Electrical Power and Energy Systems 43 (2012) 1241–1250

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

A novel multi-objective directed bee colony optimization algorithmfor multi-objective emission constrained economic power dispatch

Rajesh Kumar a,⇑, Abhinav Sadu a, Rudesh Kumar a, S.K. Panda b

a Department of Electrical Engineering, Malaviya National Institute of Technology, Jaipur, Indiab Department of Electrical and Computer Engineering, National University of Singapore, Singapore

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 March 2011Received in revised form 31 May 2012Accepted 1 June 2012Available online 20 July 2012

Keywords:Evolutionary algorithmsEconomic load dispatchMulti-objective algorithms

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.06.011

⇑ Corresponding author.E-mail address: rk.rlab@gmail.com (R. Kumar).

In this paper, a multi-objective directed bee colony optimization algorithm (MODBC) is comprehensivelydeveloped and successfully applied for solving a multi-objective problem of optimizing the conflictingeconomic dispatch and emission cost with both equality and inequality constraints is showcased.Classical optimization techniques like direct search and gradient methods fail to give the global optimumsolution. The proposed algorithm is an integration of the deterministic search, the multi-agent system(MAS) environment and the bee decision-making process. Thus making use of deterministic search,multi-agent environment and bee swarms, the MODBC realizes the purpose of optimization. The hybrid-ization makes MODBC to obtain a unique and fast solution and hence generate a better pareto front formulti-objective problems. The above mentioned multi-objective evolutionary algorithms have beenapplied to the standard IEEE 30 bus six generator test system. Results of the proposed algorithm havebeen compared with traditional methods like linear programming (LP) and multi-objective stochasticsearch technique (MOSST). The performance of the introduced algorithm is also compared with otherevolutionary algorithms like Non-dominated Sorting Genetic Algorithm (NSGA), Niched Pareto GeneticAlgorithm (NPGA) and Strength Pareto Evolutionary Algorithm (SPEA) and Particle Swarm Optimization(PSO). The results show the robustness and accuracy of the proposed algorithm over the traditionalmethods and its other multi-objective evolutionary algorithm (MOEA) counterparts.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Economic dispatch is characterized as the procedure of allocat-ing generation points to the generation units so that the entiresystem load is supplied, meeting all the system constraints eco-nomically. Real-world applications usually involve simultaneousoptimization of multiple objectives, which are generally non-commensurable and conflicting with each other. The purpose ofmulti-objective optimization algorithms in the mathematical pro-gramming framework is to optimize different objective functions,subject to a set of system constraints. There are mainly two objec-tives for any power generation industry, one is to minimize theemission of pollutants and other is to curtail the cost of generation.It becomes very necessary to find an optimal solution for both thefunctions simultaneously.

Many researchers have studied the environmental/economicdispatch problem either by considering the emission as a secondobjective function or as additional constraints. Many techniqueshave been proposed to handle the problem such as taking emissionas a constraint [1,2], goal programming [3]. If emission is taken as a

ll rights reserved.

constraint then it is difficult to obtain the relation between the costand emission. The weighting method simply assigns differentweights to each objective function based on its importance andcombines different objectives into one single objective function.The result of solving the problem using this approach is highlydependent on the assigned weights. Goal programming is imple-mented by assigning a goal or value to be achieved for each objec-tive function. These values are incorporated into the problem asadditional constraints. If prior knowledge about the solution feasi-ble space is known then only this algorithm can be applied. Goalprogramming becomes inefficient if any of the goals selected be-comes infeasible.

Evolutionary algorithms exterminate some of the above men-tioned difficulties. Some of the Algorithms discussed in this paperare: Non-dominated Sorting Genetic Algorithm (NSGA) [4], NichedPareto Genetic Algorithm (NPGA) [5], Strength Pareto EvolutionaryAlgorithm (SPEA) [6] and Particle Swarm Optimization [7]. Collec-tive behavior of decentralized, self-organized systems which mi-mic the natural behavior of organisms is the characteristics ofSwarm Intelligence (SI). The agents follow very simple rules, andalthough there is no centralized control structure dictating howindividual agents should behave, local interactions between suchagents lead to the emergence of complex global behavior [9]. A

1242 R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250

natural example of SI includes ant colonies, bird flocking, animalherding, bacterial growth, and fish schooling. Various algorithmsderive from SI are the Ant Colony Optimization (ACO), GA and Par-ticle Swarm Optimization (PSO) [10,11,16]. Particle Swarm Optimi-zation (PSO) algorithm is based on social behavior of groups likeflocking of birds or schooling of fish. It is a stochastic, popula-tion-based evolutionary computer algorithm for problem solving.It is a kind of swarm intelligence that predicts everyone solutionas ‘‘particles’’ which evolve or change their positions with time.Each particle modifies its position in search space in accordancewith its own experience and also that of neighboring particle byremembering the best position visited by itself and its neighbors,then calculating local and global positions.

In this paper, we use our own developed a new algorithm re-named as MODBC for multi-objective problems [13–15], which isa hybrid version on MAS, which mimics its structure and modifiedNelder–Mead method to find an optimal solution. The decisionmaking technique is mimicked from bee decision-making process.The decision-making process is based on the algorithm used bybees for finding a suitable place for establishing new colony. Theexperimental results show the robustness and accuracy of MODBCover genetic algorithm and PSO. Due to its hybrid nature, this algo-rithm provides only deterministic solutions. Making use of theseagents–agent interactions and evolution mechanism of beeswarms in a lattice-like environment, the proposed method canfind high-quality solutions reliably with the faster convergencecharacteristics in a reasonably good computation time.

This paper is organized as follows. The hybrid algorithm iscomprising of two parts search algorithm and other as the decisionmaking process. Section 2 details, the problem formulation and therelated issues. The development and working of the MODBC iselaborated in Section 3. The starting point and the number ofagents are important issues while handling such algorithms. Thechoice of the number of agents and the starting point of searchare also presented and discussed. The decision making process inthe honey bees makes them an interesting swarm research areato work. Section 4 also discusses the decision making method usedby the bees in the proposed algorithm. Section 5 discusses simula-tion and experimental results made on the problem formulated.Two different cases with different conditions have been consideredin this paper. Above reported techniques have been applied to thestandard IEEE 30-bus six-generator test system. A comparativestudy of the performance of the proposed algorithm with SPEANSGA, and NPGA has been carried out for solving the multi-objec-tive economic/environmental dispatch problem. MOEA techniqueshave also been compared with classical methods. The effectivenessof MODBC techniques to handle the EED problem over the classicalmethods and other MOEA counterparts is demonstrated. Finally,Section 6 concludes the paper.

2. Economic/emission power dispatch problem

The economic dispatch problem is addressed as to simulta-neously minimize the overall cost rate and meet the load demandof a power system while satisfying an equality and inequality con-straints. Assuming the power system includes N generating units.The aim of economic power dispatch is to determine the optimalshare of load demand for each unit in their operational range [8].Nomenclature of the symbols used has been tabulated in Table 11.

2.1. Minimization of fuel cost and emission

The total fuel cost function of all generating units can be mod-eled by the following quadratic function

FðPiÞ ¼XN

i¼1

ðai þ biPi þ ciP2i Þ $=h ð1Þ

where ai, bi and ci are the cost coefficients of the ith generator and Nis the number of generators committed to the operating system. Pi

is the power output of the ith generator.Many techniques were proposed to represent the emission

function. One of those is being formulated as given below [9].

EðPÞ ¼XN

i¼1

ðai þ biPi þ ciP2i Þ þ 1ekiPi ton=h ð2Þ

where ai, bi, ci, f and ki are the emission coefficients of ith generatorand N is the number of generators committed to the operating sys-tem. Pi is the power output of the ith generator.

2.2. Constraints

1. Maximum and minimum constraints in generation capacityThe power generated Pi by each generator should be bound by

upper and lower limits

Pi min 6 Pi 6 Pimax ð3Þ

where Pimax and Pimin are the upper and lower limits of the genera-tion capacity of ith generator.1. Power balance constraints

Total electric power generation (Pi) must be distributed be-tween power demand (Pd) and the power loss in the transmissionlines (Pl).

XN

i¼1

Pi � Pl � Pd ¼ 0 ð4Þ

The loss component (Pl) can be calculated using the followingformula [7].

Pl ¼XN

i¼1

XN

j¼1

ðPiBijPjÞ ð5Þ

where Bij is the element of loss coefficient.

2.3. EED multi-objective problem formulation

The economic/emission dispatch problem can be formulated asmulti-objective optimization problem as follows:

Min z ¼ ½FðPÞ; EðPÞ� ð6Þ

Subjected to

gðPiÞ ¼ 0hðPiÞ 6 0

ð7Þ

where g represents the power balance constraint and h representsthe generation capacity constraint.

2.4. Best compromise solution

The minimum and maximum values of fuel and emission costsare considered as the lower and upper bounds respectively andthese are employed then to characterize the corresponding fuzzyutility functions of the two objectives, concerned. Thereby, a fuzzysatisfaction-maximizing approach is relied on to obtain a set ofnon-inferior solutions [8]. To help the operators in determiningthe best compromise solution out of the Pareto-set obtained theith objective function (Fi) is represented as a membership functionli as given below [8].

z3

z2

z1

zT

d

d

z12

z3

z2

z1

zT

d

d

z12

d/2

ze

(a) (b)

z3

z2

z1

zT

z12

c2

c1

z3 z2

z12

z1

znew

(c) (d)Fig. 1. Agents (bees) search movements with the proposed optimization algorithm. (a) Starting of the motion in search of solution, (b) extension in the direction of goodoptimal point, (c) contraction of the movement in case optimal point quality is not good, (d) shrinking of the space towards optimistic solution.

Fig. 2. Movement of the agents for a given problem.

R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250 1243

li ¼ 1; Fi 6 Fmini

li ¼ ðFmaxi � FiÞ=ðFmax

i � Fmini Þ; Fmin

i < Fi < Fmaxi

li ¼ 0; Fi P Fmaxi

ð8Þ

where Fmini and Fmax

i are the minimum and maximum values of theith objective function respectively. For each non-dominated solu-tion, the normalized membership function (lk) is calculated as [15]:

lk ¼PN

i¼1lkiPM

j¼1

PNi¼1l

ji

ð9Þ

where N is the number of objective functions, M denotes the num-ber of non-dominated solutions. The best compromise correspondsto that having highest lk.

3. Multi-objective directed bee colony optimization (MODBC)

In the proposed algorithm different agents are being sent in thewhole search area which is divided into different fragments. Thebest solution in each fragment is being searched by its respectiveagent through modified Nelder–Mead method (NM method) [12].For this purpose total range of the independent parameters is di-vided into smaller volumes, each of which determines the startingpoint for the exploration for each agent [14]. The agent then findsits own optimized point by a developed optimization techniqueNM method. Each agent passes the information regarding the opti-mized point by a process inspired from bees, called a bee waggledance. When all the information of optimized points is obtainedthen the best among these is chosen by consensus method as incase of honey bee swarms [17,18].

3.1. Particle search methodology

For optimization of the given objective function we havemodified a very popular optimization technique usually knownas Nelder–Mead method. The methodology used is deterministicsearch methodology but in a sense similar to swarm with localsearch. Let z = (x, y) be the function that is to be minimized. Foragents this is food function. To start, we assume that agent consid-ers three vertices of a triangle as food points for a two variablesproblem as z1,z2 and z3. z1 = (x1, y1) represents the initial positionof agent z2 = (x2, y2) and z3 = (x3, y3) are the positions of probablefood points i.e. local optimal points.

The movement of agent from its initial position towards thefood position, i.e. optimization point is as follows:

The function zi = f(xi, yi) for i = 1, 2, 3 is evaluated at each ofthese three points. The obtained values of zi are recorded in away that z1 6 z2 6 z3 with corresponding agents positions and food

1244 R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250

points as from the best to the worst position. The construction pro-cess uses the midpoint of the line segment joining the two bestfood positions z1 and z2 as shown in Fig. 1a. The value of functiondecreases as an agent moves along z3 to z1 or z3 to z2. Hence it isfeasible that f ðx; yÞ takes a smaller value if agent moves towardsz12. For the further movement of the agent, a test point zT is chosenin such a way that it is a reflection of the worst food point i.e. z3 asshown in Fig. 1a.

The vector formula for zT is

zT ¼ 2� z12 � z3 ð10Þ

If the function value at zT is smaller than the function value atz3, then the agent has moved in the correct direction towards theminimum. Perhaps the minimum is just a bit further than the pointzT. So the line segment is extended further to ze through zt and z12

.The point ze is found by moving as additional distance d/2 alongthe line as shown in Fig. 1b. If the function value at ze is less thanthe function value at zt, then the agent has found a better foodpoint than zT.

ze ¼ 2� zT � z12 ð11Þ

If the function value at z12 and z3 are the same, another pointmust be tested. Two test points are considered by the agent onthe both sides of z12 at distance d/2 as shown in Fig. 1c.

The point of smaller value will frame a new triangle with othertwo best points. If the function value at the two test points is notless than the value at z3, the points z2 and z3 must be shrunktowards z1 as shown in Fig. 1d. The point z2 is replaced with z12,and z3 is replaced with the midpoint of the line segment joiningz1 and z3. Fig. 2 shows the path traced by the agents and the se-quences of triangles {Tk} converging to the optimal point for theobjective function

f ðx; yÞ ¼ x2 � 4xþ y2 � y� xy ð12Þ

3.1.1. Choice of starting point of searching in a volumeThe solution of the NM method depends upon the starting loca-

tion of the search in any volume. Experiment has been made to findthe effect over optimal solution with change in the starting point ofexploration of agents in a volume. We have tested the algorithm onmany standard functions and found the center as the best point asthe starting point [13].

3.1.2. Choice of no of agents for searchingDuring the experiments, it is found that small number of agents

gives fast and accurate results for simple problem having a lowernumber of parameters whereas, for more parameters more numberof agents should go for exploration which in turn gives a resultwith high accuracy but on the cost of time [13–15]. The numberof agents depends on the step length as discussed in the next sub-section. Experimentally, it has been found that the center of thelattice is a good starting point to get better optimal solution and30–50 agents in a number are sufficient to generate the optimalsolution [13].

3.2. Exploration

In MAS, all agents live in an environment [9,15]. An environ-ment is organized in a structure as shown in Figs. 3 and 4. In theenvironment, each agent is fixed on a lattice-point, and each circlerepresents an agent; the data in the circle represent the position ofagent and the evaluated value of the function. The size and dimen-sion of the lattice depend upon the variables and bee swarm.

The value of the objective function depends on p number ofindependent parameters. Let the range of jth parameter 2 ½Wji;

Wjf �, where Wji and Wjf represent the initial and final value of theparameter. Thus the complete domain of the objective functioncan be represented by a set of p number of axis. Each axis will bein a different dimension and will contain the total range of oneparameter.

The next step is to divide each axis into smaller parts. Each ofthese parts is known as a step. Let the jth axis be divided in nj num-ber of step each of length Sj where j = 1 to p.

This length Sj is known as step size for the jth parameter. Therelationship between nj and Sj can be given as

nj ¼Wjf �Wji

Sjð13Þ

Hence each axis is divided into their corresponding branches. If wetake one branch from each axis then these p number of brancheswill constitute a p dimensional volume. Total number of such vol-umes can be calculated as Number of volumes,

Nv ¼Yp

j¼1

nj ð14Þ

The number of volumes indicates the number of scout agentsgoing out for exploration. One point inside each volume is chosenas the starting point for the optimization, which in our approach isthe midpoint of that volume, the reason for same is also being dis-cussed here. The midpoint of total cluster can be calculated asfollows

Wi1 þWf 1

2Wi2 þWf 2

2; . . . ;

Wip þWfp

2

� �ð15Þ

For an objective function having one independent parameter,the complete domain will be given by single axis represented as h1.

Here each step will give us one volume. Let us take the follow-ing values

p ¼ 1; W1i ¼ 1; W1f ¼ 6 S1 ¼ 1

Therefore n1 = 5 and Nv = 5. Thus 5 agents are sent for explora-tion. The starting point for each agent is the midpoint of each stepas shown in Fig. 3.

For an objective function having two independent parameters,the complete domain will be given by a set of two axis representedas h1 and h2.

Let us take the following values

p ¼ 2; W1i ¼ 1; W1f ¼ 5; S1 ¼ 1 andW2i ¼ 1; W2f ¼ 5; S2 ¼ 1

Therefore n1 = 4, n2 = 4 and Nv = 16. Thus 16 agents are sent forexploration as shown in Fig. 4a. The starting point of each agentis the midpoint of each volume which is two dimensional rectanglesin this case.

For an objective function with three independent parameters,the complete domain will be given by set of three axis representedas h1, h2 and h3.

Let us take the following values

p ¼ 3; W1i ¼ 1; W1f ¼ 5; S1 ¼ 1W2i ¼ 1; W2f ¼ 4; S2 ¼ 1 andW3i ¼ 1; W3f ¼ 4; S3 ¼ 1

Therefore n1 = 4, n2 = 3, n3 = 3 and Nv = 36.Thus 36 agents aresent for exploration. The starting point for each agent is themidpoint of corresponding volume, which is 3-dimensional cu-boids in this case as shown in Fig. 4b. Objective functions withmore than three independent parameters can also be solved inthe similar manner.

R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250 1245

3.3. Bee swarms based decision process

The honey bee swarms have a highly distributed decision-mak-ing process which they used for finding out their next hive or find-ing out new source of foods. Few hundreds of bees out ofthousands work as scout bees to start a search for next possiblesite. Upon finding the site, scout informs other bees by waggledance [17]. Discovered nest sites of sufficient quality are reportedon the cluster via the scouts’ waggle dance. Depending on the wag-gle dance by scout bees quiescent bees get activated and decided torecruit or explore for nest site. If an uncommitted bee is not satis-fied with any of the scout sites than bee can go for exploring newsites. When a bee advertises a site more than once then in everynext turn, bee decreases the strength of her dance by about 15dance circuits. Once the quorum threshold reaches for any one ofthe sites, the bee start piping signals that elicit heating by the qui-escent bees in preparation for flight. There are two methods usedby the bee swarm for finding out the best nest site as consensusand quorum [20]. In consensus widespread agreement among thegroup is taken into account, whereas in quorum the decision forbest site happens when a site crosses the quorum (threshold) value.In the present paper, the consensus algorithm is used for finding outthe optimum solution i.e. best food site.

3.3.1. Waggle danceAs bee after returning from search performs waggle dance to in-

form other bees about the quality of site or food. Here in the pro-posed algorithm the agents after collecting their individual optimalsolution give to the centralized systems that choose the preferablesolution from the searched one. For optimal minimum cases, it se-lects the best optimal solution which can mathematically be statedas

Wdi ¼minðfiðXÞÞ ð16Þ

where fi(X) represent the different search value obtained by anagent. Each of these points is recorded in a table known as optimumvector table X. X is a vector containing p number of elements. Theseelements contain the value of parameters at that point. So both theoptimal solution value and the corresponding variable values arerecorded. This record is known as Personal Best i.e. Pbest in PSO.The function value gets change according to the objective functionrequirement i.e. if objective function is to be minimized then themin function is used and if we have to find maximize in an objectivefunction it will switch over to maximize function.

3.3.2. ConsensusAs bee swarms use consensus method to decide the best ob-

tained or search value. The authors mimic this event and behaviorby comparing the results obtained. Once exploration and waggledance (transmission of data) is finished the global optimized pointis chosen by comparing the fitness values of all the optimizedpoints in the optimum vector table i.e. global best, gbest as in caseof PSO. For minimization problems the point with the lowestfitness value is selected as the global optimized point. The globaloptimized point XG is found by

f ðXGÞ ¼min f ðX1Þ; f ðX2Þ; . . . ; f ðXNv Þ½ � ð17Þ

1 6

h12 3 4 5

Fig. 3. Domain of the objective function with one independent parameter.

4. Results and discussions

In this paper, the MODBC technique was tested on IEEE 30-bussystem with 6 generators and 41 interconnected transmission lines(see Fig. 5). The MOEA techniques have been compared to eachother and to classical techniques as well. The total demand of thesystem is taken to be 2.834 p.u. In the tables (Tables 1 and 2 respec-

tively) the fuel cost and the total emission function coefficientsalong with generator capacity limits of each generator are given.The line data and bus data of the IEEE 30 bus system has been tab-ulated in Tables 9 and 10. Fuel cost and emission functions wereminimized individually using MODBC, ignoring system losses (Ta-ble 3). Results obtained by minimizing both the functions individu-ally, taking losses into consideration are tabulated (Table 4).

MODBC algorithm(1) Initialize the number of parameter, p initialize the length

of steps (no of axes to be explore), Sj (j = 0 to p)(2) Initialize the range for each parameter as [Wij, Wfj] where

j = 0, 1, . . . ,p(3) Calculate the number of steps in each step

nj ¼Wfj�Wij

Sj

(4) Calculate the total number of volumesNm ¼

Qpj¼1nj

(5) For each volume, take the starting point of the exploration

as the midpoint of the volume Wi1þWf 12 ;

Wi2þWf 22 ; . . . ;

WiPþWfP

2

h i(6) Explore the volume according to modified Nelder–Mead

method.(7) Record the value of optimized point obtained

corresponding to each volume in optimum vector table infollowing way ½X1;X2; . . . ;XNv �

(8) After the exploration is being completed, the globaloptimized point in the following manner using Bee decisionapproach

FðXGÞ ¼min½FðX1Þ; FðX2Þ; . . . ; FðXNvÞ�

In the proposed algorithm, there is scope only for the steplength (Sj) to be tuned. The appropriate selection of the Sj is doneaccording to the sensitivity of the objective function to a particularparameter. For e.g.: Consider an objective function which is moresensitive to p1 than p2 then the step length selected for former willbe lesser than the latter parameter. The selection is done on arelative basis and accordingly the number of agents to be sent tosearch in the search space is determined according to theEq. (13). Unlike the latin hyper-cubes the agents are sent to allsegments of the search space and are made to follow the NM meth-od of searching rather than sampling the search space and thenchoosing the value of the variable according to probability density.Simulations have been made to find out the optimal number ofagents for MODBC as shown in Fig. 6. It is seen that 30 numbersof agents are sufficient for the problem and the same has beenverified the authors earlier [13]. Hence for MODBC number ofagents kept 30. The computational efficiency and the convergencecharacteristics are in detailed discussed in [13–15].

The objective of the paper is to show case the ability of the pro-posed algorithm to handle multi-objective functions. This specificquadratic function is considered to have a clear distinction in thecharacteristics of the two functions. Where one function increaseswith increase in the variable whereas the other function decreases.If a non-convex function is considered then such clear distinctionin the characteristics of the emission function and the cost functioncannot be made. If it was to showcase just the economic dispatchproblem then non-convex functions shall be considered. The

h1

5

4

3

2

11 2 3 4 5

h2

1 2 3 4

h2

12

3

4

h1

2

3

4

1

h3

(a) (b)Fig. 4. Domain of the objective function with (a) two and (b) three independent parameters.

21

2829

30

2423

26 20

22

27

2518

19

9

10

11

12

14

13

17

16

8 7

6

5 4

1

2

3

15

G1

G2 G3

G4

G5

Fig. 5. Single line diagram of IEEE 30 bus test system.

1246 R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250

Table 1Fuel cost coefficients and capacity limits.

Generator a b c Pgimin Pgimax

1 10 200 100 0.05 0.52 10 150 120 0.05 0.63 20 180 40 0.05 1.04 10 100 60 0.05 1.25 20 180 40 0.05 1.06 10 150 100 0.05 0.6

Table 2Emission coefficients.

Generator a b c f k

1 0.04091 �0.05554 0.06490 0.000200 2.8572 0.02543 �0.06047 0.05638 0.000500 3.3333 0.04258 �0.05094 0.04586 0.000001 8.04 0.05326 �0.03550 0.03380 0.002000 2.05 0.04258 �0.05094 0.04586 0.000001 8.06 0.06131 �0.05555 0.05151 0.000010 6.667

Table 3Best solutions (lossless).

Cost dispatch Emission dispatch

Pi (p.u.) C (Pi) Pi (p.u.) E (Pi)

0.10928 33.0502 0.40618 0.02970.29942 65.6713 0.45952 0.01190.52323 125.1322 0.53694 0.02851.01548 173.4200 0.38226 0.04890.52326 125.1388 0.53482 0.02850.35929 76.8024 0.51025 0.0467

Total cost ($/h) 599.22380 637.4649

Total emission (ton/h) 0.2221 0.19420

Table 4Best solution with losses.

Cost dispatch Emission dispatch

Pi (p.u.) C (Pi) Pi (p.u.) E (Pi)

0.11174 33.5966 0.40934 0.02970.30145 66.1222 0.46254 0.01190.52936 126.4937 0.54129 0.02851.01958 174.3306 0.38715 0.04890.52938 126.4981 0.53914 0.02850.36175 77.3488 0.51382 0.0467

Total cost ($/h) 604.3972 642.6873

Total emission (ton/h) 0.2222 0.19418

Table 5Comparison among LP, MOSST, NSGA, NPGA, SPEA, PSO, MODBC.

Optimization technique Considering fuel cost only

Cost Em

LP 606.31 0.2MOSST 605.89 0.2NSGA 600.34 0.2NPGA 600.31 0.2SPEA 600.22 0.2PSO 600.1117 0.2MODBC 599.2238 0.2

55 50 45 40 35 301.5

1.9

2.4

2.8

3.3

Number of Agents

Tim

e (s

ec)

Fig. 6. Convergence time vs number of agents.

R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250 1247

proposed algorithm was also implemented on a cost function con-sidering the valve point effects and was proved better than otherevolutionary algorithms [14].

4.1. Comparison with other techniques

The results obtained, when the cost function and emission func-tion were optimized separately using MODBC are compared withthe results obtained from the other traditional techniques likeLinear Programming (LP) and Multi-Objective Stochastic SearchTechnique (MOSST) and other MOEAs like the NSGA, NPGA, SPEA.From the results tabulated (Table 5) it is clear that MODBC hasproduced a better optimal solution for cost function than any othertechnique discussed and has given solution that is at par with thesolution given by LP, MOSST and SPEA and far better results thanNSGA and NPGA[8]. The results show the superior characteristicsof MODBC.

4.2. Optimization of the multi-objective function

Using MODBC a multi-objective problem is solved by assigninga price penalty factor to the emission function. This is done bycombining the two individual functions into one function. Thecombined objective function is formulated as given below [7].Based on the calculated value of h a single solution can be found.Here the value of h is varied to find the trade-off value betweenthe two conflicting functions. For a value h = 0 the combined func-tion boils down to a conventional fuel cost dispatch problem andwhen h =1 the problem becomes a pure emission dispatchfunction

Min z ¼ FðPÞ þ hEðPÞ$=ton ð18Þ

Considering emission only

ission Cost Emission

233 639.60 0.1942222 644.11 0.1942241 633.83 0.1946238 636.04 0.1943206 640.42 0.19422204 637.966 0.19422210 637.465 0.1942

Cost ($/hr)595 600 605 610 615 620 625 630 635 640 645

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

Emis

sion

(ton

/hr)

NSGASPEANPGAPSOMODBC

Fig. 7. Comparison of Pareto-optimal fronts (without Ploss).

Cost ($/hr)600 605 610 615 620 625 630 635 640 645 650

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

Emis

sion

(ton

/hr)

NSGASPEANPGAPSOMODBC

Fig. 8. Comparison of Pareto-optimal fronts (with Ploss).

1248 R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250

The fuel cost and the emission cost is coordinated by price pen-alty factor. The price penalty cost introduces an extra cost of oper-ation taking into account the implied emission cost. Variousmethods [19,20] have been suggested to calculate price penaltyfactor and among that the maximum price penalty factor has beenchosen for combining cost of fuel plus the implied cost of emissionas it offers a very good solution for emission restricted less cost

Table 6COMPARISON BETWEEN DIFFERENT TECHNIQUES when losses are ignored.

Pi Best cost

NSGA NPGA SPEA PSO MODBC

1 0.1038 0.1116 0.1009 0.10992 0.109282 0.3228 0.3153 0.3186 0.29995 0.299423 0.5123 0.5419 0.5400 0.52626 0.523234 1.0387 1.0415 0.9903 1.01483 1.015485 0.5324 0.4726 0.5336 0.52289 0.523266 0.3241 0.3512 0.3507 0.36012 0.35929

Cost 600.34 600.31 600.22 600.11 599.22

Emission 0.2241 0.2238 0.2206 0.2220 0.2221

condition [20]. The maximum price penalty factor h of each gener-ator is the ratio between the fuel cost and emission at its maximumpower output as given in the equation given below.

hi max ¼ ai þ biPimax þ ciP2imax

� �= ai þ biPimax þ ciP

2imax

� �þ 1ekiPimax

� �ð19Þ

The figure given (Fig. 7) shows the pareto-optimal frontsobtained when NPGA, NSGA, SPEA, PSO and MODBC were appliedand the problem constraints are power balance constraint and gen-eration capacity constraint in the lossless system (without Pl). Infigure (Fig. 8) the problem constraints are power balance con-straint with including power losses and the generation capacityconstraints. For NSGA, NPGA, SPEA the result portrayed is the bestresult obtained in 10 runs [10]. It is very evident from the graphthat the MODBC has better diversity characteristics and gives thebest cost and best emission when compared to other methods inboth the cases, when losses were considered and when they wereignored.

The results obtained by MODBC, when losses were ignored andconsidered, for best cost and best emission are tabulated in Tables6 and 7 respectively. The results are compared with recorded re-sults obtained by NSGA, NPGA and SPEA [8] and PSO .It can be in-ferred that the MODBC gave better optimal solutions than any ofthe other MOEAs considered.

4.3. Best compromise solution

Each member of the Pareto-optimal set for each technique, themembership functions as given in (Eqs. (8) and (9)) were used toevaluate. Then, the best compromise solution that has the maxi-mum value of membership function was extracted and tabulatedin Table 8. Even here the MODBC has given comparable resultswhen the losses were ignored and better results when systemlosses were considered in comparison with NSGA, NPGA andSPEA.

5. Conclusions

In this paper, the MODBC method has been proposed to showcase the ability of the proposed algorithm to handle multi-objec-tive functions. From the table (Table 5) where the comparison be-tween the different methods, used to optimize the cost andemission function individually, it is evident that the MODBC meth-od gave better optimal solutions. The optimal value of the costfunction was lesser than the values obtained by any of the otheroptimization technique when losses were ignored. Whereas thesolution for the emission function was found better than the solu-tion obtained by NSGA and NPGA and with the results obtainedfrom other techniques. The two objective functions were combinedto form the single multi-objective function as given in the equation(Eq. (18)). The optimization results, when losses are ignored and

Best emission

NSGA NPGA SPEA PSO MODBC

0.4072 0.4146 0.4055 0.40228 0.4061880.4536 0.4419 0.4499 0.45842 0.4595260.4888 0.5411 0.5431 0.54093 0.5369440.4302 0.4067 0.3875 0.38386 0.3822600.5836 0.5318 0.5385 0.53828 0.5348280.4707 0.4979 0.5095 0.51020 0.510252

633.83 636.04 640.42 637.9661 637.4649

0.1946 0.1943 0.1942 0.19420 0.19420

Table 8Best compromise solution from Pareto-optimal set.

Pi Without losses With losses

NSGA NPGA SPEA MODBC NSGA NPGA SPEA MODBC

1 0.2252 0.2663 0.2623 0.2568 0.2935 0.2976 0.2752 0.26952 0.3622 0.3700 0.3765 0.3633 0.3645 0.3956 0.3752 0.38123 0.5222 0.5222 0.5428 0.5194 0.5833 0.5673 0.5796 0.54474 0.7660 0.7202 0.6838 0.6949 0.6763 0.6928 0.6770 0.67905 0.5397 0.5256 0.5381 0.5394 0.5383 0.5201 0.5283 0.54476 0.4187 0.4296 0.4305 0.4201 0.4076 0.3904 0.4282 0.4383

Cost 606.03 608.90 610.30 606.03 617.80 617.79 617.57 616.1266

Emission 0.2041 0.2015 0.2004 0.20038 0.2002 0.2004 0.2001 0.2000

Table 9IEEE 30-bus test system line data.

Line From bus To bus Resistance (p.u.) Reactance (p.u.) Susceptance (p.u.) Rating (MVA)

1 1 2 0.0192 0.0575 0.0264 1302 1 3 0.0452 0.1852 0.0204 1303 2 4 0.057 0.1737 0.0184 654 3 4 0.0132 0.0379 0.0042 1305 2 5 0.0472 0.1983 0.0209 1306 2 6 0.0581 0.1763 0.0187 657 4 6 0.0119 0.0414 0.0045 908 5 7 0.0460 0.1160 0.0102 709 6 7 0.0267 0.0820 0.0085 130

10 6 8 0.0120 0.0420 0.0045 3211 6 9 0.0000 0.2080 0.0000 6512 6 10 0.0000 0.5560 0.0000 3213 9 11 0.0000 0.2080 0.0000 6514 9 10 0.0000 0.1100 0.0000 6515 4 12 0.0000 0.2560 0.0000 6516 12 13 0.0000 0.1400 0.0000 6517 12 14 0.1231 0.2559 0.0000 3218 12 15 0.0662 0.1304 0.0000 3219 12 16 0.0945 0.1987 0.0000 3220 14 15 0.2210 0.1997 0.0000 1621 16 17 0.0824 0.1923 0.0000 1622 15 18 0.1070 0.2185 0.0000 1623 18 19 0.0639 0.1292 0.0000 1624 19 20 0.0340 0.0680 0.0000 3225 10 20 0.0936 0.2090 0.0000 3226 10 17 0.0324 0.0845 0.0000 3227 10 21 0.0348 0.0749 0.0000 3228 10 22 0.0727 0.1499 0.0000 3229 21 22 0.0116 0.0236 0.0000 3230 15 23 0.1000 0.2020 0.0000 1631 22 24 0.1150 0.1790 0.0000 1632 23 24 0.1320 0.2700 0.0000 1633 24 25 0.1885 0.3292 0.0000 1634 25 26 0.2544 0.3800 0.0000 1635 25 27 0.1093 0.2087 0.0000 1636 28 27 0.0000 0.3960 0.0000 6537 27 29 0.2198 0.4153 0.0000 1638 27 30 0.3202 0.6027 0.0000 1639 29 30 0.2399 0.4533 0.0000 1640 8 28 0.0636 0.2000 0.0214 3241 6 28 0.0169 0.0599 0.0065 32

Table 7Comparison between different techniques when losses are considered.

Pi Best cost Best emission

NSGA NPGA SPEA PSO MODBC NSGA NPGA SPEA PSO MODBC

1 0.1447 0.1425 0.1279 0.08160 0.11174 0.3929 0.4064 0.4145 0.4108 0.4093442 0.3066 0.2693 0.3163 0.30396 0.30145 0.3937 0.4876 0.4450 0.4668 0.4625413 0.5493 0.5908 0.5803 0.58990 0.52936 0.5818 0.5251 0.5799 0.5484 0.5412894 0.9894 0.9944 0.9580 0.99763 1.01958 0.4316 0.4085 0.3847 0.3911 0.3871545 0.5244 0.5315 0.5258 0.52464 0.52938 0.5445 0.5386 0.5358 0.5515 0.5391396 0.3542 0.3392 0.3589 0.35959 0.36175 0.5192 0.4992 0.5051 0.5173 0.513824

Cost 607.98 608.06 607.86 605.562 604.397 638.98 644.23 644.77 650.044 642.6873

Emission 0.2191 0.2207 0.2176 0.2221 0.2222 0.1947 0.1943 0.1943 0.19418 0.19418

R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250 1249

1250 R. Kumar et al. / Electrical Power and Energy Systems 43 (2012) 1241–1250

considered, are tabulated in Tables 6 and 7 respectively. Fromthese tables, it can be inferred that the MODBC has provided withthe best solution among the other optimization techniques. ThePareto-optimal curves plotted in Figs. 7 and 8. The graphs comparethe performance of MODBC, NSGA, NPGA, SPEA and PSO method.The graph obtained by the MODBC lies below the graph obtainedby all the methods in both cases when losses were ignored andconsidered. This shows that when the multi-objective function atdifferent values of h or the price penalty factor was optimized byMODBC, gave better optimal points and showed better diversitycharacteristics than other methods. The best compromise solutionobtained by MODBC was better in both the cost and emission func-tion values than the ones obtained from the other MOEAs whenlosses were considered and was better than the other methodseither in the cost or in the emission function when losses were ig-nored. The results also show that MODBC removes the randomnessin the algorithm and improves significantly in global optimizationperformance. The solution of MODBC shows consistency in thesolution and hence it gives a better option to optimize real-timeand online optimization problems. Therefore, it is clear that theproposed scheme is more efficient, robust and effective than anyof the other techniques discussed.

Table 11Nomenclature.

Symbol Quantity

N Number of generating unitsPi Generating unit real power outputai, bi, ci Cost function coefficients of ith generating unitPimin,

Pimax

Minimum and maximum power output limit of ith generatingunit

Pl Overall system real power lossesPd Total system real power demandBij Element of loss function coefficientai, bi, ci, f,

ki

Emission function coefficient of ith generating unit

g(Pi) Equality constrainth(Pi) Inequality constraintxi Position vector of particle ivi Velocity vector of particle ipbesti Best position particle i based on its own experiencegbesti Best particle position based on overall swarm experienceWij, Wfj Initial and final value of jth parameter on which the objective

function dependsnj Number of steps into which the range of the jth parameter is

divided intoSj Length of each stepVi ith food pointh Price penalty factorli Membership function of ith objective functionlk Normalized membership function

Table 10IEEE 30-bus test system bus data.

Bus no. PD (MW) QD (MVA) Bus no. PD (MW) QD (MVA)

1 00.00 00.00 16 3.50 01.802 21.70 12.70 17 9.99 05.803 02.40 01.20 18 3.20 0.904 07.60 01.60 19 9.50 03.405 94.20 19.00 20 2.20 00.706 00.00 00.00 21 17.50 11.207 22.80 10.90 22 00.00 00.008 30.00 30.00 23 03.20 01.609 00.00 00.00 24 08.70 06.70

10 05.80 02.00 25 00.00 00.0011 00.00 00.00 26 03.50 02.3012 11.20 07.50 27 00.00 00.0013 00.00 00.00 28 00.00 00.0014 06.20 01.60 29 02.40 00.9015 08.20 02.50 30 10.60 01.90

Appendix A

The line and bus data of the IEEE 30-bus six-generator systemare given in Tables 9 and 10 respectively.

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