Parallel tabu search algorithm for constrained economic dispatch

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  • Parallel tabu search algorithm for constrainedeconomic dispatch

    W. Ongsakul, S. Dechanupaprittha and I. Ngamroo

    Abstract: A parallel tabu search (PTS) algorithm for solving ramp rate constrained economicdispatch (CED) problems for generating units with non-monotonically and monotonicallyincreasing incremental cost (IC) functions is proposed. To parallelise tabu search (TS) algorithmsefciently, the neighbourhood decomposition is used to balance the computing load, whereascompetitive selection is used to update the best solution reached among subneighbourhoods. Theproposed PTS is implemented on a 32-processor Beowulf cluster with an Ethernet switchingnetwork on a generating unit system size in the range 1080 units over the entire dispatch periods.With different subneighbourhood sizes, the proposed PTS compromises the experimental speedupand solution quality for the best performance. PTS is potentially viable for the onlineimplementation of CED because of the substantial generator fuel cost savings and high speedupupper bounds.

    1 Introduction

    Recently, many parallel tabu search (PTS) algorithms forpower system applications have been implemented onparallel machines. For example, Mori et al. proposed PTSfor solving many combinatorial power system optimisationproblems, including unit commitment, capacitor place-ment in distribution system, reconguration of distributionsystems and optimal allocation of FACTS [14]. Toparallelise TS, two strategies, including neighbourhooddecomposition into two or three species and multiple tabulengths, were used. The proposed PTS algorithms wereimplemented on Fujitsu S-7/7000U model parallel compu-ters with up to nine processors. However, during theparallel process, there was no solution exchange amongprocessors.A constrained economic dispatch (CED) is used to

    determine the optimal schedule of online generating units soas to meet the load demand at the minimum operating costunder various system and generator operational constraints.To obtain the exact solution for the non-monotonicallyincreasing incremental (IC) functions of generating units,the conventional equal lambda could not be used to obtainthe optimal solution [5]. Accordingly, with the advent ofheuristic techniques, evolutionary programming (EP) [6],the simulated annealing (SA) algorithm [7] and the geneticalgorithm (GA) [8] were used to solve the ED problems toobtain the optimal or near optimal solutions.In this paper, PTS is proposed to solve CED problems

    for generating units with non-monotonically and mono-tonically increasing IC functions. To speed up the

    computation, PTS was implemented on the Beowulf clusterwith 32 Intel Pentium II 266MHz processors [9]. Thealgorithm was tested on systems with different subneigh-bourhood size and the number of generating units rangingfrom 10 to 80 over the entire dispatch periods. The PTScompromises the computing time and the solution qualityfor the best performance. In addition, the speedup upperbounds on PTS with 32 species are shown for differentgenerating unit systems. The synchronisation overheadswere analysed in terms of system sizes and number ofprocessors.

    2 Constrained economic dispatch problemformulation

    The objective of CED is to schedule online generating unitsoptimally so as to meet the load demand at the minimumoperating cost under the power balance constraint andgenerator operational constraints. The ramp rates of thegenerating units are also considered to ensure the viabilityof outputs at the next time period. The CED problem isformulated as:

    minimiseCT t XNi1

    CiPit 1

    subject to a power balance constraint:


    Pit PDt PLt 2

    and inequality operating constraints of generating units,including their ramp rate limits at time period t:

    Pi;lowt Pit Pi;hight; i 1; . . . ;N ; 3where

    CT(t) total generator fuel cost at time period t (baht/timeperiod)

    Ci(Pi(t)) generator fuel cost of the ith generating unit attime period t (baht/time period)

    W. Ongsakul is with the Energy Field of Study, School of Environment,Resources and Development, Asian Institute of Technology, Pathumthani12120, Thailand

    S. Dechanupaprittha and I. Ngamroo are with the Electrical Power EngineeringProgram, Sirindhorn International Institute of Technology, ThammasatUniversity, Pathumthani 12121, Thailand

    r IEE, 2004

    IEE Proceedings online no. 20040460


    Paper rst received 30th August 2002 and in revised form 11th September 2003

    IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 157

  • Pi(t) real power output of the ith generating unit at timeperiod t (MW)

    PD(t) total real power load demand at time period t(MW)

    PL(t) total transmission loss at time period t (MW)URi up ramp limit of the ith generating unit (MW/timeperiod)

    DRi down ramp limit of the i-th generating unit (MW/time period)

    Pi,minminimum real power output of ith generating unit(MW)

    Pi,maxmaximum real power output of the ith generatingunit (MW)

    Pi,low lowest possible real power output of the ithgenerating unit at time period t (MW): max{Pi,min,(Pi(t1)DRi)}Pi,high(t) highest possible real power output of the ithgenerating unit at time period t (MW): min{Pi,max,(Pi(t1)URi)}N total number of online generating units to bedispatched.

    To achieve the true CED, transmission losses must be takeninto account. In this paper, the traditional B matrix lossformula is used to calculate transmission losses as [10]:

    PLt XNi1


    PitBijPjt XNi1

    Bi0Pit B00 4

    where Bij the ijth element of the loss coefcient squarematrix, Bi0 the ith element of the loss coefcient vectorand B00 the loss coefcient constant.

    3 Parallel tabu search for CED problems

    Parallel tabu search for CED constructs the neighbourhoodsolution space by encoding an initial feasible solutioninto a normalised binary string structure. The neighbour-hood solution space is decomposed into several equal-sizesubneighbourhoods to balance the computing load. Eachsubneighbourhood is assigned to each processor (CPU) tocarry out the tabu search process, incorporating the abilityof escaping from being trapped in local optima [11]. PTSemploys a tabu list restriction for preventing cycling ofthe solution and an aspiration level criterion for impro-ving the solution accuracy. During the parallel searchprocess, the competitive selection strategy is carried out byupdating the best solution reached among processors forevery specied epoch iteration.

    3.1 InitialisationBased on the generator operating range ratio, (5) and (6) areused to calculate the initial feasible power outputs of Ngenerating units:

    PG;reqt PDt PLP1t;

    P2t; . . . ; PN t XNi1


    Pit Pi;lowt Pi;hight Pi;lowt PG;reqtPNj1

    Pj;hight Pj;lowt;

    i 1; . . . ;N


    The Pi(t) in (7) are initially set to Pi,low(t), i 1,y, N.

    Equations (7) and (8) are then computed iteratively untilinitial feasible power outputs are obtained, satisfying boththe power balance constraint and generator operationalconstraints.

    3.2 Power balance constraintThe power outputs of N generating units at a particulartime period have to satisfy the power balance constraint,unit operating limits and ramp rate constraints. Forarbitrary free unit power outputs Pi(t), Pi,low(t)rPi,high,i 1,y, R1, R+1,y, N. The Rth dependent referenceunit power output is constrained by the power balanceequation as:

    PRt PDt PLt XNi1;i 6R

    Pit 7

    In (6), the transmission loss PL(t) can be written in terms ofPR(t) as:

    PLt AP 2Rt BPRt C 8where ABRR, B 2


    BRjPjt BR0 andC PNi1;

    i 6R


    PitBijPjt PN


    Bi0Pit B00.Substituting PL(t) in (7),

    AP 2Rt B 1PRt C PDt XNi1;i 6R

    Pit 0 9

    The Rth reference unit power output, PR(t), is obviously thesolution of the quadratic equation (9). PR(t) is regarded as afeasible solution if it satises the ramp rate operationalconstraint in (3).

    3.3 Encoding and decoding schemeThe concatenated encoding method is employed in thispaper, as shown in Fig. 1. Each unit output is encoded in abinary base string normalised over its operating range. Thisencoding method stacks each units normalised string inseries with each other to construct the string individual.Each unit string structure is assigned by the same number ofn bits.

    To obtain the actual generating power output of eachunit for objective function evaluation, each string individualis decoded to the decimal value by:

    Pit Pi;lowt Bi Pi;hight Pi;lowt2n 1 ;

    i 1; . . . ;N10

    where Bi decimal integer value of binary string of the ithunit and n number of bits representing each unit poweroutput.In this paper, 16 bits represent each unit power output.

    The more the number of bits per unit power output is, thehigher the resolution will be. Each of the N1 free unitoutputs will be decoded by (10) except that the Rth

    unit 1 unit i unit Nxxx... xxxx xxx...xxxx xxx... xxxx

    215 214 213 202122

    ... ...

    . . . . . .

    Fig. 1 16N bit concatenated encoding scheme

    158 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004

  • reference unit power output is calculated by (9) to satisfy thepower balance constraint.

    3.4 Neighbourhood decompositionTo parallelise TS efciently, the load balancing is carriedout by decomposing the neighbourhood solution spaceinto NSN equal-size subneighbourhoods, where NSN is setto the given number of processors (p). The changeableregion is referred to a subneighbourhood solution space(NS) assigned to each processor. Each NS then has a smallrestricted neighbourhood size of InN/NSNm. Fig. 2illustrates the concept of neighbourhood decompositionstrategy.

    3.5 PTS operators

    3.5.1 Trial solution generation: To generate atrial solution of a subneighbourhood, bits of the binarystring in the changeable region are ipped one at a time.Starting from an initial feasible solution, PTS performs adeterministic advanced local search [12].In this paper, the maximum number of trial solutions in

    each iteration is set to the size of NS. As an example, therst, second and third bits of an initial feasible solution areipped one at a time to yield three trial solutions, as shownin Fig. 3. Subsequently, the Rth reference unit power outputis obtained from (9) to satisfy the power balance constraint.

    3.5.2 Tabu list restriction: Tabu list (TL) isreferred to as an adaptive memory. The mechanism of TLis to keep attributes (bit positions) of the best solutions inpast iterations, in which each of the best solutions is used asan initial solution to generate the trial solutions in eachsubsequent iteration. The attributes in TL are temporarilyxed and cannot be ipped to generate the new trialsolution candidates unless the aspiration criterion is met(discussed in Section 3.5.3). As the iteration proceeds, a newattribute enters into TL as a xed attribute and the oldestattribute is released from TL and becomes a free attribute,as shown in Fig. 4. In particular, TL affects the quality ofthe solution by controlling the search directions so that thesolution is not trapped in the local optima. The length ofTL is also called the tabu length. Basically, the tabu lengthproviding good solutions usually grows with the size of theproblems. However, the appropriate size of the tabu lengthcan be identied by observing the quality of solutions. If the

    size of TL is too small, the cycling of solution occurs in thesearch process. On the other hand, if the size is too large, thesearch process will be too much restricted, which may

    deteriorate the solution. In our applications,size of NS

    p is used to determine the size of TL [12].

    3.5.3 Aspiration level criterion: To improve thesolution quality, the aspiration level (AL) criterion isincluded. The major role of AL is to allow the xed(tabued) attributes included in TL to produce the solutioncandidate if that solution candidate yields a more economic-al solution. There are many different aspiration level criteriaused [12]. The AL used in this paper is to override the tabustatus if the solution candidate of the tabued attribute yieldsa solution that is cheaper than the best solution reached.After the AL is satised, updating TL is carried out bymoving the tabued attribute back to the rst position of TL.

    3.5.4 Reference unit rotation strategy: Thereference unit rotation strategy is also used to reduce thesearch effort towards the optimal region. Regardless ofattributes in the xed region, the Rth reference unit shouldnot be xed at the particular unit. More specically, the Rthreference units of each subneighbourhood are initially set tothe rst units in the xed regions. For the second iteration,the Rth reference units are moved to the second units in thexed regions of each subneighbourhood, and so on, asshown in Fig. 5.

    3.5.5 Competitive selection strategy: To ob-tain a fast convergence rate towards the optimal solutionregion, PTS employs the competitive selection strategy bysetting the best solution reached among all subneighbour-hoods as an initial solution for all subneighbourhoods forevery specied epoch (G) of 20 iterations.

    3.6 PTS procedure for CEDThe following notation is used for the PTS procedure:

    TL: tabu list

    AL: aspiration level criterion

    NS: subneighbourhood solution space

    C(X): objective value of solution X

    X(k,0): initial feasible solution at iteration k

    X(k,m): trial solution m at iteration k

    Xkcb: current best trial solution at iteration k

    Xb: best solution reached

    kmax: maximum allowable number of iterations

    T: total number of time periods in the time horizon

    Using NSN processors, the PTS with NSN species pro-cedure can be described as follows:

    Step 1: Each processor reads the unit operating limits, heatinputoutput characteristics, ramp rate constraint limits,

    entire neighbourhood : 1001011010110010 1100011010110011 1110010011001100

    subneighbourhood 1 (CPU 1) :changeable





    fixedsubneighbourhood 2 (CPU 2) :subneighbourhood 3 (CPU 3) :

    1001011010110010 1100011010110011 1110010011001100

    1001011010110010 1100011010110011 1110010011001100

    1001011010110010 1100011010110011 1110010011001100




    Fig. 2 Example of neighbourhood decomposition

    R th

    R th

    R th1011011010110010

    initial solution 1001011010110010 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx

    trial solution 1 0001011010110010 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx



    trial solution 2 1101011010110010 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx:

    trial solution 3 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx:

    changeable fixed

    Fig. 3 Example of generating trial solutions of a subneighbourhood

    new attribute tabu list old attribute

    Fig. 4 Mechanism of tabu list

    iteration 1

    iteration 2

    iteration 3

    iteration N

    iteration N +1

    R th

    R th

    R th

    R th

    R th


    xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx

    xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx

    xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx

    changeable fixed

    xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx

    xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxx



















    Fig. 5 Example of reference unit rotation strategy

    IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 2, March 2004 159

  • fuel cost of each unit, a forecast load demand at t...


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