application of multiple tabu search algorithm to solve dynamic economic dispatch considering...

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Energy Conversion and Management 49 (2008) 506–516

Application of multiple tabu search algorithm to solvedynamic economic dispatch considering generator constraints

Saravuth Pothiya a,*, Issarachai Ngamroo b, Waree Kongprawechnon a

a School of Communication, Instrumentation and Control, Sirindhorn International Institute of Technology,

Thammasat University, P.O. Box 22, Pathumthani, Thailandb Electrical Engineering Department, Faculty of Engineering, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand

Received 16 January 2007; accepted 19 August 2007Available online 1 November 2007

Abstract

This paper presents a new optimization technique based on a multiple tabu search algorithm (MTS) to solve the dynamic economicdispatch (ED) problem with generator constraints. In the constrained dynamic ED problem, the load demand and spinning reservecapacity as well as some practical operation constraints of generators, such as ramp rate limits and prohibited operating zone are takeninto consideration. The MTS algorithm introduces additional mechanisms such as initialization, adaptive searches, multiple searches,crossover and restarting process. To show its efficiency, the MTS algorithm is applied to solve constrained dynamic ED problems ofpower systems with 6 and 15 units. The results obtained from the MTS algorithm are compared to those achieved from the conventionalapproaches, such as simulated annealing (SA), genetic algorithm (GA), tabu search (TS) algorithm and particle swarm optimization(PSO). The experimental results show that the proposed MTS algorithm approaches is able to obtain higher quality solutions efficientlyand with less computational time than the conventional approaches.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Dynamic economic dispatch; Genetic algorithm; Particle swarm optimization; Power system operation; Tabu search algorithm

1. Introduction

In power system operation, the economic dispatch (ED)problem is an important optimization problem. Moreover,it has complex and nonlinear characteristics with heavyequality and inequality constraints. Generally, there aretwo types of ED problem, i.e. static and dynamic. Solvingthe static ED problem is subject to the power balance con-straints and generator operating limits. For the dynamicED, it is an extension of the static ED problem. Thedynamic ED takes the ramp rate limits and prohibitedoperating zone of the generating units into consideration.

To solve the static ED problem, various conventionalmethods such as the lambda iteration methods, the gradi-ent method, dynamic programming (DP), etc. have been

0196-8904/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2007.08.012

* Corresponding author. Tel.: +66 2 222959; fax: +66 2 226826.E-mail address: [email protected] (S. Pothiya).

employed [1,2]. Unfortunately, for generating units withnonlinear characteristics, such as ramp rate limits, prohib-ited operating zones and non-convex cost functions, theconventional methods can hardly achieve the optimal ornear optimal solution. Furthermore, for a large-scale sys-tem, the conventional methods have oscillatory problems,resulting in a local minimum solution or a longer computa-tional time.

In the past decade, random search optimization meth-ods, such as simulated annealing (SA) [3], evolutionaryprogramming (EP) [4], genetic algorithms (GA) [5–8], tabusearch (TS) algorithm [9,10] and particle swarm optimiza-tion (PSO) [11], which are probabilistic heuristic algo-rithms, have been successfully used to solve the dynamicED problem.

A conventional TS algorithm, an iterative search algo-rithm, has been developed by Glover [12,13]. It has beenapplied to solve combinatorial optimization problems

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S. Pothiya et al. / Energy Conversion and Management 49 (2008) 506–516 507

[14–17]. The main advantages of the TS algorithm are itsability to escape from local optima and fast convergenceto the global optimum. However, a conventional TS algo-rithm might have problems with reaching the global opti-mum solution in a reasonable computational time whenthe initial solution is far away from the region where theoptimum solution exists.

In order to improve the performance of a conventionalTS algorithm, this paper proposes the multiple tabu search(MTS) algorithm. The MTS algorithm introduces addi-tional salient mechanisms for improvement of the searchprocess, i.e. initialization, adaptive searches, multiplesearches, crossover and restarting process.

The feasibility study of the MTS algorithm is demon-strated for solving the dynamic ED problem. The resultsoptimized by the MTS algorithm are compared to thoseobtained by conventional approaches such as SA, GA,TS and PSO in terms of solution quality and computa-tional efficiency.

The paper is organized as follows. Section 2 gives themathematical model of the constrained dynamic ED prob-lem. Section 3 mentions the MTS algorithm. Section 4 pre-sents the detailed procedures of using the MTS algorithmto solve the ED problem. Section 5 shows two applicationcases and gives the corresponding comparison results withthe traditional methods (SA, GA, TS and PSO). Conclu-sions are finally given in Section 6.

2. Mathematical model of the ED problem

The ED problem is a nonlinear optimization problem,which is a sub-problem of the unit commitment (UC) prob-lem. The objective of the ED problem is to find the optimalcombination of power generation that minimizes the totalgeneration cost while satisfying the system load demand,spinning reserve capacity and practical operation con-straints of generators that include ramp rate limits and pro-hibited operating zones [11].

2.1. Objective function

The main objective of the ED problem is simultaneouslyto minimize the generation cost rate and meet the load

t1−t

)1( −tPi

)(tPi

Power output (MW)

Time period (Hour)

Fig. 1. Two possible situations of

demand of a power system over some appropriate periodwhile satisfying various equality and inequality constraints.The objective function of the ED problem can be expressedas

min F t ¼Xn

i¼1

F iðP iÞ ¼Xn

i¼1

ðai þ biP i þ ciP 2i Þ ð1Þ

where Ft is the total generation cost; Fi is the generationcost function of the ith generator, which is usually ex-pressed as a quadratic polynomial; ai, bi and ci are the costcoefficients of the ith generator; Pi is the power output ofthe ith generator and n is the number of generators com-mitted to the operating system.

2.2. Constraints

2.2.1. Power balance constraints

Xn

i¼1

P i ¼ P D þ P L ð2Þ

where PD is the load demand and PL is the total transmis-sion network losses, which is a function of the unit poweroutputs that can be represented using B coefficients:

P L ¼Xn

i¼1

Xn

j¼1

P iBijP j þXn

i¼1

B0iP i þ B00 ð3Þ

2.2.2. Practical operation constraints of generators

To achieve the actual economic operation, two con-straints of generator operation are taken into account.

(1) Ramp rate limits: The ramp rate limits of generatingunits are caused by the fact that the thermal generatingoutputs cannot be adjusted instantaneously. To reflectthe actual operating process, the ramp rate limits areincluded in the ED problem to ensure the feasibility ofthe solutions. The operating range of all on line units isrestricted by their ramp rate limits. Fig. 1 shows two possi-ble situations when a unit is on line from hour t � 1 to hourt. Fig. 1a shows that the unit is in an increasing power gen-eration status. Fig. 1b shows that the unit is in a decreasingpower generation status.

t1−t

)1( −tPi

)(tPi

Power output(MW)

Time period (Hour)

the on line ith generating unit.

508 S. Pothiya et al. / Energy Conversion and Management 49 (2008) 506–516

According to Ref. [11], the inequality constraints due toramp rate limits for unit generation changes are given as

maxðP mini ; P 0

i �DRiÞ 6 P i 6 minðP maxi ; P 0

i þURiÞ ð4Þwhere P 0

i is the previous output power. P mini and P max

i arethe minimum and maximum outputs of the ith generator,respectively. URi, DRi are the up ramp and down ramplimits of the ith generator (MW/time period), respectively.

(2) Prohibited operating zone: Fig. 2 shows the input–output performance curve for a typical thermal unit. Theprohibited operating zones in the curve are due to steamvalve operation or vibration in a shaft bearing [8]. In prac-tice, the shape of the input–output curve in the neighbor-hood of the prohibited zone is difficult to determine byactual performance testing or operating records. In theactual operation, adjusting the generation output Pi of aunit must avoid the unit operating in the prohibited zones.The feasible operating zones of unit i can be described asfollows:

P i 2P min

i 6 P i 6 P li;1

P ui;j�1 6 P i 6 P l

i;j

P ui;ni6 P i 6 P max

i

j ¼ 2; 3; . . . ; ni

8><>:

ð5Þ

where ni is the number of prohibited zones of the ith gener-ator. P l

i;j; Pui;j are the lower and upper power output of the

prohibited zones j of the ith generator, respectively.

3. Multiple tabu search algorithm

3.1. Tabu search algorithm

3.1.1. Overview

In general terms, a conventional TS algorithm is an iter-ative search that starts from an initial feasible solution andattempts to determine a better solution in the manner of ahill climbing algorithm. The TS algorithm has a flexiblememory to retain the information about the past steps ofthe search. The TS algorithm uses the past search to createand exploit better solutions [14].

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

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

solution, a TL is introduced. The TL stores a set of the

miniP max

iPliP 1,

liP 2,

uiP 1,

uiP 2,

Input ($/h)

Output (MW)

Fig. 2. Typical input–output curve of a thermal unit.

tabu (prohibition) moves that cannot be applied to the cur-rent solution. The moves stored in the TL are called tabu

restrictions and are used for decreasing the possibility ofcycling because it prevents returning, in a certain numberof iterations, to a solution visited recently.

In this paper, the size of the TL is ns · 3 (row · column).ns is a number of neighborhoods around the current solu-tion. In the TL, the first column is used for storing themoves, the second column is the frequency of a move direc-tion and the last column is the recency (time to keep solu-tions) of a move [14].

3.1.3. Aspiration criterion

Another key issue of the TS algorithm arises when allmoves under consideration have been found to betabued. The tabu status of a move is not absolute, butit can be over-ruled if certain conditions are met andexpressed in the form of an AC. If an appropriate aspi-ration criterion is satisfied, the move will be accepted inspite of the tabu classification. Roughly speaking, theAC is designed to override the tabu status if a move is‘good enough’ [15].

3.1.4. Stopping criterion

There are several possible conditions for stop searching.Here, the stopping search is used if any of the followingtwo conditions are satisfied: first, the accuracy of the bestsolution is lower than the expected value, and second, themaximum allowable number of iterations is reached.

3.1.5. Procedure of tabu search algorithm

To solve a combinatorial optimization problem bytabu search, the basic idea is to choose randomly a fea-sible solution and attempt to find a best neighbor to thecurrent solution. A move to this neighbor is performed ifeither it does not belong to the TL or it passes the ACtest. During these procedures, the best solution is alwaysupdated and stored aside until the stopping criterion issatisfied.

The following notations are used through the descrip-tion of the TS algorithm for a general combinatorial opti-mization problem:

X the set of feasible solutionsx the current solution, x 2 X

xb the best solution reachedxnb the best solution among the trial solutionsE(x) the objective function of solution x

N(x) the set of neighborhood of x 2 X

TL tabu listAC aspiration criterion

The procedures of the TS algorithm are as follows:

Step 0: Set TL as empty and AC to be zero.Step 1: Set iteration counter k = 0. Select an initial solu-

tion x 2 X, and set xb = x.

Start

Generate initial solutions

Iteration = 0

Adaptive search mechanisms

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

01s 02s 10 −ms ms0

01sΔ 02sΔ 10 −Δ ms ms0Δ

Update best current solution,

1bs 2bs 1−bms bms

bests

m

Stopping criterion satisfied?

Update best final solution

Stop

Crossover criterion satisfied?

Yes

No

Crossover mechanism

Restarting criterion satisfied?

Yes

No

Restarting mechanism

Yes

No

Update best current solution, bests

Iteration = Iteration +1

Fig. 3. Procedure of multiple tabu search algorithm.


Step 2: Generate a set of trial solutions neighborhood ofx. Let xnb be the best trial solution.

Step 3: If E(xnb) > E(xb), go to Step 4, else set the bestsolution xb = xnb and go to Step 4.

Step 4: Perform the tabu test. If xnb is not in the TL, thenaccept it as a current solution, set x = xnb, andupdate the TL and AC and go to Step 6, else goto Step 5.

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

Step 6: Perform the termination test. If the stopping cri-terion is satisfied then stop, else set k = k + 1and go to Step 2.

3.2. Description of multiple tabu search algorithm

Even though the TS algorithm has many advantages(able to escape from local optima and fast convergence toglobal optimum), it might have a problem with reachingthe global optimum solution in a reasonable computationtime when an initial solution is far away from the regionwhere the optimum solution exists [17]. The convergencespeed of the TS algorithm depends on the initial solution.The convergence speed can be improved by introducing amultiple structure into the algorithm.

The MTS algorithm uses several initial solutions toincrease the probability of reaching the region where theoptimum solution exists. The procedure of the MTS algo-rithm is depicted in Fig. 3, which has several independentconventional TS algorithms. Furthermore, the additionalmechanisms, namely, initialization, adaptive searches, mul-

tiple searches, crossover and restarting process help toimprove the search process in terms of both solution qual-ity and computational time. The additional mechanismsare explained as follows:

3.2.1. InitializationTo improve searching, the MTS algorithm starts to

search from several initial solutions, which is different fromthe TS algorithm. In fact, starting with several initial solu-tions has a higher probability of reaching the optimumsolution than starting with a single initial solution. Thismechanism helps the MTS algorithm converge quickly tothe global optimum solution.

3.2.2. Adaptive searches

The step size is the range of variance at the current solu-tion, which is the important factor for the search process.Accordingly, the step size should be chosen appropriately.In general, this value is fixed. A low value of the step sizecan increase the accuracy of the solution, but it takes a longcomputational time. On the other hand, a high value of thestep size is used for decreasing the computational time, butthe search result may not be the global optimum.

Consequently, an adaptive search mechanism has beendeveloped to adjust suitably the step size during the search

process. This mechanism helps to increase the computationspeed and the accuracy of the solution.

3.2.3. Multiple searches

Nowadays, the personal microcomputer makes highspeed computations. To solve the large scale problem, sev-eral computers may be used at the same time. This methodis called parallel searches. For multiple searches, it is exe-cuted by using only one personal microcomputer. Multiplesearches help to find the promising region where the globaloptimum solution exists.

3.2.4. CrossoverAfter the search process satisfies the condition for cross-

over, all the independent TS algorithms are stopped. Thecrossover mechanism is used for comparison and exchang-ing the solutions that are found by these TS algorithms.Then, the crossover mechanism generates the best initialsolutions for the next search.

3.2.5. Restarting processWhen the search is stalled on the local solution for a

long time and the procedure of TS algorithm cannot escape


from the local solution, the restarting process is applied tokeep searching and find a new solution.

4. MTS algorithm for ED problem

In this section, the proposed five mechanisms integratedinto the MTS are described for solving dynamic ED prob-lems. Especially, a suggestion about how to deal with theequality and inequality constraints of ED problems wheneach search point is modified in the MTS algorithm isgiven. The process of the MTS algorithm can be summa-rized as follows:

In the subsequent sections, the detailed implementationstrategies of the MTS algorithm are described.

(1) Initialization and structure of solution: In the initiali-zation process, several initial solutions are created ran-domly for the individual TS algorithms. In this paper, thestructure of a solution for an ED problem is composedof a set of generation outputs. Therefore, the initial solu-tions of the individual TS algorithms at iteration 0 canbe represented as the vector of X 0

i ¼ ðP 0i1; . . . ; P 0

inÞ,i = 1, . . .,m where m is the number of multiple TS algo-rithms and n is the number of generators. Note that it isvery important to create a set of solutions that satisfy theequality and inequality constraints.

In this paper, the procedure is applied for the MTS algo-rithm to generate initial solutions as follows:

Step (1) Read system data.Step (2) The boundary of feasible solution is defined as

follows:

maxðP mini ; P 0

i �DRiÞ 6 P i 6 minðP maxi ; P 0

i þURiÞ
Step (3) Set j = 1.Step (4) Create the value of the generation output at ran-
dom satisfying its equality and inequality con-straints by

P i ¼ P mini þ randðÞ � ðP max

i � P mini Þ

where rand() is a uniform random value in therange [0, 1].

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

Fig. 4. Sequence of execution.

[1 =bx ]

[2 =bx ]

[1 =bx ]

[2 =bx ]

11P 12P 13P 14P 15P 16P

21P 22P 23P 24P 25P 26P

11P 12P 13P 14P 25P 26P

15P 16P24P23P22P21P

Before

After

Position crossover

Position crossover

Fig. 5. Crossover process.

Step (5) If j = n then go to Step 6; otherwise go to Step 2.Step (6) Stop the initialization process.

(2) Adaptive step sizes: The adaptive searches mechanismis used to generate step sizes for finding the neighborhoodof the current solution. The step size of individual TS algo-rithm at iteration 0 can be represented as the vectorD0

i ¼ ðD0i1; . . . ;D0

inÞ, i = 1, . . .,m.In this paper, the procedure is applied for the MTS algo-

rithm to generate step size as follows:

Step (1) Read system data.Step (2) The boundary of feasible solutions is defined as

follows:

P i lower ¼ maxðP mini ; P 0

i �DRiÞP i upper ¼ minðP max

i ; P 0i þURiÞ

Step (3) Set j = 1.Step (4) Create the value of the step size at random by

D0i ¼ K � randðÞ � ðP i upper � P i lowerÞ

The weight factor K is set according to the fol-lowing equation:

K ¼ kmax �kmax � kmin

itermax

� iter

where, kmax, kmin are the maximum and minimumweights. In this work, set kmax = 0.9 and kmin =0.1. Itermax is maximum iteration number, anditer is current iteration number.

Step (5) If j = n then go to Step 6; otherwise go to Step 2.Step (6) Stop the creation step size process.

(3) Multiple searches: The MTS algorithm uses themultiple searches mechanism to enhance its capacity.The multiple searches mechanism is executed by usingonly a personal microcomputer. Each step of searchingfor finding the better solution is used in the procedureof the TS algorithm, which is given in Section 3.1.5.The sequence of execution starts from TS#1 to TS#m.Fig. 4 illustrates the sequence of execution for finding bet-ter solutions.

(4) Crossover: Like the genetic algorithm (GA), the MTSalgorithm performs a crossover mechanism for improvingthe solution. After the search satisfies the condition forcrossover, it is applied to generate new solutions for thenext iteration. Fig. 5 illustrates how the new solutions arecreated by the crossover mechanism.

(5) Restarting: The restarting mechanism is used toincrease the possibility of occurrence of escaping fromthe local optimal solution. This mechanism is applied whenthe search is stalled on the local solution for a long timeand the procedure of the TS algorithm cannot escape fromthe local solution. The procedure of the restarting mecha-nism is the same as the initialization mechanism. In con-


trast, the boundary for generating new initial values isreduced as follows:

maxðP mini ; 0:9� P k�1

i Þ 6 P ki 6 minðP max

i ; 1:1� P k�1i Þ:

5. Numerical simulations and results

To assess the feasibility of the MTS algorithm method,it has been applied to solve the dynamic ED problem ontwo different power systems (6 and 15 units), and a com-parison with SA, GA, TS and PSO algorithms is made.All methods are performed with 30 trials under the sameevaluation function and individual definition in order tocompare their solution quality, convergence characteristicand computation efficiency. In these examples, the ramprate limits and prohibited zones of the units are taken intoaccount in the practical application. The software wasimplemented by the MATLAB� language, which includesthe Genetic Toolbox, on a Pentium 4, 2.6 GHz personalmicrocomputer with 256 MB RAM under Windows XP.

According to the experience of many experiments, thefollowing parameters in the MTS, PSO, TS, GA and SAalgorithms methods are used:

MTS• Number of TS = 10.• Maximum iteration = 200.• Tabu list size = 2*ns.• Restriction period = 10.• Frequency limit = 7.

PSO• 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

individual was as V maxP d¼ 0:5P max

d , V minP d¼ �0:5P min

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

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

Table 1Generating unit data of six units system

Unit P 0i ðMWÞ P min

i ðMWÞ P maxi ðMWÞ ai ($) bi ($/MW)

1 440 100 500 240 7.02 170 50 200 200 10.03 200 80 300 220 8.54 150 50 150 200 11.05 190 50 200 220 10.56 110 50 120 190 12.0

GA• Population size = 500.• Generations = 200.• Binary bits = 12.• Crossover rate = 0.8.• Mute rate = 0.01.• Crossover parameter = 0.5.

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

5.1. Numerical examples

Example 1. The system contains six thermal units, 26buses, and 46 transmission lines [11]. The characteristics ofthe six thermal units are given in Table 1. The load demandis 1263 MW. In normal operation of the system, the losscoefficients B matrices with the 100 MVA base capacity aregiven in Appendix A.

Example 2. The system contains 15 thermal units [11]whose characteristics are given in Table 2. The loaddemand is 2630 MW. The loss coefficients B matrices areshown in Appendix A.

5.2. Simulation results

Five methods (MTS, PSO, TS, GA, and SA algorithms)were employed to test the two studied systems. All methodsperformed 30 trials under the same evaluation function andindividual definition.

Example 1. In this case, each individual Pg contains sixgenerator power outputs: P1, P2, P3, P4, P5 and P6,which are generated randomly. For 1263 MW loaddemand, the best solutions, which are shown in Table3, satisfy the system constraints, such as the ramp ratelimits and prohibited zones of the units. The statisticalresults obtained with 30 trials, such as the generationcost, standard deviation, computational time and per-centage of approaching near optimal solution, are shownin Table 4.

ci ($/MW2) URi (MW/h) DRi (MW/h) Prohibited zones (MW)

0.0070 80 120 [210,240] [350,380]0.0095 50 90 [90,110] [140,160]0.0090 65 100 [150,170] [210,240]0.0090 50 90 [80,90] [110,120]0.0080 50 90 [90,110] [140,150]0.0075 50 90 [75,85] [100,105]

Table 2Generating unit data of 15 units system

Unit P 0i ðMWÞ P min

i ðMWÞ P maxi ðMWÞ ai ($) bi ($/MW) ci ($/MW2) URi (MW/h) DRi (MW/h) Prohibited zones (MW)

1 400 150 455 671 10.1 0.000299 80 120 –2 300 150 455 574 10.2 0.000183 80 120 [180,225] [305,335] [420,450]3 105 20 130 374 8.8 0.001126 130 130 –4 100 20 130 374 8.8 0.001126 130 130 –5 90 150 470 461 10.4 0.000205 80 120 [180,200] [305,335] [390,420]6 400 135 460 630 10.1 0.000301 80 120 [230,255] [365,395] [430,455]7 350 135 465 548 9.8 0.000364 80 120 –8 95 60 300 227 11.2 0.000338 65 100 –9 105 25 162 173 11.2 0.000807 60 100 –

10 110 25 160 175 10.7 0.001203 60 100 –11 60 20 80 186 10.2 0.003586 80 80 –12 40 20 80 230 9.9 0.005513 80 80 [30,40] [55,65]13 30 25 85 225 13.1 0.000371 80 80 –14 20 15 55 309 12.1 0.001929 55 55 –15 20 15 55 323 12.4 0.004447 55 55 –

Table 3Best solution of six units system

Unit power output Methods

SA GA TS PSO MTS

P1 (MW) 478.1258 462.0444 459.0753 447.5823 448.1277P2 (MW) 163.0249 189.4456 185.0675 172.8387 172.8082P3 (MW) 261.7143 254.8535 264.2094 261.3300 262.5932P4 (MW) 125.7665 127.4296 138.1222 138.6812 136.9605P5 (MW) 153.7056 151.5388 154.4716 169.6781 168.2031P6 (MW) 93.7965 90.7150 74.9900 85.8963 87.3304Total output (MW) 1276.1339 1276.0270 1275.94 1276.0066 1276.0232Ploss (MW) 13.1317 113.0268 12.9422 13.0066 13.0205Total cost ($/h) 15461.10 15457.96 15454.89 15450.14 15450.06

Table 4Comparison of MTS performance with other methods

Methods Maximum cost ($/h) Average cost ($/h) Minimum cost ($/h) Standard deviation Average CPU time (s)

SA 15545.50 15488.98 15461.10 28.3678 50.36GA 15524.69 15477.71 15457.96 17.4072 46.60TS 15498.05 15472.56 15454.89 13.7195 20.55PSO 15491.71 15465.83 15450.14 10.1502 6.82MTS 15453.64 15451.17 15450.06 0.9287 1.29


Example 2. To simulation this case, each individual Pg

contains 15 generator power outputs, which are generatedrandomly. For 2630 MW load demand, the best solutionsobtained from all methods that satisfy the system con-straints, such as the ramp rate limits and prohibited zonesof units, are shown in Table 5. Table 6 provides the statis-tical results obtained with 30 trials, such as the generationcost, standard deviation, computational time and percent-age of approaching near optimal solution.

5.3. Comparison of five methods

5.3.1. Solution quality

For the six units system, the best solutions of the fivemethods are given in Table 3 after performing 30 trials.

The results of the MTS algorithm method are comparedwith those obtained by the SA, GA, TS, and PSOalgorithms in terms of maximum, average, minimum gener-ation cost, the standard deviation and average computa-tional time as shown in Table 4. Obviously, all methodshave succeeded in finding the near optimum solution pre-sented in Ref. [11] with a high probability of satisfyingthe equality and inequality constraints.

In order to demonstrate the efficiency of the MTS algo-rithm method, the distribution outlines of the best solutionof each trial are considered. Fig. 6 shows the distributionoutlines of the best solution of each trial in the case of1263 MW load demand. Almost all generation costsobtained by the MTS algorithm method are lower. Thisverifies that the MTS algorithm method has the betterquality of solution.

Table 5Best solution of 15 units system

Unit power output Methods

SA GA TS PSO MTS

P1 (MW) 453.6646 445.5619 453.5374 454.7167 453.9922P2 (MW) 377.6091 380.0000 371.9761 376.2002 379.7434P3 (MW) 120.3744 129.0605 129.7823 129.5547 130.0000P4 (MW) 126.2668 129.5250 129.3411 129.7083 129.9232P5 (MW) 165.3048 169.9659 169.5950 169.4407 168.0877P6 (MW) 459.2455 458.7544 457.9928 458.8153 460.0000P7 (MW) 422.8619 417.9041 426.8879 427.5733 429.2253P8 (MW) 126.4025 97.8230 95.1680 67.2834 104.3097P9 (MW) 54.4742 54.2933 76.8439 75.2673 35.0358P10 (MW) 149.0879 144.2214 133.5044 155.5899 155.8829P11 (MW) 77.9594 77.3002 68.3087 79.9522 79.8994P12 (MW) 73.9489 77.0371 79.6815 79.8947 79.9037P13 (MW) 25.0022 31.1537 28.3082 25.2744 25.0220P14 (MW) 16.0636 15.0233 17.7661 16.7318 15.2586P15 (MW) 15.0196 33.6125 22.8446 15.1967 15.0796Total output (MW) 2663.29 2661.23 2661.53 2661.19 2661.36Ploss (MW) 33.2737 31.2363 31.4100 31.1697 31.3523Total cost ($/h) 32786.40 32779.81 32762.12 32724.17 32716.87

Table 6Comparison of MTS performance with other methods

Methods Maximum cost ($/h) Average cost ($/h) Minimum cost ($/h) Standard deviation Average CPU time (s)

SA 33028.95 32869.51 32786.40 112.32 71.25GA 33041.64 32841.21 32779.81 81.22 48.17TSA 32942.71 32822.84 32762.12 60.59 26.41PSO 32841.38 32807.45 32724.17 21.24 13.25MTS 32796.15 32767.21 32716.87 17.51 3.65

Fig. 6. Distribution of generation cost of five methods (6-unit system). Fig. 7. Distribution of generation cost of five methods (15-unit system).


For the 15 units system in the case of 2630 MW loaddemand, after performing 30 trials, the best solutions ofthe five methods are given in Table 5. The results of theMTS algorithm method in comparison with those of theSA, GA, TS, and PSO algorithms in terms of maximum,average, minimum generation cost, standard deviationand average computational time are provided in Table 6.

Clearly, the MTS algorithm method has always better solu-tions than those of the other methods. This signifies thehigher quality solution obtained by the MTS algorithm.

Fig. 7 shows the distribution outlines of the best solu-tion of each trial in the case of 2630 MW load demand.Almost all of the generation costs optimized by the MTS


algorithm method are lower than those obtained by theother methods.

5.3.2. Computation efficiency

The comparisons of computational time of the fivemethods in the case of an hour load demand in both stud-ied system are shown in Fig. 8. Clearly, the computationaltime of the MTS algorithm method is lowest in comparisonto those of the other methods.

Moreover, the convergence characteristics of the MTSmethod compared with the other methods are shown inFig. 9. The convergence of the MTS algorithm to theoptimum solution is faster than that of the othermethods.

5.3.3. The effects of addition mechanisms

In order to illustrate the effects of the additional mech-anisms of the MTS algorithm, the results of the 15 unitssystem in the case of 2630 load demand are demonstrated.

Fig. 8. Comparison of computation performance.

Fig. 9. Convergence characteristic of GA, TS and MTS for Example 2.

As shown in Fig. 10, the effects of several initial solu-tions and multiple searches with adaptive searches arepresented. The several initial solutions have a higherprobability of reaching the optimum solution than thesingle initial solution. For example, using four initialsolutions at the initial iteration, the MTS#1 has the low-est generation cost. On the other hand, if only the singleinitial solution is used, the generation cost at the initialiteration may be greater than that of the several initialsolutions case. For the effect of multiple searches andadaptive searches, as seen at the first and second itera-tion, the generation costs of MTS#4 are lowest, but afterthe third iteration, the generation costs of MTS#1 are

Fig. 11. Effect of crossover solution and restarting process.

Fig. 10. Effect of several initial solution, multiple searches and adaptivestep sizes.


lowest. These events are the effects of multiple searchesand adaptive searches.

The effects of crossover solutions and the restarting pro-cess are shown in Fig. 11. The crossover solution is a pro-cess of compared solutions and generation of new initialsolutions for the MTS algorithm in the next iteration. Atthe 10th iteration, the crossover process is applied toMTS#3, which has the maximum generation cost. There-fore, the crossover occurs and generates a new better solu-tion for MTS#3.

The process keeps searching until the MTS algorithmgets stuck on the local solution and cannot escape from thissolution for a long time. At the 14th iteration, the restart-ing process is applied to MTS#4. It finds a better solutionand keeps searching.

Bij¼

0:0014 0:0012 0:0007 �0:0001 �0:0003 �0:0001 �0:0001 �0:0001 �0:0003 0:0005 �0:0003 �0:0002 0:0004 0:0003 �0:0001

0:0012 0:0015 0:0013 0:0000 �0:0005 �0:0002 0:0000 0:0001 �0:0002 �0:0004 �0:0004 �0:0000 0:0004 0:0010 �0:0002

0:0007 0:0013 0:0076 �0:0001 �0:0013 �0:0009 �0:0001 0:0000 �0:0008 �0:0012 �0:0017 �0:0000 �0:0026 0:0111 �0:0028

�0:0001 0:0000 �0:0001 0:0034 �0:0007 �0:0004 0:0011 0:0050 0:0029 0:0032 �0:0011 �0:0000 0:0001 0:0001 �0:0026

�0:0003 �0:0005 �0:0013 �0:0007 0:0090 0:0014 �0:0003 �0:0012 �0:0010 �0:0013 0:0007 �0:0002 �0:0002 �0:0024 �0:0003

�0:0001 �0:0002 �0:0009 �0:0004 0:0014 0:0016 �0:0000 �0:0006 �0:0005 �0:0008 0:0011 �0:0001 �0:0002 �0:0017 0:0003

�0:0001 0:0000 �0:0001 0:0011 �0:0003 �0:0000 0:0015 0:0017 0:0015 0:0009 �0:0005 0:0007 �0:0000 �0:0002 �0:0008

�0:0001 0:0001 0:0000 0:0050 �0:0012 �0:0006 0:0017 0:0168 0:0082 0:0079 �0:0023 �0:0036 0:0001 0:0005 �0:0078

�0:0003 �0:0002 �0:0008 0:0029 �0:0010 �0:0005 0:0015 0:0082 0:0129 0:0116 �0:0021 �0:0025 0:0007 �0:0012 �0:0072

�0:0005 �0:0004 �0:0012 0:0032 �0:0013 �0:0008 0:0009 0:0079 0:0116 0:0200 �0:0027 �0:0034 0:0009 �0:0011 �0:0088

�0:0003 �0:0004 �0:0017 �0:0011 0:0007 0:0011 �0:0005 �0:0023 �0:0021 �0:0027 0:0140 0:0001 0:0004 �0:0038 0:0168

�0:0002 �0:0000 �0:0000 �0:0000 �0:0002 �0:0001 0:0007 �0:0036 �0:0025 �0:0034 0:0001 0:0054 �0:0001 �0:0004 0:0028

0:0004 0:0004 �0:0026 0:0001 �0:0002 �0:0002 �0:0000 0:0001 0:0007 0:0009 0:0004 �0:0001 0:0103 �0:0101 0:0028

0:0003 0:0010 0:0111 0:0001 �0:0024 �0:0017 �0:0002 0:0005 �0:0012 �0:0011 �0:0038 �0:0004 �0:0101 0:0578 �0:0094

�0:0001 �0:0002 �0:0028 �0:0026 �0:0003 0:0003 �0:0008 �0:0078 �0:0072 �0:0088 0:0168 0:0028 0:0028 �0:0094 0:1283

266666666666666666666666666666664

377777777777777777777777777777775

Boi¼ �0:0001 �0:0002 0:0028 �0:0001 0:0001 �0:0003 �0:0002 �0:0002 0:0006 0:0039 �0:0017 �0:0000 �0:0032 0:0067 �0:0064½ �

6. Discussion and conclusions

In this paper, the MTS algorithm method has beenapplied to solve the dynamic ED problem taking variousgenerator constraints such as ramp rate limits and pro-hibited operating zones into consideration. The MTSalgorithm shows superior features such as high qualitysolution, stable convergence characteristic and good com-putation efficiency. The studied results confirm that theproposed MTS algorithm is indeed capable of obtaininga higher quality solution efficiently and superior conver-gence characteristic and computation efficiency in com-parison with the SA, GA, TS and PSO algorithmsmethods.

Acknowledgement

This work was supported by the Thailand ResearchFund under the basic research grant.

Appendix A

B matrices of six generating units

Bij¼

0:0017 0:0012 0:0007 �0:0001 �0:0005 �0:0002

0:0012 0:0014 0:0009 0:0001 �0:0006 �0:0001

0:0007 0:0009 0:0031 0:0000 �0:0010 �0:0006

�0:0001 0:0001 0:0000 0:0024 �0:0006 �0:0008

�0:0005 �0:0006 �0:0010 �0:0006 0:0129 �0:0002

�0:0002 �0:0001 �0:0006 �0:0008 �0:0002 0:0150

2666666664

3777777775

Boi¼1:0�10�3 �0:3908 �0:1297 0:7047 0:0591 0:2161 �0:6635½ �

Boo ¼ 0:056

B matrices of 15 generating units

B00 ¼ 0:0055:

References

[1] Bakirtzis A, Petridis V, Kazarlis S. Genetic algorithm solution to theeconomic dispatch problem. Proc Inst Elect Eng-Gener TransmDistrib 1994;141(4):377–82.

[2] Lee FN, Breipohl AM. Reserve constrained economic dispatch withprohibited operating zones. IEEE Trans Power Syst 1993;8:246–54.

[3] Wang KP, Fung CC. Simulate annealing base economic dispatchalgorithm. IEE Proc C 1993;140(6):507–13.

[4] Yang HT, Yang PC, Huang CL. Evolution programming basedeconomic dispatch for units with non-smooth fuel cost functions.IEEE Trans Power Syst 1996;11(1):112–8.

[5] Walters DC, Sheble GB. Genetic algorithm solution of economicdispatch with valve point loading. IEEE Trans Power Syst1993;8:1325–32.

[6] Wong KP, Wong YW. Genetic and genetic/simulated-annealingapproaches to economic dispatch. Proc Inst Elect Eng C1994;141(5):507–13.

[7] Sheble GB, Brittig K. Refined genetic algorithm-economic dispatchexample. IEEE Trans Power Syst 1995;10:117–24.


[8] Chen P-H, Chang H-C. Large-scale economic dispatch by geneticalgorithm. IEEE Trans Power Syst 1995;10:1919–26.

[9] Ongsakul W, Dechanupapritttha S, Ngamroo I. Tabu search algo-rithm for constrained economic dispatch. In: Proceedings of theinternational conference on power systems, Wuhan, China, Septem-ber 2001. p. 428–33.

[10] Ongsakul W, Dechanupapritttha S, Ngamroo I. Parallel tabu searchalgorithm for constrained economic dispatch. IEE Proc-GenerTransm Distrib 2004;151(2):157–66.

[11] Gaing Zwe-Lee. Particle swarm optimization to solving the economicdispatch considering the generator constraints. IEEE Trans PowerSyst 2003;18(3):1187–95.

[12] Glover F. Tabu search part I. ORSA J Comput 1989;1(3):190–206.[13] Glover F. Tabu search part II. ORSA J Comput 1990;2(1):4–32.[14] Bland JA, Dawson GP. Tabu search and design optimization.

Comput-Aid Des 1991;23(3):195–201.[15] Fanni A et al. Tabu search metaheuristics for electromagnetic

problems optimization in continuous domain. IEEE Trans Mag1999;34(3):1694–7.

[16] Chelouah R, Siarray P. Tabu search applied to global optimization.Eur J Oper Res 2000;123:256–70.

[17] Kalinli A, Karaboga D. Training recurrent neural networks by usingparallel tabu search algorithm based on crossover operation. EngAppl Artif Intel 2004;17:529–42.

application of multiple tabu search algorithm to solve dynamic economic dispatch considering...

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