang, “Improved genetic algorithm for power economic dis-patch of units with valve-point effects and multiple fuels,” IEEE Trans.Power Syst., vol. 20, no. 4, pp. 1690–1699, Nov. 2005.

[28] R. Naresh, J. Dubey, and J. Sharma, “Two phase neural network basedmodeling framework of constrained economic load dispatch,” Proc.Inst. Elect. Eng., Gen., Transm., Distrib., vol. 151, no. 3, pp. 373–378,May 2004.

Aniruddha Bhattacharya (M’09) received theB.Sc. Engg. degree in electrical engineering fromthe Regional Institute of Technology, Jamshedpur,India, in 2000 and the M.E.E. degree in electricalpower system from Jadavpur University, Kolkata,India, in 2008. He is currently pursuing the Ph.D.degree at Jadavpur University.

He is a Senior Research Fellow in the Departmentof Electrical Engineering at Jadavpur University.His employment experience includes the SiemensMetering Limited, India; Jindal Steel & Power

Limited, Raigarh, India; Bankura Unnyani Institute of Engineering, Bankura,India; and Dr. B. C. Roy Engineering College, Durgapur, India. His areas ofinterest include power system load flow, optimal power flow, economic loaddispatch, and soft computing applications to different power system problems.

Pranab Kumar Chattopadhyay received theM.E.E. degree in electrical power system fromJadavpur University, Kolkata, India, in 1971.

He is currently working as a Professor in theDepartment of Electrical Engineering, JadavpurUniversity. His areas of interest include applicationof soft computing techniques to different powersystem problems.

hybrid differential evolution with biogeography-based optimization for solution of economic load...

Documents