Electrical Power and Energy Systems 33 (2011) 846–854

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Electrical Power and Energy Systems

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

Multiple tabu search algorithm for economic dispatch problem consideringvalve-point effects

Worawat Sa-ngiamvibool a,⇑, Saravuth Pothiya a, Issarachai Ngamroo b

a Department of Electrical Engineering, Faculty of Engineering, Mahasarakham University, Mahasarakham 44150, Thailandb Center of Excellence for Innovation Energy Systems, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand

a r t i c l e i n f o

Article history:Received 28 December 2007Received in revised form 24 August 2010Accepted 23 November 2010Available online 17 February 2011

Keywords:Economic dispatch problemValve-point effectsMultiple searchesOptimization methodTabu search algorithm

0142-0615/$ - see front matter Crown Copyright � 2doi:10.1016/j.ijepes.2010.11.011

⇑ Corresponding author.E-mail address: wor_nui@yahoo.com (W. Sa-ngiam

a b s t r a c t

This paper presents a multiple tabu search (MTS) algorithm to solve the economic dispatch (ED) problemby taking valve-point effects into consideration. The practical ED problem with valve-point effects is rep-resented as a non-smooth optimization problem with equality and inequality constraints that make theproblem of finding the global or near global optimum difficult. The proposed MTS algorithm is thesequential execution of individual tabu search (TS) algorithm simultaneously by only one personal micro-computer. The MTS algorithm introduces additional techniques for improvement of search process, suchas initialization, adaptive searches, multiple searches, replacing and restarting process. To show its effec-tiveness, the MTS is applied to test two studied systems consisting of 13 and 40 power generating unitswith valve-point effects. The optimized results by MTS are compared with those of conventionalapproaches, such as simulated annealing (SA), genetic algorithm (GA), TS algorithm and particle swarmoptimization (PSO). Studied results confirm that the proposed MTS approach is capable of obtaininghigher quality solution efficiently and lowest computational time.

Crown Copyright � 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

The economic dispatch (ED) problem is one of the optimizationproblems in power system operation. The objective of ED problemis to schedule the optimal combination of outputs of all generatingunits to minimize the operating cost while satisfying the loaddemand and system equality and inequality constraints. Improve-ments in scheduling the unit power outputs can lead to significantcost savings.

In the past, a number of approaches have been developed forsolving this problem using mathematical programming, such aslambda iteration method, quadratic programming, gradient meth-od, dynamic programming, linear programming, and nonlinearprogramming. However, these methods are difficult to find an opti-mal solution for ED problem. They usually get stuck at a local opti-mum. Additionally, the input–output characteristics of modernpower generating units are inherently highly nonlinear becauseof valve-point loadings, multi-fuel effects, etc. Moreover, thesecharacteristics may lead to multiple local minimum points of thecost function which make the problem of finding the global or nearglobal optimum difficult.

Recently, modern heuristic optimization techniques such assimulated annealing (SA) [1,2], genetic algorithm (GA) [2–6],

011 Published by Elsevier Ltd. All r

vibool).

evolutionary programming (EP) [7,8], tabu search (TS) algorithm[9–11], neural network approaches, differential evolution (DE)algorithm [12,13], ant colony optimization (ACO) [14], and parti-cle swarm optimization (PSO) [15–17] have been paid muchattentions by many researchers because of their abilities to findan almost global optimal solution. Among of them, the TS algo-rithm is expected as one of the advanced search technique. TheTS algorithm is able to escape from local optimal and fast con-verge to global optimum. However, a conventional TS algorithmmight have a problem of reaching the global optimum solutionin a reasonable computational time when the initial solution isfar a way from the region where optimum solution exists.

Nowadays, the personal microcomputer has high speed compu-tation. To solve the large scale problem, several computers may beused for computation at the same time. This method is called‘‘parallel searches’’. On the other hand, the ‘‘multiple search’’implemented in this paper is executed by only one personal micro-computer. The multiple searches help to find the promising regionwhere the global optimum solution exists. This idea is applied toimprove the performance of a conventional TS algorithm. This pa-per proposes the new algorithm ‘‘multiple tabu search (MTS)’’ tosolve the problems of the conventional TS. The MTS is the execu-tion of individual TS algorithm simultaneously by only single per-sonal microcomputer. The MTS introduces the additional salientmechanisms for improvement of search process, i.e. initialization,adaptive searches, multiple searches, replacing and restarting

ights reserved.

W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854 847

process. The feasibility study of the MTS is demonstrated for solv-ing the ED problem with valve-point effects. The optimized resultsby the MTS are compared to those obtained by the conventionalapproaches such as SA, GA, TS and PSO in terms of solution qualityand computational efficiency.

The paper is organized as follows. Section 2 gives the mathe-matical model of the ED problem with valve-point effects. Section3 mentions the MTS algorithm. Section 4 presents the detailed pro-cedures of the MTS algorithm for solving the ED problem. Section 5shows two application cases and gives the corresponding compar-ison results with the traditional methods. Conclusion is finally gi-ven in Section 6.

2. Problem formulation

The ED problem is a nonlinear optimization problem which is asub-problem of the unit commitment (UC) problem. The objectiveof ED problem is to find the optimal combination of power gener-ation that minimizes the total generation cost while satisfying anequality constraint and inequality constraints.

2.1. Objective function

The main objective of ED is to simultaneously minimize thegeneration cost rate and to meet the load demand of a power sys-tem over some appropriate period while satisfying various con-straints. The objective function of ED problem can be modified as

min Ft ¼Xn

i¼1

FiðPiÞ ¼Xn

i¼1

ðai þ biPi þ ciP2i Þ ð1Þ

where Ft is the total generation cost; Fi is the generation cost func-tion of ith generator is usually expressed as a quadratic polynomial;ai, bi, and ci are the cost coefficients of the ith generator; Pi is thepower output of the ith generator and n is the number of generatorscommitted to the operating system.

In actual fact, the objective function of an ED problem has non-differentiable points according to valve-point effects. For that rea-son, the objective function should be composed of a set of non-smooth cost functions. In this paper, the case with the valve-pointloading problem is considered where the objective function is gen-erally described as the superposition of sinusoidal functions andquadratic functions. The problems have multiple minima; there-fore, the task of finding the global solution still remains to be tack-led [8].

The generator with multi-valves steam turbines has very differ-ent input–output curve compared with the smooth cost function.

A

B

C

D

E

MW

$/MW

A : Primary Valve B : Secondary Valve

C : Tertiary Valve D : Quaternary Valve

E : Quinary Valve

Fig. 1. Example of cost function with five valves.

Typically, the valve point results in, as each steam valve starts toopen, the ripples like in Fig. 1, [16]. To take account for thevalve-point effects, sinusoidal functions are added to the quadraticcost functions as follows:

FiðPiÞ ¼ ai þ biPi þ ciP2i þ jei � sinðfi � ðPmin

i � PiÞÞj ð2Þ

where ei and fi are the coefficients of generator reflecting valve-point effects.

2.2. Constraints

2.2.1. Equality constraintWhile minimizing the total generation cost, the total generation

should be equal to the total system demand plus the transmissionnetwork loss. However, the network loss is not considered in thispaper for simplicity. This gives the equality constraint

Xm

i¼1

Pi ¼ PD ð3Þ

where PD is load demand

2.2.2. Inequality constraintThe generation output of each unit is between its minimum and

maximum limits. That is, the following inequality constraint foreach generator should be satisfied

Pmini 6 Pi 6 Pmax

i ð4Þ

where Pmini , Pmax

i : minimum, maximum output of ith generator.

3. Multiple tabu search algorithm

The MTS is the execution of individual TS algorithm at the sametime by a single personal microcomputer. At this point, the individ-ual TS algorithm and the MTS algorithm are explained.

3.1. Tabu search algorithm

3.1.1. OverviewIn general terms, a conventional TS algorithm is an iterative

search that starts from an initial feasible solution and attemptsto determine a better solution in the manner of a hill-climbingalgorithm. The TS has a flexible memory to keep the informationabout the past steps of the search. The TS uses the past search tocreate and exploit the better solutions [20,21].

The main two components of TS algorithm are the tabu list (TL)restrictions and the aspiration criterion (AC).

3.1.2. Tabu list restrictionsIn order to prevent cycling, repeated search at the same solu-

tion, a TL is introduced. The TL stores a set of the tabu (prohibition)moves that cannot be applied to the current solution. The movesstored in TL are called tabu restrictions and used for decreasingthe possibility of cycling because it prevents returning in a certainnumber of iterations to a solution visited recently.

In this paper, the size of TL is n � 3 (row � column), n is a num-ber of neighborhoods around current solution. In the TL, the firstcolumn is used for storing the moves, the second column is the fre-quency of a move direction, and the last column is the recency(time to keep solutions) of a move [22].

3.1.3. Aspiration criterionAnother key issue of TS algorithm arises when all moves under

consideration have been found to be tabued. The tabu status of amove is not absolute, but it can be overruled if certain conditionsare met and expressed in the form of AC. If appropriate aspiration

848 W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854

criterion is satisfied, the move will be accepted in spite of the tabuclassification. Roughly speaking, AC is designed to override tabustatus if a move is ‘good enough’ [22].

3.1.4. Stopping criterionGenerally, there are several possible conditions to stop search-

ing. Here, the stopping search is used if any of the following twoconditions are satisfied: firstly, defined by the limit time. Thatmean, if the computational time is grater than or equal to the limittime, then the search process will stop. Secondly, defined by themaximum allowable iterations, if the iteration is grater than orequal to the maximum allowable iteration, then the search processwill stop.

3.1.5. General tabu search algorithmTo solve a combinatorial optimization problem by tabu search,

the basic idea is to choose randomly a feasible solution and at-tempt to find a best neighbor to current solution. A move to thisneighbor is performed if either it does not belong to the TL or, itpasses the AC test. During these procedures, the best solution is al-ways updated and stored aside until the stopping criterion issatisfied.

The following notations are used through the description of theTS algorithm for a general combinatorial optimization problem:

X

the set of feasible solutions x the current solution, x e X xb the best solution reached xnb the best solution among of trial solutions E(x) the objective function of solution x N(x) the set of neighbourhood of x e X TL tabu list AC aspiration criterion

The procedure of TS algorithm is described as follows [22]:

Step 0: Set TL as empty and AC to be zero.Step 1: Set iteration counter k = 0. Select an initial solution x e X,

and set xb = x .Step 2: Generate a set of trial solutions neighbourhood of x. Let

xnb as the best trial solution.Step 3: If E(xnb) > E(xb), go to Step 4, else set the best solution

xb = xnb and go to Step 4.Step 4: Perform the tabu test. If xnb is NOT in the TL, then accept

it as a current solution, set x = xnb, and update the TL andAC and go to Step 6, else go to Step 5.

Step 5: Perform the AC test. If satisfied, then override the tabustate, set x = xnb, update the AC.

Step 6: Perform the termination test. If the stopping criterion(see Section 3.1.4) is satisfied then stop, else setk = k + 1 and go to Step 2.

3.2. Multiple tabu search algorithm

Although the TS algorithm is able to escape from local optimaland fast converge to global optimum. It might have a problemwith reaching the global optimum solution in a reasonable com-putation time when an initial solution is far away from the regionwhere the optimum solution exists. The convergence speed of TSalgorithm depends on an initial solution. The convergence speedcan be improved by introducing a multiple structure into thealgorithm.

The MTS algorithm uses several initial solutions to increase theprobability of reaching the region where the optimum solution ex-ists. The procedure of the MTS algorithm is depicted in Fig. 2,

which consists of several independent conventional TS algorithms(TS#1, TS#2, . . . ,TS#m). Furthermore, the additional mechanismsnamely, initialization, adaptive searches, multiple searches, replacingand restarting process help to improve the search process in termsof both solution quality and computational time. The additionalmechanisms are explained as follows.

3.2.1. InitializationTo improve searching, the MTS algorithm starts to search from

several initial solutions which are different from the TS algorithm.In fact, the starting with several initial solutions has the probabilityof reaching the optimum solution higher than the single initialsolution. This mechanism helps MTS algorithm to converge quicklyto the global optimum solution.

3.2.2. Adaptive searchesThe adaptive searches mechanism is the process of finding the

neighborhood solutions around the current solution. This processdepends on the step size. The step size is the range of variance atthe current solution, which is the important factor for searchingprocess. Accordingly, the step size should be chosen appropriately.Conventionally, this value is fixed. The step size with low value canincrease more accurate solution cause a long computational time.On the other hand, the step size with high value is used fordecreasing the computational time. Nevertheless, the searched re-sult may not reach the global optimum.

In this paper, the step size can be adjusted randomly every iter-ation and based on the changing rate of the solution. The high va-lue of step size has been set in order to find the possible solutionsfrom the wide region at the initial iteration (Iteration = 0). The stepsize will be decreased when the iteration increases. As a result,more accurate solutions can be achieved. Furthermore, the stepsize varies with the changing rate as the solution as well. The stepsize will be increased for the high changed rate and be decreasedfor the low changed rate.

Consequently, the adaptive search mechanism has been devel-oped to adjust suitably the step size during the searched process.This mechanism helps to increase the computational speed andthe accuracy of solution.

3.2.3. Multiple searchesNowadays, the personal microcomputer has high speed compu-

tation. To solve the large scale problem, several computers may beused at the same time. This method is called parallel searches. Formultiple searches, they are executed by only one personal micro-computer. The multiple searches help to find the promising regionwhere the global optimum solution exists.

3.2.4. ReplacingAfter the search process satisfies with the condition for replac-

ing, all independent TS algorithms are paused. The condition forreplacing depends on the number of multiple algorithms (m) andmaximum iteration (Itermax). For example, if m = 4 and Iter-max = 200. The number of replacing should not be grater than thenumber of multiple algorithms. In this paper, the appropriate con-dition of replacing is set for every 30 iterations.

The replacing mechanism is used for comparing and exchangingthe solutions which are found by these TS algorithms. The worstsolution is replaced by the best solution. Then, the TS algorithmwhich got the worst solution will be searched continually to thenext step with the best solution. This mechanism prevents TS algo-rithm to search in the wrong direction when seeking for the bettersolution. The searching time can be reduced. Furthermore, it alsoincreases the probability of obtaining the global solution.

01s 02s 10 −ms ms0

01sΔ 02sΔ 10 −Δ ms ms0Δ

1bs 2bs 1−bms bms

bests

m

Stopping criterion satisfied?

Replacing criterion satisfied?

Restarting criterion satisfied?

bests

Fig. 2. Procedure of multiple tabu search algorithm.

W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854 849

3.2.5. Restarting processDuring searching, the searched process gets the repeated solu-

tion for a long time (20 iterations). That means the process couldnot be found the better solutions or escaped from this solution.So this solution could be either a local or global solution. In casethe global solution is known, the searched process will be stoppedas it has been gotten the best solution. On the other hand the

global solution is unknown. The algorithm will assume that thissolution is a local solution. If any TS algorithm is stroke on localsolution for a long time (20 iterations), the restarting process willbe applied. Firstly, this process will generate a group of new initialsolutions. After that, it will select the best solution as a new solu-tion for next step. This mechanism helps the process to continuesearching and find the better solutions.

TS#1 TS#2 TS#m-1 TS#m

Fig. 3. Sequence of execution.

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4. MTS algorithm for ED problem

In this section, the five mechanisms integrated into the MTS aredescribed for solving the ED problems. Especially, a suggestionabout how to deal with the equality and inequality constraints ofthe ED problems when each search point is modified in the MTSalgorithm is also given. The process of the MTS algorithm can besummarized as follows.

4.1. Initialization and structure of solution

In the initialization process, several initial solutions are createdrandomly for individual TS algorithm. In this paper, the structure ofa solution for ED problem is composed of a set of generation out-puts. Therefore, the initial solutions of individual TS algorithm atiteration #0 can be represented as the vector of X0

i ¼ ðP0i1; . . . ; P0

inÞ,i = 1, . . ., m where m is the number of multiple TS algorithms andn is the number of generators. Note that it is very important to cre-ate a set of solution satisfying the equality and inequalityconstraints.

In this paper, the procedure is applied for MTS algorithm to gen-erate initial solutions as follows:

Step 1: Read system data.Step 2: The boundary of feasible solution is defined in (4).Step 3: Set j = 1.Step 4: Create the value of the generation output at random sat-isfying its equality and inequality constraint by

Replacing Mechanism

Pi ¼ Pmini þ randðÞ � ðPmax

i � Pmini Þ ð5Þ

where randðÞ is a uniform random value in the range [0, 1].Step 5: If j = n then go to Step 6; otherwise and go to Step 4.Step 6: Stop the initialization process.

4.2. Adaptive step sizes

The adaptive searches mechanism is used to generate step sizesfor finding neighborhood of current solution. The step size of indi-vidual TS algorithm at iteration 0 can be represented as the vectorD0

i ¼ ðD0i1; . . . ;D0

inÞ, i = 1, . . ., m.To generate the random step size of each individual TS algo-

rithm, the step sizes are modified under the following equation:

D0i ¼ K � randðÞ � ðPmax

i � Pmini Þ ð6Þ

K is set according to the following equation:

K ¼ kmax �kmax � kmin

Itermax� Iter ð7Þ

where kmax, kmin are the maximum and minimum weights. In thiswork, kmax = 0.9 and kmin = 0.1. Itermax is the maximum iterationnumber, and Iter is the current iteration number.

1bx

2bx

3bx

4bx

Solution

∗ ∗ Best costWorst cost

Before

1bx

2bx

3bx

Solution

∗

2bx∗

After

Δ

Δ

Fig. 4. Replacing mechanism.

4.3. Multiple searches

The MTS algorithm uses the multiple searches mechanism toenhance its capacity. The multiple searches mechanism is executedby using only a personal microcomputer. Each step of searching forfinding the better solution is used in the procedure of TS algorithmwhich is given in Section 3.1.5. The sequence of execution startsfrom TS#1 to TS#m. Fig. 3 illustrates the sequence of executionfor finding better solutions.

4.4. Replacing

In order to improve the computational time, this mechanism isapplied when the search process satisfied the replacing criterion.The worst solution is replaced by the best solution. Here, thereplacing is set to apply to each individual TS every 30 iterations.As a result, a new solution is generated for TS which found theworst solution. Fig. 4 illustrates how the new solutions are createdby replacing mechanism.

4.5. Restarting

The restarting mechanism is applied when the search is strokeon the local solution for 20 iterations and the procedure of TS algo-rithm cannot escape from local solution. The procedure of restart-ing mechanism is almost the same as the initialization mechanism.However, the boundary for generating new initial values is reducedas

maxðPmini ; 0:9� Pk�1

i Þ 6 Pki 6 minðPmax

i ; 1:1� Pk�1i Þ ð8Þ

5. Numerical simulations and results

To assess the feasibility of the MTS, two examples of power gen-erations with 13 and 40 thermal units of ED problems are applied.All optimization methods (SA [23], GA [24], TS, PSO [25], and MTS)were implemented in MATLAB� package and the simulation casesdone on a Pentium 4, 2.6 GHz personal microcomputer with 2-GBRAM under Windows XP. Each studied systems was run 30 timeswith differential random initial solutions. In order to evaluate theperformance of each technique, the best, worst, average, and stan-dard deviation of the generation costs and the average of computa-tional time to get near optimum solution are used for evaluation.

Unfortunately, the number of initial solutions and the computa-tional time for an iteration of these techniques are different. There-fore, the execution of search was done by the same iterations. Thismethod is not suitable for comparison.

Consequently, this work uses the time limit as the stopping cri-terion. The search processes of all techniques are executed by thesame time period; therefore, the maximum iteration of each tech-nique depends on the time limit. When the search satisfies thestopping criterion, the best, worst, average, and standard deviationof the generation costs and the average of computational time toget near optimum solution are reported. According to experience

Table 1The best results of 13 units system using SA, GA, TS, PSO and MTS.

Unit poweroutput

Load demand = 2520 MW

SA GA TS PSO MTS

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of many experiments, the following parameters in the MTS, PSO,TS, GA and SA methods are used:

MTS

P1 (MW) 627.87 627.88 628.18 628.32 628.32P2 (MW) 299.56 299.26 298.92 299.20 299.06P3 (MW) 299.40 298.89 299.13 299.19 299.17P4 (MW) 159.65 159.34 159.24 159.50 159.73P5 (MW) 159.88 158.61 159.70 159.69 159.73

� Number of TS = 10� Maximum iteration = 200� Tabu list size = 2 � ns

� Restriction period = 10� Frequency limit = 7

P6 (MW) 159.44 159.72 159.21 159.71 159.73P7 (MW) 159.35 159.67 159.94 159.65 159.73

PSO

P8 (MW) 159.21 159.29 159.79 159.72 159.72P9 (MW) 158.69 159.86 159.59 159.73 159.72P10 (MW) 77.21 75.14 77.44 77.40 75.47P11 (MW) 76.44 77.60 76.29 75.78 77.33P12 (MW) 92.39 92.49 90.56 91.13 92.10P13 (MW) 91.24 92.38 92.27 90.98 90.59Total power

output (MW)2520.00 2520.13 2520.25 2520.00 2520.00

� Population size = 500.� Generations = 200.� Inertia weight factor x ¼ xmax � xmax�xmin

itermax� iter, where

xmax = 0.9 and xmin = 0.4.� The limit of change in velocity of each member in an indi-

vidual was as VmaxPd¼ 0:5Pmax

d , VminPd¼ �0:5Pmin

d .� Acceleration constant c1 = 1.4 and c2 = 1.4.

Total generation 24187.01 24183.31 24178.65 24171.52 24169.89

TS cost ($/h)

Table 2Comparison results of 13 unit system using SA, GA, TS, PSO and MTS.

� Maximum iteration = 2000.� Tabu list size = 2 � ns.� Restriction period = 10.� Frequency limit = 7.

GA

Method Load demand = 2520 MW

Worst cost($/h)

Average cost($/h)

Best cost($/h)

Std. Average CPUtime (s)

SA 24336.10 24248.06 24187.01 46.17 108.45GA 24293.37 24225.24 24183.31 40.10 102.36TS 24305.81 24201.46 24178.65 29.50 40.32PSO 24242.57 24194.01 24171.64 20.77 29.10

� Population size = 500.� Generations = 200.� Binary bits = 8.� Crossover rate = 0.8.� Mute rate = 0.01.� Crossover parameter = 0.5.

MTS 24196.74 24179.26 24169.63 7.59 18.26

SA � Maximum iteration = 2000.� Initial temperature = 1000.� Low temperature = 0.5� Control parameter = 0.9

Fig. 5. Distribution of generation costs for 13 unit system.

5.1. Case studies and simulation results

In this paper, the MTS is applied to solve the ED problem of sys-tems with 13 and 40 thermal generation units and 40 units [8]. Theload demands of 13 units and 40 units are 2520 MW and10,500 MW, respectively.

5.2. Comparison of five methods

5.2.1. Solution qualityFor a system with 13 units, after performed 30 trials; the best

solutions of MTS, PSO, TS, GA, and SA are given in Tables 1. In addi-tion, the results of the MTS are compared with SA, GA, TS, and PSOin terms of worst, average, best generation costs, the standarddeviation, and average computational time, as provided in Table2. Obviously, all methods have succeeded in finding the near opti-mum solution. Nevertheless, the standard deviation of the MTS islowest. This implies that the probability of obtaining the best solu-tion by MTS is highest. In order to demonstrate the efficiency of theMTS method, the distributed outlines of the best solution of eachtrial are graphically shown in Fig. 5 Almost all generation costs ob-tained by the MTS method are lower.

For a system with 40 units, after performed 30 trials, the bestsolutions of five methods are given in Table 3. The comparison re-sults of the MTS with other five methods are given in Table 4.Clearly, the optimized cost by MTS is minimum. The distributionoutlines of the best solution of each trial for each method are de-picted in Fig. 6. The MTS always gives better quality of solutionsthan other methods. These results confirm that the MTS alwaysprovide better high quality of solution than other methods whenthe scale of the generating system becomes larger. Moreover, per-

centage comparison of cost and computational time of these appli-cations are shown in Table 5.

Table 6 compares the results achieved by MTS with those ofother studies reported in the literatures. The generation costs min-imized by MTS are lowest among the results from other methodsespecially in the large scale system. These comparison resultsexplicitly confirm the significant effects of the MTS on the solutionquality.

5.2.2. Computation efficiencyThe convergent characteristics of the MTS compared with other

methods for two studied systems are shown in Figs. 7 and 8.

Table 3The best results of 40 units system using SA, GA, TS, PSO and MTS.

Unit power output Load demand = 10,500 MW

SA GA TSA PSO MTS

P1 (MW) 112.41 110.60 111.41 110.65 110.80P2 (MW) 110.73 111.12 111.62 109.97 111.80P3 (MW) 119.98 117.77 119.98 97.78 97.80P4 (MW) 144.62 181.02 176.79 178.15 179.84P5 (MW) 94.68 95.16 86.00 91.78 92.70P6 (MW) 68.81 139.74 139.50 139.47 139.96P7 (MW) 261.45 299.82 298.66 299.43 299.96P8 (MW) 285.58 288.35 287.87 298.15 299.88P9 (MW) 297.05 295.26 290.29 285.92 284.48P10 (MW) 130.21 130.00 130.27 131.24 131.80P11 (MW) 94.33 94.06 95.22 169.95 168.00P12 (MW) 95.57 95.89 95.00 99.12 94.28P13 (MW) 304.61 218.84 125.99 214.48 214.22P14 (MW) 485.58 394.49 392.60 394.70 393.44P15 (MW) 326.72 326.37 484.15 303.48 304.16P16 (MW) 303.42 394.93 282.66 305.06 304.76P17 (MW) 491.02 489.39 491.14 488.15 489.00P18 (MW) 489.12 489.02 490.05 488.86 490.60P19 (MW) 513.50 511.62 509.18 512.29 511.44P20 (MW) 508.88 513.33 526.08 512.51 511.20P21 (MW) 524.69 524.00 529.19 521.63 524.60P22 (MW) 529.88 524.29 527.22 523.48 523.70P23 (MW) 529.35 531.21 524.25 522.45 523.70P24 (MW) 524.39 522.31 524.19 523.37 524.20P25 (MW) 526.84 524.01 528.29 522.97 524.00P26 (MW) 517.79 533.92 520.16 523.39 524.00P27 (MW) 10.02 10.17 10.05 13.00 13.20P28 (MW) 10.05 10.16 10.08 11.07 10.00P29 (MW) 10.05 10.00 12.18 10.59 11.60P30 (MW) 96.06 92.27 95.20 90.67 90.80P31 (MW) 189.16 189.89 189.92 189.35 189.50P32 (MW) 189.41 189.69 188.04 189.18 189.50P33 (MW) 172.69 189.53 189.81 189.95 186.50P34 (MW) 199.20 199.59 168.62 199.63 199.00P35 (MW) 198.42 199.05 198.86 199.03 197.50P36 (MW) 199.55 198.77 199.06 199.79 198.30P37 (MW) 109.93 108.04 108.17 109.99 109.70P38 (MW) 90.48 25.49 109.10 109.49 109.90P39 (MW) 109.93 109.98 109.00 109.35 108.90P40 (MW) 524.30 511.08 514.92 510.49 511.44Total power output (MW) 10500.47 10500.27 10500.79 10500.00 10500.16Total generation cost ($/h) 122946.77 122508.88 122432.28 121830.68 121532.41

Table 4Comparison results of 40 unit system using SA, GA, TS, PSO and MTS.

Method Load demand = 10,500 MW

Worst cost($/h)

Average cost($/h)

Best cost($/h)

Std. Average CPUtime

SA 124183.72 123180.70 122946.77 611.90 278.21GA 123831.61 122687.71 122508.88 466.70 243.76TS 122602.20 122473.68 122432.28 287.21 224.36PSO 122083.17 121962.57 121830.68 159.28 87.37MTS 122022.15 121798.51 121532.10 145.55 60.24

852 W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854

Clearly, the MTS converges to the optimum solution much fasterthan other methods. Furthermore, the comparisons of computationtime of five methods for two studies system are already given inTables 2 and 4. The MTS takes shorter computation time than othermethods.

Fig. 6. Distribution of generation cost of 40 unit system.

5.2.3. Evaluation effects of additional mechanisms

In order to evaluate the effects of the additional mechanisms ofthe MTS, the results of 13 units system in case of 2520 MW loaddemand are demonstrated.

Next, the effects of several initial solutions and multiplesearches with adaptive searches are considered as depicted in

Fig. 9. The several initial solutions have the probability of reachingto the optimum solution higher than the single initial solution.Here, four initial solutions are used at the initial iteration (iteration

Table 5Percentage comparison results of generation cost and computational times.

Method 13 unit system 40 unit system

Average cost ($/h) Percentage (%) CPU time (s) Percentage (%) Average cost ($/h) Percentage (%) CPU time (s) Percentage (%)

SA 24248.06 0.00 108.45 0.00 123180.70 0.00 278.21 0.00GA 24225.24 0.09 102.36 5.62 122687.71 0.40 243.76 12.38TS 24201.46 0.19 40.32 62.82 122473.68 0.57 224.36 19.36PSO 24194.00 0.22 29.10 73.17 121962.57 0.99 87.37 68.60MTS 24179.26 0.28 18.26 83.16 121798.51 1.12 60.24 78.35

Table 6Comparison results of generation cost presented in the literatures.

Optimizationtechnique

Best cost for 13 unit ($/h)

Best cost for 40 unit ($/h)

PSO–SQP [16] 24261.05 –EP–SQP[16] 24266.44 –GA [16] 24398.23 –GA–SA [16] 24275.71 –EDSA [18] 24169.92 –IFEP [8] – 122624.35FEP [8] – 122679.71PSO–SQP [15] 122252.26DEC(2)–SQP(1) [19] – 121741.98TM [18] – 122477.78Multiple tabu search 24169.63 121532.10

Fig. 7. Convergence curve of 13 unit system.

Fig. 8. Convergence curve of 40 unit system.

4 initial solutions for each TS

Effect of multiple searches and adaptive searches

24290243042432224306

24449243762443224539

24733246942458324755

24816247022474024755

Iteration # 0 Iteration # 1 Iteration # 6 Iteration # 20

... ...

Fig. 9. Effects of several initial solution, multiple searches and adaptive step sizes.

W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854 853

#0). It can be observed in the block of iteration #0 that the TS #2has the lowest generation cost at 24,702 $/h. This initial value is se-lected to be the initial solution of MTS. On the other hand, if onlysingle initial solution is used, the generation cost at the initial iter-ation may be greater than several initial solutions case.

The effects of multiple searches and adaptive search are shownas example at 1st, 6th and 20th iterations. As shown in the block ofthe 1st iteration, the generation cost of TS#3 becomes lowest at24,583 $/h. Subsequently, each TS attempts to find the best solu-tion with the different step size in the adaptive search. As a result,the generation costs of TS#2 and TS#1 become minimal at 6th and20th iterations, respectively. These results signify that the multiplesearches and adaptive search significantly enhance the efficiencyof the MTS to reach the better solutions.

The effects of replacing solution and restarting process areshowed in Fig. 10. The replacing solution is a process of comparedsolutions and generation of the new initial solutions for MTS in the

next iteration. Here, the replacing process is set every 30 iterations.At 30th iteration, the replacing is applied to TS#3, which has themaximum generation cost at 24,309 $/h. The replacing takes actionand generates a new solution of TS#3 from the best solution ofTS#1 at 24,273 $/h.

The process keeps searching until any TS# struck on the localsolution and cannot escape from this solution for 20 iterations.At 43rd and 45th iterations, the restarting process is applied toTS#2 and TS#4, respectively. The better initial solutions are gener-ated for TS#2 and TS#4 so that both TS#2 and TS#4 can keepsearching new solutions.

Crossover

24273242912430924301

24273242912427324301

Before After

Restarting at TS #2 Restarting at TS #4

24263242912427124301

24263242872427124301

24263242872427124301

24263242872426424276

retfAretfAerofeB Before

Fig. 10. Effects of replacing and restarting process.

854 W. Sa-ngiamvibool et al. / Electrical Power and Energy Systems 33 (2011) 846–854

6. Conclusions

In this paper, the MTS algorithm has been proposed to solve theED problem by taking the valve-point loading effects into consider-ation. Many sophisticated techniques such as several initial solu-tions, multiple searches, adaptive step sizes, replacing andrestarting process have been added to the MTS, in order to enhancethe search potential. Study results in 13 and 40 generating unitsconfirm that the MTS is much superior to SA, GA, TS and PSO meth-ods in terms of high-quality solution, stable convergence charac-teristic, and good computation efficiency.

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